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