p = ’Kµv , σ d = 2G µ d , (12.11)

and in matrix notation:

p = ’Kµv , σ d = 2G µ d , (12.12)

with K the compression modulus or bulk modulus of the material and G the

shear modulus (for Hooke™s law the relevant material parameters; both positive).

In the present section it is assumed, that the material is compressible, meaning that

the compression modulus K has a ¬nite value. Based on Hooke™s law the above

equations can be written as

2G

σ = K’ tr(µ) I + 2Gµ, (12.13)

3

and also

2G

σ = K’ tr( µ) I + 2Gµ, (12.14)

3

thus fully establishing the desired tensor relation σ = σ ( µ), and σ = σ ( µ) in

matrix notation. For the inverse relationship µ = µ( σ ), and µ = µ( σ ), it can

easily be derived:

1 1

1

’ tr( σ ) I + σ,

µ= (12.15)

9K 6G 2G

and also

1 1 1

µ= ’ tr( σ ) I + σ. (12.16)

9K 6G 2G

197 12.2 Elastic behaviour at small deformations

In the following an interpretation of Hooke™s law will be given. For this purpose

we focus on the matrix formulations σ = σ ( µ) and µ = µ( σ ). The symmetrical

matrices σ and µ are composed according to:

⎡ ¤ ⎡ ¤

σxx σxy σxz µxx µxy µxz

⎢ ⎥ ⎢ ⎥

σ = ⎣ σyx σyz ¦ , µ = ⎣ µyx

σyy µyy µyz ¦ . (12.17)

σzx σzy σzz µzx µzy µzz

Using µ = µ( σ ) it follows from Eq. (12.16) for the diagonal components (the

strains in the x-, y- and z-directions):

1 1 1

µxx = ’ ( σxx + σyy + σzz ) + σxx

9K 6G 2G

1 1 1

µyy = ’ ( σxx + σyy + σzz ) + σyy

9K 6G 2G

1 1 1

µzz = ’ ( σxx + σyy + σzz ) + σzz . (12.18)

9K 6G 2G

Eq. (12.18) shows that the strains in the x-, y- and z-directions are determined

solely by the normal stresses in the x-, y- and z-directions, which is a consequence

of assuming isotropy. It is common practice to use an alternative set of material

parameters, namely the Young™s modulus E and the Poisson™s ratio ν. The Young™s

modulus follows from

3K + G

1 1 1 1

= ’ + = , (12.19)

E 9K 6G 2G 9KG

and therefore

9KG

E= . (12.20)

3K + G

The Poisson™s ratio is de¬ned by

ν 3K ’ 2G

1 1

=’ ’ = , (12.21)

E 9K 6G 18KG

so

3K ’ 2G

ν= . (12.22)

6K + 2G

From K > 0 and G > 0 it can easily be derived that E > 0 and ’1 < ν < 0.5.

Using E and ν leads to the commonly used formulation for Hooke™s equations:

Constitutive modelling of solids and fluids

198

1

µxx = σxx ’ ν σyy ’ ν σzz )

(

E

1

µyy = ( ’ν σxx + σyy ’ ν σzz )

E

1

µzz = ( ’ν σxx ’ ν σyy + σzz ) . (12.23)

E

The strain in a certain direction is directly coupled to the stress in that direction

via Young™s modulus. The stresses in the other directions cause a transverse strain.

The reverse equations can also be derived:

E

σxx = ( 1 ’ ν) µxx + ν µyy + ν µzz

( 1 + ν) ( 1 ’ 2ν)

E

σyy = ν µxx + ( 1 ’ ν) µyy + ν µzz

( 1 + ν) ( 1 ’ 2ν)

E

σzz = ν µxx + ν µyy + ( 1 ’ ν) µzz .

( 1 + ν) ( 1 ’ 2ν)

(12.24)

For the shear strains, the off-diagonal components of the matrix µ, it follows from

Eq. (12.16):

1 1

µxy = µyx =

σxy = σyx

2G 2G

1 1

µxz = µzx = σxz = σzx

2G 2G

1 1

µzy = µyz = σyz = σzy . (12.25)

2G 2G

It is clear that shear strains are coupled directly to shear stresses (due to the

assumption of isotropy). The inverse relations are trivial. If required, the shear

modulus can be written as a function of the Young™s modulus E and the Poisson™s

ratio ν by means of the following equation (follows from Eqs. (12.20) and (12.21)

by eliminating K):

E

G= . (12.26)

2( 1 + ν)

12.3 The stored internal energy

It is interesting to study the stored internal energy during deformation of a material

that is described by means of Hooke™s law for linearly elastic behaviour. In Section

11.4 the balance of power was derived for an in¬nitesimally small element with

reference volume dV0 and current volume dV = JdV0 . By integrating the inter-

nally stored power dPint over the relevant time domain 0 ¤ „ ¤ t, the process

199 12.3 The stored internal energy

time between the reference state and the current state, the total internal energy

dEint that is stored by the element is

t

dEint ( t) = tr( σ · D) dV d„ . (12.27)

„ =0

For this relation, it is assumed that initially (so for „ = 0) the internal mechan-

ical energy was zero. For further elaboration we have to account for dV being a

function of time. That is why dV is transformed to dV0 resulting in

t t

dEint ( t) = tr( σ · D) JdV0 d„ = tr( σ · D) Jd„ dV0 . (12.28)

„ =0 „ =0

Subsequently, int0 is introduced as the internal mechanical energy per unit of

reference volume, also called the internal energy density (the subscript 0 refers

to the fact that the density is de¬ned with respect to the volume in the reference

state):

t

int0 ( t) = tr( σ · D) Jd„ . (12.29)

„ =0

For a deformation history, given by means of specifying the deformation tensor as

a function of time: F( „ ) ; 0 ¤ „ ¤ t (with F( „ = 0) = I) and for known material

behaviour by specifying the constitutive equations, int0 ( t) can be calculated. In

the following this will be performed using Hooke™s law.

Assuming small deformations (F ≈ I) the rate of deformation tensor D can be

written as

1™ 1™

™T ™T

F · F’1 + F’T · F ≈ ™

D= F + F = µ. (12.30)

2 2

The volume change factor J can be approximated according to

J ≈ 1 + tr( µ) ≈ 1. (12.31)

Using the de¬nition of the deviatoric form of the strain tensor µ, as de¬ned in Eq.

(7.40), and Eq. (12.11), Hooke™s law can be written as

σ = Ktr( µ) I + 2Gµd . (12.32)

Substituting these results for σ , D and J into the expression for yields

int0

t

™ ™

= Ktr( µ) tr( µ ) + 2Gtr( µd · µ ) d„

int0

„ =0

t

™ ™

= Ktr( µ) tr( µ ) + 2Gtr( µd · µd ) d„

„ =0

t

12

= Ktr ( µ) + Gtr( µd · µd ) . (12.33)

2 „ =0

Constitutive modelling of solids and fluids

200

Taking into account that µ( „ = 0) = 0 it appears that the ˜internal elastic energy

per unit volume™ int0 at time t is fully determined by the components of the

strain tensor at that speci¬c time t; this means that the indication t in this case is

redundant and without any problem it can be written:

12 1

int0 ( µ) = Ktr ( µ) + Gtr( µd · µd ) = K( µv )2 + Gtr( µd · µd ) . (12.34)

2 2

It can be established that in this energy density the hydrostatic (volumetric)

part and the deviatoric part are separately identi¬able; just like in Hooke™s law

there is a decoupling. Both parts deliver an always positive (better: non-negative)

contribution to the energy density, for every arbitrary µ.

Using Hooke™s law, the energy density int0 ( µ), according to the above given

equation, can be transformed to int0 ( σ ) resulting in

11 2 1

int0 ( σ ) = p+ tr( σ d · σ d ) . (12.35)

2K 4G

Finally, it should be remarked that from the previous it can be concluded that a

cyclic process in the deformation or in the stress will always be energetically neu-

tral. This means that the ˜postulated™ constitutive equation (Hooke™s law) indeed

gives a correct description of elastic material behaviour. Every cyclic process is

reversible. No energy is dissipated or released. If we compare the second term

on the right-hand side of Eq. (12.35), representing the ˜distortion energy density™

with Eq. (8.75), de¬ning the von Mises stress, it is clear that both are related. In

other words, if we would like to de¬ne some threshold based on the maximum

amount of distortional energy that can be stored in a material before it becomes

damaged, the von Mises stress can be used for this purpose.

12.4 Elastic behaviour at large deformations and/or large rotations

In this section the attention is focussed on constitutive equations for elastic, com-

pressible, isotropic material behaviour at large deformations. The formulation in

Sections 12.2 and 12.3 is no longer valid, because under those circumstances the

linear strain tensor µ cannot be used.

The Cauchy stress tensor σ is objective (σ transforms in a very speci¬c way

when an extra rigid body rotation is enforced, see Section 9.7). This implies that

σ can certainly not be coupled to an invariant measure for the strain, such as the

right Cauchy Green deformation tensor C or the related Green Lagrange strain

tensor E). But it is allowed to relate σ to the objective left Cauchy Green tensor

B, or the associated strain tensors A (Almansi Euler) and µ F (Finger).

201 12.4 Elastic behaviour at large deformations

Just like in the previous sections, the contribution originating from the volume

change to the stress tensor is considered separately from the distortion (shape

change). For this reason the deformation tensor F is decomposed according to:

˜

F = J 1/3 F with J = det( F) . (12.36)

˜

The tensor F is called the isochoric deformation tensor (because of the equality:

˜

det( F) = 1). The multiplication factor J 1/3 represents the volume change. Depart-

˜

ing from the isochoric deformation tensor F = J ’1/3 F, the associated objective,

˜

isochoric left Cauchy Green tensor B is de¬ned according to:

˜ ˜T

˜

B = F · F = J ’2/3 F · FT = J ’2/3 B, (12.37)

˜

and subsequently the isochoric Finger strain tensor µF according to:

1˜ 1 ’2/3

˜

µF = B’I = B’I .

J (12.38)

2 2

Analogous to the formulation in Section 12.2 linear relations for the hydrostatic

and deviatoric part of the stress tensor can be postulated:

σ h = ’pI = K( J ’ 1) I, (12.39)

˜d