a generalized failure criterion for three-dimensional ... - Science Direct

Engineering Fracture Mechanics Vol. 54, No. 1, pp. 75-90, 1996

Pergamon 0013-7944(95)00109-3

Copyright © 1996 Elsevier Science Ltd. Printed in Great Britain. All rights reserved

0013-7944/96 $15.00+ 0.00

A GENERALIZED FAILURE CRITERION FOR THREE-DIMENSIONAL BEHAVIOUR OF ISOTROPIC MATERIALS H. ALTENBACHt Institut fiir Werkstofftechnik und Werkstoffpriifung, Otto-von-Guericke-Universitfit Magdeburg, Postfach 4120, D-39 016, Magdeburg, Germany and

A. ZOLOCHEVSKY Physical Engineering Department, Kharkov Polytechnical University, Frunze 21, UKR-310 002, Kharkov-2, Ukraine Abstraet--A failure criterion for isotropic materials based on three invariants of the stress tensor and six parameters is proposed. Basic tests for the determination of the parameters are formulated. A comparison with special criteria from literature is made and recommendationsfor using the special criteria are given. It is shown that the results of calculation using the proposed criterion are in good agreement with experimental data for grey cast iron, magnesiumalloy and epoxy compound in the case of high hydrostatic pressure. The experimental data are taken from literature about failure tests in the case of moderate and high hydrostatic pressure. Thin-walled tubular specimen loaded by axial force, inside pressure and a torsional momentum in different combinations are used. Copyright © 1996 Elsevier Science Ltd.

1. I N T R O D U C T I O N THE USE of traditional metallic materials in extreme situations, e.g. in high pressure aggregates, tubes, rolling techniques, as well as the use of new materials (e.g. polymers or composites on the base of polymers) requires investigation of life time, working conditions and failure behaviour etc. of parts loaded by high hydrostatic pressure. Till now the main information in strength calculations is based on strength limits of the materials, derived from simple tests under the condition of atmospheric pressure, loaded in tension, compression and torsion. Basing on these tests it is impossible to give recommendations about the strength properties of materials working under high hydrostatic pressure conditions. Non-neglectable three-dimensional stress states are arising in the material during the working regimes of the given constructions. It is shown by tests [1-7] that the level and kind of stress state are of great influence on the strength. Strength criteria for isotropic materials are described e.g. in refs [8-13]. In dependence from the number of basic tests they are divided into different classes [14a]: strength criteria with one, two, three, four or more parameters. But in general we can obtain, that the multiparameter-criteria are not proved enough by tests in the case of a three-dimensional stress state. In this paper a generalized strength criterion is proposed, which includes as special cases well-known classical criteria. A comparison is given with data from tests under loading conditions causing three-dimensional stress states. These tests can be interpreted as the foundation of the presented method.

2. F O R M U L A T I O N OF A G E N E R A L I Z E D C R I T E R I O N W I T H SIX P A R A M E T E R S A great number of strength criteria well-known from literature can be written in the following form a~q ~< au.

(1)

"['Author presently at Institut fiir Werkstoffwissenschaft, Martin-Luther-Universit/it Halle-Wittenberg, D-06217 Merseburg, Germany.

75

76

H. A L T E N B A C H and A. ZOLOCHEVSKY

Here a,q (aeq _> 0) is a homogeneous function of the stress tensor akt and some material parameters. Using eq. (1) we describe the equivalence between the uniaxial and the (in general) three-dimensional stress states. Therefore the stress aoq is called equivalent stress. The strength limit of the material under uniaxial tension is au. If aeq < a, the material or the structural element can be used. Only for treq = au we assume a failure behaviour. In the case of isotropic materials the function treq depends upon stress invariants. Three independent invariants can be calculated, but the equations for their determination are not unique. Several authors proposed different values of invariants. Here we consider the first invariant of the stress tensor, 11, the so-called stress intensity or yon Mises-stress tr~ and the angle of the kind of stress state ~. These three invariants are determined by the following equations [14b] and [15]

11=a.,,0-~=

9 sk,s,~sl~

sktsk,,sin3~=

2

at

' {~1 < 6"

(2)

Here sk~ are the stress deviator components 1

Skt "= ffk! - -

~

Itfkl

(3)

and 6k~ is the Kronecker symbol (-

6kl= ~ 1

L0

for

k=l

for

k#l.

Let us suggest the following expression for the equivalent stress in criterion (1) aoq = 21a~ sin ~ + 22ai cos ~ + 23a~ + 2411 + 25I~ sin ~ + ;tr/~ cos ~.

(4)

The L, (m = 1. . . . . 6) are scalar factors. For practical use of the criterion, eqs (1) and (4), it is necessary to determine these parameters. 3. B A S I C E X P E R I M E N T S

The proposed criterion, eqs (1) and (4), contains six independent constants. For determination of these parameters we need six basic experiments. Therefore it is called a six-parameter-criterion. A short description of these tests is given below, for more details see e.g. refs [1, 2, 6, 13, 16]. It can be assumed that the following six basic failure tests realize a homogeneous stress state in the specimen: • uniaxial tension a,, = a o ,

(5)

a,l = a~.

(6)

• uniaxial compression

Here ac is the strength limit of the given material in the case of compression. • pure torsion o",2 = zu.

(7)

Here zu is the strength limit of the given material in the case of pure torsion. • thin-walled tubular specimen under inside pressure

In this case we obtain two stresses different from zero: the axial stress a,~ and the circumferential s t r e s s 0"22 Oil =

O'r

~',

0"22 =

fir.

(8)

Three-dimensional behaviour of isotropic materials

77

Here ar is the limit o f the circumferential stress and it can be calculated from o.~ = p R / h , p is the inside pressure in the case of failure, R, h are. the radius of the middle surface and the thickness of the thin-walled tubular specimen. • biaxial tension In this case the thin-walled tubular specimen is loaded by inside pressure and a uniaxial tensile force. So we get as non-zero stresses o'll =

T

~

+

o.t

"~-, 0"22 = o't,

(9)

Using a special combination of uniaxial force and pressure, the condition (10)

o'll = O"22 = O.•

must be fulfilled. Here o.* = o-t is the limit of the circumferential stress, o.* = p R / h , p, T are the pressure and axial force values in the tubular specimen in the case of failure, A is the cross-section area of the specimen. • uniaxial tension and superposed hydrostatic pressure In this case, stresses are different from zeroT o.. = ~ - q, o-22= o-33= - q.

(11)

Here T is the axial force, A is the cross-section area of the specimen, q is the hydrostatic pressure. In the case of failure the following conditions for the stresses should be fulfilled 2 **

o.,, = "~ o.

, o'22=o'33=

1 ** -- ~o. •

(12)

Here the value of o.** is three times the value o f the hydrostatic pressure in the case of failure. Calculation of the first invariant of the stress tensor in this case leads to L = 0. This is the reason for the specific choice o f the stresses o.., o-22, o.33 according to eq. (12). The next step in the determination of the unknown scalar factors is connected with the analysis of the strength criterion, eqs (1) and (4), in each special case of the basic tests. In the case of uniaxial tension we can calculate the following invariants from eq. (5) I1 = O.u, o'i = O.U, ¢ =

-- 6"

(13)

The results give the following equation for the criterion, eqs (1) and (4),

-- -~- + ~

42 + 43 + 4 4 -

-~- + ~

46 = 1.

(14)

In the case of uniaxial compression we can calculate the invariants from eq. (6) 7~

I, = -o.c, oi = ac, { =

(15)

and from the generalized criterion, eqs (1) and (4), we get

2 + - T -42+ 43-24

--

2

2

46 -

as. o.c

(16)

For pure torsion, eq. (7), the calculations lead to I1 = 0, al = x/~zu, ~ = 0

(17)

and the special expression of the criterion, eqs (1) and (4), is

,/ 42 +

o.u

= 7

(18)

H. ALTENBACHand A. ZOLOCHEVSKY

78

For a thin-walled tubular specimen loaded by inside pressure eq. (8) leads to

vq

(19)

and the expression of the criterion, eqs (l) and (4), is x/~ (22 + 23) + 3 ().4 + 26)

2

ou

--g"

(20)

In the case of biaxial tension caused by stresses, eq. (9), the invariants have the values (21)

Ii = 2tr*, o'i = tr*, ~ = g , and we get the special form of the criterion, eqs (1) and (4),

).1 ,/5

--2 + T

).2 + ).3 + 224 + )-5 + X//3).6 -- tr*" tru

(22)

The three-dimensional stress state, eq. (12), is determined by the invariants Il = O, trj = o**, { = - n_ 6

(23)

and finally we get ).l --

x//5

"20i---2

try

(24)

"- /~2 -)1- ).3 ~'='-" O'**"

So eqs (14), (16), (18), (20), (22) and (24) describe the dependence of the scalar parameters )., (m = 1. . . . . 6) from the six strength limits tru, tr¢, zu, ar, tr*, a** of the given material. The solution of the system of eqs (14), (16), (18), (20), (22) and (24) can be presented in the following form:

4 ~-2tr~ O'r

O'u

_3+3tr**

'~u

).6 "~"

tru

O'r

O'c

3).6

Tu

3

).s=2 -1+

~

21 = - 1 +

tT.~u 0"c

2 a-~, ).3 =

O'u

6 - 3X/~ 2 tru

).4 ~-

O'u

tr* + --

+).,+x/5 /-

+ 224 + %/3).6

tru + 2l

2 - , / Zu3

a~ _ w/~23 2 2 - ~u

(25)

Remark: the proposed basic tests are not unique. So we demonstrate only the possibilities of determination of the scalar parameters in the criterion, eqs (1) and (4), based on eq. (25). In the case of other basic tests the parameters can be calculated in a similar manner. The choice of the basic tests is connected with the technical possibilities in the laboratories, with the kind of material and kind of loading, etc.

Three-dimensionalbehaviourof isotropic materials 4. S P E C I A L

STRENGTH

79

CRITERIA

The proposed strength criterion generalizes several well-known criteria. Below we show the relations of the generalized and some special criteria. These criteria are often written in terms of the principal stresses 0"1, a2, 0"3 (0"1--- 0"2 > 0"3). The mean stresses correspond to the introduced invariants, eq. (2), by the following formulae [14b]:

0"i =

3

20"isin ~ + 11 3 , a3 =

, tr2-

3

(26)

Now it is possible to compare the different special criteria with the generalized criterion, eqs (1) and (4). The first step includes the determination of the 2,,-values for each special criterion. The second step is connected with the calculation of the relations between the strength limits 0",, 0", Vu, at, 0"* and a**, starting from the special L,-values and eq. (25). These relations can be interpreted as recommendations for the practical use of the special criteria.

4.1. One-parameter-criteria 4.1.1. Huber-von Mises-Hencky-eriterion. This criterion is also known as the criterion of distortional strain energy density. It can be given in the following form [17] 0"i = 0"~.

(27)

0"eq ~--" 0"i.

(28)

In this case the equivalent stress becomes

This expression can be derived from the generalized equivalent stress, setting )`3 = 1,

)`1 =

),2 = ),4 =

),5 =

(29)

),6 = O.

From eq. (25) we can calculate the relations between the strength limits 0"u _

0"u

O'u

~c

0-*

0-**

N/3

O'u 1.

- -

--

0-,

2

'

0-u -Tu

- -

..lb.

(30)

If we obtain that in different tests the relations (30) are fulfilled, the strength of a given material under given loading conditions can be calculated from the criterion (27) and it is not necessary to take other criteria into account. 4.1.2. Coulomb-Tresca-Saint Venant-criterion. This criterion postulates that the failure state depends upon the maximum value of shear stresses [18] 1

1

rm.x = ~ (0"1-- 0"3)-- ~ 0"u.

(31)

In this case the equivalent stress is given by 0-eq ~

al

0"3

(32)

0-i C O S ~ .

(33)

~

and from eq. (26) we can calculate

2:

-- ~

a~q-

3

Comparison with eq. (4) leads to 21=23=),4=),5=),~=0,22-

23:

(34)

and the recommendations for practical use following from eq. (25) can be expressed 0-u

0-.._~u

0-u

O'u

o-~ = 0-, = ~-~ = ~ - ~

O'u

= 1, --Tu = 2.

(35)

80

H. ALTENBACH and A. ZOLOCHEVSKY

4.1.3. Mariotte-criterion. This criterion is called the maximum principal strain criterion. The criterion was also developed by Lam6, Clebsch and Rankine. The following formulation is given in ref. [11]

1

(36)

So we get the equivalent stress

1

aoq = a , -

(37)

~ (or2 + a~)

and taking into account eq. (26) finally

1

(38)

ai(v/3 cos ~ - sin ¢).

Comparing eqs (4) and (38) 2,, can be determined 21 ~--- - - ~ ,

22 :

"--~,

23 = 24 :

25 -~" 26 :

0

(39)

and from eq. (25) we calculate the recommendations for practical use au ac

a, a*

1 au 2 ' z,

3 au 2 ' a~

3 a~ 4 ' a**

(40)

4.1.4. Galilei-Leibniz-criterion. This criterion is also called the maximum principal stress criterion. So we can write [11] al = au

(41)

aoq = al.

(42)

and for the equivalent stress we get

Taking into account eq. (26) the equivalent stress can be calculated

j

,/3

aoq = - ~ cri sin ~ + T

1

tr~ cos ~ + ~ I~.

(43)

The scalar factors in eq. (4) can be expressed for this special case as

1 22 = - x//~ 2, = - ~, 3-,

2 , = 0 , 2 , = 31 ' 25=26=0

(44)

and the recommendations for practical use follow au = 0, azu~ _ a-~

aaru _

au = 1, a** au - 23" a*

(45)

The first relation means that the Galilei-Leibniz-criterion can be used for materials with a strength limit in the case o f uniaxial compression equal to infinity (or the strength limit in uniaxial compression should be much greater than in uniaxial tension). 4.1.5. Sdobyrev-criterion. This criterion has the following expression [19]

1

(O'i + 0"1) = GU,

(46)

The equivalent stress is equal to O'eq ~

1 (ai + at)

(47)

Three-dimensional behaviour of isotropic materials

81

or taking into account eq. (26) 1

0-m

(v/3

cos ~ - ai sin ~ + 3a, + I0.

(48)

So we get the scalar factors in eq. (4) for this criterion as 2, = -

,/5

1

1

1 , 22 = - - 6 - ' 23 = ~ , 24 = g , ,% = & = 0.

(49)

On this basis we can calculate the recommendations for practical use

0-u

1

0-u

0-c

2

ru

x/

+l

0-u

2

0-r

x/3+2 4

0-°

3

0-*

2 ' 0-**

an

5 6

(50)

4.1.6. Saint Venant-Bach-criterion. This criterion is also called the criterion of the m a x i m u m linear deformation. Let us consider that E~ is the m a x i m u m linear deformation. Using H o o k e ' s law this criterion can be expressed as 0-1 - v(0-2 + 0-3) = 0-u.

(51)

The Poisson ratio v can take values from 0 to 0.5. The Saint Venant-Bach-criterion is also a one-parameter-criterion, because we introduce only one strength limit 0-u. I f we take v = 0 we get the Galilei-Leibniz-criterion, eq. (41) from eq. (51), if we take v = 0.5 we get the Mariotte-criterion, eq. (36). The equivalent stress for the Saint Venant-Bach-criterion can be expressed as [11] 0-eq ~--- 0-1 - - 1"(O"2 "~ 0-3).

(52)

Taking into account eq. (26) we get 0-¢q - -

1 1 +3 v m(x//-5 cos ~ - sin ~) + ~ (1 - 2v)I~

(53)

and from the last equation and eq. (4) follow 2~= _ T_ , 2l 2 + v

l+v3 x/~'23=25=2°=0'24=

l(l_2v).

(54)

The recommendations for practical use are 0-u 0-c

--

=v,

0-u Zu

--

=

1 +v,

0-u 0-r

-

2 - v 0-u 2 ' 0-*

-

1 0-u 2 ' 0-**

=

32 (

1 +v).

(55)

4.2. Two-parameter-criteria Let us discuss in detail some special strength criteria with two parameters. F o r the determination of these parameters it is necessary to take into account the results of two independent failure tests. For shorter writing we introduce new symbols 0-u X= ~,e=

2 - r/ 2_v/~,t/--

0-u

vT"

4.2.1. Mohr-criterion. The Coulomb-Tresca-Saint Venant-criterion, eq. (31), was modified by M o h r [11] 0-1 - Z0-3 = 0-u.

(56)

So the equivalent stress can be calculated 0-eq = 0-1 - - Z0-3

(57)

or 0-eq ~ EFM 54.I--F

1 [%//~(1 "~- •)0-i COS ~ - - ( l - - Z)0-i

sin )~ + (1 - Z ) I , ] .

(58)

82

H. ALTENBACHand A. ZOLOCHEVSKY

The Mohr-criterion is a special case of the generalized criterion (4), setting

1

2, = ~ ( Z -

1), 2 : = - - ~ - - ( X + 1), 2 3 = 2 5 = 2 6 = 0 ,

1

(59)

3(1 - X ) .

24 =

Taking into account eq. (26) we can obtain the recommendations for practical use

a. O'c

a-2 = (1 + Z), a. _

- -

~(' Tu

a. a*

O"r

a. 2 + )~ 1, a** = 3

-

(60)

4.2.2. Botkin-Mirolyubov-criterion. The criterion can be expressed according to [13]

1

[(1 + X)a~ + (1 -- X)I,] = au.

(61)

The equivalent stress becomes

1

O'eq =

(62)

~ [(1 + Z)O'i "31- (1 -- X)II].

This criterion includes the stress intensity 0"i and the first invariant L. Similar expressions based on these two values are given by Schleicher [20], Klebowski [21], Nadai [22], and Drucker and Prager [23]. From comparison of eqs (4) and (62) the factors can be derived 21= 22 = 25 = 26 = 0, ),3 =

(l+D,

1 2,= ~(1--Z),

(63)

and taking into account eq. (26) we get the recommendations for practical use

,/3(l+x),

3

o'~ = )~' Zu - 2 -

4

O'r

3-z

~(1--Z),

4.2.3. Pisarenko-Lebedev-criterion.

a* --

2

l+z 2

' a**

(64)

This criterion has the following form [13] Xai + (1 - X)al = a.

(65)

O'eq = )~0' i + (1 -- )~)0",.

(66)

aeq = ~~q//3 (1 -- X)O'i COS ~ -- ~1 (1 - ~()o'isin ~ + ;(tri + 1 (1 - X)I,

(67)

and the equivalent stress is

Taking into account eq. (26) we get

and for the scalar factors we calculate 2,-

I-X 3

'

22

x/3 ( 1 - X), 23 = X, 24 - -1--, X 25=26=0. =--~-

(68)

The last equation is the starting point to get the recommendations for practical use of the Pisarenko-Lebedev-criterion

aua¢ - Z '

anzu = I + ( w / 3 - 1 ) Z '

a~ar = 1 - ( 1 - T ) Z ,

x/~

au tru ~ = 1, a** -

2+ X 3

(69)

4.2.4. Sandel-criterion. This special criterion has the following form [24]

1 a, + ~ (1 - x)a: - Xa3 = ao.

(70)

Thus the equivalent stress can be expressed by a°q = a, + 1 (1 - X)a: - Xa3

(71)

Three-dimensional behaviour of isotropic materials

83

or taking into account eq. (26) O-~ = ~x//~ (1 + Z)o-~cos ~ + 1 (1 - X)IL.

(72)

In this ease we determine the following scalar factors for the generalized failure criterion (4) 2 2 = ~v / 3 ( l + g ) , 2 , =

2~=23=25=26=0,

~1( 1 - Z )

(73)

and from eq. (25) we calculate the r e c o m m e n d a t i o n s for practical use O-o

5 -

O-u O-~ = 1 + Z, O-~-~(' Zu a~

4

X

,

O-~

O-*

-

3 -

Z

O-u

2

' O-**

-

1 + Z

2

(74)

4.2.5. Koval'chuk-criterion. This criterion can be expressed by the following equation [25] ~o-~ + (1 - ~)(aL - O-a) = O-o.

(75)

So the equivalent stress has the formulation O-eq = ~0"i + (1 - ~)(o-, - 33)

(76)

or taking into account eq. (26) O-cq-

3

(1 -- ()o-i cos ~ + (o-i.

(77)

After c o m p a r i s o n of eqs (4) and (77) we get 2, = 24 = 2 , = 26 = o, 22 = - -

(1 -

~ ) , 2~ =

(78)

and O-u l, O-u O-~ = ~ =q'

O-u l O-u O'u O-r -- 2 q' O-* -- O-** -- 1.

(79)

In Table 1 the values o f the scalar coefficients 2,, (m = 1. . . . . 6) in the generalized criterion, eqs (1) and (4), and the relations between the strength limits are given for each special criterion (oneor two-parameter-criteria). 4.3. Three-parameter-criteria 4.3.1. Paul-criterion. F o r this failure criterion the following expression we find in ref. [26] ato-i + a20-2 + a30-3 = O-u.

(80)

Here at, az, a3 are scalar parameters. Using eq. (26) we get a new formulation o f the criterion (80) 210-i sin

~ + 220- i COS ~ "+" /~4/I ~--- O'U,

(81)

with 1

./5

1

2, = ~ (2a2 - al - a3), 22 = ---z_ a.~~ -(- - a3), 24 = 3 (a, + az + a3).

(82)

According to the Paul-criterion the equivalent stress becomes O-0q= 2zo-~sin ¢ + 2~o-icos ~ + 24Ii.

(83)

F r o m the c o m p a r i s o n o f eqs (4) and (83) we conclude that the Paul-criterion follows from the generalized criterion setting in eq. (4) 23 = 2~ = 26 = 0.

(84)

84

H. ALTENBACH and A. ZOLOCHEVSKY

Table 1. The values of the scalar coefficients 2,, (m = 1 , . . . , 6) in the generalized criterion and the relations between the several strength limits for special one- or two-parameter-criteria Relations between strength limits Scalar factors O'u

Criterion

21

22

23

24

,~5

26

0

1

0

0

o

3

0

0

0

0

l

O'u

O'u

O'u

O'u

2

,

1

2

l

1

1

1

i

One-parameter-criteria (27)

0

2fi

(31)

0

(36)

- 2

2

0

0

0

o

~

1

~

3

~

3

~

(41)

1 - ~

,/7 ~-

0

1 ~

0

o

0

1

1

1

(46)

1 - 6

x/~ 6

1 ~

1 ~

0

0

~

2

4

2

x/3 (1 + v) 3

0

1-2v 3

0

0

v

l+v

2--v 2

1 ~

3

0

0

Z

1+~(

1

1

1--Z 2

0

0

;(

1

FS_

(51)

l+v 3

1

2

3

5

2

(l+v)

Two-parameter-criteria (56) (61)

x- 1

~/7(x + l) 3

0

0

0

1+ g 2

3

l-x

2+;( 3

43(I + z)

v6(l + x)

3-z

2

4

2

1 + ;( 2

1--(1-- ~x -~) X

1

2 3+ x

5--;( 4

3--;( 3

~

1

+ 3(l-x) 4

(65)

X-1 3

x/~ -~- (1 - Z)

~

(70)

0

x/-j -~- (1 + ,~

(75)

0

--

(1 - ~)

1 -;( 3

0

0

X

1 + (x/3-- 1)X

0

1-;( 2

0

0

~(

l+z

~

0

0

0

1

r/

2

1+ 2 1

F r o m eq. (25) we get the r e l a t i o n s ~=17~+1, Zu

17u=2~-1,

O'c

~

17,,-

17r

3

4-2--+-17r

17¢

.

(85)

In the case o f the use o f the P a u l - c r i t e r i o n the i n d e p e n d e n t tests are u n i a x i a l t e n s i o n , u n i a x i a l c o m p r e s s i o n a n d t u b u l a r s p e c i m e n l o a d e d b y inside p r e s s u r e . F o r the o t h e r tests the r e l a t i o n s (85) s h o u l d be fulfilled. 4.3.2. Tsvelodub-criterion. T h e f o r m u l a t i o n o f this c r i t e r i o n is given in ref. [27] Z~17i sin ~ + •2ai COS ~ + ~-317i=

O'u.

(86)

So we get the f o l l o w i n g e q u i v a l e n t stress tr0q = 2tai sin ~ + 220"i COS ~ "]- /~-30"i.

(87)

T h e T s v e l o d u b - c r i t e r i o n is a special case o f the p r o p o s e d g e n e r a l i z e d c r i t e r i o n (4) setting )~4 = ~'5 = ~'6 = 0,

(88)

T h e o t h e r s t r e n g t h limits s h o u l d be c o n n e c t e d b y the r e l a t i o n s 2~ 17r

= tru Tu'

tru

17"

17"

(89)

0 ' * * = 1, 17" - - O'c'

In this case the i n d e p e n d e n t tests are u n i a x i a l t e n s i o n , u n i a x i a l c o m p r e s s i o n a n d p u r e t o r s i o n . 4.4. Four-parameter-criteria 4.4.1. Birger-criterion. T h i s c r i t e r i o n is f o r m u l a t e d in ref. [28]

all71 + a20"2 q- a3173 q- a417i = 17u.

(90)

Three-dimensionalbehaviour of isotropic materials Here am (m = 1. . . . .

85

4) are scalar parameters. From eqs (90) and (26) we get 210"i sin ~ + 2 : i cos ~ + 23al + 2dj = au,

(91)

with 21= ~1 ( 2 a 2 - a l - a 3 ) , 2 2 = T

1 x//3 (al - a3), 23 = a4, 2, = ~ (a, + a2 + a3).

(92)

Thus we have the following equivalent stress O'eq = 210"i sin ~ + 220"i COS ~ "~ •30"i "q- 2411-

(93)

From the comparison of eqs (4) and (91) follows that the Birger-criterion is a special case of the generalized criterion setting 25 = 26 = 0.

(94)

Using eq. (25) we get finally a. - 1 a**

~1 ( 2 -au - + .a o ) . O'r

a..

"Cu ~ ~

. 2 a.,

O'r

a. + -ou -. Tu

(95)

O'c

That means that the independent tests for the Birger-criterion are uniaxial tension, uniaxial compression, pure torsion and tubular specimen loaded by inside pressure. For the other tests the relations (95) should be fulfilled. 4.4.2. T a r a s e n k o - c r i t e r i o n . This criterion can be formulated as [29] 2:~ cos ~ + 24/1 + 251l sin ~ + 261t cos ~ = au.

(96)

The equivalent stress becomes O'eq : 220"iCOS ~ -'~ 24/i + 2511 sin ~ + 2611cos 4.

(97)

The comparison of eqs (4) and (97) leads to 21 = 23 = 0

(98)

2 a *au* - a~ zu' 2 aa* . =3--z~ a. - 4 aac" .

(99)

and using eq. (25) we get

These values formulate the limits of the use of the Tarasenko-criterion. For this criterion we obtain the same independent tests (uniaxial tension, uniaxial compression, pure torsion and tubular specimens loaded by inside pressure) as for the Birger-criterion. For the other tests eq. (99) should be fulfilled. 5. GREY CAST IRON UNDER M O D E R A T E H Y D R O S T A T I C C O M P R E S S I O N For the first comparison of theoretical and experimental results, the level of hydrostatic pressure is chosen to be moderate. In ref. [16] the experimental data of the basic tests are given (grey cast iron, room temperature): au = 253 MPa, ac = 6 2 4 MPa, zu = 168 MPa, CTr 2 2 2 MPa, a* = 195 MPa, a** = 592 MPa. "~-

From these values and eq. (25) we can calculate the 2j = 0.2753,

22 = 2 . 2 7 2 , 23 =

-

2,,

1.402, 24 = 1.580, 25 = - 0.2752, 26 =

-

1.322.

It can be shown that all special failure criteria are not able to describe all data from the basic tests for the given grey iron, e.g. the special criteria could not predict the value a , / a * * on the basis of strength properties for grey iron evaluated in tests with atmospheric pressure (cf. Table 2). So we conclude that it is necessary to use the proposed six-parameter-criterion and basic tests, particularly taking into account loading conditions which cause a three-dimensional stress state.

H. A L T E N B A C H and A. Z O L O C H E V S K Y

86

Table 2. Comparison of theoretical and experimental strength values (material: grey cast iron) Relations between strength limits Test or strength criterion Experiment (27) (31) (36) (41) (46) (56) (61) (65) (70) (75) (80)

flu O'c

O'u ~u

O'u O'r

O'a (7*

O'u O'**

0.405 1.000 1.000 0.500 0.000 0.500 0.405 0.405 0.405 0.405 1.000 0.405

1.51 1.73 2.000 1.500 1.000 1.37 1.410 1.22 1.30 1.41 1.51 1.41

1.14 0.866 1.000 0.750 1.000 0.933 1.000 1.05 0.946 1.15 0.755 1.14

1.30 1.000 1.000 0.500 1.000 1.500 1.000 1.30 1.000 1.30 1.000 1.28

0.427 1.000 1.000 1.000 0.667 0.833 0.802 0.703 0.802 0.703 1.000 0.709

Now we discuss several failure test results [16] for thin-walled tubular specimens under moderate hydrostatic pressure loaded by an axial tensile or compressive force, a torsional momentum and inside pressure. In Table 3 the results of the equivalent stress calculations by eq. (4) and the principal stresses trl, tr2, a3 in the failure state are shown. A satisfactory strength prediction on the basis of the proposed generalized criterion in the case of moderate hydrostatic pressure and grey cast iron can be obtained. 6. GREY CAST IRON U N D E R HIGH HYDROSTATIC C O M P R E S S I O N Now we discuss the test results of thin-waUed tubular specimen failure in a high hydrostatic pressure cabin [16]. The tubular specimen are made of grey cast iron. In this case the generalized criterion, eqs (1) and (4), with the determined scalar parameters does not give a good correspondence. The reason for this behaviour is the determination method for the parameters: the used data in connection with eq. (12) are related to a moderate level of hydrostatic compression. So it is necessary for the use of the generalized criterion to include data of basic tests with three-dimensional stress states related to a high level of three-dimensional compression. We are taking into account following data for failure under atmospheric pressure: uniaxial tension (5), uniaxial compression (6), pure torsion (7). Moreover, we analyse three new failure tests related to three-dimensional stress states: • uniaxial tension in a high hydrostatic pressure cabin In this case we calculate the principal stresses tr~

=

-

klS,

(72

-~" I~3

=

--

(lOO)

~,

Here kl is a material parameter (0 < kl < 1).

Table 3. Calculation o f the relative error 6 = ( I t r ~ - t~.I/a.) 100%. The equivalent stress is calculated by criterion, eqs (1) and (4). Material: grey cast iron under moderate hydrostatic pressure Type of test in the high pressure cabin

Inside pressure Inside pressure Inside pressure Biaxial tension Biaxial tension Torsion Uniaxial compression

Mean stresses, MPa Hydrostatic . . . . . . . . . . . . . . . pressure q, M P a trh tr., a3 100 200 300 100 200 100 100

176 136 82 130 97 106 - 100

38 - 32 - 109 130 97 - 100 - 100

- 100 - 200 -300 - 100 -200 - 306 -767

6, % 6.24 9.78 19.80 17.40 18.60 7.95 28.50

Three-dimensional behaviour of isotropic materials

87

• u n i a x i a l c o m p r e s s i o n in a h i g h h y d r o s t a t i c p r e s s u r e c a b i n . N o w t h e p r i n c i p a l stresses b e c o m e 0-1 = 02 --- - ~, 03 = k2 is a m a t e r i a l p a r a m e t e r (k:

- k2~,

(101)

> 1).

• p u r e t o r s i o n in a h i g h h y d r o s t a t i c p r e s s u r e cabin. N o w we get the f o l l o w i n g p r i n c i p a l stresses o"l =

- f , 0"2 =

- k3f, 0"3 =

- k4f.

(102)

H e r e k3, k4 a r e m a t e r i a l p a r a m e t e r s (0 < k3 < 1, k~ > 1). I n eqs (100)-(102) the v a l u e s 0, ~ a n d ~ a r e the m a x i m u m v a l u e s o f 0": in the tests. N o w we c a n d e t e r m i n e b y a n a l o g y the six s c a l a r p a r a m e t e r s 2,, (m = 1 . . . . . 6) for the g e n e r a l i z e d c r i t e r i o n , eqs (1) a n d (4), b a s e d o n s t r e n g t h limits o f a g i v e n m a t e r i a l , eqs (5)-(7) a n d (100)-(102). I n this c a s e we get m o r e difficult r e l a t i o n s in c o m p a r i s o n w i t h eq. (25) a n d we leave o u t the n e w e q u a t i o n s . F o r t h e g i v e n g r e y cast i r o n we are u s i n g the f o l l o w i n g d a t a f r o m basic tests [16]: 0"u = 253 M P a , 0"c = 624 M P a , zu = 168 M P a , = ~ = f = 500 M P a , k~ = 0.16, k2 = 2.492, k3 = 0.396, k4 = 1.604. F r o m these v a l u e s w e c a n c a l c u l a t e the Am 21 =

-0.4502,

22 = 1.537, 23 =

-0.6675,

24 =

-0.3311,

25 =

-0.0784,

26 = 0.4656.

(103)

T h e results o f c a l c u l a t i o n s b a s e d o n the g e n e r a l i z e d c r i t e r i o n , eqs (1) a n d (4), a n d the coefficients (103) f o r several cases o f t h r e e - d i m e n s i o n a l stress states (different f r o m the basic tests) are s h o w n in T a b l e 4. T h e e x a m p l e s s h o w a g o o d c o r r e s p o n d e n c e b e t w e e n t h e o r e t i c a l a n d e x p e r i m e n t a l results. S o w e c a n r e c o m m e n d the g e n e r a l i z e d failure c r i t e r i o n , eqs (1) a n d (4), b a s e d o n the p r o p o s e d b a s i c tests f o r p r e d i c t i n g the f a i l u r e o f m a t e r i a l s u n d e r h i g h h y d r o s t a t i c p r e s s u r e conditions.

7. M A G N E S I U M

ALLOY

UNDER

HIGH

HYDROSTATIC

COMPRESSION

T h e n e x t e x a m p l e is c o n n e c t e d w i t h t h e failure tests o f m a g n e s i u m alloys ( r o o m t e m p e r a t u r e ) [16]. T h e tests w e r e c a r r i e d o u t w i t h t h i n - w a l l e d t u b e s l o a d e d by u n i a x i a l f o r c e a n d

Table 4. Calculation of the relative error 6 = (laeq-trul/~,)100%. Equivalent stress is given by criterion, eqs (1) and (4). Material: grey cast iron under high hydrostatic pressure Type of test in the high pressure cabin Uniaxial tension Uniaxial tension Uniaxial tension Uniaxial tension Inside pressure Inside pressure Inside pressure Inside pressure Inside pressure Inside pressure Torsion Torsion Torsion Torsion Uniaxial compression Uniaxial compression Uniaxial compression Uniaxial compression

Principal stresses, MPa Hydrostatic . . . . . . . . pressure q, MPa tr~ a., a3 100 200 300 400 100 200 300 400 500 0.1 100 200 300 400 100 200 300 400

187 128 64 - 15 176 136 82 10 - 50 222 106 40 - 43 - 122 - 100 -200 - 300 -400

-

100 200 300 400 38 - 32 - 109 - 195 - 275 111 - 100 -200 - 300 -400 - 100 -200 - 300 -400

- 100 - 200 - 300 - 400 - 100 - 200 -300 -400 - 500 0 - 306 -440 - 557 -678 -767 -905 - 1022 - 1148

6, % 0.40 3.16 4.35 0.79 11.90 5.14 3.56 9.09 9.88 16.20 2.77 11.I0 5.14 1.58 3.16 5.14 3.95 4.35

88

H. ALTENBACH and A. ZOLOCHEVSKY

inside pressure in different combinations in a high hydrostatic pressure cabin. In this case a special form of the generalized criterion, eqs (1) and (4), can be used for the determination of the equivalent stress (104)

O'eq = 230"i + 24/1.

This special criterion is derived from the generalized criterion setting 21 = 22 = 25 = 26 = 0.

The basic tests for the determination of the scalar coefficients are uniaxial tension under atmospheric pressure (5) and uniaxial pressure in a high pressure cabin with principal stresses (100). In our example we have the following test data [16] tru = 181 MPa, 6=500 MPa, k1=0.302

(105)

and the scalar coeffÉcients in the failure criterion follow 23 = 0.888, 24 = 0.112.

(106)

The results of calculations based on the generalized criterion, eqs (1) and (104) and the coefficients (106) for several cases of three-dimensional stress states (different from the basic tests) are shown in Table 5. The correspondence between theoretical and experimental results is satisfying. The special failure criterion, eqs (1) and (104), can be used for magnesium alloys under high hydrostatic pressure. 8. EPOXY C O M P O U N D UNDER H I G H HYDROSTATIC COMPRESSION The last example is related to failure tests of an epoxy compound (room temperature) [30]. The tests were carried out with thin-walled tubular specimen loaded by uniaxial force and a torsional momentum in different combinations in a high hydrostatic pressure cabin. In this case the special criterion, eqs (1) and (104), was first used for determination of the equivalent stress. The basic tests for the determination of the scalar coefficients are the same as for magnesium alloys. Then we have [30] au = 26 MPa, 6 = 200 MPa, k~ = 0.605

(107)

and the scalar coefficients in the failure criterion follow 23 = 0.9117,

24 =

0.08833.

(108)

Table 5. Calculation of the relative error 6 = (lacq-aul/ go) 100%. The equivalent stress is determined by the criterion, eqs (1) and (104). Material: magnesium alloy under high hydrostatic pressure Principal stresses, MPa ~l

~2

~3

3, %

126 59 --20 -74 174 120 56 - 16 --96 - 172 150 95 20 -48 - 130 -212

- 100 -200 -300 --400 87 10 -72 - 158 -248 -336 150 95 20 -48 - 130 -212

- 100 -200 -300 -400 0 - 133 -238 -343 -446 -549 0 - 129 -233 -338 -440 -543

6.08 6.08 1.10 6.08 0.00 7.73 9.39 7.18 0.00 4.97 1.66 13.80 12.60 15.50 8.84 2.76

Three-dimensional behaviour of isotropic materials

89

Table 6. Calculation of the relative error 6 = ([~r~-truf/tru)100%. The equivalent stress was calculated by criterions, eqs (l) and (104), and eqs (l) and (109). Material: epoxy compound under high hydrostatic pressure Type of test in the high pressure cabin Tension Tension Tension Torsion Torsion Torsion Torsion Torsion

Principal stresses, MPa Hydrostatic pressure q, MPa 50 100 150 0.1 25 50 100 150

tr~

o2

o3

6, % criterion (1), (104)

6, % criterion (1), (109)

-2 -39 -80 20 -0.3 -22 -64 - 109

-50 - 100 -150 0 -25 -50 - 100 - 150

-50 - I00 - 150 -20 -50.3 -78 - 136 - 191

33.7 32.7 16.4 21.5 26.2 19.1 16.7 3.9

33.7 32.7 16.4 21.5 22.2 11.2 0.9 27.5

T h e c o m p a r i s o n o f the results o f calculations based on the special criterion, eqs (1) a n d (104), a n d the coefficients (108) a n d e x p e r i m e n t a l d a t a for several cases o f t h r e e - d i m e n s i o n a l stress states [30] (different f r o m the basic tests) are shown in T a b l e 6. E q u a t i o n s for the equivalent stress differing f r o m eq. (104) are possible, e.g. we can a s s u m e the following expression aeq = 230'i -'[- 2611 COS ~.

(109)

This e q u a t i o n follows f r o m the generalized criterion (4) setting 21 = 22 = 24 = 25 = 0. F r o m the results o f basic tests (107) we can d e t e r m i n e the scalar p a r a m e t e r s in eq. (109) ~,3 = 0.9117, ~,6 = 0.1020.

(110)

In T a b l e 6 the c o m p a r i s o n o f c a l c u l a t i o n results, b a s e d on the strength criterion, eqs (1) a n d (109), a n d coefficients (110), with e x p e r i m e n t a l d a t a derived by t h r e e - d i m e n s i o n a l stress state (different f r o m basic tests) is given. The use o f t w o - p a r a m e t e r - c r i t e r i a for the equivalent stress, eqs (104) a n d (109), gives a satisfying d e s c r i p t i o n o f the e p o x y c o m p o u n d strength in the case o f high h y d r o s t a t i c c o m p r e s s i o n . 9. S U M M A R Y

AND CONCLUSIONS

T h e p r o p o s e d failure criterion is one k i n d o f the classical strength criteria g e n e r a l i z a t i o n a n d c o n t a i n s several classical criteria as special cases. F o r the generalized criterion some basic tests p e r m i t t i n g r e c o m m e n d a t i o n s o f using the classical criteria are defined. The p o w e r o f the generalized criterion is d e m o n s t r a t e d for t h r e e - d i m e n s i o n a l stress states. The c o m p a r i s o n with e x p e r i m e n t a l d a t a for different m a t e r i a l s is also satisfying in the case o f high h y d r o s t a t i c pressure. A n o t h e r direction o f strength criterion f o r m u l a t i o n b a s e d on the expression F(a~q) = 1 is discussed in the literature (e.g. refs [31-34]). Here F(aeq) is a m o r e c o m p l i c a t e d n o n l i n e a r function. It can be shown t h a t expression (4) with a c o r r e s p o n d i n g n u m b e r o f scalar p a r a m e t e r s can be used for this direction. Acknowledgements--This work was carried out during the visit of A. Zolochevsky at Magdeburg, financed by the Alexander

von Humboldt Foundation (Germany).

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90

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[9] N. S. Ottosen, A failure criterion for concrete. Trans. ASCE, J. Engng Mech. 103, 527-535 (1977). [10] D. Gross, Bruchmechanik 1. Springer, Berlin (1992). [11] S. Shhn, H. G61dner, J. Nickel and K.-F. Fischer, Bruch- und Beurteilungskriterien in der Festigkeitslehre. Fachbuchverlag, Leipzig (1993). [12] H. Altenbach, Werkstoffmechanik. Deutscher Verlag fiir Grundstoffindustrie, Leipzig (1993). [13] G. S. Pisarenko and A. A. Lebedev, Deformation and Strength of Materials under Multiaxial Loading. Naukova Dumka (1976). [14a] M. Zyczkowski, Combined Loadings in the Theory of Plasticity. PWN-Polish Scientific, Warsaw (1981). [14b] V. V. Novozhilov, On the principles of analysis of results of static tests of isotropic materials. Prikl. Mat. i. Mekh. 15, 709-722 (1951). [15] H. Altenbach, U. Lauschke and A. Zolochevsky, Ein verallgemeinertes versagenskriterium und seine gegeniiberstellung mit versuchsergebnissen. Z A M M 73, T372-T375 (1993). [16] O. E. Ol'khovik, On influence of volume deformations on strength of construction materials. VINITI 2444-85, Leningrad (1984). [17] R. yon Mises, Mechanik desfesten K6rpers im plastischen Zustand. Nachrichten der GeseUschaft der Wissenschaften, G6ttingen (1913). [18] H. Tresca, M6moire sur l'ecoulement des corps solides. M~moires Pres. par Div. Say. 18, 733-799 (1868). [19] V. P. Sdobyrev, Criterion of long-time strength for some alloys under multiaxial stress state. Izv. A N SSSR. OTN. Mekh. i Mashinostroenie 6, 93-99 (1959). [20] F. Schleicher, Der Spannungszustand an der FlieBrenze (Plastizithtsbedingung). Z A M M 6, 199-216 (1926). [21] Z. Klebowski, Obecny stan wytrzymato~ciowego obliczenia material6w o wlasno~cia-chuog61nionych. Przeglad Techniczny 11 (1934). [22] A. Nadai, Theory of Flow and Fracture of Solids. McGraw-Hill, New York (1950). [23] D. C. Drucker and W. Prager, Soil mechanics and plastic analysis or limit design. Q. appl. Math. 10, 157-165 (1952). [24] G. Sandel, Ober die Festigkeitsbedingungen. Ph.D. Thesis, TH Stuttgart (1920). [25] B. I. Koval'chuk, On criterion of limit state of some steels under multiaxial stress state on room or higher temperatures. Probl. Prochnosti 5, 10-15 (1981). [26] B. Paul, Macroscopic criteria of plastic flow and brittle fracture, in Fracture: an Advanced Treatise (Mathematical Fundamentals) (Edited by H. Liebowitz), Vol. II. Academic Press, New York (1968). [27] I. Yu. Tsvelodub, The Stability Postulate and its Applications in Creep Theory of Metallic Materials. Institute of Hydrodynamics, Novosibirsk (1991). [28] I. A. Birger, On criterion of fracture and plasticity, lzv. A N SSSR. Mech. Tv. Tela. 4, 143-150 (1977). [29] I. I. Tarasenko, On criterion of brittle strength of metals. Sb. Nauch. Tr. Leningr. Inzh.-stroit. lnstit. 26, 161-168 (1957). [30] O. E. Ol'khovik, Static strength of an epoxy compound under hydrostatic stress state. Izv. vuzov. Mashinostroenie 9, 3-7 (1986). [31] P. V. Lade, Elasto-plastic stress-strain theory for cohesionless soil with curved yield surfaces. Int. J. Solids Structures 13, 1019-1035 (1977). [32] M. Langer, Entwurf und Dimensionierung eines Endlagerbergwerkes fiir radioaktive Abfiille im Salzgebirge, Proc. 6th Int. Congress on Rock Mech. (Edited by G. Herget and S. Vongpaisal). Balkema, Rotterdam (1987). [33] R. de Boer and H. T. Dresenkamp, Constitutive equations for concrete in failure state. Trans. ASCE, J. Engng Mech. 115, 1591-1608 (1989). [34] D. F. Hambley, C. J. Fordham and P. E. Senseny, General failure criteria for salt rock. Proc. 30th U.S. Syrup. on Rock Mech. as Guide Effic. Util. Natur. Resour. (Edited by Khair). Balkema, Rotterdam (1989). (Received 2 September 1994)