1
2
Dissipative propagation of pressure waves along the slip-lines of yielding material Jack Moorea,b,c,1,, Ali Karrechd , Michael Smalla,b , Emmanouil Veveakise,b , Klaus Regenauer-Liebe
3 4
a
School of Mathematics and Statistics, The University of Western Australia b Mineral Resources, CSIRO c School of Earth and Environment, The University of Western Australia d School of Civil, Environmental and Mining Engineering, The University of Western Australia e School of Petroleum Engineering, The University of New South Wales
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Abstract
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Traditional slip-line field theory considers steady-state pressure distributions. This work
12
examines the evolution of pressure at yield. We show that, for a two phase material in plane
13
strain at the point of plastic yield of the main constituent, pressure develops according to
14
dissipative wave mechanics. The pressure wave equations are examined in more detail for the
15
special case of radially symmetric plane strain, and for Von Mises and Drucker-Prager yield
16
envelopes. The positions of pressure troughs are compared to the separation of slip-lines
17
formed during the expansion by internal pressurisation of a cylinder of mild steel.
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Keywords: slip-line field theory, wave mechanics, two phase material
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Acknowledgements
20
JMM thanks Martin Paesold, Neville Fowkes and Thomas Poulet. JMM was supported
21
by the Prescott Postgraduate Scholarship of the University of Western Australia and the
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OCE PhD Scholarship of CSIRO.
1
[email protected]
Preprint submitted to International Journal of Engineering Science
June 8, 2016
1
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Dissipative propagation of pressure waves along the slip-lines of yielding material
3
Abstract
4
Traditional slip-line field theory considers steady-state pressure distributions. This work
5
examines the evolution of pressure at yield. We show that, for a two phase material in plane
6
strain at the point of plastic yield of the main constituent, pressure develops according to
7
dissipative wave mechanics. The pressure wave equations are examined in more detail for the
8
special case of radially symmetric plane strain, and for Von Mises and Drucker-Prager yield
9
envelopes. The positions of pressure troughs are compared to the separation of slip-lines
10
formed during the expansion by internal pressurisation of a cylinder of mild steel.
11
Keywords: slip-line field theory, wave mechanics, two phase material
12
List of symbols
13
Latin. Symbol
Description
a/b
Constant coefficients used in separation of variables.
A/B
The matrices describing the part of the rate of change of momentum which is proportional to the derivative of (p, ✓)T with respect to x1/2 .
B
The Biot coefficient.
c
The cohesion of a Drucker-Prager material.
C
The drag coefficient.
cn /dn /fn
Fourier coefficients.
cs
The speed of second sound.
cu
The wave speed under undrained conditions.
Preprint submitted to International Journal of Engineering Science
June 8, 2016
d
Half the distance separating wide parallel plates. (This symbol appears only in the appendices.)
d
The negative of the constant of proportionality between the -volume fraction
and specific interaction force f . (This symbol appears only
in a footnote.) D(↵/ ) Dt
The ↵/ material derivative.
f
The interaction force (exerted by
on ↵).
g(p)
The shear stress at which yield occurs at hydrostatic pressure p.
G↵
The shear modulus of ↵. (This symbol appears only in the appendices.)
H
A signature time scale.
k
The shear strength of a Von Mises material.
K
The bulk modulus relating pressure and volumetric strain.
Ks
The bulk modulus of the skeleton.
K↵/
The bulk modulus of ↵/ . (This symbol appears only in the appendices.)
l
The distance of steady-state laminar flow of an incompressible Newtonian fluid. (This symbol appears only in the appendices.)
L
A geometric length scale; the total length of a slip-line.
n
A positive integer.
n⇤
An upper/lower bound of the range of n corresponding to hyperbolic/trigonometric pressure eigenmodes pn .
nmax
The index of the highest pressure eigenmode used to produce plots.
p
The hydrostatic component of the e↵ective stress.
P
The pressure on an incompressible Newtonian fluid. (This symbol appears only in the appendices.)
pD
The discontinuous part of the pressure.
pf
The pore fluid pressure. (This symbol appears only in a footnote.)
2
po
The constant total pressure at the outer wall of the cylinder.
pQ
The steady state solution to the pressure for vanishing specific discharge, V.
pˆ
The overpressure, p
p˜
The dimensionless overpressure.
pˇ
The rescaled dimensionless overpressure.
\
The continuous part of the overpressure.
p
\
|ˆ p|max / p
max
pQ .
Attenuation factors used to plot overpressures/the continuous parts of overpressures.
q
The shear component of the e↵ective stress.
r
The radial polar coordinate.
r0
The value of r where s1 = s10 .
ri
The value of the radial coordinate at the inner wall of the cylinder.
ro
The value of the radial coordinate at the outer wall of the cylinder.
S
A function of s˜1 used in separation of variables.
s10
The value of s1 where r = r0 .
si
Coordinates the level sets of which are characteristics of the momentum balance.
s˜i
The dimensionless coordinates.
T
A function of t˜ used in separation of variables.
t˜
The dimensionless time.
u
The displacement.
v¯
The mean speed of the incompressible Newtonian fluid. (This symbol appears only in the appendices.)
v (1/2)
The velocity of ↵/ .
V
The specific discharge of .
w
The width of wide parallel plates. (This symbol appears only in the appendices.)
3
w(J)
A left eigenvector corresponding to the left eigenvalue µ(J) .
xi
Cartesian coordinates.
y
A coordinate describing location between wide parallel plates. (This symbol appears only in the appendices.)
y
The column vector of coordinates of the stress state, (p, ✓)T .
z/z
A variable used to express identities or introduce notation.
zs
The expression in coordinates (s1 , s2 ) of a vector written z in coordinates (x1 , x2 ).
14
Greek. Symbol
Description
↵/
The phases. The di↵usivity.
ij
s1
The Kronecker delta. A signature distance between characteristics.
✏ij
The Eulerian infinitesimal strain.
"
The dilatation.
⇣
The proportionality between the initial dimensionless rate of change of dimensionless overpressure and the initial dimensionless pressure.
⌘
The dynamic viscosity of the incompressible Newtonian fluid. (This symbol appears only in the appendices.)
✓
An angle such that ✓ + ⇡/4 is the angle counter-clockwise from the positive x1 -axis to the direction of the maximum tensile stress.
⇥
The Heaviside theta function.
The dimensionless parameter defined as the ratio of the total length of a slip-line L to the material length scale 2 /cs . n
The inverse of the time constant corresponding to the hyperbolic pressure eigenmode pn .
µ(J)
A left eigenvalue corresponding to the left eigenvector w(J) . 4
⌫↵
The Poisson ratio of ↵. (This symbol appears only in the appendices.)
⇠
The sinuosity. (This symbol appears only in the appendices.)
$
A constant used in separation of variables.
⇢1/2
The relative densities of ↵/ .
⇢↵/
The densities of ↵/ .
1
The maximum tensile stress.
(1/2)
The partial stress of ↵/ .
00
The e↵ective stress (experienced by ↵). The stress in .
&
Half the angle by which the s1 and s2 axes di↵er from normality. The volume fraction of . The azimuthal polar coordinate. A di↵erence between angles, ✓
!n
.
The angular frequency corresponding to the trigonometric pressure eigenmode pn .
⌦t
15
Other. Symbol
16
A material volume.
Description
r·
The di↵erential operator
rs ·
The di↵erential operator
⇣ ⇣
@ , @ @x1 @x1 @ , @ @s1 @s1
⌘
⌘
·. ·.
1. Introduction
17
This work examines the evolution of pressure of plastically yielding material. It will
18
show that, for a two phase material in plane strain, pressure propagates along slip-lines in
19
dissipative waves.
20
Slip-lines are the characteristics of the quasistatic momentum balance and are used in
21
the design or description of, among other things, welds (Khan et al., 2014), hardness tests 5
22
(Jackson et al., 2015), retaining walls (Vo and Russell, 2014) and pipeline foundations (Gao
23
et al., 2015). A particularly elegant application of slip-line field theory (SFT) for Von Mises
24
and Drucker-Prager materials, and one relevant to the plastic design of pressure vessels
25
(Sowerby and Johnson, 1970), arises for radial stress of a cylinder in plane-strain. As Hill
26
(1950) and Shield (1953) discuss, slip-lines are both the only curves across which pressure
27
and velocity fields can have discontinuities, and are directed at a constant angle to the
28
direction of maximal tensile stress. Thus SFT predicts that any strain localisations will
29
occur along logarithmic spirals. Sowerby and Johnson (1970) and Papamichos et al. (2001)
30
verify these predictions via pressurisation experiments from which come beautiful networks
31
of intersecting failure lines following logarithmic spirals. Figure 9.3a of Sowerby and Johnson
32
(1970) appears here as figure 4a.
33
Thus the quasistatic momentum balance r ·
= 0 determines the geometry of lines of
34
failure for a steel cylinder internally pressurised to yield. However, the analysis does not
35
identify the spacing of failure lines which, at least in figure 4a, is intriguingly regular. There
36
are a few di↵erent approaches with the potential to add information about the spacing of
37
slip-lines to the existing, correct prediction of their logarithmic spiral shape.
38
An example is the dynamic strain ageing model of McCormick (1988). He considers the
39
interplay of local solute concentration, which limits the dislocation motion which enables
40
plastic deformation, and strain rate, an increase of which allows more rapid di↵usion of
41
solute particles. Feedback between the two local properties can lead to sinusoidal strain
42
oscillations. (McCormick, 1988) From a dynamic strain ageing model, Estrin et al. (1991)
43
deduce that in tensile extension these spatially homogenous strain rate oscillations can lead
44
to strain inhomogeneous in space.
45
Any material deformation description which leads to predictions of spatial undulations
46
of strain rate has the potential, if incorporated into SFT, to resolve the spacing between
47
logarithmic spirals formed during radially symmetric plane strain. However, lines of strain
48
localisation in mild steel are an upper yield point phenomenon (Hertzberg et al., 2012, pp.
49
99-100), and the upper yield point is sensitive to pressure (Capp et al., 1973). It follows
6
50
that the locations of lines of strain localisation are sensitive to pressure inhomogeneities. For
51
this reason we seek to understand regular formation of the logarithmic spirals of figure 4a
52
by describing pressure.
53
The focus on pressure emulates Veveakis and Regenauer-Lieb (2015), who identified spa-
54
tially periodic pressure singularities in a two phase material in which the primary component
55
underwent plastic volumetric changes at a rate proportional to a power of the pressure. In-
56
stead of following Veveakis and Regenauer-Lieb (2015) in considering only the plastic com-
57
ponent of the volumetric response, we consider only the e↵ect of the elastic part of the
58
volumetric response. For a Von Mises material, this leads to a telegraphy equation in the
59
pressure. In many ways our approach follows the derivation for an elastic saturated material
60
of a telegraphy equation in the dilatation of Vardoulakis and Sulem (2005, Chapter 5).
61
The structure of this paper follows. Section 2.1 derives an approximate momentum
62
balance for a two phase material. Assuming a state of plane strain, section 2.2 identifies the
63
characteristics of this equation at the point of plastic yield. Along these curves the evolution
64
of pressure can be reduced from a two to a one-dimensional problem. Section 3 applies the
65
results of section 2, derived (with some simplifying assumptions) for a two phase material in
66
a state of plane strain, to a convenient material type and geometry. Section 4 uses separation
67
of variables to describe the evolution of the pressure distribution for a Von Mises cylinder
68
expanded via internal pressurisation. Section 4.2 employs parameters and initial conditions
69
pertaining to expansion of a cylinder of mild steel. As Appendix A discusses, the viscosity is
70
difficult to estimate precisely. Hence the section considers three distinct values of viscosity
71
spanning two orders of magnitude. It compares the output of each realisation of the model
72
to experimental observations. The final section, section 5, discusses results and suggests
73
ways this work could be continued.
74
2. Describing pressure evolution using conservation equations
75
76
Section 2.1 ends with a relationship between e↵ective stress and the specific discharge of the less prominent phase,
, which is a form of the momentum balance. Section 2.2 uses
7
77
the method of characteristics to reduce the spatial dimension of the relationship from two
78
to one.
79
2.1. The mass and momentum balance 1
Consider a composite of two phases ↵ and
. The volume fraction of the second phase
will be denoted . The relative densities ⇢1 and ⇢2 are related to the densities ⇢↵ and ⇢ of constituents ↵ and
by ⇢1 = (1
Balances of the mass of phases ⇢↵ and
)⇢↵ ,
⇢2 = ⇢ .
lead to
@⇢1 + r · ⇢1 v (1) = 0, @t @⇢2 + r · ⇢2 v (2) = 0, @t
(2) (3)
where v (1) and v (2) denote the velocities of phases ↵ and . Using the definition (1), 1 ⇢
(1)
1 (2) + ⇢↵
(3) can be written r·V =
r · v (1)
⇢↵
D(1) ⇢↵ Dt
v (2)
v (1)
1
D(2) ⇢ , ⇢ Dt
(4)
where V=
D(J) Dt
=
@ @t
+ v (J) · r is a material derivative.
80
is the specific discharge of
81
The material derivative indexed J is the rate of change which would be measured by an
82
observer travelling with the phase indexed J. The next section (section 2.2) uses the mass
83
balance (4). 1
and for J = 1, 2,
(5)
The phases do not need to be chemically distinct. For example, Vardoulakis (1989) partitions a granular
material into a “frail” phase and a “competent” phase, which contribute di↵erently to the total stress, according to number of grain contacts.
8
In the absence of body forces, balances of the linear momentum of ↵ and r·
(1)
r·
(2)
lead to
D(1) v (1) Dt
(6)
D(2) v (2) f = ⇢2 , Dt
(7)
+ f = ⇢1
and
where f is the specific interaction force exerted on the primary constituent ↵ by the secondary constituent , and
(1)
and
(2)
are the partial stresses of ↵ and . The interaction force
f is assumed to oppose relative motion between the phases. Specifically, it is assumed that the interaction force is linear in the relative velocity; f=
C v (1)
v (2) ,
(8)
where C ⌘ C( ) is a function of the volume fraction of .2 Appendix A considers estimation of the drag coefficient C. Decompose the total stress (1)
where
00
+
(2)
=
00
+B
,
(9)
is the e↵ective stress which ↵ experiences, B is the Biot coe↵cient and
stress in constituent , related to the partial stress of (2)
=
is the
by
.
(10)
84
Coussy (2004, pp. 78–79) shows that the Biot coefficient of a linear isotropic material can
85
be estimated from the total bulk modulus K and the skeleton bulk modulus Ks . Appendix
86
B, upon which relies the application comprising section 4.2, uses this result. 2
Vardoulakis and Sulem (2005) consider a situation in which the secondary component is a fluid. In this
case shear stresses are negligible, so that the partial stress tensor is
(2)
=
pf 1. They show that if spatial
variations of -volume fraction and convective and inertial terms can be neglected then the choice C = d where d is constant recovers, via (7), Darcy’s law V = pressure.
9
1 d rpf .
In this expression, pf is the fluid pore
The fields
(1)
(2)
,
, f and v (2) will be eliminated from (5), (6), (7), (8), (9) and
,
(10). Substitution of (10) into (7) gives r · the -volume fraction
D(2) (2) v . Dt
f = ⇢
Hence, assuming that
varies slowly in space and neglecting convective terms, r·
Together, (9) and (10) show that
= (1)
1
=
f +⇢
00
@v (2) . @t
+ (B
(11)
)
. Using (11), treating the Biot
coefficient B as constant, and again assuming that the -volume fraction space, the divergence of this equation can be written r· Substituting this expression for r·
(1)
into (6), r·
By (8), assuming that the -volume fraction
00
(1)
= r·
+ B f +(B
00
+B
)⇢
varies slowly in f +(B
@v (2) @t
= (1
)⇢
@v (2) . @t (1)
)⇢↵ @v@t .
varies slowly in time and using the definition
(5) of the specific discharge, this can be written r·
00
=
C
B
V 2
B
⇢
@ V + ((1 @t
)⇢↵
(B
)⇢ )
@v (1) . @t
(12)
87
Via the method of characteristics and a choice of constitutive relationship, the next
88
section (section 2.2) reduces the momentum balance (12) from two to one spatial variables.
89
2.2. Pressure at plastic yield
90
This section (section 2.2) considers a material the yield condition of which is an alge-
91
braic relationship between the hydrostatic and the shear components of the Biot e↵ective
92
pressure. Following Paesold et al. (2016), it derives a description of the characteristics of
93
(12), the momentum balance, at the point of plastic yield. As the section will show, along
94
the characteristics3 of the momentum balance, termed slip-lines, pressure evolves in a single
95
dimension. Assume plane strain in the plane parameterised by Cartesian coordinates (x1 , x2 ) (see figure 1a or figure 1 of Shield (1953)). Assume that ↵ is in a state of yield described4 by q = g(p), 3
Note that if the right hand side of the momentum balance (12) is considered a function not only of s1 ,
s2 , t, p (s1 , s2 , t) and
@p @t
(s1 , s2 , t) but also of the spatial derivatives
@p @s1
(s1 , s2 , t) and
@p @s2
(s1 , s2 , t) then the
coming considerations would not necessarily identify characteristics. 4 Here arises a major assumption of this work: that the isotropic and deviatoric components of the Biot
10
1 3
where p =
P3
i=1
00
ii
is the hydrostatic and q =
qP 3
i,j=1
(
00
ij
+p
00
ij ) (
ij
+p
ij )
is the
shear component of the e↵ective stress. Then (because conservation of angular momentum implies symmetry of the stress tensor), as Johnson et al. (1982, p. 31) show, 00 11
=
p
00 12
g(p) sin(2✓),
= g(p) cos(2✓) =
00 21 ,
00 22
=
p + g(p) sin(2✓),
where ✓ + ⇡/4 is the angle counter-clockwise from the positive x1 -axis to the direction of the maximum principal stress
1
(see figure 1a). Following Paesold et al. (2016)5 , the momentum
balance (12) can be written A
@y @y +B = @x1 @x2
C
B
V 2
B
⇢
@ V + ((1 @t
)⇢↵
(B
)⇢ )
@ (1) v , @t
(13)
where 0
A,@
g 0 (p) sin(2✓)
1 0
g (p) cos(2✓)
2g(p) cos(2✓) 2g(p) sin(2✓)
1
0
A, B , @
g 0 (p) cos(2✓) 0
2g(p) sin(2✓)
1 + g (p) sin(2✓)
2g(p) cos(2✓)
1
A,
(14)
y1 , p and y2 , ✓. Let w(1) , w(2) denote two linearly independent left eigenvectors with corresponding eigenvalues µ(1) , µ(2) which satisfy, for J = 1, 2, µ(J) w(J) T · A = w(J) T · B. Assuming that |g 0 (p)| < 1, solutions to µ(J) A µ(1) = tan(✓
&),
B = 0 are
µ(2) = tan(✓ + ⇡/2 + &),
where the angle & 2 ( ⇡/4, ⇡/4) is defined by sin(2&) = g 0 (p). The corresponding eigenvectors are 0
w(1) = sec(2&) @
sin (✓ + ⇡/2 + &) cos (✓ + ⇡/2 + &)
1
A,
0
w(2) = sec(2&) @
sin (✓ cos (✓
&) &)
1
A.
e↵ective stress given by (9) describe (via a yield envelope) yield of the material ↵. Of course, there are more general descriptions of yield in a two phase material. For example, stress could be partitioned according to a version of (9) with the scalar B replaced by a tensor. 5 Paesold et al. (2016) actually considered a steady state momentum balance of a single phase material, for which the right hand sides of (12) and (13) would be zero.
11
Of course, any multiple of an eigenvector is also an eigenvector. However, these choices 0 1w(J) 0 1 have the convenient property @ defined such that directed at ✓
@ @s1
and
@ @s2
w(1) T T
A=@
@s1 @x1
@s1 @x2
@s2 @x1
@s2 @x2
A, where (s1 , s2 ) are coordinates
w(2) have the same Euclidean magnitude as
@ @x1
and
@ , @x2
but are
& and ✓ + ⇡/2 + & to the positive x1 -direction (see figure 1a); 0 1 0 1 @x1 @x1 cos(✓ &) cos(✓ + ⇡/2 + &) @ @s1 @s2 A = @ A. @x2 @x2 sin(✓ &) sin(✓ + ⇡/2 + &) @s1 @s2
(15)
96
If & = 0 then this transformation is a rotation. The si axes intersect at an angle ⇡/2 + 2&
97
and are oriented symmetrically about the direction of maximum stress (see figure 1a).
Figure 1: Coordinate systems. (a). The angle ✓ is defined such that the maximum tensile stress
1
occurs
at ✓ + ⇡/4 counter-clockwise from the positive x1 axis, where (x1 , x2 ) is a Cartesian coordinate system. The si axes intersect at an angle ⇡/2 + 2& and are oriented symmetrically about the direction of maximum tensile stress. (b). Coordinate directions for the Von Mises cylinder of section 4. Since & = 0, the s1 - and s2 -directions are everywhere orthogonal. The direction of maximum tensile stress is everywhere orthogonal to the radial direction. At the point considered in the diagram s1 = 0 and r = ro .
For any vector written z in coordinates (x1 , x2 ), let z s be its expression in coordinates
12
0
w(1) T
(s1 , s2 ). With this notation, the momentum balance (13) left-multiplied by @ 0 @
@p @s1 @p @s2
1
@✓ 2g(p) sec(2&) @s 1
A=
@✓ + 2g(p) sec(2&) @s 2
C
B 2
B
Vs
w(2)
T
1
A is
@V s @t
⇢
(1)
+ ((1
)⇢↵
(B
)⇢ )
@v s . @t
(16)
98
Thus the transformation (15) decouples the momentum balance. Lines parameterised by
99
s1 and s2 according to (15) are the characteristics of the momentum balance (12), which
100
are called slip-lines. As shown in (15) or figure 1a, the orientations of the characteristics
101
parameterised by s1 and s2 are ✓
102
g 0 (p) = 0, characteristics intersect at right angles.
103
& and ✓ + ⇡/2 + & respectively. In particular, where
Where the fields comprising the right hand side of (16) ( , V s ,
@V s @t
and
(1)
@v s @t
) are known
104
as functions of space and time the decoupled momentum balance (16) would represent an
105
informative constraint between geometry and pressure. As will be seen, spatial di↵erentiation
106
allows more general use of the decoupled momentum balance (16). The chain rule shows that 0
r · z = rs · z s + sec(2&)z s T · @ where rs · ,
⇣
@ , @ @s1 @s2
⌘
⇣
⇣
sin(2&) @s@ 1
@ @s1
P
⌘
⌘
(✓
&)
+ sin(2&) @s@ 2 (✓ + ⇡/2 + &)
1
A,
(17)
· is a di↵erential operator. Hence, neglecting convective terms, rs · v (1) s =
where ✏ ,
+
@ @s2
i "ii is the dilatation, "ij ,
1 2
⇣
@ui @xj
@✏ , @t
+
@uj @xj
(18) ⌘
is the Eulerian infinitesimal strain,
and u is displacement. So, by (4) and (17), neglecting convective terms, rs · V s =
@✏ @t
Assuming that the -volume fraction
1 ⇢↵
@⇢↵ @t
@⇢ . ⇢ @t
(19)
and densities ⇢a and ⇢ vary slowly in space, rs · 13
(16) can be written 2
rs p
2rs ·
g(p) sec(2&)
✓
@✓ , @s1
@✓ @s2
◆T !
=
C
B 2
B
rs · V s
⇢ rs ·
@V s @t (1)
+ ((1
)⇢↵
)⇢ ) rs ·
(B
Hence, assuming commutativity of spatial derivatives rs and the time derivative
@v s . @t (20)
@ , @t
by the
approximations (18) and (19), ◆T ! @✓ @✓ rs p =2rs · g(p) sec(2&) , @s1 @s2 ✓ ✓ ◆ ◆ 2 B @✏ B @ ✏ +C 2 + (1 ) ⇢↵ + ⇢ @t @t2 ✓ ◆ ✓ B 1 @⇢↵ @⇢ B +C 2 + + (1 ⇢↵ @t ⇢ @t ✓
2
107
Some geometries imply relationships between @ @si
@ @s1
and
⇢ @ 2 ⇢↵ @ 2⇢ ) + ⇢↵ @t2 @t2
@ @s2
◆
.
(21)
which allow elimination of the
108
dependence on one of the
in the above. The next section (section 3) shows how assuming
109
radial symmetry and particular forms for the yield envelope q = g(p) assists further analysis.
110
3. Radially symmetric plane strain For convenience, introduce polar coordinates (x1 , x2 ) = (r cos , r sin ). With this definition and the matrix of derivatives (15), the chain rule shows that 0 1 0 1 @r @r cos( &) cos( + ⇡/2 + &) @ @s1 @s2 A = @ A, @ @ 1 1 sin( &) r sin ( + ⇡/2 + &) @s1 @s2 r ,✓
(22)
111
where
. As Johnson et al. (1982, p. 68) observe, for a homogeneous and isotropic
112
material in plane strain under radially symmetric loading, the directions of principal stress
113
are radial and azimuthal. Hence the angle
114
with the instance depending on whether the azimuthal or radial tensile stress is greater.
115
Note that in either case, tan2
is either
= 1. 14
=
3⇡/4 or
=
⇡/4 (a constant),
If g 0 (p) is constant then & is constant. Hence the orientations of characteristics relative to the radial direction, which by the coordinate transformation (15) are + ⇡/2 + & = ✓ + ⇡/2 + &
&=✓
&
and
, are constant. In this case, by (22), along the characteristic
parameterised by s1 the radial coordinate is r = cos( 116
&) (s1
s10 ) + r0 ,
(23)
where s10 is the value of s1 at r = r0 . A computation using (15) and (22) shows that the derivative in the azimuthal direction is
@ @
= r sec 2&(sin( + &) @s@ 1 + cos(
&) @s@ 2 ). Since the radial symmetry of the problem
means that the pressure does not change in the azimuthal direction, ✓ ◆ @p sin( + &) @p sin2 ( + &) @ 2 p 2 = , rs p = 1 + . @s2 cos( &) @s1 cos2 ( &) @s1 2 117
(24)
3.1. Quasistatic pressure of Von Mises and Drucker-Prager materials Consider the steady-state form with vanishing specific discharge (V s = 0) of the momentum balance (16). Letting pQ denote this quasistatic pressure solution, 0 1 0 1 @pQ @✓ 2g(pQ ) sec(2&) @s1 0 @ @s1 A = @ A. @pQ @✓ + 2g(pQ ) sec(2&) @s 0 @s2 2
(25)
118
This section (section 3.1) identifies the pressure profile pQ along the characteristic parame-
119
terised by s1 of this momentum balance for the two classes of materials for which the slope
120
of the yield envelope g 0 (p) is constant: Von Mises and Drucker-Prager. The yield envelope of a non-hardening Von Mises material satisfies g(p) = k, where k is constant.6 Observing that since g 0 (p) = 0, & = 0, equations (22), (23) and (25) show that along the slip-line parameterised by s1 , (2k tan ) Hence, by (23) with & = 0, the pressure is ✓ ✓ s1 pQ = p0 2k tan ln cos 6
and
s10 r0
◆
+1
◆
1
dp =
= p0
(s1
2k tan
s10 + sec r0 )
ln
✓
In this case, (25) reduces to Hencky’s equations (see, for example, Hill (1950, p. 135)) @ @s2
(pQ
2k) = 0.
15
r r0
◆
@ @s1
,
1
ds1 .
(26)
(pQ + 2k) = 0
121
122
123
where p0 denotes the pressure at s1 = s10 (and thus, at r = r0 ). Recall that since either =
3⇡/4 or
=
= ±1. Sowerby and Johnson (1970) and Johnson et al.
⇡/4, tan
(1982) reached results equivalent to (26), also via SFT.
124
The yield envelope of a Drucker-Prager material satisfies g(p) = c+sin(2&)p, where c and &
125
are constant and sin(2&) 6= 0. In this case, (22) and (25) show that along the slip-line param-
126
eterised by s1 , (c csc(2&) + p)
129
2 tan(2&) tan( &) (s1 s10 + sec( &)r0 ) 1 ds1 . ⇣ ⇣ ⌘ ⌘ 2 tan(2&) tan( &) s1 s10 Hence the pressure is pQ = (c csc(2&) + p0 ) cos + 1 c csc(2&) = r0 ⇣ ⌘ 2 tan(2&) tan( &) (c csc(2&) + p0 ) rr0 c csc(2&), where p0 denotes the pressure at s1 = s10
130
results from which, in the case of radial symmetry, this expression for pQ would easily follow.
131
3.2. Pressure evolution of Von Mises and Drucker-Prager materials
127
128
1
dp =
(and thus, at r = r0 ), and the second equality follows from (23). Gao et al. (2015) reach
132
This section (section 3.2) and those which succeed it rely upon (21). This equation is
133
a derivative of the reformulation along its characteristics of the momentum balance (12).
134
The momentum balance (12) was derived by neglecting convective terms (that is, making
135
the approximation
136
The derivation also assumes that
137
remainder of the document it will be assumed that the drag coefficient C is constant in
138
space.
D(J) Dt
=
@ ) @t
and assuming that the Biot coefficient B is constant in space. varies slowly in space and time. Additionally, for the
Let pˆ denote the overpressure pˆ (s1 , s2 , t) , p (s1 , s2 , t)
pQ (s1 , s2 ), where pQ satisfies the
quasistatic momentum balance (25). The derivative (21) of the momentum balance can be written 0
rs 2 pˆ =rs · @
@pQ @s1 @pQ @s2
+
@✓ 2g(pQ ) sec 2& @s 1 @✓ 2g(pQ ) sec 2& @s 2
1
0
A + tan 2&rs · pˆ @
✓ ✓ ◆ ◆ 2 B @✏ B @ ✏ +C 2 + (1 ) ⇢↵ + ⇢ @t @t2 ✓ ◆ ✓ B 1 @⇢↵ @⇢ B +C 2 + + (1 ⇢↵ @t ⇢ @t
@✓ @s2
1 A
⇢ @ 2 ⇢↵ @ 2⇢ ) + ⇢↵ @t2 @t2
Using (24), twice using (22), and noting that because either 16
@✓ @s1
=
3⇡/4 or
◆
.
(27)
=
⇡/4,
cos 2 = 0, it can be shown that 0 0 11 rs · @pˆ @
@✓ @s1
@✓ @s2
AA = 2 sin( r
&)
@ pˆ @s1
sin 2 cos 2&
pˆ . r2
(28)
139
Classically, increments of plastic deformation can be resolved as normal to the yield en-
140
velope when it is written as a function of the elastic component of strain (Hosford, 2010, p.
141
73). For a Von Mises material this corresponds to vanishing plastic volumetric strain (Rud-
142
nicki and Rice, 1975). Spitzig et al. (1975) investigated for martensic steel the relationships
143
between volumes and stresses at yield. They found volumetric changes orders of magnitude
144
less than those predicted by normality. Also, as Rice (1976) notes, to describe localisation
145
conditions for a range of geological materials, Rudnicki and Rice (1975) consider dilatancy
146
factors which are always less than or equal to, and usually less than, the corresponding co-
147
efficients of internal friction g 0 (p). This choice corresponds to plastic volumetric change less
148
than that associated with a normal flow rule (Rudnicki and Rice, 1975). Cases such that plastic volumetric changes can be neglected admit the volumetrically linear and isotropic volumetric relationship @ pˆ = @t
K
@✏ , @t
(29)
where K is the bulk modulus. In such cases and in radial symmetry, by (24) and (28), (27) can be written ✓ ◆ 1 sin2 ( + &) @ 2 pˆ 1 @ pˆ 1 pˆ 1+ tan 2& sin( &) + sin 2 sin 2& 2 2 2 2 cos ( &) @s1 r @s1 2 r ✓ ✓ ◆ ◆ 2 C B @ pˆ 1 B @ pˆ = (1 ) ⇢↵ + ⇢ 2 2K @t 2K @t2 ✓ ◆ ✓ ◆ B 1 @⇢↵ @⇢ B 1 @ 2 ⇢↵ @ 2⇢ +C 2 + + ⇢ + . 2 ⇢↵ @t ⇢ @t 2 ⇢↵ @t2 ⇢ @t2 149
(30)
If desired then (23) could be used to write the above in terms of s1 instead of r.
150
Von Mises materials correspond to & = 0 and so to a form of (29) the left hand side of
151
which has a single term instead of three. The next two sections (3.2.1 and 3.2.2) consider
152
ways to eliminate the densities ⇢↵ , ⇢ from the above, and write it in terms of the overpressure 17
153
pˆ alone. For simplicity only Von Mises materials are considered. The same assumptions and
154
very similar methods could be used to eliminate densities from (30) for a Drucker-Prager
155
material.
156
3.2.1. Neglecting changes in density Following the derivation of a telegraphy equation in the dilatation ✏ of Vardoulakis and Sulem (2005), consider the case of negligible changes in time of the densities ⇢↵ and ⇢ . For a Von Mises material, the spatial derivative (30) of the momentum balance becomes @ 2 pˆ 1 @ pˆ 1 @ 2 pˆ = + , @t cs 2 @t2 @s1 2 where
,
2K 2 C B
is defined by
1 cs 2
(31)
is a di↵usivity and the positive quantity cs , called the speed of second sound, ⇣ ⇣ ⌘ ⌘ 1 = 2K (1 ) ⇢↵ + B ⇢ . The right hand side of the equation defining
cs is assumed to be positive. Equation (31) is a telegraphy equation in the overpressure pˆ. To express (31) dimensionlessly, define the dimensionless variables p˜, t˜ and s˜i by pˆ = k p˜,
t = H t˜ and si = L˜ si , where H , 2 /cs 2 , and L is a geometric length scale which will be chosen in section 4.2. Assuming
> 0, the telegraphy equation (31) becomes 1 @ 2 p˜ @ p˜ @ 2 p˜ = 2 + , 2 @˜ @ t˜ @ t˜2 s1 2
(32)
157
where ,
158
Table 4 lists values of the scales H, 2 /cs and for the physical parameters considered in
159
section 4.2.
L 2 /cs
is the ratio of the geometric length scale L to the material length scale.
The dimensionless overpressure p˜ has greater physical significance, but since (32) is to solved by separation of variables it will be easier to consider pˇ defined by p˜ s˜1 , s˜2 , t˜ = exp
t˜ pˇ s˜1 , s˜2 , t˜ .
(33)
In terms of pˇ the telegraphy equation (32) can be written as the Klein-Gordon equation 1 @ 2 pˇ = 2 @˜ s1 2
pˇ +
@ 2 pˇ 2. @ t˜
(34)
160
Section 4.2 considers the solution of (34) under boundary conditions which correspond to
161
sudden application of internal pressure. As section 4.2 explains, this is pertinent to the 18
162
expansion of a cylinder of mild steel. Appendix C supplements this solution with the outcome
163
of a very di↵erent sort of initial condition; one corresponding to a particular choice of spatially
164
smooth pressure perturbation.
165
3.2.2. Neglecting changes in -content
166
This section (section 3.2.2) describes pressure evolution for the case of small changes
167
in the mass composition. The assumption is analogous to that of undrained conditions
168
in poromechanics, which occur when in any reference volume of the skeleton ↵ there is a
169
constant mass of . The rate of change apparent to phase ↵ of the amount of phase in a material volume R (1) R (1) ⌦t is DDt ⌦t ⇢2 d⌦t = ⌦t DDt ⇢2 d⌦t . This integral is zero for every material volume ⌦t if and only if the integrand is everywhere zero. Coussy (2004) shows, and expresses in his equation (1.42), that the material derivative of a material volume is
D(1) d⌦t Dt
the condition for -content to remain fixed can be written
@⇢2 @t
= r · v (1) . It follows that
+ r · ⇢2 v (1) = 0. So, by the
fluid mass balance (3), r · (⇢ V) = 0,
(35)
170
where one of the definitions of the relative densities (1) and the definition of the specific
171
discharge (5) have been used. Hence, by (17) and (19), neglecting convective terms,
172
@⇢ @t
⇢
=
1 @⇢↵ ⇢↵ @t
+
@✏ . @t
By this result and the constitutive equation (29), assuming that the -volume fraction and densities ⇢↵ and ⇢ change slowly in time, the derivative (30) of the momentum balance ⇣ ⌘ 2 (1 )⇢↵ (B )⇢ @ 2 pˆ @ pˆ can be written (for a Von Mises material) @s . Assuming that 2 = 2K @t2 1 (1
)⇢↵
(B
)⇢ > 0, this leads to a wave equation in the overpressure pˆ , p cu 2
q
173
where cu ,
174
to the wave equation (36) is pˆ = F (s1
(1
2K )⇢↵ (B
)⇢
@ 2 pˆ @ 2 pˆ = . @s1 2 @t2
pQ ;
(36)
is a wave speed for undrained conditions. The general solution cu t) + G(s1 + cu t) for arbitrary F and G.
175
In contrast to (31), (36) predicts non-dissipative propagation of pressure waves. When
176
-content is constant the divergence of the specific discharge is small. Hence the dissipative 19
177
interactive force (8) does not enter the spatial derivative of the balance of momentum. Of
178
course, conservative pressure propagation is only possible because the volumetric pressure-
179
strain relationship (29) neglects any plastic contribution to volumetric strain.
180
4. A Von Mises cylinder internally pressurised to yield
181
Consider a cylinder of a Von Mises material held in plane strain and internally pressurised
182
to the point of yield. Note that because the azimuthal is greater than the radial tensile stress,
183
=
3⇡/4, and because the material is Von Mises, & = 0.
184
Section 4.1 uses separation of variables to find a form of the pressure distribution con-
185
sistent with overpressure pˆ vanishing at the inner and outer wall of the cylinder. Section
186
4.2 considers instances of the pressure distribution corresponding to parameter values and
187
initial conditions relevant to the experiment with mild steel of Sowerby and Johnson (1970),
188
a product of which is shown in figures 4a and C.7a.
189
4.1. The solution modulo an initial condition To identify boundary conditions, assume that, from time t = 0, a constant pressure is applied at the inner wall (r = ri ) of the cylinder. Choosing s1 = 0 at the outer wall (r = ro ) and denoting the pressure p there po , it follows from (23) that 1 p s1 2
r = ro
and hence from (26) that the quasistatic pressure pQ = po
(37) ⇣ 2k ln 1
p1 s1 2 ro
⌘
. Following
Sowerby and Johnson (1970), consider the minimum pressure at the inner wall to establish quasistatic plastic yield. Equivalently, assume zero overpressure at the inner and outer walls (r = ri and r = ro ); pˇ|s˜1 =1 = 0 = pˇ|s˜1 =0 .
(38) p
190
Note that, as figure 1b indicates, the geometric length scale L ,
191
the arclength of a slip-line. For this reason s˜1 = 1 at the inner wall of the cylinder.
20
2 (ro
ri ) is set equal to
Following the method of separation of variables, consider solutions of the form pˇ s˜1 , t˜ = 00 s ) T 00 (t˜) 1 S (˜ s1 ) T t˜ . With this substitution, (ˇ p) 1 ⇥ (34) is 12 SS(˜(˜ = 1 + = $2 , where $ s1 ) T (t˜) is a real constant. Hence S (˜ s1 ) = a cos ($˜ s1 ) + b sin ($˜ s1 ), for constant a and b. The boundary conditions (38) are satisfied by a = 0, $ = n⇡/, n 2 Z. So the form of the solution of T 00 t˜ = (1
where
$2 )T t˜ depends on n as 8 < c cosh ˜ ˜ n n t + dn sinh nt T t˜ = : c cos ! t˜ + f sin ! t˜ n n n n n
q , 1
!n ,
(n/n⇤ )2 ,
q
, n < n⇤ , n > n⇤
(n/n⇤ )2
,
1
(39)
and n⇤ , /⇡. The full solution for the dimensionless overpressure p˜ = exp
t˜ pˇ is
⇤
p˜ = exp
t˜
bn c X
sin (n⇡˜ s1 ) cn cosh
˜ + dn sinh
nt
˜
nt
n=1
+
1 X
sin (n⇡˜ s1 ) cn cos !n t˜ + fn sin !n t˜
n=bn⇤ c+1
!
,
(40)
where for z 2 R, bzc denotes the greatest integer less than or equal to z and for z 2 / R,
bzc , 0. By (40) and orthonormality of the odd functions sin (n⇡x), the coefficients cn are given by cn = 2
Z
1 0
sin (n⇡˜ s1 ) ( pˇ|t˜=0 ) d˜ s1 .
(41)
Assume that the initial dimensionless rate of change of dimensionless overpressure is proportional to the initial dimensionless pressure; @ p˜ @ t˜
t˜=0
= ⇣ p˜|t˜=0 .
(42)
192
The choice ⇣ = 0, which is perhaps most natural, describes the case for which the initial
193
rate of change of pressure is zero. A choice ⇣ < 0 could reflect the fact that the application
194
of pressure can not be instantaneous, and that the material will begin to react against its
195
perturbation from equilibrium. A choice ⇣ > 0 could describe the continuation of the external 21
196
process which brought the material to its t˜ = 0 disequilibrium state. By the definition (33)
197
of pˇ, (42) can be written
@ pˇ @ t˜ t˜=0
= (⇣
Assume n⇤ 2 / Z, so that for n 2 Z,
1) pˇ|t˜=0 . n
and !n are non-zero. In this case it follows, by the
definition (33) of pˇ, (40), and L2 ([ 1, 1]) orthonormality of the odd functions sin (n⇡x), that the coefficients dn and fn are dn = ⇣ n1 cn , fn = ⇣!n1 cn , where cn is given by (41). Hence (40) P becomes p˜ = 1 ˜n , where p˜n is the component of overpressure with an s˜1 -wavelength of n=1 p 2/n,
p˜n = cn exp
With the expressions (39) for
198
199
200
201
8 ⇣ ⌘ < cosh n t˜ + ⇣ 1 sinh n t˜ , n < bn⇤ c n ⌘ t˜ sin (n⇡˜ s1 ) ⇣ . : cos ! t˜ + ⇣ 1 sin ! t˜ ⇤ , n > bn c n n !n n
(43)
and !n , the eigenmode solution (43) shows that if ⇣ 6=
1
then there can be a local peak in the amplitude of the components of pressure around n = n⇤ . ⇣ ⌘ n⇤ n⇤ 2 ⇤ n ˜ q The coefficient of sin (n⇡˜ s1 ) has a stationary point at t , arccos 2 n 1 7 and ⇤ 2 1 (n ) n so limn!n⇤ t˜⇤ = 2. Hence, if the components of overpressure around n = n⇤ are of interest
202
then it will be worthwhile considering the pressure profile at dimensionless time t˜ = 2. Figure
203
2a includes the profile at this time, as do several plots of figures 3 and C.6.
204
4.2. A mild steel cylinder suddenly loaded to yield
205
This section (section 4.2) identifies the pressure evolution during expansion of a mild steel
206
cylinder. The following ideas and observations motivate the modelling approach. Firstly,
207
note that the main constituent is assumed to be a Von Mises material, a useful description
208
for steel8 (Hill, 1950). Next, note that the experiment of Sowerby and Johnson (1970)
209
sought strain localisation in the form of the L¨ uders bands pictured in figure 4a. At the
210
stage of deformation in which localisation is observed, the preponderance of total plastic
211
deformation can be taking place within the L¨ uders bands. (Hertzberg et al., 2012) This
212
suggests importance of the bands to the overall deformation process. At least, a complete
213
description of the experiment would e↵ectively identify L¨ uders lines. Following are reasons Since cos(ix) = cosh(x), this formula holds both for n < n⇤ and n > n⇤ . 8 Admittedly, the major component of steel considered as a two phase material is alpha-ferrite. 7
22
214
supporting the working hypothesis that the chosen model is suitable for describing L¨ uders
215
lines.
216
The experiments and observations of Price and Kelly (1964), producing L¨ uders bands in
217
single crystals of aluminium alloys, helped to delineate the simplest framework which could
218
encapsulate the physics integral to L¨ uders bands. In their experiments deformation localised
219
despite small volumes and slow deformation, showing that shear band formation need not
220
hinge upon temperature e↵ects. Also, repeated deformation to the point of strain localisation
221
of the same crystal did not result in shear bands localising in the same places, showing that
222
localisation was not the result of permanent work softening. These results admit neglect of
223
work hardening and softening, and of any dependence of constitutive laws on temperature.
224
The results of Cox and Low (1974) suggested that the critical mechanism leading to
225
strain localisation during their loading of steel was di↵erential deformation of metallic and
226
non-metallic material elements. As Rice (1976) observes from the figures of Cox and Low
227
(1974), the cavities which nucleate around inclusions are prominent in localisation bands
228
but not visible elsewhere. The influential role of material disparity suggests a description
229
which accounts separately for the two phases of steel. Admittedly, the saturated mixtures
230
description of this work does not explicitly account for cavities.
231
Pressure impedes shear strain (Frost and Ashby, 1982), and so the maxima of shear
232
strain, L¨ uders bands, may be sought via the minima of pressure. An additional motivation
233
for seeking to describe pressure is the sensitivity to small stress inhomogeneities of the
234
positions of L¨ uders bands (Hertzberg et al., 2012).
235
This section (section 4.2) considers the pressure solution p˜ of the last section (section 4.1)
236
for the expansion of a mild steel cylinder. Figure 4a shows the cylinder after expansion and
237
etching by Sowerby and Johnson (1970). Appendix B discusses estimation of the physical
238
parameters corresponding to their experiment. Because the drag coefficient C is difficult to
239
estimate, values of C around an order of magnitude greater than and less than that roughly
240
estimated in Appendix B are also considered. Table 4 shows signature quantities which
241
correspond to these choices of parameters.
23
C
⇥
kg m3 s
⇤
220 · 106 2.2 · 109 22 · 109
H [s] 2.0 · 10 200 · 10 20 · 10
6 9 9
2 /cs [m]
n⇤
3
1
0.318
3
10
3.18
6
100
31.8
22 · 10 2.2 · 10 220 · 10
Table 4: Signature quantities of the computations of section 4.2. The first, second and third rows come from the parameter values estimated for mild steel in Appendix B, with the drag coefficient chosen so that the dimensionless parameter is 1, 10 and 100 respectively. Note that, by definition, n⇤ = /⇡. 242
243
244
To compute pressure profiles at particular times this section (section 4.2) employs the P max truncation p˜ = nn=1 p˜n , where nmax = 105 . The calculations of this section assume that expansion began from static equilibrium. That is, the parameter ⇣ of (42) is chosen ⇣ = 0.
245
The initial condition to be considered corresponds to sudden loading. As will be seen,
246
the sudden application of pressure leads to a propagating discontinuity. Appendix C instead
247
considers an initial condition for which both the overpressure and the spatial derivative of
248
overpressure at the inner and outer wall are zero. It is included to show that results do not
249
depend essentially upon the discontinuous application of pressure. Consider the case in which the cylinder is suddenly subjected to internal pressure. To identify initial conditions, assume that prior to time t = 0 the pressure is zero. By (44), the definitions of overpressure pˆ and the dimensionless fields p˜ and pˇ and the dimensionless variable t˜,
✓
pˇ|t˜=0 = 2 ln 1
✓
ro
ri ro
◆
◆
s˜1 .
(44)
This initial rescaled overpressure pˇ is zero at the outer wall of the cylinder (˜ s1 = 0) and negative between the inner and outer wall. By (41) it follows from (44) that ✓ ✓ ◆ ◆ Z 1 ro r i cn = 4 sin (n⇡˜ s1 ) ln 1 s˜1 d˜ s1 . ro 0
(45)
250
Computation of the coefficients dn and fn requires a choice of the dimensionless parameter
251
.
252
Figures 2a and 2b show, for sudden loading and the choice = 10, the dimensionless
253
total pressure and overpressure profiles at a range of dimensionless times in 0 t˜ 16. In 24
1
p
= 10 (n⇤ = 3.18)
k ( p
max )
-1
p
1.0
1
1.5 0.5
0.5
1
0.5 -0.5
0.5
0
1/4
1
1
s1
p
max
1 0.50
1.0
t˜ =
p
k
s1
0.10 0.05 0.01
-1.0
2
0.5
2
p 2 2
4
8
1
5
10
˜ t
16
Figure 2: Time evolution of dimensionless (a) total pressures and (b) overpressures of a hollow cylinder suddenly loaded to yield. Parameters are those estimated for mild steel in Appendix B and listed in table B.5, and which correspond to the signature quantities listed in the second row of table 4. The factors (|ˆ p|max )
1
are chosen so that the maximum magnitude of the scaled overpressure profiles of (b) are about unity. The coincidence of the discontinuity in and the local maximum of the overpressure at t˜ = 2 distinguishes this time. (c) Values of the inverse of the scaling factor |ˆ p|max at di↵erent dimensionless times t˜. 254
particular, they include the profile at the dimensionless time t˜ = 2. As section 4.1 showed,
255
this time was a critical point of the amplitude of the component of pressure p˜n as n ! n⇤ .
256
However, as this section (section 4.2) will discuss, figure 3c, a plot of the continuous parts
257
of the overpressures, indicates more clearly the potential significance of the time t˜ = 2.
258
The plots of figures 2, 3 and C.6 correspond to the parameters for mild steel estimated in
259
Appendix B and listed in table B.5.
260
The profiles of figures 2a and 2b, corresponding to = 10, show that the initial pressure
261
discontinuity at the inner wall of the cylinder (where s˜1 = 1) propagates and attenuates.
262
The same occurs for = 1 and = 100. The considerations of Zauderer (2006, pp. 700
263
– 716) allow identification of the discontinuity. It travels at the speed of second sound cs ,
264
attenuates as exp
265
(where s˜1 = 0) wall of the cylinder. Let pD denote the discontinuous part of the pressure.
t˜ and is reflected when it reaches the inner (where s˜1 = 1) or outer
25
266
Specifically, let pD 9 denote the spatially piecewise constant function satisfying pD |s˜1 =0 = 0
267
such that p , pˆ
268
overpressure.
\
\
pD is continuous. The function p will be called the continuous part of the
269
The left column of figure 3 shows scaled dimensionless overpressure profiles at a range of
270
dimensionless times for sudden loading and choices = 1, 10 and 100. The scaling factors
271
\
p
1 max
are chosen so that the maximum magnitude of the scaled pressure profiles are \
272
about unity. The right column of figure 3 shows values of p
273
times t˜. The factor exp
274
that, as the right column suggests, as time passes the magnitude of the pressure profile
275
approaches zero.
max
at di↵erent dimensionless
t˜ in the expression (43) of the components of pressure means
278
The expression (43) of overpressure components shows that for 1 n < n⇤ , p˜n has a ⇣ ⌘ factor cosh n t˜ + ⇣ n1 sinh n t˜ . Also, the expressions (39) of time constants shows that
279
shape of the pressure profile approaches that of the spatial factor of p˜1 , sin (n⇡˜ s1 ). This is
280
seen in figures 3c and 3e.
276
277
for 1 < n < n⇤ ,
1
>
n.
Hence, as time passes the mode p˜1 comes to dominate and so the
The profiles of figure 3a, which also pertains to sudden loading but to the choice of the
281
t˜
282
dimensionless parameter = 1, show di↵erent behaviour. Of course, the factor exp
283
in the expression (43) of the components of pressure means that, as figure 3b suggests, the
284
magnitude of the pressure profile approaches 0 as time approaches infinity. However, because
285
n⇤ = 0.318 < 1, no components of overpressure have additional hyperbolic factors. Hence,
286
no single term comes to dominate the shape of the profile. The experiments inspiring this work produced L¨ uders bands. How and where does the
287
9
The discontinuous, piecewise constant function pD can be written (regrettably, fairly opaquely) ⇣ ⌘ pD = pQ |s˜1 =1 exp
t˜ ⇥ s˜1
1+
t˜/ + 1
8 < 0 where ⇥ is the Heaviside theta function satisfying ⇥(z) = : 1
,
mod 2
z0
1 ,
and for real a and positive n, a , z>0 mod n (which can be read ‘a modulo n’) is the unique real number r satisfying 0 r < n for which there exists an integer q such that a = nq + r.
26
288
preceding analysis suggest they will form? Towards answering this question, note that ob-
289
served sensitivity of the position of L¨ uders bands to small stress inhomogeneities (Hertzberg
290
et al., 2012) justifies consideration of critical points of the pressure distribution. By (29),
291
consideration of minima of pressure is equivalent to examining the maxima of volumetric
292
strain. The peaks of volumetric expansion may provide a more intuitive way to think about
293
shear strain localisation than do the minima of pressure.
294
For a constant shear the rate of shear strain increases with decreasing pressure. (Frost
295
and Ashby, 1982) However, material which has reached a quasistatic pressure distribution
= 1 (n⇤ = 0.318)
( p⋂
max )
p
1.0
1
p⋂
max
k
0.5
2 1 0.5
1
s1
0.5
-0.5
0.2 0.1
-1.0
t˜ =
0.05
0 ( p⋂
= 10 (n⇤ = 3.18)
-1 ⋂
max )
1/2
1
p
2
0.5
p 2 2
2
1
2
64
256
˜ t
-1 ⋂
p
1.0
1
p⋂
max
k 0.5
1 0.50 0.5
1
s1 0.10 0.05
-0.5
0.01 -1.0
t˜ =
0.5
0
1/4
1
p
2
2
p 2 2
1
5
4
8
10
˜ t
16
Figure 3: Continues on a later page which contains a descriptive caption.
27
( p⋂
max )
-1 ⋂
p
= 100 (n⇤ = 31.8)
1.0
1
0.5
max
1.
0.5
1
0.1
s1
0.01
-0.5
0.001 0.0001
-1.0
t˜ =
p⋂
k
0
1
2
16
256
1
10
100
1000
˜ t
1024
Figure 3: Continued from an earlier page. Left column. Time evolution of the scaled dimensionless continuous \
parts of the overpressures p of a hollow cylinder suddenly loaded to yield, using the parameters for mild steel estimated in Appendix B and listed in table B.5. For = 10 and 100, the coincidence of the discontinuity in and the local maximum of the overpressure pˆ at about t˜ = 2 distinguishes this time. Right column. Values \
of the inverse of the scaling factor p
max
at di↵erent dimensionless times t˜. Both abscissa and ordinate use
a logarithmic scale. The initial profile is scaled by the inverse of
1 k
\
p
max
t˜=0
= 1.91. Any times t˜ > 0 \
which are represented in a pressure profile on the left but absent on the right have values of p
max
which are
cumbersomely many orders of magnitude less than those which are shown. Each row of the figure pertains to the signature quantities listed in the corresponding row of table 4.
296
p = pQ is not expected to deform further. A simple way to affirm the latter result is by
297
assuming that the tendency towards shear strain depends upon the overpressure pˆ. Assume
298
that a point favourable for strain localisation is one for which the left and right hand limit
299
of the overpressure, pˆ, is less than at points in a neighbourhood just to its left and right
300
respectively. Then the discontinuous part of the overpressure which was identified does not
301
a↵ect the location of points favourable for strain localisation. Hence, for clarity, consider the
302
continuous part of the overpressure p rather than the total overpressure pˆ.
\
The following paragraph applies for n⇤ > 1. For positive times t˜ < 2, there is no local \
spatial minimum of the continuous part of the overpressure p of pressure in the interior of the cylinder. For an open interval of times with infimum t˜ = 2, the continuous part of the
28
\
overpressure p exhibits a local minimum at the site of the discontinuity in pˇ, s˜1 = 1
t˜/.
Figure 3c suggests this but, unfortunately, scale means that it is not possible to discern for = 100 from figure 3e. The scaled arc-length s˜1 of this relationship for t˜ = 2, around which the interior pressure minimum is forecast, is a reasonable location to anticipate initialisation of strain localisation. So is s˜ = 1 (which corresponds to the inner wall of the cylinder), the location of the global minimum of the continuous part of the overpressure. The distance s1 between characteristics intersecting at these locations can be written in terms of the physical parameters of the problem p
s1 2 (ro
ri )
=1
p
2p
4 K⇢↵ C B
1
s
1+
B
⇢ . ⇢↵
(46)
303
Figure 4b shows characteristics which intersect at the values s˜1 = 1 and s˜1 = (1
304
0.8. It corresponds reasonably closely to figure 4a, which shows the product of the operation
305
of Sowerby and Johnson (1970). However, it is coincidence that for the relevant parameters
306
the choice = 10 corresponds so closely to the experiment. Still, the process suggests a way
307
to use (46) to estimate the drag coefficient C relevant to the experiment of Sowerby and
308
Johnson (1970) and to similar applications.
309
For = 100 the location of the discontinuity at t˜ = 2 is s˜1 = (1
2/)|=10 =
2/)|=100 = 0.980,
310
which would correspond to curves of strain localisation far more closely spaced than those
311
indicated by figure 4b or seen in figure 4a.
312
Several links in the chain of reasoning which locates lines of strain concentration are
313
insecure. The initial condition of section 4.2 (that is, (44)) may not conform precisely with
314
the experiment of Sowerby and Johnson (1970), some details of which we do not know.
315
Having obtained from this initial condition a pressure profile, and aware that that spatially
316
regular strain localisation indeed occurs during the expansion, the reasoning of section 4.2
317
represents one way to identify the spacing of lines of concentrated strain. It may be that
318
the identified pressure profiles are not, after all, accurate, or that in the expansion pressure
319
is not the most appropriate property whereby to locate the lines of strain concentration.
320
Even if pressure is, as the observations of Capp et al. (1973) and Hertzberg et al. (2012)
321
might suggest, a fitting way to pinpoint strain localisation, it may not be the case that the 29
Sudden loading, = 10
x2 [in.] 1.0 0.5
-1.0 -0.5
0.5
1.0
x1 [in.]
-0.5 -1.0 Figure 4: Characteristics and slip-lines. (a) Figure 9.3a of Sowerby and Johnson (1970). Slip-lines in a mild steel cylinder of outer diameter ro = 2.2500 and inner diameter ri = 100 . The inner diameter was inferred from the image and knowledge of the outer diameter. (b) Characteristics which intersect at s˜1 = 1 and s˜1 = 0.8, positions which section 4.2 argues are favourable for strain localisation if = 10. \
322
continuous part of the overpressure, p, should be used. For example, either the continuous
323
part of the total pressure, p
324
However, their local minima (where strain would tend to concentrate) might need to be
325
found numerically.
pD , or the total pressure, p, may play a more important role.
326
Pressure minima also develop for the smoother initial condition of Appendix C. Lines of
327
strain localisation to which these could correspond are shown in figure C.7b. Their locations
328
had to be found numerically, but the section shows that results do not depend essentially
329
upon the discontinuous application of pressure.
330
5. Conclusion
331
This work focused on expanded cylinders of mild steel, but the relationship (21) between
332
overpressure and its spatial and time derivatives could be applied to describe pressure evo-
333
lution in other conditions of plane strain. The relationship is not restricted to Von Mises 30
334
and Drucker-Prager materials.
335
It would be interesting to assess the role of dissipative volumetric deformation by mod-
336
ifying the volumetric pressure-strain relationship (29) to include a plastic contribution to
337
volumetric strain. Although, as discussed in section 4.2, strain localisation need not de-
338
pend upon temperature e↵ects, more precise realism could also be sought by including heat
339
di↵usion and the dependence of material properties on temperature. These dissipative phe-
340
nomena could be useful in describing the width of deformation bands. Another possible
341
extension could be to materials containing more than two phases. Since, as discussed in 4.2,
342
in deforming steel cavities can nucleate around inclusions in localisation bands more than
343
elsewhere, it would be particularly interesting to consider a component comprising void.
344
Applying SFT to yielding two phase materials identified dissipative pressure waves. The
345
case of radially symmetric plane strain and a Von Mises or Drucker-Prager yield envelopes
346
allowed particularly simple expression of the wave equations along a slip-line. When the main
347
phase is Von Mises this was a telegraphy equation. The pressure solution associated with
348
radially symmetric, sudden loading of the alpha-ferrite-cementite mixture which comprises
349
steel uncovered pressure troughs which may be linked with strain concentrations. For one of
350
the surveyed values of an imprecisely known physical parameter called the drag coefficient,
351
the pressure troughs would lead to strain concentrations similar to those of figure 4a. The
352
process could be used to estimate drag coefficients for this and other systems.
353
It would be interesting to load rapidly to yield cylinders of di↵erent sizes and di↵erent
354
compositions, and see how closely the observed variation of the spacing of lines of strain
355
concentration matches that predicted by (46). Likewise, it would be interesting to record
356
pressure during loading and use it as the boundary condition at the inner wall of the cylinder
357
for numerical integration of the pressure. The resulting pressure profiles could be compared
358
with the observed spacings of lines of strain concentration. Pressurisation of confined rather
359
than free materials would also provide extra sets of boundary conditions to test.
31
360
References
361
Aboudi, J., Arnold, S. M., Bednarcyk, B. A., 2013. Micromechanics of Composite Materials:
362
363
364
A Generalized Multiscale Analysis Approach. Butterworth-Heinemann, Oxford. Bhadeshia, H., Honeycombe, R., 2006. Steels: Microstructure and Properties. ButterworthHeinemann, Oxford.
365
Biot, M. A., 1956a. Theory of propagation of elastic waves in a fluid-saturated porous solid.
366
I. Low-frequency range. The Journal of the Acoustical Society of America 28 (2), 168–178.
367
Biot, M. A., 1956b. Theory of propagation of elastic waves in a fluid-saturated porous solid.
368
II. Higher frequency range. The Journal of the Acoustical Society of America 28 (2),
369
179–191.
370
371
372
373
Callister, W. D., Rethwisch, D. G., 2010. Materials Science and Engineering: An Introduction. Wiley, New York. Capp, D. J., McCormick, P., Muir, H., 1973. The e↵ect of pressurization and quenching on the yield behaviour in low carbon steel. Acta Metallurgica 21 (1), 43–47.
374
Coussy, O., 2004. Poromechanics. John Wiley & Sons.
375
Cox, T., Low, J. R., 1974. An investigation of the plastic fracture of AISI 4340 and 18
376
377
378
379
380
381
382
Nickel-200 grade maraging steels. Metallurgical Transactions 5 (6), 1457–1470. Estrin, Y., Ling, C., McCormick, P., 1991. Localization of plastic flow: Spatial vs temporal instabilities. Acta Metallurgica et Materialia 39 (11), 2943–2949. Frost, H. J., Ashby, M. F., 1982. Deformation Mechanism Maps: The Plasticity and Creep of Metals and Ceramics. Pergamon Press, Exeter. Gao, F.-P., Wang, N., Zhao, B., 2015. A general slip-line field solution for the ultimate bearing capacity of a pipeline on drained soils. Ocean Engineering 104, 405–413.
32
383
Godunov, S. K., Deribas, A. A., Zakharenko, I. D., Mali, V. I., 1971. Investigation of the
384
viscosity of metals in high-velocity collisions. Combustion, Explosion and Shock Waves
385
7 (1), 114–118.
386
387
Hertzberg, R. W., Vinci, R. P., Hertzberg, J. L., 2012. Deformation and Fracture Mechanics of Engineering Materials. Wiley, Hoboken.
388
Hill, R., 1950. The Mathematical Theory of Plasticity. Clarendon Press, Oxford.
389
Hosford, W. F., 2010. Solid Mechanics. Cambridge University Press.
390
Jackson, R., Ghaednia, H., Pope, S., 2015. A solution of rigid-perfectly plastic deep spherical
391
392
393
394
395
indentation based on slip-line theory. Tribology Letters 58 (3), 1–7. Johnson, J. N., 1981. Dynamic fracture and spallation in ductile solids. Journal of Applied Physics 52 (4), 2812–2852. Johnson, W., Sowerby, R., Venter, R., 1982. Plane-Strain Slip-Line Fields for MetalDeformation Processes: A Source Book and Bibliography. Pergamon Press, Exeter.
396
Khan, I. A., Bhasin, V., Chattopadhyay, J., Singh, R. K., Vaze, K. K., Ghosh, A. K.,
397
2014. An insight of the structure of stress fields for stationary crack in strength mismatch
398
weld under plane strain mode–I loading - Part ii: Compact tension and middle tension
399
specimens. International Journal of Mechanical Sciences 87, 281–296.
400
401
402
403
404
405
Koot, L., Dumberry, M., 2011. Viscosity of the earth’s inner core: Constraints from nutation observations. Earth and Planetary Science Letters 308 (3), 343–349. Laszlo, F., Nolle, H., 1959. On some physical properties of cementite. Journal of the Mechanics and Physics of Solids 7 (3), 193–208. McCormick, P., 1988. Theory of flow localisation due to dynamic strain ageing. Acta Metallurgica 36 (12), 3061–3067.
33
406
407
Miodownik, A. P., 1994. Young’s modulus for carbides of 3d elements (with particular reference to Fe3 C). Materials Science and Technology 10 (3), 190–192.
408
Paesold, M. K., Bassom, A. P., Veveakis, E., Regenauer-Lieb, K., 2016. Conditions for the
409
localization of plastic deformation in temperature sensitive visco-plastic materials. Journal
410
of Mechanics of Materials and Structures 11 (2), 113–136.
411
Papamichos, E., Vardoulakis, I., Tronvoll, J., Skjaerstein, A., 2001. Volumetric sand produc-
412
tion model and experiment. International Journal for Numerical and Analytical Methods
413
in Geomechanics 25 (8), 789–808.
414
415
Price, R., Kelly, A., 1964. Deformation of age-hardened aluminium alloy crystals-ii. Fracture. Acta Metallurgica 12 (9), 979–992.
416
Rice, J. R., 1976. The localization of plastic deformation. In: Theoretical and Applied Me-
417
chanics (Proceedings of the 14th International Congress on Theoretical and Applied Me-
418
chanics). pp. 207–220.
419
Rudnicki, J. W., Rice, J. R., 1975. Conditions for the localization of deformation in pressure-
420
sensitive dilatant materials. Journal of the Mechanics and Physics of Solids 23 (6), 371–394.
421
Savenkov, G. G., 2010. Dynamic viscosity and material relaxation time during shock loading.
422
423
424
425
426
Journal of Applied Mechanics and Technical Physics 51 (2), 148–154. Savenkov, G. G., Meshcheryakov, Y. I., 2002. Structural viscosity of solids. Combustion, Explosion and Shock Waves 38 (3), 352–357. Shield, R., 1953. Mixed boundary value problems in soil mechanics. Quarterly of Applied Mathematics 11 (1), 61–75.
427
Sowerby, R., Johnson, W., 1970. Use of slip line field theory for the plastic design of pressure
428
vessels. In: Experimental stress analysis and its influence on design : Proceedings of the
429
Fourth International Conference on Experimental Stress Analysis. pp. 303–317.
34
430
Spitzig, W. A., Sober, R. J., Richmond, O., 1975. Pressure dependence of yielding and
431
associated volume expansion in tempered martensite. Acta Metallurgica 23 (7), 885–893.
432
Vardoulakis, I., 1989. Shear-banding and liquefaction in granular materials on the basis of a
433
434
435
436
437
438
439
440
441
442
Cosserat continuum theory. Ingenieur-Archiv 59 (2), 106–113. Vardoulakis, I., Sulem, J., 2005. Bifurcation Analysis in Geomechanics. Blackie Academic & Professional, Glasgow. Veveakis, E., Regenauer-Lieb, K., 2015. Cnoidal waves in solids. Journal of the Mechanics and Physics of Solids 78, 231–248. Vo, T., Russell, A. R., 2014. Slip line theory applied to a retaining wall-unsaturated soil interaction problem. Computers and Geotechnics 55, 416–428. Zauderer, E., 2006. Partial Di↵erential Equations of Applied Mathematics. John Wiley & Sons, Hoboken.
Appendix A. Estimating the interaction force
443
Section 2.1 considers an interaction force between cementite and alpha-ferrite which op-
444
poses relative motion of the phases; the force, given by (8), is proportional to the relative
445
velocity. This appendix considers a method of estimation of the coefficient of proportionality
446
C, and shows that estimates of C could range across several orders of magnitude.
447
As figure A.5a suggests, mild steel consists of a mixture of grains of alpha-ferrite (nearly
448
pure iron) and grains of the two phase material pearlite. Pearlite comprises alternating layers
449
of alpha-ferrite and cementite (a ceramic). The lamellar structure of mild steel motivates
450
investigation of flow between layers.
451
Consider steady-state laminar flow over a distance l of an incompressible Newtonian
452
fluid between stationary wide, long parallel plates separated by a distance 2d. Figure
453
A.5b illustrates the flow. Denote the width of the plates by w. Since the cross-section
35
Figure A.5: Lamellae and laminar flow in mild steel. (a). Following figure 9.29 of Callister and Rethwisch (2010). Mild steel is composed of grains of metallic alpha-ferrite and grains of the mixture pearlite. Pearlite is composed of alternating layers of alpha-ferrite and the ceramic cementite. Alpha-ferrite (↵) in grey and cementite (Fe3 C) in black. (b). Following Biot (1956a). Laminar flow between parallel plates.
454
is constant and the phase is assumed to be incompressible, the material does not accel-
455
erate. It follows that the force balance on the layer of material between y and y + dy is
456
0 = P |x=0 wdy
⌘
dv dy
y
wl
P |x=l wdy + ⌘
dv dy
y+dy
wl, where P is pressure ⌘ is the vis-
457
cosity of the phase. In the limit of small dy this is
458
which can be written
( P |x=0
2
( P |x=0
2
d v P |x=l ) wdy = ⌘wl dy 2 dy,
d v P |x=l ) /l = ⌘ dy 2 . Assuming that the material is stationary
462
where it contacts the plates, v|y= d = 0 = v|y=d , and so the solution to this di↵eren1 ( P |x=0 P |x=l ) tial equation is v(y) = 2⌘ (d2 y 2 ). Hence the average speed of the phase is l R y=d 1 ( P |x=0 P |x=l ) 2 1 1 ( P |x=0 P |x=l ) 2 v¯ = 2d (d y 2 ) dy = 3⌘ d. l l y= d 2⌘
463
cementite v (1)
464
force per unit volume of alpha-ferrite
465
Hence, the proportionality factor between specific force and discharge, C =
466
estimated from the viscosity ⌘ and the plate separation distance d.
459
460
461
Associating the average speed v¯ with the relative motion between the alpha-ferrite and v (2) and the spatial rate of pressure drop ( P |x=0 f /(1
) gives f =
P |x=l ) /l with the drag
(3⌘)/d2 (1
) v (1)
3⌘(1 d2
)
v (2) .
, can be
467
Of course, not all lamellae will be parallel to the flow. Biot (1956b) accounted for the
468
anisotropy of flow channels by multiplying his result for the drag coefficient by a factor ⇠ > 1
36
469
which he called the sinuosity. Incorporating this, the result of this appendix for the drag
470
coefficient would be C = ⇠ 3⌘(1d2
471
drag coefficient which depended on the frequency of the relative motion between the phases.
472
For simplicity this work considers a constant drag coefficient and does not explicitly consider
473
the sinuosity ⇠. A greater viscosity ⌘ or lesser plate separation d achieves the e↵ect of a
474
sinuosity ⇠ greater than unity.
)
. Biot (1956a,b) further increased accuracy by considering a
475
Estimated viscosities of solid metals vary, and depend not just upon temperature pressure
476
and loading conditions but upon the models used. (Savenkov, 2010) Godunov et al. (1971)
477
cite the use and estimation of viscosities of steel in the range ⌘ = 10
478
Koot and Dumberry (2011) mention that calculations of the viscosity of the iron core of
479
the Earth range between ⌘ = 1011
480
that viscosity decreased with the length scale of the flow. At the smallest length scale they
481
consider, 10
482
figure 5), but they do not indicate a positive lower bound. To reproduce observed spalling
483
under plate impact of copper Johnson (1981) identifies a viscosity of 1.0 Pa s. He notes that
484
this is orders of magnitude lower than the value observed for macroscopic metallic flow, and
485
speculates that the discrepancy may be due to localisation e↵ects.
7
3±1
106 Pa s, while
1021 Pa s. Savenkov and Meshcheryakov (2002) found
10 6 m, the pertinent viscosity is bounded above by 100 Pa s (see their
486
The e↵ective interplate distance d will lie somewhere between the separation of neigh-
487
bouring cementite plates within a pearlite grain and the separation of cementite plates of
488
neighbouring pearlite grains (separated by ferrite). Figure 3.18 of Bhadeshia and Honey-
489
combe (2006) shows interlamellar spacings of carbon steels spanning the range d = 4. ⇥
490
10
491
d = 1. ⇥ 10
492
8
5. ⇥ 10 7 m, while their figure 11.6 shows grain sizes of mild steel spanning the range 5
1. ⇥ 10 3 m.
With viscosity in the range ⌘ = 10 8
4
1021 Pa s and plate spacing in the range d =
1. ⇥ 10 3 m, it would be easier to estimate C =
493
4. ⇥ 10
494
constituent factors.
37
3⌘(1 d2
)
directly than from its
Property
ri [m] 2.9 · 10
Value
ro [m] 2
1.3 · 10
⇢↵ 2
1.3 · 10
2
⇥ kg ⇤ m3
7.69 · 103
⇢
⇥ kg ⇤ m3
7.85 · 103
K [Pa] 1.70 · 1011
B 2.2 · 10
3
Table B.5: Estimates of the parameters of the cylinder of mild steel, used for the computations of section 4.2. Appendix B discusses the choice of parameters.
495
Appendix B. Estimating parameters for a cylinder of mild steel
496
Inner radius, ri and outer radius, ro . The radii considered are those pertaining to
497
figure 9.3a of Sowerby and Johnson (1970) (figure 4a of this document); ro = 1.12500 and
498
inner diameter ri = 0.5000 . The inner radius was inferred from their figure 9.3a and knowledge
499
of the outer radius.
500
Volume fraction of
,
. Densities, ⇢↵ and ⇢ . The
-volume fraction
was es-
501
timated assuming the carbon content of the mild steel of Laszlo and Nolle (1959), 0.088%
502
(by weight), along with the densities of alpha-ferrite ⇢↵ and cementite ⇢ they used. These
503
densities are those listed in table B.5. The calculation treats the alpha-ferrite fraction as
504
pure Fe and the cementite fraction as pure Fe3 C. The molecular weight of Fe and C used
505
were 55.85g/mol and 12.01g/mol.
506
Bulk modulus, K. Biot coefficient, B. The -volume fraction
of table B.5 is orders
507
of magnitude less than unity, motivating use of the formula for dilute bulk modulus of Aboudi
508
et al. (2013, pp. 109–110). This is K = K↵
509
is the shear modulus and ⌫↵ is the Poisson ratio of phase ↵.
510
(K↵
↵ +4G↵ K ) 3K , where G↵ = 3K +4G↵
3 1 2⌫↵ K↵ 2 1+⌫↵
The Biot coeffient was calculated using (4.35) of Coussy (2004) for a linear isotropic mateK . Ks
511
rial; B = 1
512
the skeleton, Ks , with the bulk modulus of alpha-ferrite, K↵ . With the formula above for the
513
dilute bulk modulus K , the formula of Coussy (2004) becomes B =
514
The
Any occluded
-volume fraction
was neglected, allowing identification of the bulk modulus of
(1
↵ +4G↵ K /K↵ ) 3K . 3K +4G↵
is that discussed earlier in this appendix and given in table
515
B.5. The other values in the formulae are those of Miodownik (1994); bulk moduli K↵ =
516
1.71 ⇥ 109 Pa and K = 1.45 ⇥ 109 Pa, and Poisson ratio ⌫↵ = 0.29.
517
Coefficient of the interaction force, C. Appendix A shows that the drag coefficient
38
518
C can be estimated from the viscosity ⌘, characteristic distance d and -volume fraction .
519
However, it also demonstrates the difficulty of choosing the parameters precisely, and in this
520
way precisely estimating the drag coefficient C.
521
As figure 4b shows, the choice
= 10 produces curves of strain concentration with spacing
522
similar to those observed in the experiment of Sowerby and Johnson (1970). As table 4
523
indicates,
524
there is strong resistance to relative motion between alpha-ferrite and cementite. As table 4
525
suggests, this work also considers values of C an order of magnitude lesser and greater.
526
= 10 corresponds to C = 2.2⇥109 kg m 3 s 1 . This suggests that, unsurprisingly,
The process of choosing
and C suggests a method by which (46), an expression for
527
the predicted distance along slip-lines between the first and second intersection of slip-lines,
528
could be used to estimate the drag coefficient C from experimental or real world observations.
529
Appendix C. A C 1 pressure perturbation in a yielding cylinder of mild steel
530
As seen in section 4.2, the sudden application of pressure leads to a propagating dis-
531
continuity. This appendix supplements section 4.2 by considering an initial condition very
532
di↵erent from sudden loading: one for which both the overpressure and the spatial derivative
533
of overpressure at both the inner and the outer wall vanish. It shows that the local pressure
534
minima which could locate lines of strain concentration do not depend essentially upon the
535
discontinuous application of pressure. Consider an initial pressure perturbation which will be referred to as a C 1 perturbation; ◆ 2✓ ◆2 ✓ 1 1 s˜1 1 s˜1 . (C.1) pˇ|t˜=0 = By (41) it follows that cn = 2
Z
1
sin (n⇡˜ s1 ) 0
s˜12 (1
s˜1 )2 d˜ s1 .
(C.2)
536
The left column of figure C.6 shows scaled dimensionless overpressure profiles at a range of
537
dimensionless times for a C 1 perturbation and choices = 1, 10 and 100. The scaling factors
538
(|ˆ p|max )
539
unity. The right column of figure C.6 shows values of |ˆ p|max at di↵erent dimensionless times
1
are chosen so that the maximum magnitude of the scaled pressure profiles are about
39
540
t˜. The factor exp
t˜ in the expression (43) of the components of pressure means that,
541
as this column suggests, as time passes the magnitude of the pressure profile approaches
542
0. For = 10 and 100, as time passes the mode p˜1 begins to dominate and so the shape
543
of the pressure profile approaches that of the spatial factor of p˜1 , sin (n⇡˜ s1 ). For = 1,
544
n⇤ , /⇡ = 0.318 < 1, and so no single term comes to dominate the shape of the profile.
545
Figures C.6c and C.6e show a di↵erence from the case of sudden loading; the dimensionless
546
time t˜ = 2 is not distinguished by being the time at which the discontinuity in and the
547
local maximum of the overpressure coincide. It follows that for a C 1 perturbation the
= 1 (n⇤ = 0.318)
( p
max )
-1
p
1.0
1
p
k
max
0.5 10-2 0.5
1
s1
10-5
-0.5
10-8 0.5
-1.0
t˜ =
0
= 10 (n⇤ = 3.18)
( p
max )
1/2 -1
1
p
2
8
16
1
2
5
10
20
1024
p
1.0
1
p
k
0.5
0.5
1
max
s1
-0.5 0.001 1
-1.0
t˜ =
˜ t
0
1
p
2
2
p 2 2
4
2
5
10
˜ t
8
Figure C.6: Continues on a later page which contains a descriptive caption.
40
( p
max )
-1
p
= 100 (n⇤ = 31.8)
1.0 1
p
k
0.5
max
0.001 0.5
1
s1 0.0001
-0.5 0.00001 1
-1.0
t˜ =
0
2
p 2 2
4
64
256
10
100
˜ t
1024
Figure C.6: Continued from an earlier page. Left column. Time evolution of the scaled dimensionless overpressures of a yielding hollow cylinder the pressure profile of which is perturbed from equilibrium according to the initial condition (C.1). The plots apply to the parameters for mild steel estimated in Appendix B and listed in table B.5. For = 10, the troughs are particularly clear at t˜ = 2. Right column. Values of the inverse of the scaling factor |ˆ p|max at di↵erent dimensionless times t˜. Both abscissa and ordinate use a logarithmic scale. The initial profile is scaled by the inverse of
1 k
(|ˆ p|max )|t˜=0 = 6.25 ⇥ 10
2
, the maximum
magnitude of the initial pressure profile (C.1) (attained at s˜1 = 0.5). Any times t˜ > 0 which are represented in a pressure profile on the left but are absent on the right have values of |ˆ p|max which are cumbersomely many orders of magnitude less than those which are shown. Each row of the figure pertains to the signature quantities listed in the corresponding row of table 4.
548
dimensionless time t˜ = 2 is not the earliest positive dimensionless time for which there is an
549
interior point most favourable for strain in its local neighbourhood. However, figure C.6c
550
shows that when = 10 the troughs of pressure, which correspond to regions more favourable
551
for strain, are particularly prominent when t˜ = 2. Figure C.7b shows characteristics which
552
first intersect at the inner wall of the cylinder (where s˜1 = 1), and next at the pressure
553
trough for t˜ = 2 which is closer to the inner wall of the cylinder; s˜1 = 0.860.
554
Interestingly, although figure C.6e, corresponding to sudden loading and the choice =
555
100, exhibits sharp troughs in overpressure, the dimensionless time when they are most
556
prominent is not t˜ = 2. Note that the pressure extrema evident after a C 1 perturbation are
557
only troughs because the initial perturbation involves a initial reduction in pressure relative 41
C 1 perturbation, = 10
x2 [in.] 1.0 0.5
-1.0 -0.5
0.5
1.0
x1 [in.]
-0.5 -1.0 Figure C.7: Characteristics and slip-lines. (a) Figure 9.3a of Sowerby and Johnson (1970). Slip-lines in a mild steel cylinder of outer diameter ro = 2.2500 and inner diameter ri = 100 . The inner diameter was inferred from the image and knowledge of the outer diameter. (b) Characteristics which intersect at s˜1 = 1 and s˜1 = 0.860. The latter is a prominent trough of the overpressure after the C 1 perturbation which the initial condition (C.1) describes if = 10.
558
to quasistatic pressure. If the negative of this perturbation were considered then peaks in
559
overpressure would be identified in place of troughs.
42
