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inelastic constitutive models such as plasticity because several of the key
equations to be introduced will not be fully justi¬ed but loosely based on
similar expressions that are known to apply to small strain theory. More
in-depth discussions can be found in the bibliography.


1
2 LARGE INELASTIC DEFORMATIONS



X3, x3



`

dx

p




dX
time = t
P
X1, x 1
X2, x 2


time = 0

FIGURE A.1 Deformation in a small neighborhood.




A.2 THE MULTIPLICATIVE DECOMPOSITION

Consider the deformation of a given initial volume V into the current volume
v as shown in Figure A.1. An elemental vector dX in the local neighborhood
of a given initial particle P will deform into the spatial vector dx in the
neighborhood of p shown in the ¬gure. If the neighborhood of p could be
isolated and freed from all forces, the material would reach a new unloaded
con¬guration characterized by the spatial vector dxp (see Figure A.2). Of
course if the material is elastic, this unloaded con¬guration will di¬er from
the initial undeformed state only by a rigid body rotation. In the case
of inelastic materials, however, this is not true and a certain amount of
permanent deformation is possible. Note also that the unloaded state can
only be de¬ned locally, as the removal of all forces acting on v may not lead
to a stress-free state but to a complex self-equilibrating stress distribution.
As a result of the local elastic unloading the spatial vector dx becomes
dxp . The relationship between dX and dx is given by the deformation
gradient F as explained in Chapter 3, Section 4. Similarly, the relationship
between dxp and dx is given by the elastic component of the deformation
gradient F e , and dX and dxp are related by the permanent or inelastic
3
A.2 THE MULTIPLICATIVE DECOMPOSITION



dx



F
Fe

dxp


dX


time = t


time = 0
Fp

FIGURE A.2 Multiplicative decomposition.


component F p . These relationships are summarized as,
dx = F dX (A.1a)
dx = F e dxp (A.1b)
dxp = F p dX (A.1c)
for any arbitrary elemental vector dX. Combining Equations (A.1b) and
(A.1c) and comparing with (A.1a) gives,
F = F eF p (A.2)
This equation is known as the multiplicative decomposition of the deforma-
tion gradient into elastic and permanent components and constitutes the
kinematic foundation for the theory that follows.
Strain measures that are independent of rigid body rotations can now
be derived from F and its elastic and inelastic or permanent components.
For instance the inelastic and total right Cauchy“Green tensors given as,
Cp = F T F p ; C = FTF (A.3a,b)
p

are often used for the development of inelastic constitutive equations. Note
that because Cp measures the amount of permanent strain whereas C gives
the total strain, both these tensors are needed in order to fully describe the
current state of the material. As a consequence the stored elastic energy
function Ψ must be a function of both C and Cp , and following similar
arguments to those used in Equation (5.7), an equation for the stresses is
4 LARGE INELASTIC DEFORMATIONS



obtained as,
‚Ψ
Jσ = F SF T
Ψ = Ψ(C, Cp ); S=2
; (A.4a,b,c)
‚C
This constitutive model must be completed with an equation describing the
evolution of Cp in terms of C and Cp or the stresses in the material. This
can, for instance, be given by a plastic ¬‚ow rule or similar relationship.
For isotropic materials it is possible and often simpler to formulate the
constitutive equations in the current con¬guration by using the elastic left
Cauchy“Green tensor be given as,
be = F e F T
e
= F Fp Fp F T
’1 ’T

= F C’1 F T (A.5)
p

Given that the invariants of be contain all the information needed to evalu-
ate the stored elastic energy function and recalling Equation (5.25) for the
Cauchy stress gives,
„ = Jσ = 2ΨI be + 4ΨIIb2 + 2IIIbe ΨIII I
Ψ(be ) = Ψ Ibe , IIbe , IIIbe ; e
(A.6a,b)
where the Kirchho¬ stress tensor „ has been introduced in this equation
and will be used in the following equations in order to avoid the repeated
appearance of the term J ’1 .


A.3 PRINCIPAL DIRECTIONS

As discussed in Chapter 5, many hyperelastic equations are presented in
terms of the principal stretches. In the more general case of inelastic consti-
tutive models, the strain energy becomes a function of the elastic stretches
»e,± . These stretches are obtained from F e in the usual manner, by ¬rst
evaluating the principal directions of the tensor be to give,
3 3
»2 n±
be = — n± ; Fe = »e,± n± — n± (A.7a,b)
˜
e,±
±=1 ±=1

Note that since F e maps vectors from the unloaded state to the current
reference con¬guration, the vectors n± are unit vectors in the local unloaded
˜
con¬guration (see Figure A.3).
Expressing the hyperelastic energy function in terms of the elastic stretches
and using algebra similar to that employed in Section 5.6 enables Equa-
5
A.3 PRINCIPAL DIRECTIONS



n2
0
n2 0
n1
n1
F
Fe


˜0
n ˜
2 n
0 ˜ 1
N2 n ˜0
2 n
1
N1
N2
0
N1


Fp

FIGURE A.3 Principal directions of Fe (solid line), F (dotted), and Fp (dashed).


tion (A.6b) to be rewritten as,
3
‚Ψ
„= (A.8)
n± — n±
‚ ln »e,±
±=1
It is crucial to observe at this point that the principal directions of be and
„ are in no way related to the principal directions that would be obtained
from the total deformation gradient F or the inelastic component F p . In
fact these two tensors could be expressed in terms of entirely di¬erent sets
of unit vectors and stretches as (see Figure A.3),
3 3
F= »± n± — N ± ; Fp = »p,± n± — N ± (A.9a,b)
˜
±=1 ±=1

In general, the principal directions n± obtained from the total deforma-
tion gradient do not coincide with the principal directions of the Cauchy or
Kirchho¬ stress tensors. A rare but important exception to this statement
occurs when the permanent deformation is colinear with (that is, has the
same principal directions as) the elastic deformation. In this case the vec-
tors n± coincide with n± and the multiplicative decomposition F = F e F p
˜ ˜
combined with Equations (A.9a,b) gives n± = n± , N ± = N ± , and,
»± = »e,± »p,± or ln »± = ln »e,± + ln »p,± (A.10a,b)
Equation (A.10b) is the large strain equivalent of the well-known additive
decomposition µ = µe + µp used in the small strain regime. Although Equa-
tions (A.10a,b) are only valid in exceptional circumstances, a similar but
more general and useful expression will be obtained in the next section
6 LARGE INELASTIC DEFORMATIONS



when we consider a small incremental motion.


A.4 INCREMENTAL KINEMATICS

The constitutive equations describing inelastic materials invariably depend
upon the path followed to reach a given state of deformation and stress.
It is therefore essential to be able to closely follow this path in order to
accurately obtain current stress values. This is implemented by taking a
su¬cient number of load increments, which may be related to an arti¬cial
or real time parameter. Consider the motion between two arbitrary consec-
utive increments as shown in Figure A.4. At increment n the deformation
gradient F n has known elastic and permanent components F e,n and F p,n
respectively that determine the state of stress at this con¬guration. In order
to proceed to the next con¬guration at n + 1 a standard Newton“Raphson
process is employed. At each iteration the deformation gradient F n+1 can
be obtained from the current geometry. In order, however, to obtain the cor-
responding stresses at this increment n + 1 and thus check for equilibrium, it
is ¬rst necessary to determine the elastic and permanent components of the
current deformation gradient F n+1 = F e,n+1 F p,n+1 . Clearly, this is not an
obvious process, because during the increment an as yet unknown amount
of additional inelastic deformation may take place.
It is however possible that during the motion from n to n + 1 no further
permanent deformation takes place. Making this preliminary assumption
F p,n+1 = F p,n and a trial elastic component of the deformation gradient
can be obtained as,
F trial = F n+1 F p,n
’1
(A.11)
e,n+1

from which the left Cauchy“Green tensor can be found as,
T
btrial = F trial F trial
e,n+1 e,n+1 e,n+1

= F n+1 C ’1 F T (A.12)
p,n n+1

Using this trial strain tensor, principal directions and a preliminary state of
stress can now be evaluated as,
3 3
‚Ψ

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