B II

.....

- .................

•

B

II

FORE._WORD

Thi_

|_,_;_oarchwaI_ crlrr_o_] out by thr_ Ar,ro_lastic

Ro.t;r;,:|rch

].]L_ratory, ]hl:_titu£,,

Ma;_sachusetts NASA

Gr;,nt No.

NGR

Ac_ronaut]_.'sand Rotor

Burst

from

discussion,

out

these

advice,

and

Meredith,

was

waived

Lewis This

in August C.C.

Report

to their in this

Center

submitted

1979.

The

authors

of NASA-Lewis

suggestions

out necessary reproduction

0

,

valuable

Mrs.

Solomon

advice

most

effort:

of the NASA Weiss

and

gratefully. for stimulating

Dr. Robert Susan

L. Spilker,

E. French,

computations

were

carried

of MIT. NPD

2220.4,

in accordance authority

September

with

14,

provisions

of the Director

of

Center.

on

report followed

to NASA-Lewis wish

to thank

for this

their

revisions immediately

,

,

for pre-publication Dr. M.S.

thoughtful

report.

ii

.............

llolms and Mr.

Directions

by the

National

as part

colleagues

The

document

Directive

was

constructive carried

the present

Ohio

is acknowledged

(NASA Policy

report

Chamis

The

Merlis.

Processing

5d of that

Research

monitors.

R. Stagliano,

and Mr. Fred

for

Cleveland, G.

assistance

Mr. Thomas

Massachusett_:_ undL_r

Lowi.,_Re:zuarch Center,

Dr. Arthur

indebted

use of SI units

of paragraph

Dr.

are also

StructureH

,%nd Ar;tronautics°

cambridge,,

the

individuals

at the Information

1970)

the

Program.

as technical

W. Leech,

The

from

S[,ac:,] A¢_ninistratioll,

The authors

Dr. John

22-009-339

served

cooperation

Denny

of Tochnoloqy,

Protection

of NASA-LeRC

Dr.

Dc:_partm_nt ef Aeronautics

and

Hirschbein review

Subsequently,

which

were

and for

the authors

completed

thereafter.

and

review

in July

1980.

i J

I I CON_I%',N'PS P_P_]o_.

th_etlon I

2

IN'I'I_OI]OCTION

i

1.1

Background

1

1,2

I_urpo_o

1.3

SylXO_I_irlof

of

t,ho

Stmly

6

the Pr_sont

6

GI_NRNAI+ FORMULA'PION

9

2. I

Ihtroduction

9

2.2

Notation

_.3

Review

of Tensor

2.3.1

Vectors

ii

2.3.2

Tensors

15

2.4

2.4.2

2.4.3

2.5

i0 Analysis

ll

2.3.2.1

Linear

Vector

2.3.2.2

Dyadic

Representation

2.3.2.3

Covariant

Kinematics 2.4.1

of

General

Functions

15 of a Tensor

Differentiation

a Deformable

19

of a Tensor

Medium

24

2. _. I. 1

Double

2.4.1.2

The Unit

2.4.1.3

The Displacement

2.4.1.4

The Velocity

Deformation

Tensors Tensor

The Deformation

2.4.2.2

The Spatial

2.4.2.3

Rotation,

Vector

29

Deformation Stretch,

30

Gradient

Tensor

Tensors

Tensors

2.4.3.3

The Spin

2.4.3. •

The

between

Tensor

36

Strain

Rate

44 Tensors

Tensor

Spatial

33

44

Relations

The Cauchy

Tensor

and Strain

2.4.3.2

2.5.1

30

Gradient

The Rate-of-Deformation

Tunsors

28

Tensors

2.4.3.1

Stress

28

Vector

2.4.2.1

Rate

28

(Metric)

and Strain

Deformation

21 24

Descriptiozl

Velocity

45 49

Gradient

Tensor

50 51

Stress

iii

I

Study

Tensor

52

CONTENgS

CONTINUED

Section

p/_e Pc

2.6

2.5.2

The

Kirehhoff

Stress

_nnnor

2.5.3

Th_

Second

2.5.4

The First

Piola-Kirchho_f

i.5.5

Relations

between

Stress

Rates

52

Piola-Kir_hhoff

and Rates

Stress Stress

Stress

Tensor

54

Tensor

57

Tensors

of Second

58

Order

Tensors

in General

60

2.6.1

Rates

of

2.6.2

Rates

of the Cauchy

2.6.3

the Unit

Fixed-Obse%ver

2.6.2.2

Convected

2.6.2.3

Co-Rotational

Rates

61

Stress

2.6.2.1

66

Rate

66 68

Rate

Order

70

Tensor

71

Rate

71

2.6.3.]

Fixed-Observer

2.6.3.2

Co-Rotatlonal

2.6.3.3

Convected

Rates

2.6.3.4

Relations

between

Co-Rotational

Tensor

Rates

of a Second

Order 2.6.4

Tensor

Rate

71 72 Rates

of Second

Tensors Rate

73

of the Kirchhoff

Stress

Tensor 2.6.4.1

74 Co-Rotational

Expressed

Second

Piola-Kirchhoff Strain

Coordinate Energy

2.8

Specialization:

of the Kirchhoff

Stress

Green

2.7

Rate

in Terms

of

Stress

the and

the

for a Convected

System

75

Equation

76 Homogeneous

uniaxial

Irrotational

Deformation

80

2.8.1

Deformation

and

Strain

Tensors

83

2.8.2

Deformation

Rate

2.8.3

Stress

Tensoz_

90

2.8.4

Stress

Rates

95

Tensors

88

iv

..........

, ,,

l

I i

!

C()NT.I Nt]ED

CONTENTS

2.. If.5 3

l,]n_Lrqy Eqllati.on

till

C(.)N,q TI']'[I'I' TVE I,',QUAT.T()N_I

102

3.1

]lit[

1(12

'3,:I

l¢ov.!.ow

c_f Sma]]~lItrai.n

3,2.1

lh_Vi_W of l?r_ncIp_l Conc_ptl_

102

3.2.2

The

109

3.3

odlletitln f,1,.mtkcity

Plant lcity Theory

for Finite

Strains

II0

3.3.1

Introduction

1 i0

3.3.2

General

iii

3.3.3

A Finite-Strain

3.3.4

Concepts Elastlc-Plastic

Computation

of Mcchanical-Sublayer-Model

3.3.4.2

Comments

to Uniaxial

Stress-

Conditions

Application Strain

AND

123

Application Strain

BEAMS

i 16

Factors

3.3.4.1

3.3.5

Strain-Rate-

Theory

Weighting

CURVED

loft

M_chanJeal-Sublayor-Mod_l

Dependent

4

Tho.o}."y

123

to Multiaxial

Stress-

Conditions

125

on Strain-Rate-Behavior

Modeling

127

RINGS

135

4.1

Introduction

135

4.2

Strain-Displacement

Relations

for Finite

Strains

and Rotations 4.2.1

135

Strain-Displacement Bernouilli-Euler

Relations

for

Displacement

4.2. i. 1

Formulation

4.2. i. 2

Membrane,

the

Field

IA5 135

Bending,

and Polar

Decomposi tlons 4.2. I. 3

Specialization

143 to Small

Membrane

S trains 4.2.2

Inclusion Finite

4.2.3

Summary

of Thickness

[51 Change

Associated

Strains of Strafn-Displacement

v

with 157

Equations

163

CONTENTS

CONTINUED

Suction

P__

4,2,3,1

Straln-Displaeem_nt Small

4.2.3.2

5

PLATES

Constitutive

for

Strains

163

Straln-Displacement Finito

4.3

Relations

Strains

Equations

Relations

and Finite

for Finite

for

Rotations

Strains

164

and

Rotations

166

4.3.1

Introduction

166

4.3.2

Constitutive

AND

Equations

166

SHELLS

173

5.1

Introduction

173

5.2

Strain-Displacement and

5.3

Relations

for Finite

Strains

Rotations

174

5.2.1

Formulation

5.2.2

Strain-Displacement

Constitutive

for General

Equations

Shells

Relations

for Finite

174 for Plates

Strains

189

and

Rotations

195

5.3.1

Introduction

195

5.3.2

Constitutive 5.3.2.1

Equations

Plane

Stress

Shells 5.3.2.2

yon

Mises

Function

195

Stress

Strains

200

Part

Relations

for Plane

of the Constitutive Stress

and

Strains

205

"Plastic"

Part

Relations

for Plane

Finite 5.3.2.5

for Plane

"Elastic"

Finite

for Thin

Strains

Strain-Rate-Dependent

and Finite

5.3.2.4

Assumption

at Finite

Loading

5.3.2.3

195

of

the constitutive Stress

and

Strains

Incremental Evaluation

214

Procedure

for the

of Stresses

221

vi

................

,

,

i

i

|

I I i

t:_JHTI_NT8 CCINTI NUPp ['1

G

GOVERNING b]QUAT[ON_ 6. i

Introdact

6.2

Equatiolln of

AND SOLUTION

6.2.2

229 Motion

Fillit,_ Element

6.2.3

ForiBulatlon

Strategies

Puru Vector

6.2.3.2

Constant

6.2.3.3

Tangent

6.3.1

Linear

6.3.2

Nonlinear

for the Ailsumed 234

6.2.3.1

Difference

'230

Model

Computatiollal

Finite

238

Form

242

Stiffness

Form

Stiffness

249

Operators

Dynamic

252

Systems

252

Systems

6.3.2.1

Implicit

Methods

without

6.3.2.2

Implicit

Methods

with

of tho

Governing

Explicit

246

Form

Dynamic

Solution 6.4.1

230 Formulatlc_n

Displacement

6.4

;]29

l.on

6.2. '_ Variational

6.3

PROCEDUREL_

Solution

256 Iteration

258

Iteration

261

Equations Process

263

of

the Equations

of Motion 6.4.2

264

Implicit

Solution

Process

of

the Equations

of Motion

7

EVALUATION

269

6.4.2.1

Extrapolation

272

6.4.2.2

Iteration

274

AND

and Convergence

DISCUSSION

7.1

Introduction

7.2

Impulsively-Loaded

280 280

7.2.1

Problem

7.2.2

Comparison

Narrow

Plate

Definition

281

of Small-Straln

Predictions

for Structural

Beam

Elements

Finite

i

281

vs Finite-Strain Modeling

by 282

vii

I ...........

,,.

II

I

'_i|i

i

I

III

.........

II

I

I

I

I

I

CONTENTS

CONTINUED

Section

P___'! 7.2.3

Modelling b? Plate

Finito

7.2.3.1

D_s_ription

Modoling

El¢_montn

289

nnd O1|tllne

of Analysis 7.2,3.2

2R5

Slngl_Proois_oll

vl_ Doublo-Procl

_llon

Prodietions

293

7.2.3.3

Time

Incremont

7.2._.4

Small-Strain

Sizo

vs.

Eff0ctt}

297

Fin lto-Straln

Predictions 7.3

Impulsively-Loaded 7.3.1

Problem

7.3.2

Comparison

7.3.3 7.4

Free

302

Circular

Ring

306

Definition

307

of Small-Strain

vs Finite-Strain

Predictions

307

Comments

309

Impulsively-Loaded 7.4.1

Problem

7.4.2

Comparison

Square

Thin

Flat

Plate

310

Definition

310

of Finite-Strain

Predictions

vs. Experiment 7.4.2.1

311

Finite-Strain Analysis

7.4.2.2

7.4.2.3 7.5

Model

311

Transient

Strain

Transient

Displacements

Permanent

Deflections

Containment-Ring Tri-Hub-Burst

and Finite-Element

Response

to T58

Comparisons

and 313

and Strains

Turbine

320

Rotor

Attack

7.5.1

Problem

7.5.2

Comparison

323

Definition

323

of Small-Strain

vs. Finite-Strain

Predictions 7.6

324

Steel-Sphere-lmpacted

Narrow

7.6.1

Problem

Definition

7.6.2

Modeling

by Beam

7.6.3

Modeling

by Plate

Plate

326 326

Finite Finite

Elements

327

Elements

329

viii

..............

,

, ,,

, ,,,,,,

,

,

,

,,,

I

I

I I

II I

I IIII

I

7,6.4

Compi+ril_on

of

[+n,_ll-+Modol

+t-;.

Pl.itt¢+-blodnl

l"r,+dJ.et].ont+ '1._.4,

I

330 Str-ain

Compi_rinon_

:._i_]

7. [+.4. '2, [Jt+_l+:,l.o+-'t inn t'nmpnr;l[iCll}.fl ].1.;,5

|'+illJt'lt"+it.l°illll

Pr" .__,_ ] .g_--(F-'_'_ )._ _ _j-r-' Hence

-I

L •

_i I

(2.1101

i

ThQre fore e

CF)" G"_ =

34

(2.111)

AtlaJ.nt

_

fr¢_lll Eqtl.

F

-

Vt:om Eqs.

"|

'

2.105

G_ "_

2.7t.),2.77,

alitl

=

2.66

2,.1.06:

"' :K '

z

"

alld 2.52-2.55,

ono

,

-

IZ'''!

4" kt

2,_'"

(2.114)

ubta:i|Lq=

Then

Then

( sS- ST., _)G,::(F"):. i G,

,,..,_,,

If'here fore

liF>:-' • u- I =

--

(2.118)

Hence,

(F"),,. =G:,_(6} - EI_,_,)-G=,,- U,,,, ,_.,,,,

(r-')'"=G'"(st-TJ:':.)-G `°-u:',° ,_.,_o, (r");:-G,,G _'' _' ,_.,,. (6,U,,,): S_:-U:, 35

.......

,

'

,,

,

,

ii

I

I

II

I

I II

I

"

It can be shown

that

the

following

relations

5 t. --F-F ? VTFF"

=_

U

Cauch[-Green The

'

_

is called

square

of the right

See, J.L.

the right

stretch

"

=

•

stretch

tensor T

C auchy-Groon

deformation

12.1241

tensor.

The

square

of the

tensor

for example, SeetioD Eriek_en (Ref. 67)..

83 of P.R.

Halmos

(kef. 69) and

S_ction

36

•"

(2.123)

Deform at_Ion Tensor s

=

left

are valid:

i

i

i

i

i

4_ of

,

.t._ called

''

_hown

I

thu

left_

tonsar

of

dofarmation.

.T,t can

be

,)anily.

that

The

riqht

reference

Cauchy-Grecn

configuration,

differential

defozT,ation

and it gives

line element

dR

tensor

C is associated

the new squared

into which

the given

length

differential

with

%/_o

(d8)2 of

the

element

d_

is deformed"

The inverse of the left Cauchy-Green deformation tensor _, denoted =-i 2 by B gives the initial squared lengtJ_ (ds) of a deformed differential i

line element

dR

(:,_4' °J_.a_-(a_.(_-'T). (_-'._).-,;t.(_'. Ft'._. -j_.g"-a_

(2.128)

The

i

and

right

from Eqs.

in terms

i

I

2.124,

2.88,

deformation

2.90-2.95

of t/_e displacement

vector

tensor

and

has components

2.100-2.103,

components,

one

can express

as follows:

_X__

?_._j-(_,4__,,_ - t,,,r__ _ _

i

Caachy-Green

_\I__

^

_

^ ^ _._ _&

,_.,_o,

: _},j. ug + tLj,_+

u.t,_

C'J_-(F!LyF J_F/,;.FLJ,a'_tG_ •

i.,/

_tj

(2.13a)

_.....

U_

M

Notice,

that

although

following

components

The Green

Strain

The Green*

C _ F, are

from Eqs.

2.95,

2.132,

2.94,

and

2.131,

the

equal.

Tensor strain

tensor

_ is defined

as follows-

"accordi'gg to +ruesdell (Ref. 15, page 266), this strain measure was la_troduced by Green in 1841, and by St. Venant in 1844; since its components are usually referred to a fiwea reference configuration, it goes by the name of "Lagrangian strain" in the older engineerlng llterature. IB

From

Eq.

This

strain

vector

2. 124, it is easily

measure

gives

d_ as follows

shown

that

expressions

in tile .B_qU_.r_od 10n@th

the _

from Eqs.

equivalent

2.127

and

arel

of the material

2.1281

(as)_-(_4_- a_,._ - _._ Expressing Eq.

this

in terms

o£ the material

vector

,_.,-,

d_, one

obtains

from

2.127:

Defining

1 _ - _

(_-_),

one

obtains

(JS)_- (d =d_.)f. g_

c,._o_

Z

Components

of

the Green

strain

tensor

_ are:

__----^

These

components

components,

can be expressed

from Eqs.

2.136

and

in terms

of the displacement

2.130-2.133,

vector

obtaining.

(2. 142a)

(2,142a)

¢1_

'"

U

_J )

&

_' j U.& ' ' U._, ) b)/_ (_..142d)

39

.......

•.

. ,

. ..... i

i

ii

I

I

I

II

The Almansi Th_

Strain

TenBor

AlmanBi*

strain

t_nsor

a iB d_fin_d

as followBs

(2.143)

Equivalent

oxpr_sion_

_or the Almansi

strain

o are obtained

from

tho

r_

de£initlon

The

of the

Almansi

loft

strain

material

vector

Defining

_, i , , _.

Cauchy-Groon

also

gives

deformation

the _

d_ as follows,

tensor

in the squared

from Eqs.

2.138,

B, Eq.

length

2.125s

of the

and 2.128:

i (_._-1)one obtains = , ,

J 'l

(ds)'_d,= Z

._.

,

(_.,,o,

, ,,,

m

Components

of the Almansi

(Eulerian)

strain

tensor

e are:

.e.. ^ _,_ .e_, g'_.e"g._,e,g,g""""_" where

^

,

_x_

'_X

r

According to Truesdell (Ref. 15, page 266), this strain measure was introduced by Alma_si in 1911 and Ha_el An 1912; since its components are usually referred to the present configuration, it goes by the name "Eulerian strain" in the older engineering lite_atuEe.

4O

..........

,

, ,

,

' m,

,

, ii

i

I

II

I

I

I II

II I II

" ±( ;-G'" e"='(G'" observe, that

the ¢ova_imlt

ZOllSOrS with

respect

bas_: vectors

_I,

compon_)nts

of the Green

rr4

r_

y and Almanfd

e ntrain

-i

ThereZorot

to thai reference

rospootlvoly,

from Eqs.

2.140,

ba_o

are tlm

2.64,

and

voctorl_ g

and

to tile pres,)nt

smno_

2.141:

(ds)'(d_)' -?..

'

-_.

_. • j

or

r

Also,

from. EqS.

i

2.145,

2.64,

and

ii

2.146;

Z

i

. of,

course,

equality

p_ although

does

i

, |

I

= e_:4r Tij

not hold

= eij,

these

in the absolute

are different tensor

tensors,

and this

notation.

41

1 .........

,

i

i i

'1

i

i

I

other

Strain Since

Mnasures in _uclidean

the Cauchy_Gr_en tnnsQrs

ar_ by

Weiss_nherg directions principal other olongatlon E

space

deformation far the most

tensors, popular

[70_ has ob_]_rv_d, _* of the prlncil?a] elongations

may

tensors with

_

nxon

th_ name

tonsors

_ and _

strain

tonsors

are

strain

with

_,oasurns. su_ficlent and

the

with

However,

strain as

of

the

the

g_noral.

of Blot

[72]) as well the name

form,

to detnrmino

the magnltud_s

in fully

name

a quadratic

and Almansl

In the present

of swalngor

(associated

defined

strain moasur_

of,

by

and the Green

of interest

(associated

strain

are measured

be employcld and

_-_t_ainmeasures

(associated

with

distances

ana].ysil_ are the [71, page as the

of lloncky

118])

and

logarithmic

[73]).

These

as follows:

components

_ * As a matter of measures which "but there can usually led to

f_ct, it is possible to describe strain _orr_etly by are not tensors; Truesdell points out (Ref. 15, page 269): har41y _e any advantage, and attempts of this kind have _onfusion if not disaster".

42

Ho

i

a.i Hj

=

- -

RI_]-_t&[Qnl-I blt t:Wl:ll_n B_:;IE_I1 Tt_nl.ll-i_l-t Th(_l tonflo_rn tll-rilill

;-it

rp

al_d _ all

at

t

_: into

line

£h{i

_ t,

nrm_o, pr,lnelpal

'['lle

ten,_ors

The

t_nsors

principal

values

Tile principal

V llavo

tl_e

principal

are equal

values

e

of the

strain

E

of the elongation

e are== e

= £n I .

is, _

in t/le reference

= He,

and

principal

are

th_

_w Os |'] prollont

axe_] of

strain

squares

?he_e

ratios

of

principal

tl_o

defor_ed

to the undeformed _ dS(_ la = d-s--" values

line

(I(_)5 and

of the principal

those

stretches

successively,

each

deformation for

= _2 (i -

(le)'2) .

The principal

l

to the prinCipal

E c_ of _

stretches

the elongation

tensor =

and He of the tensors

related

components

to the principal

of the tensor base

values

_ are

s_retches

_ and the

respectively,

by

_ and

tensor

are equal:

4_

defor_%ation

e.omponents

b

same principal

t/le present

=

resultant

VI tile

values.

directions,

values

The mixed

base

axes of

in

p,rJn_ipa]

princit_al la'

and are

H; H in the

u_mlo

Values

The prinoipal

by _

tl_nllorl] RI

at _.

_ are related

the priheipal

that

cub

the

to the

tensor

are equal;

earried

at

of

the strain tensor _ are related to the 1 2 Y_ = _ ((la) - I) , while the principal values

tensor

(l)'l.

If the

Tim

ntraln

allnn

y@ of by:

_e = _e - i, while

= H

_ t o.

I_ carries

str_tchoo

_ and B have

stretches

1 -

t

in the prineil_al directions

principal

=

rennet

axes of strain

tile dS

at

axo,_l of

_ dst_ in the same principal

elements

E

rotat£on

U and

called elements

[1 all haw _ tile ncu_l_prJncil_n],

,r_if, orl111g_1 nhapo

the.

Principal

Thu valuers,

ill

h:ivo

z_hap_ at

U ne C _ t _ i l_w and

(2.162)

are

fixed

principal is equal

the several 43

and

several

component to

of the

the sum of

successive

deformations tensors

are _ and

the corresponding

deformations.

For

tile

E, components

of

_,.V,

functions

of the

it i_ usually

and H as moasurnf_

2.4.3

o£

where,

componei%ts

b_tt_r

Rate

rato-of=doformatioB

ehangu

rospoot

_, _, _H, _ are

to use

in t/_is notation has been

this

[22] :

notation

tnnnar

D

complJ.catsd

of _, and hence

components

The

irrational

or

in the solution

of y, _ and o,

(also _a_..l,¢_d ntrotohlng)

reference

limit

the subscript

t denotes

configuration

is used,

i

that

of _ are:

44

_nto

[_.2]z

that

the present

Ci I l

it can be shown

as t;_ 0

in thn

t + c with

c_hosen as thc_ ref_renc:e configuration.

"

Components

exist.

'_'nn_9___r

at t.imo t, in the

configuration

Also,

not

of the nt_'ol;ch _ or _ at R J.l%th,._uhak_o at time

to that

If a fixed

does

,I'onf-_o_.'n

'I.'ho Rato_of-bofo_matlon

2.4.3.1

property

of strain.

D_£armat&an

The

_t 5, _ this

the tossers

transcendental of problems

and _,

then

ii

(time t) Also,

in

Since _t in] symm_trlc, parm_t_r I

_o_,].llnq

that

so is _, bluing its

th(_ vol(_llt7 v_ctnr

dnrlvativn

with

rnsp_.ict to a

_ _ u " _ _ o..anbo oxpr_D,nd

nn

Th0n

A

^

i) -- 4"_- a'_a'_,:_,_. ,VT),_..,, _-""GK_ . ±K (v=,_ +v. ,") I2.1val D " _. -z(a 2.4.3.2 Observe tensor

Relatxons that

the

_ in conveoted

covaria_t

components

betw99.n Strain covariant coordinates

Yij

of

Rate

components are equal

the Green 45

Tensors DIj of

the rate-of-deformation

to the mat_rlal

(Lagrangian)

strain

_ate tensor

of thu _

an_

also arn nqual to tho mater.lal rate Almansi (Euh_rian) strain t_nsQr _I

[Jut, thin donn

llot at all

imp)_y that

n¢lqal to th, m(itrlrial rat(_s of fa(_t, thr) rnp.tmlq_]ar

covariant

cQmponnnts

_'!J nf the

tlm ratn-of-(_eforIBat.lon tl_lIFIQr iFl

t]l_ Grnnn

and Alm_lnSi Mifflin tunnc_rn.

/%

0

_

tho colivoc%od mlxud

A

(2. 176)

o_mI_QllUllt{] aro (-|l£_o_ollts

"%j _"

' c_

(s_ _)D_

Then,

Also_

The material

or

rate

=

of

i

In

Oartl]fll,qno¢)B1ponnntfla_'o (]Ifforolltl e

and

Qf the

the Green

i"

" l

strain

tensor

= I.,L'J1. ' •

i

i

46

dan be expressed

' i

ii

"1

"

as

with

components

A

Tho rolation

bntiln_n th¢_ l_Ix_d compori¢_nt_ of

the de,formation

I an__]tll_ rat¢_ of tho Gr¢_on ntraln _onfi.quratiol] Dj

present _ft t pr

) wall. |.1¢_ of

impartane_

i_qu,%t|_-Ji,11 for buo fiMod c,btitlnr_d atJ follows

in tgo

formulation

rata

t_,nsor (eltho.r

nf thn

oonatitutiva

rsf_ron_t_ oollfi.qi_ratioll. _hifl rolation

rlin_o,

from Eq,

in th¢_

_,nn bo

2,1.751

_cj'" _' ---

i

If the unit

tho _amo. fnr

paJ:tiol_).

If the original then

not functioi_l of timc_ (they romain

vector

X_,

directed

present

lenqth

is £,

-

along

___i_L --

the axis

of deformation

is [l'

--

then

(2.370)

and

_L The

deformed

--_

base

'&=

vectors

L

(2.371)

are:

1 = and "the

metric

of

the

deformed

G - --

G_

,,-

so

that

the unit.second

The position

vectors

order

;

T_.

(2.372)

configuration:

tensor

--y°

is:

are:

r :q L_= xIg_= _ _ _

r_

_ - q __i

, %=_

L:ri=

xi

81

......

,

,,,,

,'

,, I,

,I,

I

II

II

II

I II

^

]_-t2.375)

ThQ diaplacem_nt voator

i_,

_'._,a,-. (t._),,.,. (2,,376)

Ig _- _.-

x,

T_-

t-

x,

The velocity vector is:

_t IE_.x,=©,,,_T.

Xt= ¢o_sT,

Vi

to

^ i X, - V_." V_ " _ D.

_. i_c,. v' _,-v, _'-V' !L r,-V,{r_ l,

82

'l'hott, m_ rates

of thL_ d_forme.d bane

voctorn

ares

"ax_

dt _

__

2.8.1 The

__

_

Doformatlon

--

and Strain

=

_I

12+._79)

TonsorH L_

components

Cartesian

of

the

defo_ation

gradient

tensor

I,'are:

Com_: _X._

II "

Double

--

Tensor

_

_

_ X_ _

(2. 380)

,

_"

--_"0

Components:

' si

F'*i+i

F'='i"

F+j"=l

F'J= F.=&_l j'j

£

% +F_jG,,+.

r,+:F-:++,,p

components

in the Convected

configuration

F; + '++"+"

coordinate

System

in the Reference

are :

F._= S_ + u_,_. F+.+.- j+

,)x_

.:L+

F._;:F" F,_:F;+.-- -_ 83

.....

,

,

l

II

I

I I

I I

II

I

III

[

-

Th_

eomponQnts

cartesian

Qf th_-_spatial

deformatiQn

C_mponc_nt,

Tensor

_"

ar_s

!

Componentnl

(F")_.: - 6k

(F")ti --

(F .'),L . (r) -,

:;G,_

L " -(r" )'r

(r"); t =(F')_"_,_

(F");t. _J.t F

(F")_.--(F")!; #'a Com_x_nents

tens_r

L "

Doubla

gradient

in

the

Convected

(2. 384)

(F"),, - Z

Coordinate

System

in

the

Present

Configuration

-(r-'):;G

- _.

(r")_, --(F"):' G

(F'

(r-');t- (F-'),_ a'_

(F");:- __-_ '_"_'

Since orthogonal

an

irrotational

rotation

tensor

deformation is the unit

£_

is being

considered,

then

tensorz

%" :L And

for

this special

case,

the

the

(_.3._)

right

and

left

stretch

tensors

U and V

li

become

bot/1 equal

to the deformation

gradient

tensor

. •

F:

84

.........

" ,

;it

"

'"

_

""

',

"

-I

_

"1 iT"

'"

i1 ii

1

T

I I

I I

fl

I

I I IIIIII II I

Compollnntf_ of in tho

rofnronen

th_ right

_onfi_juration

^ Com,pononta present

A

that

strotoh of the

and

tonner

eonvnetod

cartnslan

coordinate

=

system nyntnm

c_rdlnato

(2.3a8)

_yntc_m and

system

the possible

range

tensors

is equal

in the

arol

v'_-T_° V,_-_' of the stretch

and

aro l

-"

In the Cnrtcmlan

convected

_0

the value

deformation,

of the

in thn

=

of thn loft

v,,=Vl--

tensor

I/'_°LL' I/

=

configuration

Observe

stretch

(_.3891

to un£ty

for no

is

A

o< LL,,.Ut- V,,-V:2,

_'_

I

i

.,;:/ /t-r_3:,_>.)' •

I

'A.=9o-_('_:,m ti+t-',_

(3.s2) Equation plastic

3.52

impliesthat

dissipation

positive

s-_ of

semidefinite

summarize,

s, which

s_ characterizes

in turn rQstricts

the s_ to be

o .,.oo

Finally,

to

plastic,

strain-hardening, ,,

sublayer

parameter

i

'l as:

the scalar

one

can express

these

finite

strain-rate-dependent

.......... i_.,, -',,,:'.._ , "

strain,

constitutive

"elastic"equations

,

£:i.A.,'_ 'G,'>"_-½m-£)/.

_'>._l- _(+,-_)i

"B : _. ".5"+'5" ,_,:'_':5

! I !

if

_I 0 ,.I {"_o

121

"i < 0

(3.54)

wheroz

s_ |

is the

fourth

order

"elasticity"

tensor

of

S.ublayer s As

i_ the wuighting

Sd and

sp

are material

I

sublayer sTY u

factor

strain-rate

uniaxial of

s%

stress

loading,

sublayer

scalar

_ublayer

s

aonstants

of

s

is the Kirchhoff o

of

at yield

in static

in

conditions,

s

factor

dissipation

that

characterizes

of sublayer

the

s.

m

It is evident constants

Sd and

s, a very

complex

the present to be

that by considering Sp,

and of the

material

numerical

the same

"elasticity"

behavior

calculations

for each

different

sublayer

could

values

tensor

_E for each

be represented.

these parameters s/; that

of the material

have

sublayer

However, been

considered

is_

d _ 'd - _a - "d - 3d ,".... ="_d for the present

to be

It should 3.36,

and

ca1 model,

Impact

for a few numerical strain-rate

also

3.54 are and

analysis

¢3._s_

analysis.

In addition, considered

in

independent,

be mentioned not

the actual

for these,

the

of 6061-T651

calculations

that

the

loading

reader

aluminum 122

the material

in which

loading

case:

conditions

conditions

should

alloy

turn

has been

used

3.35,

in the numerl-

to Sections

structures.

of Eq.

4 and

5.

J 3.33.4_utation !

to Uniaxial

The

the mo_hanioal-sublayer-model

will s assumed

A

s

Weighting.

3.3.4.1!_ A__lioation

A

as

of Me_hanicpi-Sublay_erTM0dul

determination be eonsldered that

the

of

in the

Kirohhoff

following.

stress

the sum of , components

Str6ss-Strain

As

Conditions Weighting

indleated

Y at a material

(s_, s _ l,

Faators

in Eq.

point

..., n) with

fa_tor

3.29,

it is

can _a considered

weighting

factors

:

The weighting

factors

A s may

be

dimensional,

or three-dimensional

dimensional

stress

static

conditions,

stress-straln

antisymmetrlc

curve

in Kirchhoff

space,

as shown

[I14],

among

stress

Eq.

2.402,

Considering

(denoted

by subscript

is assumed

(Tu ) versus o

by the

ons-dimenslonal,

conditions.

of the material stress

From

for either

the unlaxlal

approximately

others.

selected

classic

two-

oneu)

to be perfectly

logarithmic experiments

the logarithmic

strain of G.I.

strain

(e u) Taylor

is

(3.58) where

£(£

) is the final

o

E-

is the relative extensometers

(original)

gage

length

and

1-I0 _ o

elongation,

(3.591 or

"engineering

strain"

that

strain

Eq.

This

static

2.424,

the uniaxial

stress-strain

Kirchhoff

curve

stress

is first

is:

approximated s

segments

which

..., hi; see Fig. any point

2a.

are

defined

Next,

in the material,

at coordinates

the material

[s(Tuo ) ,

is envisioned

of n egually-strained

123

I

or

can provide.

From

linear

gages

by n+l piecewise* i, 2, (Eu) , s =

as consisting,

sublayers

of elastic,

at

+ _erfeotly[plastie modulus yield

E as the idealized stress

script By

matgrial,

(denoted

hy

o in T u ) yield o

(see Nig.

with

eaQh

sublayer

material,

but

superscript

y).

stresu

(superscript

having

the

same

an appropriately Fez example, y) of thQ

the

history

with

and

collectively the

stress

(_)

may

(sub-

is given

_b)"

the Kirchhoff

associated

static

s sublayer

I Then,

elastic

different

stress

the

sth sublayor

the value with (T u

value

of the

under

an appropriate

) at the material

static

conditions,

can be defined

strain

uniquely

_u at that material

weighting point

,.o,,

s

factor

(Tu ), o by

the strain

point.

A s for each

corresponding

Taken sublayer,

to logarithmic

strain

o

he expressed

as,

where the uniaxial weighting be confirmed to be:

factor

A S for the

E2 S

sth sublayer

may

readily

"

E

(3.63)

where E:"

E

(Young's

modulus

S-l_#

of the material)

Is : 2, 3, ..., n)

EI,-o The

elastic

constitutive

perfectly-plastic

relations

may

and

be treated

the

elastic

as special

linear cases.

strain-hardenlng In the case

of

+_s previously mentioned this assumption is not necessaryl by employing different elastic modulilSE, more complicated material behavior can be represented. 124

elastic

perfectly-plastic

case

lineal

of

limit

of the

formation of

strain-hardening

second

in that

sublayer

sublayer

(Tu

since

)Y of the

there

is taken

proper

sublayers,

there

one

are

sufficiently

elastic.

method

with

is only

material

remains

the mechanical-sublayer

are utilized, s

behavior,

complet_ material

sublayern

high

so that

the doadvantage

or more

sublayers

the yield

behavior

and the

the main

if throe

of

in the

two

However,

is realized adjustment

sublayer;

stresses

can be

represented,

o

including sisl

elastlc-plastic

see Fig.

unloading,

the Bauschinger

dependent, elastic + is described by s

rate dependence

strain-hardening

+I D

and hystere-

2c.

For a strain-rate

where

effect,

is the uniaxial U

component

--

of the

material,

)

-

rate-of-deformation

_

the

tensor_

e

(3.65)

i+E W

that

is equal

shoWn

previously yield

to the material

stress

the

constants

logarithmic

and

S(ru)Y

is the

university

stress-strain

_onstants

strain

E u , as

strain-rate

dependent

s. strain-rate

to represent

response

d and p are obtained

strain-rate

of the

3.65 is the Cowper-Symonds

[139] at Brown

uniaxial

2.405,

of sublayer

Equation in 1957

in Eq.

rate

the

of metals.

from

developed

strain-rate

effect

material

straln-rate

The

experiments.

d and p are chosen

equation

When

to be equal

on

the material

for each

sublayer,

0W

the

stress-strain

stant

curve

magnification

from the

Kirchhoff

3.3.4.2 Generally, sublayer

a_ a given

of the stress

static versus

A_lication

deformation stress-strain

logarithmic

to Multia_ial

a somewhat

different

model

is needed

when

occur.

Fowler

[140] has

derived

biaxial

stress

state

using

rate

emanating

(see Fig. 3).

Conditions

conditions

coefficients by

rays

a oon-

for the mechanical-

stress-straln

given

along

origin

Stress-Strain

the weighting

expressions

curve

strain

description

multiaxial

eu is simply

Plan

based

on a

[141] in 1966.

In

+As p:eviously mentioned the material straln-rate constants d and p, can be assumed to ue different for each sublayer s, thereby representing very complicated strain-rate material behavior. 125

1974

Stalk

stress

[142] derived

Fowler

stress-strain a

unlaxlal

concluded

state

based

than

approximation _e

duced

by

states

based

on a triaxial

comparisons

of

However, a two

Hunsaker

sublayer

Hunsaker

which

determination

stress-strain

of the state

weights

when

et

al.

cent

of

produced

[144] obtained

a closed-form

differences order

of the materlal [97] in 1953 properties

of stress-strain. Hunsaker's

are

between of the

that

the errors

the

stress

from

states

hardening)

[]40]

intro-

or

The

the tmiaxial typical

stress

calculation

of

are present.

No

coefficients

even

solution

model.

to

weights.

weighting

shown,

the

not lead

in the sublayer

states

the

should

for three-dimensional

and multiaxlal

strain

does

that

[143] disuussed

multiaxial curves

aertalnly,

model

|141] concluded

of 1 to 4 per

Fowler

... it is concluded

weights

from

of a stralght-lino-scgmont

in a blaxlal

Stalk

are of the

sublayer

are present),

curve,

the

based

obtained

small.

the use

(linear

Besseling

very

from

Hunsaker

[143]shows

cedures

wore

resulting

stress-strain

on unlaxlal

the diagrams

state

the one-dimensional

weights

and

between

coefficients

this difference,

model

recently,

the w_ightlng

from

is of the order

the sublayer

from

the differences

resulting

errors".

using

that

of stress-strain

of the

unlaxial

More

based

obtained

that

significant

concluded

on a multlaxlal

"th(_ error

be smal%er

use of

and Stalk

dlaqrams

coefflc_onts

any

coefficients

state.

Both

on

the weighting

discussed.

for the

example

case

shown

5Y

and multiaxlal

experimental

errors

of

proin the

properties. had already

obtained

(for an_number It is easy

of sublayers)

to show

closed-form

a closed

that

solutlon

form for

(when only

coincides

with

solution

a general two sublayers Bessellng's

formula. One and

can readily

stresses

by the

show, total

that

upon

strains

become:

126

and

replacing stresses,

the deviatoric Besseling's

strains formulae

I ....

Z --

£r

soon $ from

It iS easily

assuming

elastic

identical (Eqs.

with

3.61 and

between

the

those

derived

3.63).

Also,

3.66a

that

and

3.66b)

properties

stress-strain

it is interesting derived

(3. _6b)

for U _ 1/2

the sublayer

from unlaxial

properties

(Eqs.

,/_

equations,

incompressibility),

s_blayer

and multiaxial

those

--

from

to pote uniaxlal

(i.o.,

becomo

conditions that

the difference

(Eqs.

3.61

conditions

is directly

the difference

between

and

related

3.63) to

l the

factor

(_-

(V) and plastic ratios.

(assumed

to be equal

in the present ++

and

3.63)

for the plate

3.3.5

for example,

(and beam)

the total

strain

condition

(Eq. 3.45)

elastic

rate

(under

the

Behavior

if the initial

9 is to be used

++Both for plastic and elastic assumed to be small.

in Eqs.

strains,

127

!

strain

3.66a

and

rate

rather

is to remain

3.66b.

the first apply.

since

(as indicated,

(non-stationary)

condition

equations hold for s > 11 for s = i, only and only the first two terms of Eq. 3.66b

assump-

report.

reasons

the dynamic yield

procedure

Modeling

as theoretical

govern

in thickness;

the uniaxial

of this

[145-152] ), the plastic should

and shells,

incompressibility

FE calculations

as well

Poisson's

plates,

the changes from

the elastic +

analysis)

for beams,

properties

on Strain-Rate

by Perzyna

in the

in calculating

is consistent

of physical

to 1/2

analysis

of sublayer

Comments

Because

+The

is assumed

the calculation

(Eqs. 3.61 tion)

expresses

Moreover,

incompressibility hence,

9), which

Note

than

yield the

that

these

term of Eq.

3.65a

the elastic

strains

are

same

as in plasticity

dynamical [147],

yield

theory.

condition

it is conveninnt

In order

of this work to express

Eq.

to relate

th_ _quatlon

with

thn equations

3.49

in terms

for the

Qf perzyna

of the

following

invariantsl

and

the yield

stress

J , Then,

in shear,

as:

Y

_

from Eq.

defined

°_

(

,j_)z

( r,, 3 ....

=

3.49 +

(3.e9) i

'

--D:

Z '

'_ °:'?"=z s:J.=_-,,., "_ _f'Z. q'[I +i.......

(_,o)

I

or

s

which

would

invariant

be

_

identical

of the plastic

t

with

Eq.

strain

2.68 of Perzyna

(3.72)

[147] if the

second

rate

-.B

+Since superscript "p" is used here to denote plastic components, the strain*rate constant sp for sublayer "s" (see Eq. 3.49) is replaced only in S_bsection 3.3.5 by tSe symbol s@ to avoid confusion. 128

I i

I

I i

were

used

inst¢_ad af the

ID.

Also

obsnrve,

that

tension

Sd and

i _

in simple

'1

same

'

aft th,_ relation

(a_mpar,_ with

Eq.

equation

tho

plnstle

strain

can

also be

related

using

relntion

the

of the dnviatoric

between

the viscosity

between

[3,52]

fnr rate

strobe

strain

th_ viscosity

coefficient

yield

the rl_ttla_ £aetor to

thai

coefficient

in shear

in tnnsion

rate

s7 is the

and It, shear

of propartianallty

a]_ i:olating

dov,_,atori¢__,t,ronnt

to the equations

(viscoplastic)

fo Then,

the

invariant

3.691 !

The

the dissipated

second

work

of P0rzyna

[147],

by

per. unit mass

as..

and

obtains

expr_mslng

-n

Eqs.

3.75,

3.72,

3.69p

3.52, one

376 or

$

_,

,

which

S_

is identical

with

Eq.

2.77 of Perzyna

slp.

(3.771

D is replaced [147] if 12

I The

strain

rate

equation +

-_

_, +This

by

equation

• applies

(3.78) only

for

_/ (_)P 129

# O; thenj

T u = ,Tlt U > TyU " o

used

in the present

work

can represent

socondaEy

creep,

since

for

cQn_t_I_t str_ss

(3.78)

Thus,Eg. 3.78 tax,bu Qxpr_suodau

+ which

iu the powur

However,

th_ strain

ruprescnt strain

law

• m

e_fects.

E

u

in the present

In effect,

used

condition

static

in Eq.

yield

f0_

_

if the plastiC Were

_f secondary work

for relaxation,

creep.

cannot the total

(3.83)

the

tO the

"[_ =

rate

usud

law)

= 0

3.78 expresses

instantaneously

However,

_quation

aL_ Norton's

is zero"

E

and Eq.

rate

relaxation

rate

(alsu known

strain

_ate

that stress

the Tyu

stress

Tu relaxes

o

"_ --- O e_

(_u;

(3.841

rather

than

the

for T u

> Ty _ • o

total

strain

3.78,

+However, secondary creep is present ar_ temperature dependent, 130

only

Also,

d and

Thon,

for rf_laxations

ff_ = O For example, I

=

this

exponential

E

equation

can be

the

solvsd

13.8_)

for @ = i, yislding

an

relaxation:

=

where

-----,._y - I

+ d

+

relaxation

constant

Uo

R is

T,No y

(3.88)

E If many that

creep

r_covery,

represented Only usually

sublayers

by Eq. total

that

each

result

curve

in strain

straln-rate

magnification

along

rays

as well

(rather

rate

strain (page

constants

the present

stress-strain

constant

rates

by Campbell

sublayer,

rather

it can be shown

as secondary

curves o_

emanating

than plastic

tests;

therefore,

rates

are small

52 of

[153]),

d and p are

mechanical

creep, can be

the static from

strain

strain

When

to be equal

model rate

are

produces

that

to

experiments,

for example,

(rate-independent)

the origin

rates)

it is necessary

in those

chosen

sublayer

at a given

131

!

than one,

3.85.

the elastic

as indicated material

and primary

strain

measured

assume

are present

are

the

for as a

simply

a

stress-strain

of the Kirchhoff

stress

vs.

Iogarlthmle

strain

Hac_regor

curv_.

Thln

[154] and by Wulf

is the b_havior

[155]

in n number

that was

observed

of experlmsntn,

by"

among

others. In any case, rato_

the

can be d_ducod

difference _om

the

between

following

the

total

argument

and

the plastlc

for a unlaxial

strain

tent:

£_.I-L t--_ "

_

=

_)

total

I(')

_)"

_

(_+

relative

_

,

"_.

C_ _

_)

E the

total

logarithmic

rate of

and plastic

uniaxial

Kirchhoff

strain

(elastic)

rate

=

strain

E* u

elastic

Young's

Decomposing

where

unJ axial

material

_P

elongation

parts

of £u' respectively

stress modulus

gU into elastlc

and plastic

F..,u,)

parts:

1a.89)

since

± "'"- E T_ the elastic

strain

rate (6U)e is related

one

to the

stress

rate

(e:.)P as

£:

one obtains

Hence,

(3.9o_

can express

f)., P-"

F.,..,.

.

Since

the

versus

tangent

modulus

logarithmic

quantity

ET/E

=

materials

For 6061-T651

ET, = (dTu)/(de)u of the Kirchhoff stress strain £ curve is small for most metals, the , u

(dTu/d_u)]E

the ualculatisns lowing

1+.').'+I

ii

in the

is small

present

work

compared have been

with

unity.

carried

For

Tu

example,

out with

the fol-

•

aluminum:

: _.de++

;o_

.00_-xCa'-o>, a°[N. O, the

that

is not valid.

the actual

bohavlor

function

timo

str[,_, inoromont_

|lolw.n, the act_a.I strnnn

the contrary,

of the

fo_: this

strains

during (C-l)

a finite

time

increment

At by

taking

ij

i

, Cij to be

I

225

! .

-

"

_.....

:

::i 2:

.

3:,.

,

•........

..

Therefore,

on_

incromont

(c-'yJ.(c")_

'_'_'"

Cti : (C_j)t

(°""'

the

obtainu

following

eXp_Ol_aion

£0=

the

aotual

atro_o

SgtJ

AlssiJ), m |.

m

,

,,

.

ii

i

i

(5.320)

L

due _¢

due +o

A'Y_j

A s y_

where

The

actual

stress

at time

t is

-_(,,_,_),_(,_,)t3es'_),.,,(c-.)_J's'}_ """' •,L

The parameter degree

polynomial

from th_ satisfy stress

A(B_ *) will

in _(sl*).

condition th_

loading

(SsiJ) t at

Expressing

be obtained

that

the

This actual

function timu

t is,

from

second

the solution degree

stresses

(s_) t = O. indeed,

"226

Of a second

polynomial

(SsiJ) t at time

This

located

this mathematically:

'

condition exactly

is obtained t must

_.nsures that

on the yield

the

surface.

(c,,_j'+.._ (c,,_, (_,,_!_ [('s'0Z +{3[(c,,_,l'-(c,,],(c,a,_ (,s"),(,s'% - [('zb,]" where (STY) ut

is obtainea _rom Bq. 5,310.

Substituting Eq. 5.322 into

Bq. 5.323 and solving fez A(B_ *) one obtains the physically valid value_

_5-_

(,x,)- .......A.

,

G

cs324,

_ B*-_F'6'r:-A'_"

where z

* {_[(c,aZ-(c,,>_ (c,,>,_ ('m")('_-) --[(c,,>j(,_")_(,_"),-[(c.),]'(,_"),('_")

227

! .......

TSE1

p

T'

Tho

coo£_Iclont

roqulromont8

must

During problems,

C wa_

instances value

this difficulty

the size

At,

of the

a positive

real

successively,

of

of large of sA*.

strain A

increments

into

is chosen

value

for each

of _sA,

The value

correct.

procedure,

a valid

calculated

for each

strains

are

(a) the information negative

value

Then,

subinterval, kept

updated.

needed

(t - At)

can be derived henceforth

foll_wlng

using

successfully,

set to unity

_ntil

the

be derived

increments

latter

value

of L.

solution

an imaginary

encountered.

228

_7i j during

_7ij/L. It is change in strain mentioned

increments

case,

or

the

until

and either

(b) a complex

the process

continues

or negative

d(Ss ij) are

stresses

If the stresses

procedure

The basic

smal _ so that

is continued

t is calculated

In the

a larger

can

in the meanwhile,

to

sublncrements;

the previously stress

lead

to circumvent

into L equal parts, of length dt/L this

with

loadlng

which

sufficiently

of the strain

The process

at time

may occur

subincrement

along and

intense

say L, of equal

to be

by employing

for _sA*

AsA * is encountered.

from time

Tho

[165] is employed.

a number,

subincrements

for

procedure

by Hufflngton

the time interval At are also divided assumed that during each subincrement

plastic

5.308-5.312.

process

subincremental

is divided

as follows.

is approximately

An _q_.

the solution

as developed

increment

di_playod

bo _ti_fiod8

the operation

an imaginary

time

already

or

is repeated at

time

with

L

AsA * is again

t

_I'_¢,",P]'ON G llOVl,:lh%llNU I,:QIIATTONII AND

H(H,IIq'TtIN

PROt I,,DIII,II,,,,

_!__ A...J:!F:r?dn-'!J:,!:P_ I_ 'Ill

Hit n 'inv,_;t

¢l._ljl]ll!.}_

o|:

S[.I'UCtLII'o_I

t|loS_'arit_inq from

consideration called

is the

"material

al_d effects impulsive

haps the of

of

to several

structural

response

comp01_ents

responses

material

as well

with

tile time

tioll eol%dltion of tral_sient

the

D'A]t,II_Dt, Ft'r;

lUt'thodH

(a)

is

strains,

structure

of _evere

termed

of

"structural

here;

z_ro

span

impact

I to i00 microseconds.

from

such

response"

responses or per-

type of res_xgnse I_rtains

behavior rings,

of overall plates,

study

strains

u_x)n transient

large

and time-dependent

information primary

oli both interest

aft or subsidence

of

or

shells.

centers

including

to

structures

and

or

interest

to 1 millisecond

of that p o.akkand

the l ] k,'

peru tilt,

flni.tu ulun_:nt equatlonl_

statement Pr]nclph,. vectcn" Ct,lltral-d|

cenz;istilm The

rotations elastic-

the _k

transient

ell stra.lns)

tile permanent

deforma-

the externally-app].led

form

_,qtlationH

(characttur.tstlt;

florence

t_f 111oi:1o11 are del'Ived

of the PY1.nc_ph,

remlltJn(]

opel'ator)

229

1

prepaqat.ion,

of

time

with

is often

loadil_q.

from a variat|ona]

wayt_:

Sought

from

t.he tlme

ill this

finite

such

rouqhly

this

of occurrence

Ill thi_; s[,etiou, the

and

from

nature,

as a result

is discussed

as path-deI_ndent

and

to the

is usually

int0rest

behavior.

(deflections

together

pertains

.t'ri]l_.,.!.10_;_

]oadz_

ret_ponso which

st retqhing

involving

t,x't-_rnal

time"

such as beams,

principal

t'h,,

"early

tile order

(:xtendh_g

and/or

trans._eut

to

,:lll,:i:lyI':lllq

l.:xpllc.it]yexcluded

or

which

dew'_tod

for

e.te.

structure;

is of

{:o Illlc_thodl'l

Impact,

hl_,dred milliseconds;

deflections,

responses

to

in tile material

to the

interest

Furthermore,

plastic

sub'joctod

and which

waves

trallsient bending

and

atl_,l_l"ion

wihl_ '%natorial response")

times

structural

p'rino.'ipa]

arc,

time"

r'esponse",

time"

:|,n IN_t'd',,ct,t;t-od

blast,

"short

type of respollSe

(in contrast involw:

gunts,

of stress

the "late

with

whl.¢:h

loads app1_od

_or this Only

lit |:onl:iol)

',_tl:l)_!J.UJJ/i[:l. l"vnpt_rl;u,,

1"esl_onlleS as

I.qat |(_lI,

of ,

(b)

of Vlrtual

can exl,lictt tilt,

be

Wo;k

._,_lved t_olutton

I't_U,_It,lllt

in

thrt,t_

by

;;t i fflle,q,H,

and

(_:) tha tanqent

with

Impllelt

do _Mpllcit Per terest

operators

offleiont

however,

motion,

the

tional"

form and

_xhlbit

lant

bettor

path-dependent, work,

resulting

two

formn

stabillty

are often

prop_rtli_s

for small-strain,

St takes

more

"pure

unod thfln

developed

formsz

form.

storage)

The

for _

is the

best

of "conven-

new

"modified

kind

of ma-

is valid

In addition,

than

the

(a) the

formulation

is more

they

the equations

to be applicable

formulation"

in-

form of

formulation

in two

"aonventional"

of

slnco

vector"

form of

materials.

computer

problems

are used,

unconventional"

elastic-plastic

uncouventioml

the

stlffnes_"

is shown

the usua_

forms

"u_Iconvontloral"

ec_latlons are

while

two For

"_onstant

formulation

ti_,-do}_ndont

first

(b) the "modified

behavior,

"modified

the

the so-called

for th_

unconventional"

(although

those

computatlonally.

of motion,

to usol

terial

which

transient,

in the present

equations

forml

o_]rators.

the

are more

the

stiffness

only

it is shown

efficient

that

and economical

is the conventional

formu-

lation. A brief tors

review

suitable

equations

Equations 6.2.1

Variational

most

element

(a) they that are

invarlant,

any particular d[fferentlal

used.

investigated.

Also,

the

opera-

solution

of

is discussed.

..thin the Plan

of physical

about

they

description,

as a whole,

refer

to derive

(c) they

that

can be developed

Variational

following

rather

than

imply

bounda=¥

the parts

as well

thb effects

reactions

they

appropriate

eondltions

Include

advantagest

of a scalar,

tllc si_uelal forms

the corresl_3ndlng

230

the

of the

of variational

[166].

to the uxtremum

(d) they automatically

requiring

have

version

method

framework

and Tong

laws,

a system

and n_uy bc used

without

by

assumed-dlsplacement

finlte-element

conveniently

(b) since

equations,

the The

for example,

statements

it _omprises,

straints,

and

as expressions

are

flnite-difference

Formulation

n_ethod was

as shown,

principles,

of motion

investigation,

systematically

principles

being

timewise

of MotioD

In %he present finite

of different

for the problem

the governing

6.2

is made

to as

of con-

be known,

I I (_,)

t:h_,y

haw,

l_tun:'t,_ftc,

vnlut,

G_r _luqq,,_f:ln,l

_l_norali_:atlon:_,

(F)

th_W

nrt_

t i only.

I

CoI_II'I i|_,_r _I cod eitlllUlll it%O.qlI_ ] _br :I nllllltldL_r the a¢_.t _.on of bo,.ly for_4,1l, e.xtol'll_ll ] y-app] 'h,d l_ur£a(_.t , t:_'at_i:Jnn_l, azl(lwl th arbitrary de format ion {_Iilrdlt].ons _c_n_lII1l'_Ilt wi.th the,

I

t'hI.:_ _qul lJbr st,t of

I

UIllllc.olli_|._Turat:[oll be sub

(IOCIIIIO.tl?._C! bO_l_]aYy

The

conditions.

cause

they

need

under

tile given

loads,

d_splacements.

The

[50],

[168])

[167] and (body

displaoen_nt

not be actual

forces

} are not the

(G. 4o)

nodal

completely nodal

generalized

generalized

displacements

independent,

displacements

{q)

a transformation {q} to independent

displacements

{q*}

for different is required global

to

(o_

for the discrete-element

by

fq}- JJf '; The quantity coordinates for the

of the

from each

system

Applying

[ J]

includes individual

the effect element

of transferring

to global

from

reference

local

coordinates

as a whole. Eq.

independent

6.44

to Eqs.

global

6.38-6.41

generalized

to describe

displacements

the system {q*}, one

in terms

obtains..

L_p (r_'J I_'I, I_'i+ r_'j I¢ I-[_'I); o

(6.45)

I

I

243

I

wh_ro

[h']- [j]" [hJ[j] Since

the

[h] involve

matrix

nonlinear

is no practical analysis,

square

geometric

reason

and this

[hi is not effects

to calculate

is not done.

as well

It is more

and

since

as plastic

convenient

Eqs.

6.37

and

6.4_

5ecome:

244

bot.__±h {p} and

effects,

[h] explicitly

there

in the

to express

ja{_1-L_'aJE_alf} :X_j ( o)e

Therefore,

a constant,

the matrix

LD _

and hence

,o.,_,

,o._o,

I Pt:rformLnB funot

the

I.o1_ gonoral.t

tlrt_ :lndt,pondont nro

ohtainod _

i

t_ummatlon

zod and

for _

dl_pJ aaomont arbttrary,

the I

fn

ec_mpleto

tnvoktn_; the ;_pproprblto

eompat tht It I:I o_1, and

tim. fol.lowfng a_mt, mbled

•

!

Eq0 L_,_3_

vectors"

d lnc'r_t:tzod

l_ocaum,

nquatlont_

the

o.]om_nt {'&l_ }

of mt_tlon

ntructure:

i

- -{z?,,,,,,, ,.iF} l,,

,o_,

where [_I]:istl_eglobal mass matrix, {I} Is a vector of internal foroes assoe.iated with llnuar and nonlinear terms of the strain displacement relations as well as elastic and plastic foroesl and {F} represents the generalized load vector accounting for externally-applied distributed or concentrated

loads.

Xn terms of element information,

[M], {X] and {F}

may be expressed as.

OQOIb

°

(6.56)

(6.571

:

245

i

11

6.2.3.2 Two The be

first

type

Eijk_

of the

valid

only

elastic

parts).

strain

For

be a quantity By means Eq.

6.59

not

affected

have

be presented.

formulation,

S ij in Eq.

and

given

y_

6.33 by

since

parts

strains

that

this

for finite

on the total

strains

represents

initial

be evident

strains,

finite

will

which the

may

follow-

7k E

constants

depend

force

tensor

(or other

and plastic

will

pseudo

it should

it will

formulations

the strains

strain

int_ elastic

"elastic"

of

Of course,

but

stiffness

the stress

for infinitesimal

and plastic

express

of

plastic

Form

"conventional"

in terms

a constantl

will

is the

etc.).

strain

of constant

consists

total

strain,

Stiffness

by replacing

expression

where

be

types

obtained

ing

Constant

strain

the

usual

meaning

by the

total

deformation°

of the strain-displacement

as thermal

formulation

is

E i3k£" cannot

(both _he elastic of

the total

concept,

of elastic

equations

components

strains

the decomposition is not a useful

the such

(6.28)_

since

strains,

one

Green

the Green but

can

as:

+Since experiments that the constant

on polycrystals elastic modulus

logarithmic strain, (Lagrangian) strain 5. 297-5. 299.%.

with a cubic c_ystal structure confirm relates the Kirchhoff stress and the

and not the 2nd Piola-Kirchhoff when finite strains are present

stress and the Green (see Lqs. 4.167 and

246

00000003-TSGC

Performing

the

compatibility, arbitrary, onlz, for

summation

invoking

and because

interelcment

the variation

the £ollowin_j conventional small

strains

[_q*]

generalized ean be

equilibrium

displacement

independent

equation,

which

and is valid

is obtained:

E_J{_,'J+_KJ{¢]-tr_+tq "_7_t*IF, } ,_._, 247

whore

[M] is th_ global

olastlc

global

Vector

(aonstant)

ropres_ntlng

IFNL}

represents

vectors with

due'to

only

to

evident

than The

onQ

terms

"unconventional" other kind

hand,

However,

matrix fore, the

it is

formulation"

unconventional"

constant

elastic,

of the

is an exact

assumptions

whatsoever

about

the

side

to have

method,

matrix

made

only

is _%at

in the

as Eq. 6.59,

materials. by Eq.

gradient

the mass

6.55,

matrix matrix.

properties time

On the

for the special

given

as expressed

is

preserving

the small-strain

[K] is added

tO both

of motion_

248

., ..

sides

form of

has

(the There-

similar

formulation,

unconventional

and

of the Principle

equation

the only

next.

for small

equations.

is valid

at the same

modified

•

been

and convergence

while

"unconventional"

the following

since

o_mputational

for this

expression

of the equation)

stability

is valid

of elastic-plastic

fQrmulation,

more

to be presented

constitutive

the constitutive

problems,

much

reason

have

formulation

strains

6.55

The

formulation

constant-stlffness

to obtain

of Eq.

of materlal.

for finite

stiffness

involves

formulation

formulation

left hand

to be able

the structural

plates, and shells),

convergence

on the

properties

6.55

and

Is applicable

of beams,

the "unconventional"

stability

oE

equations.

rotations

that obeys

valid

that

description

in

respectively,

relations

the "conventional"

is not

the drawback

tennn

strain-displacement

formulation

of material

which

No

the nonllnear

lo_ds,

nonlinear

finlt______eestrains, for anykind

Work.

from

load

concentrated

the strain-displacement

have

llnnar-

to employ

"unconventional"

of Virtual

of

formulation

"modified

the "unconventional"

arising

or

straln_i and are aosoalato_,

the "conventional

the

small_stroln

{F} is the g_nernli_d

dlstributod

but if an adequate

for finite

that

matrix,

load vector

nonllnoar this

[K] is the u_ual

applied

(small)

strains,

requires

(specially

work

and does

small

behavior

externally

plastic

only

matrl_,

ntlffnoss

a pseudo

the _inonr Not

mass

to

the useful linear-

of the Eq. the equations

r'

I Observe j

that

this oqnatlon

of material,

_ pi_uud_-furee

as well

dlsplaeument

This

formulation

tangent

the reader

j

!

implicit 6.55

nonlinear) Eq.

"modified

terms

6.69

made.

Dnfining

ntr,%_l%_lal_tlc-plastlc of the

ntrain-

as.

unconventional"

form

of the

6.72b

of the equations

while explicit

of the equations Work

the tangent

report

for any

equations

Work)

the

formulation

"unconventional"

time

operators.

Form

of Virtual

of Virtual

Eq.

unconventional"

time operators,

form

form of the

respectively,

"modified

is to be used with

stiffness

Principle

vector

this

Stiffness

in the present

Subtracting form

the

is re/ninded that

the Principle written,

with

Tangent

from the

The

and

_an express

subsection,

of _q.

6.2.3.3

utilized

havrJ b_on

for any l_ind

of motion.

is to be used

here

one

asflumpt_onn

_%r_sing i.Tromfinite

(lln_a_

is called

In the next

The

as all

equations,

expression

equations

for finit_l ntr¢_in_], and

slncL1 no _onntitutivo

wher¢_ {VN_'} is b_havior

i_] valid

at

for

of motion

completeness

stiffness

time instants

be derived

purposes,

formulation

computations of motion

will

but

is not

or predictions.

(Eq. 6.55 t and

derived

t-At may

from

be

aF

from Eo

_.72a,

of motion.

249

one

obtains

the

following

incremental

Next,

th_

increment

of the

internal

force

vector

{_I}

is treated

as a

differential:

[w1{A{] ,o.,,, Hence,

one obtains

the

following

"tangent

stiffness"

form of the equations

of motion: i

where and

This

the "unbalanced consists

error

current

of writing

term

consists

increment

the error correction

force"

would

I

{fu}

the residual

equal

to

in the equations

for Eq.

the terms

had been

zero).

to the error

equation

of evaluating

(if no errors be

is due

of motion,

25O

one may

in Eq.

bQfore

by previous

this obtain

6.73

6.55:

at the stat_

introduced

By including

implicit

residual convergent

the

increments load solutions

U_]iI1qtim_l incromonts _]oluti_n_] obtain_d

and,

that

without

P_'om Eqf_. _,37,

6.39,

slnuo,

6.73

from Eq.

_r_ r_l_tivel¥ tho

Inrge

in eompazJflon

with

_orr_letlon.

6.40,

._nd 6.51e

ono obtaills

EA.'3--_ it follows

th_

_°.-,

that

-(_'_. By means

of

the strain-displa=ement

equations,

6.28,

one

tan write.

_'_ - L:_._+_k_{_o_}_4_I_{_}_o_ ,o.,,, Placing

Nq.

for finite

6.79

into Eq.

element

"e"..

6.78,

one obtalna

251

I

the following

tangent

stiffness

)

16.'/9) [k_] I

It be 0mphasized this tangont matrix uponshould the current state ofthat displacement {q} stiffness and stress. Also, Eqs.

6.79,

involvsd

in the

formation for

6.63,

6.76, and 6.70, formulation

of the

internal

the unconventional

6.3

Difference

6.3.1

Linear

the

structural

timewise

their

many

how

termed

"unconditionally

critical

introduce

the

value

this

time

increment

stable"l

-- and

a phase-shi_t

upon

size of

the

error

matrix

than

in the

unconventional

than others

of undamped

and/or

or false All

/'t.

explored

are stable

to be -- and hence for _t

damping

Of these

in the predicted -- some

dynamic been

schemes

te._med "conditionally

error

for a given

Some

have

are unstable

feature.

_t used

linear

operators

is chosen

others

artificial

undesirable

finite

At

thus are

produce

shift

solution

and shortcomings.

usually* the

for the modified

finite-dlfference

(unintentionally)

do not exhibit

stiffness

are

Systems

attributes

large

tangsnt

aal0ulatlons

Operators

numerical

matter

some

forces

Dynamic

problems,

to assess

of ths

that more

formulations.

Finite

For

it is _vldent

depends comparing

schemes

A concise

larger

stable". whereas

than Some

however,

response,

tabulation

are

others

methods,

exhibit

no

depending

more

phase-

[177] of

*An exception, however, has been noted in Ref. 176 wherein the 3-point centra]-di=ferenee formula was used t.o solve the one dimensional wave equation. When At was chosen such that (At)/(Ax) = i, a solution w_ieh was exact in both a_plitdde and phase was obtained. Second, the Gurtin averaging operator with c = 0 exhibits no phase shift error but only with one

(much

too large)

value

of At;

false 252

damping

also

is Present.

I I som_ of thn fuatures of the morn commonly-usQd "vari_tios of thin mothod I

aro qiv_:n bo]ow,

FOI_ IINDAMP_D L_NEAR DYNAMIC SYSTEMS

I

h11owablo At _Or Condition-

I Motl_od II I I I

1 i

(MATII,MODESt)

I

i

Phaso Lh%oon4£t4,on- li'almo @lly Stablo __ i

ally Stable Mothod ii i i

im

Shi_t 9rro___.rr , ii,

_XPLICIT Contral Diff. 3 Pt.

_t < _ --(_max

No

No

Yo_

3rd Order Rungs Kutta

At _ _/_max

No

Yes

Yos

4U_ Order Rungs Kutta

At !

No

Yos

Yes

No

Yes

Yes

---

Yes

"as

Yes

...

Yes

NO

YeS

No

No

Yes

de Vog_laere

(I)

2/_/__a x''

At < 2_/w - --max

IMPLICIT I

Houbolt +Newmark 8

"I y=l_8=

! 4

1 0 1.00

finite

at the

(0, 1.50

in_ 90 deg),

as shown

in Fig_

making

_oeation

from

of upper-surface

strain

cent.

are unreliable

tb_se pez_manent-strain

EL-S____.HHpermanent relative

elongations

5 _t

stations coarse

accordingly,

(x,y,8)

3_

relative

r_gion.

elongation

y-dlrection

region

that

larger

relative

of evident

are not

It As seen to be

the _L-SH

gage

and, hence,

tend

_or

in this

in the

row 300

are mad_

a p_rmanent

estimates.

about

rolatlvuly

transient

Strains

For

unreliabls!

8trains

of 6.5 per

deterioration

estimate.

strains

to estimate

by

thn

l through

th_ asoociatod

predicted

in

atat_on_

at nodon

strain

ease

600 _Inoc end o_

nodal

the EL-SH

_tate

nodal

to the

the predictod

29a was used

there

at

EL-SH

olomnntn

final

elongations

that

thn

_mall

f_n

unchanged

estimates

for

thn

the at_aina

the permanent

makes

However,

of

re_ohod

in, At is bslievod mesh

prodi_tion_

onn_o¢

relative

_or

no permanentstr_In

numerical

th_

almost

the

chosen

_loment

elongation

at

3 nhow_,

_.00 in _omain

calculation.

ntrain

on_ntially

£n fast a_ T_o

0 _ x _

with

thQ

employed these

than

in

predicted

the measured

values. For

the EL-SH-SR

the permanent strain

relative

estimate"

Included

also

than

permanent

from

measured

values

of element

relative

the EL-SH

was

this

relative 6.

time was

used

elongation

are

out

stations

_t is seen

and

a tendency

carried

center

elongations

prediction

with

at

and element

th_ permanent

center

which

elongatlon

for nodal

was

outer-surface EL-SH-SR

calculation

300 _suc,

as the

"pezmanent-

at 0 < x < 1.00 in.

(at 300 _sec)

that

these

in the mean,

at

the

predicted

(1) considerably

(2) in reasonably

of being

to only

good

smaller

agreement

perhaps,

with

somewhat

smaller. It should provide

b_ noted

displacemenh

that

gradient_

4ircction, and displacement y direction. This element finer cost

i

l

mesh

of the LLC

of greater

storage

the LLC u,x,

assumed-dlsplacement V x which

are

elements

constant

in the x

gradients u,y, v,y which arc constant i_ much too stiff! however, the use of

elements

could

and computing

321

improve expense.

the prediction,

used

but

in th_ a much at the

An ovaluation

was mad_

of thQ principal

strains

and associated

dlrsctions (Sp) on th_ upps): surfa_.s at the Qontor o_ thQ "qmall" _olnments (_¢_0 Fig. 28h) for both the ?L-filla/_d th_ _5_8|i-SR calculation. ,_, illustr_tlon

o_

_OS_

r_s1_Its are gIvon that th. moat in aach row_

£iDito-straln_pred_utod

in Tsb%o

oxtromo

5.

values

An

maxlm_m

Inspo_tlon

occur

at

o£ thosc_ v_luos

tho _ontor

EL-SH Row

Element

Value (Per Cent)

38.0

23

i0.6

4

38

30.4

38

9.5

5

57

12.3

56

8.8

it is of interest

relative

cent.

el_ngation

hi-

CP-2

close

(x -" 0.65

tests",

direction

to

value

aseful

rupture

conditions

incipient

rupture

criterion

state

preparation.

whereas

the

in the

"u/%iaxial

averaged

the experimental

matter

26.4

elongation

to an independent

This

the location

about

T, directiun)

type of aluminum

grids

that

at)

of the relative

r_pect

for this

determining

< y < 0.7) was

to assess

with

and post-test

lightly-scribed

'but not exactly

(the transverse,

incipient

mill

the pre-test

permitted

strain

the rupture

be

that

in and -0.7

or tri-axial

It would

attendant

to note

of the mechanically

o_ specimen

rupture

for this

static

its

14.3

27

the coz'respondinu

and

1

3

of incipient

40 per

37.2

13 •0

surface

olomonts

Value (Per Cent)

12

ca the upper

coupon

Element

40.0

of the spacing

cent

tho _ollowlng

16

Finally,

per

indlc_ltot_

2

measurements

permanent

oZ

strain

EL-SH-SR

5 "

1

prln_Ipal

allo).

is left

for

in about

CP-2 strain

based

6061-T651 future

study. The

computing

Houbolt-MULE impulsed

predictions

6061-T651

the following

times

required

to carry

of the transient

aluminum

for the EL-SH

thin panel

out

responses

specimen

and the EL-SH-SR

322

the .finite-strai__._________n

CP-2

of exploslvelyare

calculatlons.

summarized These

in

l _mI,_,,tatlon_ w_r,_ _nrf_rmnd

_atl. _ohav_o_

."

_o. of Plat_ _

CPU (m_nJ.,_., ............... DOF Cy_1o0

200.49

_5G.8

_ lO "6

BL-_H-SR

121

661

3_0

131.88

665._

X l0 "6

oomparisons

for othor

7.6._.I

and

Contalnmont-Rinq

At have

resulting

been

used

impact

permanent

Propulsion

in spin

Center,

chamber

while

fragments

have

impacted

multi-material

quite

strain

Rotor

the late

_tudF

are given

Tri-Hub

rotating

the

at high

containment

ring-fragment

Burst

and

in

A_tack

rp_

rotors

has been

[208-210].

photography from

Transient

strain

have

made

ring

The

of single-layer

interaction

locations

deformed

rotor

speed

history.

at various

engine

rings

High

impact

in the _esponse

the permanenbly

aircraft

in which

construction.

measurements Also,

various

tests

ways

of the rings. bee_

Air

to observe

until

pre_en_

of TS.STuzbino

in various

rotor

or multi-layer

in _u

Definition

_mployed

to fail

examp_os

7.5.6.

Response

Problem

the Naval

beuh

ca_ed

been

configurations

has initlal and

on some have

measured, Selected

cast

for analysis

steel

containment

15.00-1n

inside

on smooth to fail impact

support

in three against

geometric

properties,

!

CPU T_m_ (mln)

at

600

7,5_i

I

No. _£ Cych_

IBM 370/169

661

7.5

i

Total Unknown _OF

on tha

121

Subsectlons

and

prea_ion

EL_SH

Simila_

and

in dQ.bla

From

ACIPCO

ring

diameter

wires equal

this data

here

is NAPTC

of 0.625-in

120-degree

containment

and the

test

billet

No.2.

201 in which

thickness,

and weig?,ing 12.83

and encircled

defining

Test

the

ring.

pounds

ring_

for NAPTC

323

Test

l_ngth,

rpm and

caused to

2 are the weight

the rotor 201

spln-

horizontally

whi_hwas

19,859

in Table

axial

rested

rotor

at about

Given

containment

condition_

1.50-in

a T58 turbine

segments

a 4130

burst

[208,209].

_rag_ent

_ach t_on

fraqmnnt

attachod

consksted

blndosl

to tho cG of

tho

wail

an_

rouU]ts

sma11-n_rai.n attacking

only

In particular, "cylindrical inortia from

disk"

matching

and

The entire

£ragmenu

the

its rotatLonal

ring

trannla_ion(_l

analysis

radius

ring elements.

Local

p_rfectly

elastic_

a coeffielent

Further,

at is assumed

between

sash

7.5.2 For billet used

fragment Comparison

static

tensii_

ring material those

approximated

o

201 ring identical

vs Yu stati_

(80,950

for

+Material

is almost Tu

calculations

p_i,

0.00279);

use in the mechanical

rupture

occurred

at >u

and

30 will

the

b_

used.

o£ a rigid a mass

momont

of pre-impact

of

reluase

at the CG of the idealized of the

actual

in Fig.

33) by

48 _qual-length

impact

was

of restitution that

fragment.

treated

as being

_ = 1 was

used.

the impact-interaction

vs _inite-Strain

Pzedisticns

reported

30, National

data since

in Ref.

supplied

by

according

tensile

segments

sublayer

billet.

by

material

eena.

model.

[209] were 201

Accord-

data + were

(Tu ,yu ) =

0.o225)7

Fo;ge

R09 the Test

stress-strain

psi,

= 52.3 peE

324

Forge

defined

(105,300

the NAPTC

to Ref.

to the National

uniaxLa:

by piecewise-linuar

0 In/in)) 0. 2000)

the Test

and

ring

is frictionless.

of Small-Strain

stress-strain

of

those

purposes

the zing

the small-straln

to analyze

ingly,

and

in Re_.

a m_ss

_'ing-fragment

for present

in-lb

in_Ib

_or pr{Js(mt puryos(_s,

velocity

(as depicted

43R,054

158,922

as con._istihg

match

was

t_;-t and of various

at its instant

velocity

and

19,850, In/sac.

£ragmontn

the containmont

30.

having

translation

rpm of

:t4netlo ono_'gy.

mod(_is consldored

4SOY/node

h_nce,

nominally

of thls

in Ref.

fragment

was modeled

ro_oaood

snvnn-

the) rotor

waf] 14,S57.2

was

o£ rotational

o£

burst

In-lb

is idoalized

of 2.555-in

also,

freer)at

tho rosponse.b of

the actual

the rotor)

fragment

Ear

rotor

throo

thc_ analysis

each

At tho

had

rim wlth

of rotation

tho

fraqmon_

In-lb

of th_

of

are reporto_

one of

tho axle

In.

onorgy

o_ an oxt{,nsivo

pTodictions

soctQr

tho CG of oauh

476,766

144,018

fzagmonts

however,

2.707

llanos, oath

rotational,

The

kln_tlo

of whloh

translatlonal

was

w_].oc_ty at

ronultln(! total

_]08,8R0 In-lb,

thn distal,so from

fragmont

tho trannl_t_onal Tho

of a 120-dnHrnn

and

(0 psi, (121,000

The material

psi,

4

is asm_,ed

to bn strain

in r_pQrted most

to be

relation

employs

applicabln

Type

B and

the tlmcwise

straia

displacement uniaxial

[201] to mild

d = 40.4 Scc "I and p _ steel.

Also,

4B computer

program

central-difference

operator

static

following

Type

tensile

was

,e_) pairs,

unit

FE-modeled

initial

it was

ring mod_l

the same

used.

fitted

5 which

strain-displace[27] which

used

ring

for the

small _

0.4121789

x i06

consists

of 196

the highest

tad/see.

_t ~ < 0.8

(2/_max)

central-difference

operator

form of the

equations

numerically

with

It was

found

that

locations

in this

the same

at a given

prediction are omitted

and

spanwise

is used

and four

the deformed

ring

time

after

(b) the finite-strain

here.

the circumferential

HoweVer,

a _t of

Gaussian

inner-surface

greater

_max

=

was

select used.

(unconventional) are evaluated stations.

and

fragment are very

nearly

(a) the small-strain Hence,

interest

and outer-surface

cast steel

one must

depthwise

for

the mass

4130

was

properties

impact

0.557).

2.50 _sec

problem

the

this mathematical

element

prediction.

of much

the

the vector

configuration

with

psi,

instabillt_

impact-response initial

of

(I +

0.002890),

behavior

to solve

Finite

4 for

billet

= _E

Taking

frequency

calcuiation

Tu

segments

(172,700

(ib-sec2)/in

for convenience

two-dimensional

into

DeF.

linear-elastlc

of motion.

three

and

unknown

natural

To avoid

= 3.88 _sec;

recast

and

strain-

Forgo

(84 240 psi,

0.0600),

mQthod

However,

the Natlonal

were

(0,0),

psi,

for small-displac_ent

element

by piecewise-linear

Do as 0.000733

that

finite

as before.

Also, data

(Tu ,e_) =

(118,008

volume

found

the basic

stress-st;ain

psi,°0.0225),

This

were

F was

(l + E u) , and

(Tu

(107,500

analysis +

conditions

relation

E u) vs 8*u HEn

The

with

t_o CIVM_JET

the' _inito-straln

Impact-interaction

ring,

sQnsitlvo

analysis.

For

per

rats

such

comparisons

and

importance are 2 strain___.__s72 . Small-

strain

[30], vs 2 finite-straln predictions ++ of the inner-surface and the outer-surface 72 strains at the midspan stations of elements i, 4, 6, 9,

ii, A

and

47 are shown_

respectively,

finlte-strain-modified

is called

CIVM-JE_

4C

version

in Figs. of CIVM-JET

34a,

34b,

4B was

34c,

34d,

employed;

34e,

and

this version

I L4].

++For I this the I

!

the present fiDite 8trai_ calculation, L .. = 0.497 in was chosen since value was used for the small attain cal_ations of Ref. 30. otherwise, "_or_ plausible" value Lef f = 2h _ 1.25 in would have been preferred. 325

34f.

Shown

Jn Fig.

the tlmall-strain

36 for a time

initial

impact

of 1180 _soc + ar_]

pr_d_otions of the circumferential 2 distribution of Deter-surface strains Y2' lloro it is seen that _,or¢, are distinct differences between ths finlt,_-straln and

very

llttlo

the same 7.6

with

should

impact

found

plates;

clamped

of

2870

[27] and/or

CIVM-JET

realistic from

the plate

procedure

if

procedures.

6061-T651 each

by a l-inch

1893

to 3075 1.5-in

in the range

2485

to large was

except

observed

aluminum

diameter

width,

steel These

to about

sphere

span.

2800

deformations

for steel

sphere

narrow

and 8.0-in

in/see

plates

to perpendicular

in/see.

permanent

in the near

5B

in/see

in the

velocities

finite

predictions

calculations

(2) 3-D plato

mod_llng

is essentially

modeling

appear

the

of

essentially

location. carried

codes

useful

would

to permit

structural

2-D CIVM-JET

are present of

appear

of

4B

approximate

the behavior

the actu_l

were

to provide

3-D deformations

elements

of

exhibited

the location

2-D impact-response

significant

hence,

than beam

the

[29] would

HoWever,

vic;Inity of

specimens

regions,

the inltia]-impach

flnite-strain

narrow

thickness,

moderate

those

location";

rather

from

the na=row-plat_

for

"impact

f_nlte-straln

the small-strain

subjected

location

0.l-in

_at

deflections;

predictions.

which

in/sec.

noted

impact,

procedure,

_ho

for both

strains

Plate

have been

ranging

to produce

about

as input

i, initially-flat

velocities

rupture

initial

+This

Narrow

of nominal

It was

and

i_ used

than

larger

fO_lnulation-and-

small-strain strains

locat_ons

Dofinitlon

velocities,

pre-impact

far

strains

at its midwidth-midspan

Sphere

more

data

ideally

were

plate

larger

in Ref.

plates

above

for tensile

Problem

ends

at various

were

that

at some

however,

finito-straln

_o

Steel-Sphere-Impacted

both

and valid

the former

fact

prodlctlonn

Generally,

with

predict

As reported with

in others.

compared

stress-strain

7.'%.i

the smnll-tltraln

the consistent

procedure

procedure

and

dlf_oronce

by

is consistent !

the finite-strain

prodlctions

are predicted solution

and

after

near

the structure

response

one both

the by

to make near

and

Accordingly,

small-strain

out

(I) 2-D bca_modeling

for both

and

of the structure. time

of occurrence

of peak

straining.

326

,

|

! I To

illu_trat_l

narrow-plat_

t_p_elmon CB-_8

of O.097-1n steel sec

thlckn_l_, woighlng

66.810

impacted

spoclmon

CB-18

is giw)n

in Fig.

mea0urod

location)

J6.

(upper

both

uniaxial

(44,200 psi,

model.

For

this

data

30 and

the

small-straln

for EL-SH will

includ_

specimen

mainly

interaction

el_m_nts sphere than

in Subsection modeling

and

the narrow-plate to accommodate fragment

will

7.6.2

Mod_e!_iDg by

In modullng

Hence,

found and

for

in/in),

sublayer

level

predictions

>u

were

of specimen

element

to b'e presented

the material

strain

105 percent.

response

2-D beam

(Tu ,Eu) = o

material

[2] to be about

for

to be the

and

of Green

finite-straln

prediotions

taken

the

namely,

0.075

transient

were made

consistent

speu

modeling

in this

and

report

of narrow-plate

Beam

that

is, the

Next

also,

as a spherlcal as in the

2-D impact-

in Subsection

with

plate

t/lu attacking fragment

2-D modeling

solid-

(rather case).

Elements

plate

by bcmu

as _olnq

strict_

the attacking

fragment

idealized

be modeled

response;

fragment

Finite

narrow

this,

will

faithfully

cylindrical

and

be discussed.

.den (CB-18)

3-D structural

approximated with

2-D beam. element

rus_,onse will

the CB-18

is being

fragment;

7.6.2,

be modeled

as an "equivalent

response

was

behavior

surfaces).

CB-18.

First,

7.6.3,

EL-SH

surface),

lower

7.2.3.2;

rupture

impact-lnduced

behavior, only--the

data

(midwidth

(upper

and

in the mechanical

both

calculations

olongatlon

y = 1.2-1n

psi,

in/

diroctlons

the y-axls

(upper

2794

the plato-center

_oordinato

along

(49,200

Finally', since

those

of

were

specimens

CB-18--and

gages

waI_

A l-ln dlam,_t_ir

for this material

material,

30 for the

plato

calculations,

test

in Ref.

This

the finlte-strain

static

reported

£rom

relative

in Subsechlon

for use

t_xporimont,

span.

glgbal

transient

in/in),

6061-T651

0.06-£n

surface),

and

with

a pro-.Impa(_t vo]ocltF

and y = 1.5-in

0.00442 in/in)

8.O02-in

showing

strain

(upper

in R_f.

0.615

for unlaxial

2-D

with

surface),

psi,

with

the model

stress-strain

as described

(76,400

qr_,s

compari_1on

he analytical.

wldtll, and

the small-strain

static

I will

t/lis test

at y _ + 0.6-in

For

(0,0),

of In

tholr

approximately

successfully

y = -1.5-in

same

1.498-in

A schematic

and

of Rof.

sphere

location.

were

tl%_So prodlctio_ts

elements,

two-dlmunslonal

fragment

rather

the structural

is dlso

than belnq

(2-D).

idealized

a 1-inch

as a

diameter

327

! •

I[I ......

--

II I

_Iphere in Jdeallzed eylindrlanl width Imve

of

nnd vinuallzc_d

fragment

of

l-lneh

the narrow-plate

the

Nam_

The

entire

equal-length

total

mass of

span

upon

_ipnelmen. actual

oxt_nsive

non-dofo_able

na oss

Idealized

the entire

fragraent .In deflnnd

to

fragment.

specimen

cubic-cubic

a nolxd

and ext_nding Thin

narrow-plate

(0.186-in)

41X_F/node -- based

diameter

ns the

a_

conceptually

CB-I8

hen boon

assumed-dinplacoment

studies

reported

modeled

beam

in Ref.

by 43

el_ments

30.

with

The ma_s

per

unit initial volume Po of the CB-18 material is assumed to be 0.25384 M 10 -3 (ib-sec2)/in 4. As a result, the finite-element model consists of 157 unknown

DOF

rad/sec.

Accordingly,

modified

versions

one must or

choose

less

_sec

element. half

of

Each

on

each

with

side

that

For'the displacement was

element each

these

were

element

used

nodes

full

This

B

for the volume Also,

of

since the center

program

logic)

of that for mass"

frOm

hence., the

element

length region

or 0.186-in is consistent

arguments

calculation,

spanwise

F

as

strain-

(Eq. 4.146),

and four integration

a diagonalized

328

elementl

propagation

and Type

nu_erlcal

used.

"account

length

inch.

(Eq. 4.90) three

of

computer

effective

the finite-strain

cases,

matrices.

is one

of impactl

impact

to receive

a spanwise

station

_Isec

of 0.50

at each

two end nodes

of the next

= 0.194

and

Type

In both

to the

30 on stress-wave

(_.097)

small-straln

was

half

of impact.

in Ref.

relation

of the station

(with the resident

= 0.688

is assumed

27) on

at the mldspan

of impac.t influence

2h = 2

property

side

operator,

a At

Finally,

x 107

(and

(2/_max)

the structure

increments

and

0.8

results.

assembled-structure

element

program

for convenience

(see Ref.

occurred

resulted

of the station

employed.

stations

of

estimated

approximately

on either

of velocity

the _enter

effectiv e re_ion

increment

impact

criterion

impa_ting

converged

= O. 2326

central-difference

At of about

instabilityl

momentum

= 0.0993-in

this

size

is _max

4B computer

the timewise

and the structure,

perpendicular

element, in the

increment

and provided

the fragment

frequency

the CIVM-JET

calculation

an impact-imparted

initial

since

a time

employed

At(E/Po )I/2

linear-system

tdlereof) utilize

to avoid

was

b_tween

and its maximum

depthwise for

(lumped)

respectively, Gausslan

the finitemass

matrix

for

r, 1

! t Thuao

calculations

the flnit_-strain match

prndictions.

t/lo oxperlmontal

structural

response

these

predictions 7.6.3 To

actual

narrow

plate

more

o_ plate

modeled both

seen

in the CB-18 was

lation;

namely,

wise

lengths finite

same For

lations 5.123

given

as reported

the quarter

plate

each

in Fig. shown

plate

employed

displacemezlt cases,

the van

attendant

utilized 5.118

in Fig.

5.123

three

for

by

calculations The

the LLC

Gaussian

comprehensive

point

as

elements

used

the

spanwise "refined"

elements

relations the

used

were

v, since

is bilinear spanwise

(Eqs.

calcu5.118

-

finlte-strain

involving

U and

calcu-

of iI span-

the small-strain

while

terms

in each

329

each with

strain-displacement

finite-element

stations

assumed

7.2.3.1.

CB-18,

paragraph)

displacements

was

two rows

flat plate

in Subsection

the

along

perpendicular

this

of flat-plate

widtl_ and

37b.

(without

from

was

[210] for the small-st_ain

of specimen

following

CB-18

and ideally-

fragment

represented

later

(0,0),

spherical

Karma,] strain-displacement

the in-plane

field

was

For

imposed

Initial

mesh

was

6DOF/node.

were

0.06-in

of

CB-18

end.

element

of 0.375-]n

37a;

the more -

clamped

earlier

3-D type

specimen

(x,y) =

than about The

of

conditions

non-deformable

modeling

of

near

steel-sphere-

the

type with

quarter

station:

at the

the FE

derivatives

!

imposed

as described

in Eqs.

3-D

between

dominant--specimen

one

symmetry

LLC elements

calculation

distinct

agreom[_nt

accommodate

arc

only

experiment.

mesh

and the

can not

possibly

of the CB-18

of t_e LLG

(0,0)--rather

as depicted element

elements

elements

situation

which

the midwidth

the same

flat plate

where

(except

reasonable

faithfully--to

diameter

(x,y) =

employed

and

Rlements

physical

ele/nents;

were

of a 1-inch at

Finite

and economy,

_nd

conditions

tc occur

i

thrift

the midspan

impact

both

finite

by flat-plate

clamped

small-strain

station

oltlowhero

to find

deformations

plate

computational

the

t]]_se 2-D predictions

the impact

However,

by Plate

the

with

to both

and experiment.

ModQling

structural

near

can expect

simulate

impacted

modeled

results

one

apply

Accordingly,

occurred.

tlle 01craped ends),

the

and modeling

relations

the second

order

the asstLmed " Jn u and

direction

and

v). four

In

I

f d_p_,wiso

Gausslan

evaluate,

by volume

Also,

stations

worm

numerical

a diagonalized

uncd

in each

integration,

(It_ped)

mass

flat-plate

element

t_o proportion

matrix

was used

to

of oath

for each

element.

element.

The maximum linear-system fruqucncy _max of the Fig. 37a fin_teelement model was found to be 0.2372 x 107 rad/sec. Thus, if one were 0omputo

the impact-induced

transient

response

by using

the tlmowiso

difference operator, a At of about 0.8 (2/_ma x) = 0.67 _sec required to avoid calculation instability. However, these were

carried

out

by usin$

operator

is employed.

employed

which

"reliable

At each momentum

Accordingly,

earlier

converged

the CIVM-PLATE

experience

program

in which

a convenient

and discussion

_t of

central-

would be predictions

the Houbolt

1.0 _sec

indicated

to

was

would

provide

predictions".

impact

between

is transferred

the

fragment

and

the plate,

by a perfectly-elastic

it is assumed

collision

to a plate

that region

M

(from

the fragment)

0.1985-In

centered

for Lef f could purposes

be

defined

employed,

Comparison

First,

it is useful

for

impacted case

the 2-D narrow

in which

plate-type proper

but

of radius

location; this

Lef f = At[E-

other

one is used

more

o rational

for present

]1/2 selections

illustrative

.'

7.6.4

tions

by a circle

at the impact

3-D predictions flnlte-straln

Next,

the proper

and

only

Plate-Model

small-strain

(with beam similar

response

and a spherical

Finally, for the

finite

Vs.

lnalysis.

330

finite-strain

will

of

impacting

be made

formulated

fragment to compare and

predic-

the CB-18 for

is accommodated

it is illuminating

consistently

Predictions

el_ments)

comparisons

3-D structural

elements

Jhape.

vs.

to compare

idealization

plate.

finite

size

of Beam-Model

the

by

of

the

2-D vs.

implemented

! I I 7.G.4.1 }

SincQ

Strain

primary

com pari_on_ Interost

contor_

on tho predictod

and moasured

strains,

2 comparisons of longit,dinal Groon indicated figures at the spocimen 1ocatlons

on the uppor

,

i

iii

FE Model .,,,

I

Pigure

I

strain y2 ar_ mado in tho following midwidth location at various spanwlsQ

(non-impacted) i

iii

or

Plate Small At ......... 1.0

_sec

_sec

Finite

_

, ,

,

,

Data

_.,

Prediction Upper Lower

•

Experiment Upper Lower

X

-

X

X

0

X

-

-

-

38b

X

-

X

X

0

-

X

-

-

38c

X

-

X

X

0.3

X

-

-

-

38d

X

-

X

X

0.3

-

X

-

-

38e

X

-

X

X

0.6

X

-

X

-

38f

X

-

X

X

1.20

X

-

X

-

38g

X

-

X

X

i.50

X

-

X

-

38h

X

-

X

X

1.50

-

X

-

X

38i

X

-

X

X

3.00

X

-

P

-

38j

X

-

X

X

3. O0

-

X

-

P

38k

X

-

X

X

3.70

X

-

P

-

38£

X

-

X

X

3.70

-

X

-

P

38m

X

-

X

X

4.00

X

-

-

-

X

-

X

4.00

-

X

-

"

I

I

,

P denotes

,I

X ii

that

only permanent

I

strain

331

I

.,..

Station I_'--

i

2 of 72 strain

Location .......

surface.

|m

38a

38n

i

(impacted) ii

Analysis Strain Type

Beam At 0.5

lower

ii

information

was

i

obtained.

Location

y _

Gauusian

tnteqration

in

at

the

(at

the

end

of

_lamp_d-ond

v and w and

stations

ocour

of a finite

the

at

and

figures

available.

these

stations

show

4.0-in

well

y = 3.7 in the

clamped

end

bee&

_arried

have

current

out

strain

long enough

in time)

than by

Figure

39 shows

that

each

from

other.

and F

Finally,

at statlon

Y respectively,

(associated

ences

part whil_

with

for the

caused

by

of the the

the

two predictions. good

for

support

support

are valid

strain

of

strains), reaction

reaction the

expressions

only

version

bending

for

small

that

very

o_t

close

MM .

S z,

these

reaction some

S z and

of CIVM-JET

Mx,

to

and 40c,

between

force

does

strain

lateral

one observes

rotations

the

reactions

support

moment

of the program

332

are

40a, 40b,

(shear)

at which

proced_e".

agreement

but

station"

(if carried

CB-18

the longitudinal

at

permanent

the support

The

at

calculations

of the midspan

in Figs.

of

do occur

element

the

better

for beam

shown

at all

"nearby

stations

histories

the current proced_tre"

it appears

"small-straln

time histories

fact that

strain

finite

the time

the membrane

in the transverse

differences are

is very

finite

spanwise

two predlct%ons

the

to the

provide

the former

x = 4.0 in are

for these

two predictions Fy

these

at

Although

i_togra-

strains

effect

displa_o-

on thesu

strains

large

node

All other

Gausslan

experiment

of

in

the midspan

"small-straln

900 microseconds,

at all

and

(a) by

Large

imposed).

would

_oro).

are indicated

4.0 in.

adjacent

procedure"

_ are

spanwisc

with

and pronounced

for o,ly

element

finite

the end

the occurrence

has been

a

2 Y2 predicted

and/or

midnpan

(n_tmoiy that the

between with

the

y _ + 4,0

gradlont

the forme;

m%d

in the element

wlthmeasurements

w

imposed

with

I_cation with

strains

other

also,

comparisons

deflection

(b) by

y _ 0, 3.7,

condition

"finite

boon

¢o_nc_de

each

a distinct

(located

coincides

the strains

and

with

except

exerts

and

permanent

that

_oincidos

element.

intormedlato

y = 0 and y = 4.0 in;

y =

beam)

finite

have

do not

procedure"

reasonably

a

beam

measured

figures

agree

both

thc_

locations

Also,

"finite-strain

of

the

lateral-dlsplacomont

tlon points.

These

of

conditions

element,

where

midnpan

station

_lamped

at which monte

0 in

force differ-

large

These

differences

4B for the bending and

not

small

have

strains,

this

I ! |

rostrlction, inglu_nced The

of

course,

tho support

by the bending computing

time

part

r_action

of the

required

bending

moment

is most

"

atrain_di_plaoemant

to analy_o

M

rolatlonn.

stool-sphere-impacted

beam

CB-18 by the two procedures, under oth_rwise-idontical conditions, is conveniently displayed in th0 following t_ulatlon (for a tlmo step of

I

0.50 mlcrosccondl

•

all runs

wore

conducted

No. of Beam FE

Formulation

on

an IBM

370/168

No. of Gausslan Sta. per Elcm.

Small

Strain

Finite

I

Strain

Small

Here

170

43

3

4

170

Mass Matrix

No. of Cycles

B

DM

F

DM

Strain

again,

the finite-strain-formulation

(DOF)(cycle)

than

time per

(DOF)(cycle)

noted

specimen

CB-18

arises

compared

from mass

the use matrices

vs.

and

the use of

for

calculations.

It appears that stress tensor in the tions

of certain

stress

and

the monotonic

I

compression)

mechanical

sublayer

model,

and

2250

5.11

13.4 x 10 -6

1850

6.81

21.7

x 10 -6

require

more

CPU

formulation.

for steel-sphere-impacted

latter

of

diagonalized 3 rather

(a) the use constitutive

representing

cPu(min) (DOF)(Cycles)

explosively-impulsed

in the

calculations, the CB-18

here

CPU Time (mln)

calculations

the small-strain

with

I •

!

4

Finite

consistent

!I

3

time per

CB-4

I

Strain

DOF

43

Strain-Displ. Relation Type

Formulation

_an

matrices

4 spanwise

The

smaller

narrow plate

CPU

plate

specimen

populated for the CB-18

Gaussian

stations

of the prope.r (second Piola-Kirehheff) equations by making proper transforma-

strain

measures,

strain-hardening

behavior

narrow

the more-heavily mass

z

Total No. of Unknown

spanwise Depth I

computor)

(b) the use antis_nmetric

of the material

(c) the use of a finite-strain

333

of Tu

u vs. e* for (in tension and

by the mechanical strain-displacement

equation, improved

and

(d) ths inclusion

predietlons

of

of trannlent

thickness strains

changes

(the most

provids

significantly

important

and

s_nsi£ive

quantitiQs). Next,

consldor

the plato-model

predictions#

s_e Fig.

411

l

FE Model

Fiquro B£am

Analysis St_'aln TwDp.

Piate Small

0.25 _/soc

|

Location of y_ Strain Data Along the Plato Midwi,dth Station Station Prediction Experime{_t y (in) Upper iower! Upper Lower

Finite

1.0 Usec

41a

-

X

X

X

0

X

-

-

-

41b

-

X

X

X

0

-

X

-

-

41o

-

X

X

X

3.40

X

-

-

-

41d

-

X

X

X

3.70

X

-

P*

-

41e

-

X

X

X

3.70

-

X

-

P*

41f

-

X

X

X

4. O0

X

-

-

-

-

X

X

X

4. O0

-

X

-

41g i

At

the plate-center

it is seen strain

that

prediction

unreliable) station near

m.

location

the transient

and at the

which

is more

level

of

in)

clamped

end.

"remote"

from

impact-lnduced

significant

provided larger

calculation

(0, 3.70

(0,0) where

strain

is substantially

small-strain

(x,y) =

(xty) =

and

than

exists

that

However,

at station

the clamped

end,

between

impact

consistent given

result

in) which

structural-response

difference

by the

+. A similar

(0, 4.00

initial

by

are,

the

(now at

respectively,

one observes

the strains

finite-

is observed

(x,y) =

strain!

occurs,

(0,3.40 a much

a lesser

predicted

but

in) smaller still

by these

two

schemes.

7.6.4.2 Since measured

Only

Deflection

only

in the

permanent

Comparisons

permanent CB-18

strain

deflection

experiment,

was

data

(no transient

only permanent

recorded

+Note that the static-test uniaxial [2] is about 1.05 or 105 per cent.

at this rupture

334

deflections)

deflections

were

can be used

location. level

for Yu for

this matezial

to cQmparo

predictions

to comparo

varloun

Accordingly, in

with

nxper_mont.

trannient

dlspl_cn,lent

such deflection

thcl following

Hownv_r,

comparisons

it in instructive

prodictlons are

shown

with

each

on figures

other.

indleatnd

ti_lbulations

....

r_pt.'_-.

Analysis ,, _%rsln Type Small Finite

FE Model Beam Plato

Figure

Stress Strain Approx.

,

42a

X

alsc_

-

X

Predicted w-Displ. Location (x,y)

r

-

i

EL-SIi

Displ. Looatlon (x,y), in

i

y m 1.00

and

AVg.

at

y _

1.00

EL-SH- SR 42b

X

-

X

X

EL-SH

42c

-

×

x

x

EL-S_

42d

-

X

X

-

EL-SH

X _ 0

(0,0)

(0,0)

At

Ix=0 I

Along

X -- 0

Along

x = 0

Along

x = 0

840 _see x=0. 375 vs.

y

Along

x=0.75

(Estimated Permanent) 42e

-

X

-

X

EL-SH

At

_ x=0

840 _sec _=0. 375 VS.

y

Along

x=0.75

(Estimated Pennanent) 42f

-

X

X

X

EL-SH

Along

x = 0

at t = 840 _sec '

In Fig. for the i

i

t

larger material

42a

translunt peak

it is seen

w-displacement

for the EL-SH

behavior;

that

it

than

is seen

the FE beam at "2-D

iI u

model

location

for the EL-SH-SR also

that

335

small-strain y = 1.00

prediction

In" exhibits

_epresentation

the EL-S}b,gR prediction

r

of the for the

a

permanent

di_plac_Qnt

at

oxporlmcntally-ob_ervod Finite-strain transient compared

predi_tioDo

compare

r_sponso,

as noted

versus

_-D w.-dlsplsoomont

prodlotlons

for

"_mpa@t

well with and

_arlior,

important

regions

which

the

at the plate-center finite

strain

Again

these

permanent

modeled

in Fig.

oth@r

42h,

in overall

transient vs.

with

location

the

calculation

predictions

are

compare

and near

by the

of the predicted

response,, Howev,_r,

the clamped modeling

of the

each

predictions

small-strain

42c for EL-SH other

but

finite-strain

at the

end.

w-displacement

for the

in Fig.

two mid[Ipnn

calculation

finite-element

w_il with

is predicted

3-D character

midspan

shown

Those

arc

arc signi_icsntly

flnlte-straln

3-D Houbolt-MULE

the

NtructUrS

ostlmato.

strains

(x,y) = (0,0)

for

transient

the

flat plate

transient

deflection

The

each

are near

the more-rualistic

s_zucture,

agreement

p_di_tlonn

th_ beam-l_iomont station"

the p_odlcted

for the small-utraln

CB-18

small-straln

in the pormanont-doformatlon

different

For

in in the better

result.

at the mldspan

in peak

t_i_ ntati_n

vs.

material a larger

the behavior.

peak

and

calculation.

w-displacement

for the small-strain

i

plate-element _sec,

model

calculation

the w-displacement

midspan

to the clamped

(centerline)

station,

Beyond

station

about

identical

along

essentially Closer

is shown

is shown end

along

half-way y = 1.50

these

impact

is clearly

plate-element-model

qualitatively

and

in Fig.

Finally_

Pig.

42f.

a slightly

quantitatlvely

"bulgy"

and

along

plot

region

is shown

edge.

indicates

of the the

free

to be nearly

structure.

3-D character

modQl only

finite

are

similar

small-strain along

strain

than

in Fig.

42e for

at t = 840 _sec.

these profiles

profile

and thus

from

midwidth

the

is seen

however,

calculation

the FE plate

realistic

distance

evident.

for w is compared

The more more

location,

profile"

finite-strain

prediction

in this

at t = 840

at the plate

edge,

stations,

Here

of spanwlse

lines

free

42d.

in, the w-displaoemunt

behavior

"displacement

node

to the

2-D displacement

A similar

strain

the

widthwise

of the w-displacement

42d.

as a function

tkree

to the plate-center

in Fig.

prediction

the small-strain

_s

Both

to those

Vs. the

th_ midwidth

the

finite-

location seen

shown

in

to exhibit

pEediction.

As

336

O0000004-#SFI

I noted

earl_ir,

haw_v_r,

the ntraln

b_tw(l_n th_ fi_Lit_-strain fo_r

I

bolng

in much

and

he,tier

predictions

th_ ,mall-straln

aq_o_,mon_ wltb

S_nc_ thn finite o_omoni_ modol±ng quarte_ 09 narraW_pla_o npocinmn CB-18 llmi_ad

the response

to omplo_ quarter

a "refined

plate

model,

detail

elements

station"

(0,0);

length

are

in which

pronounced

those

gradients

are

The lateral

plate

the

y = 4.00

two sides

Hence,

calculations maximum found induced

13_19775

(lumped)

impact

in),

ewe rows

two reg_o_s

and pronounced

strain

0.8

provlde

"converged

37c

Xf one were timewise

(2/U_max! e 0.12 _se_ However,

the CIVM-PLATE

a convenient

experience

with

= 478.

was used.

are invoked imposed

along symmetry

For

these

Thus,

the

fi,_ite-element to _ompute

model

the

was

impact-

centra)-dlfference would

in which

At of

with

6 at 6 clamped-end

the present

program

is

4uadrl-

5 _rom double

and

model

75 LLC

96 nodes

clamping

nodes,

the

of

conditions

DOF are:

the Fig.

by using

instability.

computational

of

_equired

to

D redictions

were

the Houbolt

operator

1.0 _@ec

Hot,bolt-MULE

be

h_d

wa_

employed,

indicag_a

which

wou!d

predictions", between

the fragment

337

!

These

has

Symmetry

mass

lO6 tad/see.

hceOrdingly,

impact

the

re£1ned-me_h

(y = 4.00

37c consists

structure

the restrained

response

out by using

each

in Fig.

assembled

frequency x

earlier

At

shown

19 single=symmetry

a At of about

is employed,

doc_dod

"initia_

ar_ employed.

of 576 DeF.

a diagonalized

calculation

carried

the

the t%nknown DOF -- 576-5-(3)(19)-6(6)

transient

operator, avoid

at

linear-system to be

_n this

near

at x = 0 and y = O, while

in; accordingly,

nodes.

used

it wan

to represent

3..D respons,_ effects

The

'giving a total

I, 3 each

37d.

cl,%mped end

LLC _lements

model

elements.

at node

are

the

the

expected.

refined-mesh

6 DOF/Node, along

to be

near

with

me_nuremo_tn,

be a_commodat_3d,

of LLC [_lato ol_mu_.s

also

spanwise

eal_ulat_nn,

dlffnrent

_lhoWh _n P£gfl, 37a and 37b for one was r_thur _oars0 and thereby

could

by 0.l-in

o£ 0.1-1n

qignifi_anLly

axp_rimontal

in [.'igs.37c and

of 0.1-in

(x,y) _

which

P_ mesh"

as depicted

arn

add the pl._te, it is asstt_ed that

_%olnentum is trannf_rr_,d (from

_,n

imp_c_ t$on

a pnrfeotly_lantlo

_ra_Tm_,nt) d_finnd

location.

From

2._ of. Bof.

Fo_

by

thSa

by a _irol_

str,ss-wav_

30, L_f

tl_u [t_mo a_| for

_rq_onts

chss¢_n to be

P_ l,odol, all ot.h_"

manned-mash

twice

_t the.

glvnn

strain

in Subsoe-h of,

fo_ulatlon-

at,el othor

relations,

£_nltu-ol(_mont

raglon

th,_ thieknnns

f_nlto

utrain-dislplaoomont

th,_ coarse-mush

to a plat_

of rfldl,s L¢_ff e_ntorod

p_opflgntiQn

h_s be_n

and..c_alc_uJ, at.._orl procedures, were

collision

plata

model

data

compute-

tlon. Shown

flnite-straln

EL-SH

displacement speulmen peak

43

in Fig.

are

the coarse-.mooh

prodietions

As expected,

d_-_lection

latur

_E Plate

Model

ro£1r_d, mo_h

o_ the plota,-c_ntor,

w of stuol-sphor_-Im_ao_ed

CB-18.

vs.

6061-T651

the reEined-mcsh

in tlm_

compared

with

(x,y) m

(0,0),

ah,.I_inua;narrow-plate

model

the

plato-ulomont

_xhlbits

coarse-mesh

a larg'uz model

pr_dlc-

tion:

6oarse

Mash

Refined

However,

tion.

Mesh

as noted

a sensitive

on

a much

adequacy.

strain

in Figs.

plate-elemmnt-model (Lagrangian)

_ble, number

and

750

(o_ pemmanent)

hand

and/or

and a_eaningful

predictions

indication

are not

of the predic-

inte_'est and

are examined 440 are

displacements

reliability

are of primary

(_sec)

cOncern,

end

of prediction

next.

coarse-mesh

vs.

zofined-mmsh

finitm-strain predictions of transient longitudinal 2 strain "(2 on the surface at various spanwise stations

transient

am_ included

0°987

44a through

of steel-sphere-impacted Experimental

at Peak 690

o_ the accuracy

sensitive

Time

O. 970

transient

the other

more

_en_o,

Compared

Grmen

earlier,

indicator

Strains

provide

Peak w(in)

6051-T651 and/or

also.

associated

aluminum

permanent

S_m_arIzud

station/s_rface

narrow-p]at_

strains,

as appropriat_

in the following at which

specimen

these

are

CB-18. and

avail-

_/Im _igure

2 strains 72

are i

compared;

338

I

_,.,_,,,,,_...........

........... _ .......__V,;_

_z._

_

....

I I Plate Pigur_

Locatlon of y2_ Strain Data Alonq Plato Midwidth (x_0) Station

VE Mod_l coarse

Rofinnd

Station y (in)

PrQdiction Upper Lower

the

Rxp_riment Upper Lower

44a

X

X

0

X

-

-

-

44b

X

X

0

-

X

-

-

440

X

X

O. 30

X

-

-

-

44d

X

X

0.30

-

X

-

-

440

X

X

0.60

X

-

X

-

44f

X

X

0.60

-

X

-

-

44g

X

X

i. 20

X

-

X

-

44h

X

X

i. 50

X

-

X

-

44i

X

X

i. 50

-

X

-

X

44j

X

X

3.00

X

-

P*

-

44k

X

X

3. O0

-

X

-

P*

44_

X

X

3.70

X

-

P*

-

44m

X

X

3.70

-

X

-

P*

44n

X

X

4. O0

X

-

-

-

44o

X

X

4.00

-

X

-

-

*Only

permanent

Figure

strain

44 shows

initial-impact

that

station

was

at

recorded

at

the upper

(x,y) =

(0,0),

this

location.

(non-impacted) the

refined-mesh

su=face

at the

plate-element

model predicts initial

impact

predicts

a peak

_2 strain

of about

TAII=

750 _seu,

While

a peak

y2

sinlilar disparity (x,y) peak

=

of much

Hence,

(x,y)

(0,0),

the

strain

of about

is seen

but

(Fig.

44b) at the

maqnitude

reflned-mesh

model

model

than

that

predicts

predicts from

larger

after

coarse-mesh

cent at TAIl lower

time

= 690 _sec.

surface

A

at station

a compressive

the coarse-mesh membrane

model

strain

model.

strains

at

= (o,o). At station

prediction

(x,y)

= (0,

of Y2 differs

0.3D in),

significantly

339

|

cent at

corresponding

35.6 per

the reflned-mesh

smaller

the

59.7 per

the more accurate from

refined-mesh

the coarse-mesh

model

model

! pr_dlctlon,

as Pigs.

is presser does

_

the

th_ upper

44c and

low_r

surface,

surface

station,

_videnco

_xperi_nc_s

and both

are

(0, 0.60

in) which

At ntatJ.on (x,y) _ i._[tlal-impact

44d sh_w.

of "reversed

a larger

curvature"

pulik strai_

tonsilo. is more

r_mote

from

2 (longitudinal) peak 72

the predicted

and

17.5 per

surface, upper

whore

(Fic

well

prediction" It should

be

economy

reasons,

modeled

'by finite

experiment, in).

only

impact

the

location

initial-impact

are

strains

compared

from

the

initial

impact

ible

in part

distant

at those

stations,

from

(x,y) =

measured

especially

of strain those

(0,0).

wi_h

are actually

point.

Accordingly,

discrepancies

however,

near this

between

factor

to these

and

was

An these

calcula-

in the

actual

(+.057,

-.019

to the "asstu_ed"

effect

and the

different should

location.

a lesser

distances be

and predicted

impact

values.

CB-18

the computed

measured

"refined-

to the actual

at somewhat

assumes

the

(x,y) =

relative

this

when

pe_manent

(0,0);

respect

the initial

agrees

efficiency

assumed

Therefore,

here

stations

close

the

at respective

that

at about

gages

trace

obtained

(x,y) =

occurred

Per

surface,

specimen

it was

at station

impact

different

be

of narrow-plate

occurred

locations

upper

for computational

Furthermore,

initial

location

for the

that

strain

it is evident would

is about

and t/%e lower

500 microseconds

cent were

in);

strain

one quarter

impact

On the

2.36 per

however,

about

are

the poak

as a reference.

transient

until

(0, -0.60

elements.

however,

Therefore,

and

the upper

is used

lost.

the permanent

noted,

initial

result

trace was

in) and

of

respectively,

predictions

of 2.24

(0, +0.60

t.lons that

for,

the experimental

both

strain

measurements

stations mesh

44e),

with

the experimental strain

hlghor

the refined-mesh

suzface

reasonably

cent

the

strains

of tensile character on both sur£acos (see Pigs. 440 and 44f)! 2 72 strain for thu reflned-mesh model vs. t/%e coarse-mesh model 9.0

than

responsstrains,

At more

to negligible

importance. On that

the upper surface at station (x,y) = (0, 1.20 in), Pig. _4g shows 2 the peak 72 strain from the coarse-mesh calculation is about 36 p_r

cent

smaller

From

0 to 209 _sec,

than

that

for the refined-mesh

the measured

strain

340

prediction

trace

(3.13 per

agr.ees very well

cent).

with

both

I I }

prodletionfl;

from

,.I

rc_sult; and

beyond

bettor

aqr_oment

strain

aqre_s

Although I

that

m_asurod;

effect

well

t/_at (x,y) = this

relatively

a rather

region

small

permanent

(0.50-1n in)

likely

out

would

long)

improve

is in

be

800 psoc,

larger

particular

porh]an_nt

calculation.

to only

finite

station--

strain

coarse-mesh

carrlod

at this

coarse-mesh

The measured

the

strain

unexpected

(0_ 1.20

would

was

th_

trnnsiont

pr_diction.

prediction

large

with

than

(less

clement

important)

was used

the use of smaller

the prediction

it

in this

and

elements region

of

strains.

distant (0, 1.50

from

the initial-impact 2 in) where 72 predictions

(x,y)

=

Figs.

44h and

44i,

this

location,

the

on both

cases

better

(and btJst) with

is not

contains

both

the rnfinod-mosh

"indicated"

since

strains

,.

the

it agrees

475 Dflee, thn mnasurnd

reasonably

this

More i

with

location

to span ,:

abQut

the refined-mesh

appears

..

300 to 475 Dsec,

respectively, coarse-mesh

values

for the

given

are

by

less

is station

and measurements upper

calcslation

s_Irfaces than

the peak

location

and

the

indicates

2.5 per

lower

larger

the refined-mesh

than

are shown

in

surface. 2 peak 72

prediction;

cent.

At

in

The measured

transient _2 strain on the. upper surface is larger than either prediction, but at the lower surface the measured information is in reasonably good agreement

with

predictions.

(I) upper-surface was

1.48

and

stations per

stations

1.13 per

(x,y) =

cent,

agreement

Coarse-mesh at station tively,

(0, 1.50

those and

(x,y) =

for the and

values

_laan does

refined-mesh

the measured

(0, 1.50 in)

respectively

in) and

and

(x,y) =

the refined-mesh

permanent

and

strain

(x,y) =

(2) the

(0, -1.50 prediction

at

(0, -1.50

in)

lower-surface in) was

1.31 and

is seen

to be

1.27

in good

measurements. refined-mesh

predictions

(0, 3.00 in) are

upper

small,

(x,y) =

coaL,

respectively; with

Finally,

and the

the coarse-mesh

lower

indicates

in Figs.

surface.

calculation

the refined-mesh

prediction

shown

for

Here

predicts

computation. th_

closer

the transient 44j and

On

44k, respec-

the peak

somewhat

strains

larger

the upper

agreement

with

2 Y2 strain

are

peak

surface, the the measured

strain. Of greater to the

clamped

importance end.

Here

and interest significant 341

i

I

are spatial

the strains strain

at stations

gEadients

and

close strain

values

themselves

(_,y) = mush

must

(0, 4.00

transient

station

occur.

Y2 strain

predictions

(0, 3.70

_

(0, 4.00

in) at,

for each

station.

£or

(x,y)

-

(0, 3.70

calculation measured and

the upper

are much

permanent

(x,y)

=

(2) lower

surface

predictions

occur.

than

on

44n

prediction

transient

rather

than

lower

surface

in this

region

model,

the

predictions,

surface

0.68 per and

prediction.

at

(x,y) =

cent,

(x,y) = _%t

surface

the

sequence

the tension

that

predicted

bending

this station compression_

The

the final by

and

contribution

prediction.

except

strains.

severe

the be_ding

refined-mesh

and

refined-mesh

very at

coarse-

state

is

the refined-mesh

calculation. On strains

the lower surface at (x,y) = (0, 4.00 in), very large tension 2 72 are expected from the additive effect of membrane and severe

bending_ Fig.

this

440.

2 strain 72

Note

(x,y)

per

to be the

cent

no strain

(0, 4.0 in),

case

the coarse-mesh

computation

Although =

that

of 11.5

refined-mesh cent.

is seen

from

the predictions

calculation

at _/_is locatlon

predicts

a peak

measurements

it is evident

tensile

were

342

a peak

the more _2

made

from visual

shown

predicts

while

in)

in) were

permanent

tension,

overwhelms

(0, 3.70

(0, -3.70

(0, 4.00 in),

The

respectively,

these measured

the upper

effect

the refined-model

coarse-mesh

uompression,

a similar

by

It is seen

(more reliable)

shows

of compression

and

(x,y) =

as the membrane

to the

the

with

shows,

44m for

station

reliable

predicted

respectively. agreement

fine-

at station

(i) upper

0.56

the

is used

more

surface

from

the

and

and

(0, 4.00 in).

strains

station

sequential

-- according

one

cent,

upper

and

44£ and

44o for

the coarse-mesh

(x,y) _ (0, 3.70 in)

As Pig.

tension

lower

in) ware

clamped-end

experiences

mesh

strains

are in close

At'the

finally

smaller

at

per

the

in F_gs.

mesh

with

(x,y) _

the peak

(0, -3.70

1.07 and 0.47

strains

in),

the

(0, 3.70 in)

coarso-mcnh

44n and

substantially

end

and

shown

element

compared

to provide

at the clamped

On both

are

in Figs.

a finer

(x,y) _

interest,

respectively,

model

ks expected

especially

in) and

S_nae,

the refined-mesh

former

stations

in) are of particular

(x,y) _

(x,y)

Hence,

strain

at the

inspection

in tensile

reliable of

lower of

22.6 per surface the

at

]

! I specimens

the pert_nent

that

thorn .(at

strains

the clamped-end

lower

i

surface) pool

are

kind

tensile plate that

tests

Nonhomogenoous this k_nd

the same

batch

for tensile

of 6061-T651

strains

is present

was

noticed

aluminum

o_ about

18 per

with

an orang_ _

in static for tho

u_d

oont

uniaxial CS-18

or more.

Recall

inltial-impact station, the ro_ined-mesh calculation predi£ts 2 tensile 72 strain of 59.7 per cent. Hence, it is apparent that

3-D structural model

response

would

behavior

result

predicted

by

the 2-D model

and 4.00

in,

respectively). of

to steel-sphere

impact

impact

station (compare The

rather

with

by the plat0-finlte-

incipient

6061-T651

Initial-impact

the point

accommodated

in predicting

of steel-sphere-impacted

at the midspan

near

doformatlon

of surface

the

element type

Qf

specimen

a peak the

largn.

of nurface;

rupture

al%m_inu_ narrow rather

Figs.

experimental than at

plate

than at the

38a and

the clamped slightly

further

investigation

to occur

clamped

end

38n at stations

specimen

a velocity

of the present

CB-16

end,

higher

y = 0

did break

when

than

as

[i]

subjected

than

the CB-18

velocity. One

point

that

t.lon" of strain

in the impact

indicate

that

midpoint

of the plate),

strain

takes

the maximum

place

this

discrepancy

that

location

into

account).

details Wave

propagation

described, The MULE

computing

6061-T651

sphere that

a high

precision

in Refs.

t}e transient

in both

and

at MIT are

show

of transverse do not take

be that

the

and the plate

23, 27, and to carry

specimen

summarized

the refined-mesh

finite

(the

this

reason

for

strains

type

at

of straining

impact-interactlon contact

procedure model

and

does

stress

not

is used

take

-- as

30.

out

responses

one

involve

impac%

point

the maXimum

shear

local

distribu-

predictions

that

location,

simplified-interaction

time required of

that

the present

alt_ninum narrow-plate

coarse-mesh used

details

for example,

predictlons

double

steel

instead

in from

may

"exact

at the initial-impact

predictions

reason

is the

the computer

experiments

be the presence

Another the

0.2

While

occurs

the actual

at about

might

region.

strain

(the computer

between

into account;

I

deserves

the finite-straln

Houbolt-

of steel-sphere-impacted.

GB-18 in the elem_Rt

on the following model;

IBM

370/168

for both At

in the

= 1 _sec

was

cases:

343

! .

. .

FE Model

Coarse Mash

Mesh Roflned

No, of Plato FE

Total Unknown DOP

No. of Cycl_

CPU Tim_ (mln)

22

157

900

65.4

75

478

800

202.6

cPU(mln__ DOF Cycl_s

462.8 x 10"6

529.8 x i0.6

As pointed out in S_bsectlon 7.6.4.1, the computing time in terms of CPU time per (DOF) (cycle) for the finite-strain prediction

of specimen

CB-18's response when modeled by (2-D) beam elements was 21.7 x 10"6. Thus, it is seen that the plate-element

finite-strain 3-D structural

response is about 24 times "more expensive" than the simpler, less reliable 2-D model and calculation.

344

,

I SECTION SUMMARY

8. I

a method that

incorporates

finite

rings,

strains of

(see Section

and

is that

the continuum,

and

and constant

Tensors

considered

components),

i

I

l

!

time-

is consistently

valid

finlte-element

deformations

of beams,

have

the present

(containment)

these

necessary

flnlte-straln material

dependence

of

plasticity

theory

ring

been

demonstrated

method

of analysis

response

to engine

used

concise

that

space-fixed books

functions,

of simply

materials

vectors

by most

vector

plasticity

behavior

tools

The

detail,

and

in common

mechanics. of the

as a collection

to clarify

kinematics

of

of variable

use is made

tensors

carefully

deform

Continuum

are helpful

deform.

system

system

on

considering

theory

of Hill

(llke elastic-plastic

the

of a deformable

defining

is extended unloading,

by of the "mechanical and means Dtlwez. Strain-hardening are

is referred

by means

in a body-fixed

precisely

for the analysis.

_/_e material

the present

traditional

in considerable

effect, and hysteresis) by Prandtl, Timoshenko,

with

to which

shells)

accurate

functions

materially-embedded

vectors

which

is tzeated

configuration

transient

systematically

in the

because

all quantities

complex

problem

(instead

laws under

The

and which

forcing

m%d

to a flxe_____d reference

As a result,

large

to known

as linear

d_____representation

continuum

and

of structural

with

coordinates

physical

rotations.

pl_tes,

validating

strain-hardening,

respect

formulation)

is formulated

coordinates

are

with

and

impact.

theory

convected

rings,

elastic-plastlc,

strains

A practical

applied

The

finite

t_ developing

(beams,

implemented

subjected

7).

principally

structures

Lagrangian

transient

rotor-fragment

with

behavior

and plates

has been

for thin

(total

predictions

is devoted

flnite-s_rain,

material

configuration for

study

of analysis

dependent

i•

AND CONCLUSIONS

_arZ The present

of

8

easily

to quantities

of proper

and with

accommodated

the reference

345

very

the Bauschi_g,_r

sublayer method" pioneered and complex straln-rate by this

associated

transformations

to include

with

between

configuration.

model.

This"

a fixed

the tensors

reference associated

i

Straln-dlsplaeomont rotations

and which

platost

_quations

include

which

thinning

are valid

effects

are

for finltn

derived

and

strains

for bo_Imn, ring_,

and _holls.

Thn

flnito

of Virtual

clement

Work

concept

is used

and D'Alombort's

in conjunction

Prlnciplo

with

to obtain

the Prinelplo

the equations

of motion

of a general continuum is permitted undergo arbitrarily largo rotations and solid strains A now which constant stiffness toformulation of the finite element

equations

efficient

of motion

eomputationally

conventional

pseudo-force

is valid

flnite-straln

for

conventional plastic

and better

second

behavior

order

formulation

numerically

Furthermore,

of any

formulation

to be

assemblage stepwise

equations

ordinary

kind

is valid

is more

than

this new

of material, only

which

in time by

the

formulation while

the

for small-strain

elastic-

impulsive

using

loading are

or

of

at

the nodes

with

of the

scheme

of the

of freedom flnlte-element

set of equations

integration

The

predictions

with

is solved

an appropriate

with

data.

(steel-sphere sides

missile, ideally

the

finite-element

elastic-plastic The

when

or

clamped)

pose

346

and

the

Houbolt

computer

Either

well-defined

linear

operator that

developed

and

finite-

of the nonlinear

introduced

thin beams,

small-strain-

Houbolt

pKograms

theory

to initial

finite-strain

and with

iteration

(implicit)

and targets

c._,iped-end

subjected

ealculatlons.

tlme-dep_ndent

missiles

data

operator

forces

are

of a sequence

The present

experimental

internal

by means

which

fragments.

for the timewise

is employed of the

is made

structures

central-differenQe

the fin_te-qrtrain experimental

of analysis

reliable

the nonlinear

of motion

four

gradients)

by rigid

with

are used

equations

all

the degrees

and plate

to impact

The

operator

extrapolation

of

This

system

equations

the values

a numerical

ring,

compared

predictions.

difference

differential

continuum.

of this method

for beam,

predictions

the

of a finite-size

time operator.

assessment

9f problems

being

represents

consist

nonlinear

and displacement

finite-dlfference An

of motion

(coupled)

determined

(displacements

with

now

conditioned

formulation.

pseudo-force

resulting

unknowns

ments

This

materials.

The

_eory

is developed.

thin

is used.

incorporate are compared

in these square

configurations

experipanels and

,

: _.,_,.,,_. :....... ; • i._--_---" _m'_._"_"_

.._

-_ ,., ,_ .....,.......... _ _:_

b

! I conditions

for which

dafloeti_n

data

The_1_ test wit/1 velocities and

Incl_dlng

th{_ "small From strain tions thin

8.2

c_nd_tions

have

included

suf.fici_nt

to produce

ware

comparisons

transient 3-D

compared

strains

and permanent

obtained. Lmpulse

_oadinq

oY. fraq_ent

re;]pannes of various

it appoa_n

oft_n

which

that

finite

the use of

ca** provide

(the most

structures

strain,

impact

severitlus strains

up

well

to

beyond

observed.

formulation

with

bonn

ruptur(_ conditions;

i]traln" range

2-D and

permanent

have

threshold

these

strain,

of hi_jh quality

elastlc-plastic of

loads,

transient

important

are

finite-

significantly

improved

and sensitive

quantities)

subjected

the prevlously-employed

the present

to severe

impulse

small-strain

predicin

or impact

procedure.

Conclusions on the

basis

of

the presQnt

study,

the following

conclusions

may

be stated: (i)

For

general

strain

theory

response levels

by

Large

should computer

of strain

defined _2}

application,

small

differences

(3)

the

between theory

(b) at regions

The

use

the peak

and superior

comparisons

show.

the present

the present cost ove_ types

are

small-

of transient

for only

where

of

about

larger

for

all

a poorly-

results

347

strain

than

cent

and

gradients

about for

i0 per

cent).

thin

compared

with

small-strain

theoretical-experimental

formulation

transient

and

studied

physically

use of the

af nonlinear

csses

provides

as the present

flnlte-strain

results

5 per

formulation

and plates)

the

in the

significant

are

strain

theory

found

finite-strain

rings,

predictions,

use of

the finite-strain

strains

fo_mulation

The

than

is valid

is valid

of the order

of the present (beams,

analysis

the former

latter

results

and

(where

rather

of strain.

larger

no additional

I

since

(a) strains

realistic

J

methods

for

structures

(4)

in nonlinear

herein

occur

theory

be used

whereas

level

the small-strain

finlte-strain

involves

small-strain structural

practically

formulation response

for

problems.

(5)

(6)

Finito_strain

ola_tlc_pla_tic

impl_mentod

_asily

appears

not

to have

Whereas

the use

and procedur_ vQry

strains (7)

The

in a tot_l be_n

compared

(_le most

subjected

be

Lagranglnn

domonntrated

to affect wi_h

(and han

rofnrnncn

and

important

data)

show

good

to explosive-impulse

,frame! thin heretofore,

finlts-strain translont

calculations,

are affected

comparisons

gonerally

b_on)

implemented

the predicted

small-straln

theorotieal-experim0ntal

calculations

_

of the proper-and-oonslstent

appears

little

th_orF

analysis

41spla_omonts

the predicted

slgni_icantly.

for the finite-strain

agreement

loadlngs

for thin

structures

or to impact

by

a rigid

fragment. (8)

The

Kirchhoff

stress

Piola-Kirohhoff strain

stress)

plasticity

(a)

confused be used

because

considerations

thermodynamic

(b)

should

problems

theoretical

as well

(not to be

equations

numerical variational

principle

in the

-- based

formulation

2nd

of finite-

on the simplicity

employ

of a rate

considerations

the ist or the

of:

which

as the exlsuence

with

-- the

the Kirchhoff

potential,

existence

and a symmetric

of the

stress,

and

of an incremental

tangent

stiffness

matrix. Additional (o)

(d)

merits

the Kirchhoff

The

for e_ample_

and

A. Nadai,

the Kirchhoff

mechanical

to the popular The present strain describe

stress

as,

material (9)

include:

the

is easily

measured

classical

experiments

of G.I.

such

Taylor

and stress

in simpler sublayer isotropic

strain-rate

represents terms

model

348

of impact

stress

hardening

rules

sublayer

a very and

behavior

powerful

expl6sive

of the

measures.

is superior

mechanical

provides

structures.

other

of plasticity

sensitive

problems

the actual

than by

and kinematic

elasto-viscoplastlcity the complex

in experiments

theoretically of 'plasticity.

model

of

tool

to

loading

finite

of

(I0)

Th_ now finite

(finlt_ _lomont

tlonally

and

_quationn bottnr

(nmall-_traln) problems (Ii)

Tho

and

are of the

actual

to consider

these

widely

different

the

course

the

_train-rate

in the

of

actual

cumputa_

thn oonvn[it_on_i

formulation

though

f_r

material

of

th_

dependence

was

this

strain

much

alloy

wau

strain-rate

constants these

values

appropriate

it is

as being

constant

strain

encountered

and remain

considered

and

is considerable

and how

"constants"

response

as

representative

strain-rate

the transient

were

the strain-rate

properties,

strain-rate levels

there

(2-D

analysi_

the aluminum than

of

far as how

tho

_tructuroa

when

sensitive

_vQn

As

material

than

of

aluminum

apprgprlateness

in the analysis.

offlciont

ntlffnoss

str_nn)

results

is so,

in %he

of tho

work.

strain-rate

This

to he _oro

numorleally

explo_ively-lmpuln_d

insensitive.

used

in this

as being

uncertainty

shown

constant

to the exporlmental

analyzed

Htlffnosfl formulation

conditionnd

(dlsplacomonts

3-D) of _ho closer

wan

pn_udo_foruo

tested

r_nultn

conntant

strain_

uncertain.

to be

dependence

impacted

narrow

in

Moreover,

isotrcpic,

rate

over

could

while be

anisotropic. (12)

The 2-D

analysis

satisfactory tions

are

strain

of steel-sphere

as far

as the transient

concerned.

information

Howe%er,

if detailed

is needed,

and

of ruptttre is to be predicted necessary.

In effect,

of

the narrc_

analysis impact,

beams

predicts which

that

will

that

agrees

largest both

tions.

349

!

and permanent

a 3-D analysis

strains

experimental

is

(2-D structure

and

(and hence

rupture)

strains

at the clamped

predic-

if the occurrence

analysis

the highes_

occur

the

With

the 2-D

is quite

response

transient

in particular

ade_,ately,

while

2-D frag_ient) predicts

displacement

b_ams

ends,

occur

the

3-D

at the region

results

and

expecta-

of

8.3

Su99_ntion_ for Put_e

Roee_rch

It le advln_btn _o purnun the inclu_len e£ the fcllewing a_po_t_ i_ _ut_rn _n_ly_i_ _ove_opmnnt_. i.

To study th_ Impl$_It Dark op_rato_, that appnarM to pof_sosflb_tter _nl_o-d_b%plng nnd _r_quonoy-d_stor_on

_._ro8

thnn thos_ _f the

Moubolt ope_ato_, but its performance co_t, hav_ n_t boon eomp_e£oIy assessed fo_ the present _ato_o_y _ 2,

_oblemD.

To invost_g_to tho utilization of qua_i-Nuwton

itoratlon mo_hods

(liko Broyd_n's mothod o_ the BPGS mothod) within each t_mo stop as required to achlevo convorgencc in accord with speciEi_d critoria o£ the nonlinear equations that have _o be solved with impl_cit operators llke the Houbolt or Park operators. 3.

The development and impl_msntatlon of an uf£iulent shell £initeelement analysis of finite-strain elastlc-visooplastic

4.

The inolusion of transverse shear deformations,

5,

The inoluslon of anisotroplc material e£gects.

350

problems.

|j

•.... ....... ......

•

i RF,PF,_E_CF,,q

351

I I

Ii. Cauolly, A.L., "_ur _ns dlverflns m_thodes _ ]'a|ca_ deflqu(;_llnson pout _tabllr lon f_qumtJons qul rapr_sontant los lots d'_qul].lhro, ou h_ m,mv_mont Jnt_riti_r don ,_orpn _olldnn ou fl-uido_", Bull. noi. math. _no. prop. conn,, Vol. ,1,3S In30, pp. 169_176. 12. TrU,v_d_ll, C., "_hn Ratignal )J!i!O//1711A". L. ;.hl!¢,ri Oporo

M_?Ob_nlqn o___f_P_%ox!hl____rJ[laD_tig_Bod_of!, ('J'nl_lin, Volllmo I], Part 2, lq60.

].3, K.Ol_rIJ?,W.._.., "()h tho NhI|_,i,l,oar 'Pllr)or_, o._ [Ph'/,n |_],no_,|,o _]holln", Pro = c_odlu_In of the K_n_|_R%yka 11. 69, 1966, ])]_.I_54.

No¢l(",,'].nu(/r_

Akad_m_o

]4. l,c_o_h, {_.W., "}?_.n_t;o=D_ffor(_ncoCalculation I'l, at_tl,¢by_mm_c;all,y-lnducod Do_o_._%iono of A_.'FDL-TR-66-171, Do_. 1966.

van

Mo_hoa _nu_a_

Wot_mi_oh_:_l,l_on,

_ot.'_a_9(, l.:im;t_.c_hh; ,_,hollu " ",

15. L_o_h, J.W., Witmo;, E.A., and Plan, _._.II., "Numerical _al_ulation T_chnlquu for Largo Elastic-Pla_4tlc Translont Dofo%_ations of Thin Shulls", ATAA Journal, Vol. 6, No. 12, Dec. 1968, pp. 2352-2359. 16. M_ino, L., L_h, d.W., and W£tmer, E,A., "PETRO8 _ A New Pinit_Difference Method and Program f_ the Calculation of Largo ElasticPlastic Dynamically-lnduced De_ormatlons o£ Ounu, al Thin Shells", BRL CR 12 (MIT-AS_L TR 152-i), December 1969 (In two parts. AD 708773 and AD 708774). 17. Morlno, L., Leech, J.W., and Witmer, E.A , "An Improved Num_rlcal Calculatlon Technique for Large Elastlc-Plastlc Transient Deformations of Thin Shells", Journal of Applied Mechanics, Sure 1971, Parts i and 2, pp. 423-436. 18. Atl_rl, S., Wit/,er, E.A., Le_ch, J.W., and Merino, L., "PETROS 3, A Finlte-Differen_ M_thod and Program _o1_ the Czl_ulat_on of Large Elast_c-Plastlc Dynamically-lnduced Deformations of Multilayer, Variable-Thlckness Shells' . BRL CR 60 (MIT-_SRL TR 152-2), Novambor 1971. AD #890200L. 19. Pirotin, S.D., Berg, B.A. and Witmer, E.A., "PETROS 3.5: "New Developmenta and Program Manual for the Fil_i_e-Difference calculation of Large Elastis-Plastlc Transient Deformations cf MUit.ilayer Vaziable-Thlckness Shells", BRL CR 211 (MIT ASRL TR 152-4), February _.975. 20. Pi_otin, S.D., Merino, L., Wltme_, E.A., and Leech, J.W., "F_nlteDifference Analysis for Predicting Large Elastic-Plastle "2ransient Def_rmatlons of Varlabl_-Thi_kness Kirchheff_ Soft-Bonded Th_n and Transverse-Shear-D_formable Thicker Sh_lls", BPJ, CR 315 (MIT-ASRL TR 152-3), September 1976.

352

21. Pirotln, S.D., Barg, R.A._ _nd wirer, E.A., "PETROS 4s New D_v_lopmohts and Pro_ra_ Manual for the Finlte-DifforQnce Calculation of Large Rla_tio-Plastie, and/or Vlscoolastic Transient De£ormations of MultilayQr VarlablQ-Thiakness (_} Thin Hard-Bonded, (2) ModeratelyThick Hard-Bonded, or (3) Thin Soft-Bonded Shells", BRL CR 316 (MITASRL TR 152-6), September 1976. 22. Truesdell, C.s "A First Co_rs9 'in Rational General Concepts, Academic Press, New York

Continuum 1977.

Mechanics"

Vol.l,

23. Wu, R.W.-H. and Witmer, E.A._ "Finlte-Element Analysis of Large Transient Elastic-Plastic Deformations of Simple Structures, with Application to the Engine Rotor Fragment Contalnment/Deflection Problem", ASRL TR 154-4, Aeroelastic and Structures Research Laboratory, Massachusetts Institute of Technology, Jan. 1972. (Available as NASA CR120886. ) 24. Wu, R.W.-H. and Witmer, E.A., "Computer Program - JET 3 - to Calculate the Large Elastlc-Plastlc D_namically-lnduced Deformations of Free and Restrained, Partial and/or Complete Suructural Rings", ASRL TR 154-7, Aeroelastic and Structures Research Laboratory, Massachusetts Institute of Technology, Aug. 1972. (Available as NASA CR-120993.) 25.

Ziri_, R.M. and Witmer, E.A., "Examination of the Collision Force Method for Analyslng the Responses of Simple Containment/Deflection Stz-_ctures to Impact by One Engine l_tor Blade Fragment", ASRL TR 154-6, Aeroelastic and Structures R_search LabOratory, Massachusetts Institute of. Technology, May 1972. (Available as NASA CR-120952.)

26. Wu, Richard W-H. and Witmer, E.A., "Finite Element Predictions of TZansient Elastic-Plastlc Large Deflections of Stiffened and/or Unstiffened Rings and Cylindrical Shells", AMMRC CTR 74-31 (MIT ASRL TR 171-4), April 1974. 27. Stagliano, T.P., Spilker, R.L. and wirer, E.A., "User's Guide to Computer Program CIVM-JET _B to Calculate the Transient Structural Responses of Partial and/or Complete Structural Rings to Engine Rotor Fragment Impact", MIT ASRL TR 154-9, March 1976 (also available as NASA CR-134907). 28. 8mdal_ J.J.A. and Witmer, E.A., "Finite Element Nonlinear Transient Response Analysis of Simple 2-D Structures to Impulse or Impact Loads", MIT ASRL TR 182-1, June 1976. 29. Wu, R.W.-:I., Stagliano, T.R., Witmer, E.A. and Spilker, R.L., "User's Guide he Computer Programs JET 5A and CIVM-JET 5B to Calculate the Lei_ge Elastic-Plastlc Dynamically-Induced Deformations of Multilayer _,artial and/or Complete Structural Rings", MIT ASRL TR 154-10, November 1978. (Available as NASA CR-159484).

i

I

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A_qyrin, J.ll., D_f|o, P._. and _nqolopoulos, T., "Nonl_noar 6se_lln_ ti_nn usinq the _n_to Elom_n_ To_hnlqun", J. comp. Mnths. App]. MQoh. Eng., _, 1973, pp. _03-2,50.

172. Key, _.W., "h ?Inlto Element Procodu=e _or the Largo Deformatlon Dynami_ But]ponsu o_ Axlsymn_)trlo Solids", _. comp. Moths. hppl. Mech. _ng., 4, _974, pp. 195-218. 173. Strloklin, J.A., Martinez, J._., Tillsrson, J.R., llong, J.H., and Haisler, W.E., "Nonlinear Dynamic Analysis of Shells of Revolution Matrix Displacem_nt Method", NASA CR 1_0405. Natlonal Aeronautlcs Spaoe Admi_.istration, Washington, D.C_, 1971. 174.

by and

Strioklin, J.A., }_aisler, W.E., and Riesemann, W.A., "Evaluation of Solution Procedures for Material and/or Geometrically Non.l.inear Structural Analysis by the Direct Stiffness Method", AIAA/ASME/SAE Structures, Structural Dynamics and Materials, 13th Conference, San Antonio, Texas, April 1972.

175. MeNamara, J.F. and Ma_cal, P.V., "Incremental Finite Element Analysis of Nonlinear Dynamic and Computer Methods in Struet.ur_l Mechanics Academic Press, New Yolk, 1973, pp. 353-376.

Stiffness Method for Problems", in Numerical (Edited by S.J. ?enves),

176. Leech, J.W., "Some Experiments with Institute of Teohnology, May 1964.

Equation",

the Wave

Massachusetts

177. Wu, R. and Witmer, E.A., "Stabillty of the De Vogelaere Method for Timewise Numerical Integration", AIAA Journal, Vol. 11, No. i0, Oct. 1973, pp. 1432-1436. 178.

Lax, P.D. and Richtmyer, R.D., "Survey of the Stability of Linear Finite Difference Equations", Communications On Pure and Applied Mathematics, Vol. 9, 1956, pp. 267-293.

179. O' Brlen, Solution Physics,

G.G., Hyman, M.A. and Kaplan, S., "A Study of the Numerical of Partial Differential Equations," ._. Mathematics and Vol.

29, 1950,

pp.

223"251.

180. Eddy_ E.P., "Sterility in the Numberical Solution of Initial Value Problems in Partial Differential Equations," HOLM 10232, NaVal O_dnance Laboratory, White Oak, Silver Spring, Maryland, 1949. 151.

Roache, P.J., "Computational Albuquerque_ N.M., 1976.

Fluid

364

Mechanics",

Hermosa

Publishers,

_82. Kiot_tmyo_, R,D., "_ Su_.voy of D_,_fo_onoo Mnthodn _o_ Nonntn_dy P1uld Dyn_m£on", NeAR Toohnlenl Noto 63-_, _ouldo_, Colorado, 196_. IN3, Krlo_, R.D., "OIi_on_ition_1 _tabilit_ in Numo_ioal T_.mo _ntog_atlon Mothods", J. Ap91. Much., Juno 19_3, pp. 417_4_I. 184. Morlno, _,., LQoah, JoW. and Wi_mor, _.A._ "Optimal P_odlc_'o_Cor_octor Mothod _or Systoms of S_cond-Or_r Di_£ozon%i_ Equations", A%AA Journol, Vol, 12, No. i0, Oct. 1974, pp. 1343-1347. 185. Hicks, D.L., "Ono-Dimenslonal Lagrangian Hydrodyn_,ics and the IDLH Hydrocode", SC-RR-60-728, Sandia Laboratorios, Albuquerqu_, New M_xico, 1969. 186, Phillips, N.A., "An Example o_ Non-Linear Computational _nstability", _t,_Sph_reand_Sea in Motioq, Rossby Memorial Volume, (B.Bolin Ed.), Rockefeller Institute Press, New York, 1959. 187. Hirt, C.W., "Heuristic Stability Theor_ for Finite Difference Equations", Journal of Computational Physics, Vol. 2, 19_8, pp31_'_ 188. Go,flay, A. R. and Morris, J.L., Finite Differ_nr_ _' ds for _onlinear Hyperbolic Systems", Mathematics of C_ . _(.'_h, Vol. 22, No. 101, 1968, pp. 28-39.

_ ,

189. Lilly, D.K., "On the Computational Stability of Numerical Solutions Of Time-Dependent "1o_Linear Geophysical Fluid Dynamics Problems", U.S. Weather B_ea_ Monthly Weather Review, Vol. 93, _o._, 1965,pp.]I-26. 190. Stricklin, J.a., MartlneZ, J.E., Tillerson, J.R., HOng, J.H. and Haisler, W.E., "Nonlinear Dynamic Analysis of Shells of Revolution b> Matrix Displacement Method," AIAA Journal, Vol_ 9, No. 4, April 1971 pp. 629-636. 191. Strlcklin, J.A. et el., "Large Deflection Elastic-Plastic Dynamic Response of Stiffened Shells of Revolution", Department of Aerospace Engineering, Texas A & M University, Report 72-25, Dec. 1972. 192. Weeks, G., "Temporal Operators for Nonlinear Structural Dynamics Problems", Journal of the Engineering Mechanics Division_ Proc. Amer. Soc. Civil Eng., October, 1972_ pp. 1087-1104. 193. McNamara, J.F., "Solution Schemes f_r Problems of Nonlinear Structural Dynamics", Journal of Pressure Vessel Technology, May 1974, pp.96-102.

J

I

194. Park, K.C., "Evaluatlng Time Integration Methuds for Nonllnear Dynamic Analysis", in Finite Element Analysis of Transient _onlinear StructuFal Behavior, (Edited by T. Belytschko et el) ASME Applied Mechanics Symposia Series, AMD-14, 1975, pp.35-58.

355

195. Bolyt_hko, T. _nd Sohooborlo, D.F., "On tho Unoondltionol St_b£11_y o_ an Implicit Algor_thm Eor Nonlinoar Structur_l Dyn_micn", Journal of _ppllod Mochanlc8, Dec. 1975, pp. BGS-F69. 195. Dahl_iut, G. and BJorok, /_.,NumoEiual M_thod_, translated by Nod Andorson, Prentico-H_ll, New _erBey, 1975. 197. DQeai, C.8. and Ab¢l, Jo, Introduction to ,ha PinitQ Elemont Method, Von Nostr_uld Reinhold Company, 1972. 198: Strlcklln, J.A. and Hal%let, W.E., "Formulations and Solution Procedures for Nonlinear Structural Analysis", Computers & Structures, Vol. 7, _, 125-136i Feb. 1977. 199. Bushnell, D., "A Subincremental Strategy for olving roblems nvolving Large Deflections, Plasticity and Creep", in "Constitutive Equations in Vlgcoplastlcltyt Computational and Engineering Aspects", AMD Vol. 20, 1976, pp. 171-199. 200. Bathe, K.J., "Static and Dynamic Geometric and Material Nonlinear Analysis Using ADINA", Report 82448-2, Acoustics and Vibrations Laboratory, Department of Mechanical Engineering, MIT, May 1977. 201. Ting, T.C.T., "The Plastlc Deformation of a Cantilever Beam with Strain Rate Sensitivity under Impulsive Loading", Brown University TR 70, Contract Monr-562 {10), July 1961. 202. Bodner, S.R. and S_nnonds, P.S., "Experimental and Theoretlcal InVestigation of the Plastic Deformation of Cantilever Beams SubJeuted to Impulsive Loading", Jo_rnal of Applied _hanics, December, 1962, pp. 7!9-728. 203. Bathe, K.J. and Ci_nto, A.P., "Some Practical Procedures for the Solution of Nonlinear Finite Element Equations", J. Computer Meth. in Appl. M_ch. and Eng., in press. 204. Mar,hies, H. and Strang, G., "The Solution of Nonlinear Finite Element Equations", Int. J. Num. Meth. in EDg., Vol. 14, 1979, pp. 1613-1626. 205. Newmark, NoM., "A Method of Computation for Structural Dynamics", Journal of the Eng. Mech. Div., Proc. Am. Soc. Cir. Eng., Vol. 85, July 1959, pp. 67-94. 205. Houbolt, J.C., "A Recurrence Matrix Solu. ion for the Dynamic Response of Elastic Aircraft", Journal of the Aero. Sciences, Vol. 17, 19500 pp. 540-550.

366

! 207.

Clark, of Structures, E.N., Suhmitt, II. Plastic F.II. and Deformation Nioolaidos,of Rings", S., "Plastic FDL-TDN Deformation 64-64 Vol. II, Eng. March 1968.

S_i.

Lab.,

Pleatinny

Arsenal,

Dover,

New Je[soy,

208. Anon., "Rotor Burst Protection Program". (Study for NASA Lowls Research Center on NASA DPR-ID5 and NASA Interagency Agreement C-4158_-B), U.S. Naval Air Propulsion Test Center, Auronautlcal Engine Dept. Progress Reports Sept. 1969-Jan. 1977. 209.

Private com_nt_nications from G.J. Mangano, R. DeLucla U.S. Naval Air Propulsion Test Center, Philadelphia, 1975-76.

and J. Salvino, Pennsylvania,

210.

Ma/_gano, G.J., Salvino, J.T. and DeLucia, R.A., "Rotor Burst Protection Program -- Experimentation to Provide Guidelines for the Design of Turbine Rotor Burst Fragment Containment Rings", Proceedings of the Work_hop on An Assessment of Technology for Turbojet Engine Rotor Failures, NASA CP-2017, 1977, pp. 107-149.

211.

Spilker, R.L. and Witmer, E.A., "Theoretical and Experimental Studies of the Nonlinear Transient R_sponses of Plates Subjected to Fraglnent %mpact", Paper presented at the Fourth International Conference on Structural Mechanics in Reactor Technology, San Francisco, Calif., 15-19, August 197#.

212.

Epstein, M. and Murray, D.W., "Large Deformation In-Plane Analysis of Elastic Beams", Computers and Structures, Vol. 6, 1976, pp. 1-9.

213.

S_rang, G. and Fix, G., An Analysis Prentice-Hall, New Jersey, 1973.

214.

Rodal, J.J.A., French, S.E., Witmer, E.A. and Stagliano, T.R., "Instructions for the Use of the CIVM-JET 4C Finite-Strain Computer Code to Calculate the Transient Responses of Partial and/or Complete Arbitrarily-Curved Rings Subjected to Fragment Impact", MIT ASRL MR 154-1, December 1979. (AVailable as NASA CR-159873.)

of

the Finite

215. Dennis, J.E. and Mor_, J., "_uasi-Newton Theory", SIAM ROy. 19, 1977, pp. 46-89. 216. Bell, J.F., dur Physik, 217. Bridgman, Specimen", 218.

Distribution at the Neck 32, 1944, p. 553.

and Felgar, R.P., pp. I14-127_

367

p!ast!city

.1

Method,

Motivation

The Experimental Foundations of Solid Mgchanics Volume VI 2/1, Springer Verlag, Berlin, 1973.

P.W,, "The Stress Trans. ASM, Vol.

Lubahn, J.D. Wiley, 1961,

Methodsz

Element

and

, Handbuch

of a Tehsion

and Creel_ of Metals_

John

219.

Norris, D., Mor_n, B., Sauddor, J. and Quinono_, D., "A Computer Simulation o£ the Ton_ion Teat't, Journ_l of th(_ M_chanics _nd Fhy_ic_ of Solids, Vol. 26, 1978, pp. 1-19.

220. Sajo, M., "NQcklng of a Cylindrical Bar in Tension", Solids and Structures, Vol. 15, 1979, pp. 731-742.

Int. Journal

of

221. Hutchinson, J.W., "Survey of Some Recent Work on the Mechanics of Necking", Proceedings of the Eight U.S. National Congress of Applied Mechanics, Los Angeles, June 26-30, 1978, Editor: R.E. Kelly, Western Periodicals Co., North Hollywood, 1979.

368

t

369

!

370

J

I I I

TABLE 2 DATA CHARACTERIZING NAPTC TEST 201 FOR T58 TURBINE ROTOR TRI-HUB BURST AGAINST A STEEL CONTAINMENT RING

Containment

Rin_ Data

Inside Diameter (in) Radial Thickness (in) Axial Length (in) Material Elastlc Modulus (psi) 4130 Cast Steel

15.00 0.625 1.50 4130 cast steel 29 x 106

Fragment Data* Type T58 Tri-Hub Bladed Disk Fragments Material Disk: A-286 Blades: SEL-15

i

Outer Radius (in) Fragment Centroid fzom Rotor Axis (in) Frag_nt Pre-Test Tip Clearance from Ring (in} Fragment CG to Blade Tip Distance (in) Fragment Weight Each (Ibs) Fragment Mass Moment of Inertia about its CG (in Ib sec2) Rotor Burst Speed (rpm) Fragment Tip Velocity (ips) Fragment CG Velocity (ips) Fragment Initial Angular Velocity (rad/sec) Fragment Translational KE (in-lb) Each Fragment Total for Thzee Fragments Fragment Rotational KE (in ib) Each Fragment Total for Three Fragments

Applies to each fragment unless specified otherwise.

! I

JT1

7.00 2.797 0.50 4.203 3.627 666XI0 "4 19,859 14,557.2 5816.7 2079.6 158,922 476,766 144,018 432,054

)

372

TABLE 3 --

CONCLUDED (_L-SH)

i

iml.

_

-

UPPER-SHIt_CE PRINCIPAL GP_EN STRAIN ,

lemon t Canter LOU., x(in)

iii

1 .111

i

ii

2 .333

i

3

4

.555

.7_7 ,

(PER C_NT)

iii

i

5

i

6

.999

i._98

i

Tiros

(psec) 20

.35

1.01

5.15

10.29

34.50

.12

40

3.92

7.93

12.10

21.16

35.54

10.06

60

16.73

20.94

24.61

23.24

37.02

10.81

80

21.06

26.21

23.57

22.16

37.17

11.02

100

21.13

26.12

23.17

21.40

36.99

11.44

120

21.11

26.05

23.24

21.28

36.12

10.93

140

21.08

26.11

23.35

21.43

35.18

10.94

160

21.07

26.15

23.38

21.56

34.96

10.60

180

21.06

26.20

23.42

21.52

34.58

10._7

200

21.06

26.20 •

23.37

21.48

34.63

10.04

220

21.06

26.22

23.42

21.57

34.69

10.17

240

21.05

26.18

23.36

21.52

34.72

10.25

260

21.05

26.18

23.37

21.50

34.66

10.22

280

21.05

26.18

23.38

21.49

34.64

10.18

300

21.06

26.20

23.37

21.46

34.63

10.20

350

21.07

26.20

23.37

21.47

34.68

10.27

400

21.08

26.20

23.38

21.44

34.65

10.29

450

21.08

26.21

23.38

21.43

34.68

10.34

500

21.09

26.23

23.41

21.47

34.74

10.42

550

21.09

26.26

23.43

21.56

34.79

10.61

600

21.10

26.26

23.42

21.57

34.82

10.71

......

373

I

I I _ASl'_

I

4 --

CONCLUDED

UPPER-SURFAC_ il

im

i

1

_,lomont

i

i

(R_-SH-SR)

PRINCXPAL i

I

2

i

3

GR_N

8TiffIN (P_R CENt) ii

i

pl

4

5

6

.777

.999

1.298

Conte_

Lo=,, x(In) 'Time

.111

.333

,555

....................

{_sec) 20

.89

3 .O0

6,12

6 .88

4.90

i. 39

40

10.28

10.93

i0.5t_

8.98

6.30

4.69

60

13.87

12.73

10.03

7..65

8.44

5.99

80

14.27

12.90

9,44

7.27

8.60

6.25

i00

14.07

12.86

9._7

7.3?

7.41

6.78

120

14.01

12.91

9.92

7.50

7.04

7.25

140

14.13

12.87

9.59

7.30

7.39

7.22

160

14.08

12.90

9.77

7.50

7.28

7.36

180

14.14

13. O0

9 •88

7.45

7.23

7.56

200

14.05

12.82

9.63

7.44

7.49

7.61

220

14.15

12.97

9 76

7.42

7.55

7.94

240

14.05

12 .87

9 .74

7 •51

7.49

8.24

260

14.09

12.84

9.59

7.35

7.56

8.38

280

14.11

12.88

9,63

7.34

7.59

8.66

300

14.11

12.87

, 65

7.39

7.59

8.81

I I I I

375

TABLE

5

FINITE_STKASN PIIED_C_ON OF THE MAXIMUM PRINCIPAL STRAINS AND A_SOCIAT_D D_R_CT_ONS ON Tl_ [_P_R SURFACE AT Tl_ CEN_R OF CBRTA_N _I.4_MBNTB OF _XPZ,OSIV_5¥_ £MPULS_D 6061-T651 THIN ALURIN(JM PANB,L CP-2 Pri.nolpol Greon No.

Contor

Location

1 2 3 4 5 6

x(in) .iii .333 .555 .777 .999 1.298

y(in) .Iii " " " " "

O(aog) 45.00 18.43 11.31 8,13 6.34 4.89

12 13 14 15 16 17

.111 .333 ,555 .777 .999 1.298

.333 " " " " "

23 24 25 26 27 28

.Iii .333 .555 .777 .999 1.298

34 35 36 37 38 39

Strain

VaZuo

O_£ont.

(in/in) .2113 .2626 .2461 .2324 .3717 .1144

Op(dog) 45.08 13.86 4.59 1,28 1.87 1.70

(In/in) .1427 .1300 .1055 .0898 .0860 .0881

Op(dog) 45.00 9.91 3.59 2.14 16.40 37.51

71.57 45.00 30.96 23.20 18_43 14.40

.2719 .1910 .1611 .2179 .3997 ,1234

76.78 45.00 11.06 3.44 5.72 26.74

,1301 .09tl .08_9 .0850 .0880 ,0803

80.09 45.00 2.87 2.54 28.00 -32.78

.555 " " " " "

78.69 59.04 45.00 35.54 29.05 23.17

.2466 .1611 .1345 .1776 .3798 .0722

85.15 78.94 45.00 8.72 8.09 -41.32

.i055 .0899 .0688 .0828 .0864 .O712

86.42 87.13 45.00 -6.62 16.95 31.15

.111 .333 .555 .777 .999 1.298

.777 " " " " "

81.87 66.80 54.46 45.00 37.87 30.93

.2324 .2179 .1776 .1103 .3036 .0995

88.72 86.56 81.28 45.03 14.23 20.93

.0898 .0850 ,0828 .0700 .0946 .0661

87.86 87.46 -83.38 -45.00 26.92 9.39

45 46 47 48 49 50

.111 .333 .555 .777 .999 1.298

.999 " " " " "

83.66 71.57 60.95 52,13 45.00 37.61

.3733 .3997 .3798 .3035 .2155 .1053

88.15 84.28 81.91 75.77 44.79 ;_.61

.0860 .0880 .0864 .0946 .0738 .0809

73.60 62.00 73.05 63.11 45,04 31.79

56 57 58 59 _0 61

.iii .333 .555 .777 .999 1.298

1.298 " " " " "

85.11 75.60 66.83 59.07 52.39 45.00

.1142 .i_31 .0715 .1002 .I059 .0710

F8.31 63.19 -43.97 68.43 52.48 46.6]

.0882 .0806 .0668 .0651 .0806 .0482

52.53 -57.24 72.83 80.52 58.09 -44.88

37_

Vaiuo

... 0rlont.

!

o

i

'I

FIG.I

NOMENCLATURE

FOR SPACE COORDZNATES

377

A_D DEFORMATION

+

_I_°

",

"J %"_ =_ %,,

_'

I 1

/I

-----APPROX, "----ACTUAL

,

I i

(a)

.... Actual

LOGARITHMIC

I I

I I

I

|

I

I

I

I

_l,_o

I

I

_oo

:

I

I

i

..= +

li

E

=

_n(l

+ E u)

I

,

_

...... 4. _' SUBIA_R "4"

I

I ,

STRAIN I CuzVes

lulE "Approximated

and

I

I

/ /

II

,, /

.........

I

I

I

I

_%,,

I

I_

2,_Y

,, , Z.,_SLOPE

=E

.........

LOGARITHMIC Properties FIG.

of 2

th_

Elastic,

APPROXIMATION BY

THE

"2"

SUBLAYER

"i"

"K'V "'

(b)

SUBLAYER

"

W

STRAIN

£u

.........

=

OF

A UNIAXIAL

378

_

.....

9_n(l + E u)

Perfectly-Plastic

MEC}L_NICAL-SUBLAYER

" "

SUblayers STRESS-STRAIN MODEL

....

CURVE

ml,_ ° ii ST [_A _N_IAI(IDNN _NG

I/

.....

-

I / /I

'

I' I

" ?" '__

_j_r.

.....

.... _

LOCAP/THM_C STRAIN _U

l J I I/S

L-

BAUSCHINGER EFFECT

= £n(l+ Eu)

/

(c)

FIG. 2

Schematic of Loading, Unloading, and Reloading Paths

CONCLUDED

I ! 379

"

II_0

•

II

/-............

_:.o

i

O