1 Elasticity, Hooke’s Law, Strain Energy

FIGURE 6.1

Uniaxial loading-unloading stress-strain curves for (a) linear elastic; (b) nonlinear elastic; and

(c) inelastic behavior.



Within the context of the above assumptions, we write the constitutive

equation for linear elastic behavior as



σij = Cijkm εkm or



␴ = Cε

ε



(6.1-3)



where the tensor of elastic coefficients Cijkm has 34 = 81 components. However,

due to the symmetry of both the stress and strain tensors, it is clear that

Cijkm = Cjikm = Cijmk

which reduces the 81 possibilities to 36 distinct coefficients at most.

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(6.1-4)



We may demonstrate the tensor character of C by a consideration of the

elastic constitutive equation when expressed in a rotated (primed) coordinate

system in which it has the form



σ ij = Cijpnε ′pn

′

′



(6.1-5)



But by the transformation laws for second-order tensors, along with Eq 6.1-3,



σ ij = aiq a jsσ qs = aiq a jsCqskmε km

′

= aiq a jsCqskma pk anmε ′

pn

which by a direct comparison with Eq 6.1-5 provides the result

Cijpn = aiq a jsa pk anmCqskm

′



(6.1-6)



that is, the transformation rule for a fourth-order Cartesian tensor.

In general, the Cijkm coefficients may depend upon temperature, but here

we assume adiabatic (no heat gain or loss) and isothermal (constant temperature) conditions. We also shall ignore strain-rate effects and consider the

components Cijkm to be at most a function of position. If the elastic coefficients

are constants, the material is said to be homogeneous. These constants are

those describing the elastic properties of the material. The constitutive law

given by Eq 6.1-3 is known as the generalized Hooke’s law.

For certain purposes it is convenient to write Hooke’s law using a single

subscript on the stress and strain components and double subscripts on the

elastic constants. To this end, we define



σ11 = σ1



σ23 = σ32 = σ4



σ22 = σ2



σ31 = σ13 = σ5



σ33 = σ3



σ12 = σ21 = σ6



(6.1-7a)



and



ε11 = ε1

ε22 = ε2



2ε31 = 2ε13 = ε5



ε33 = ε3



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2ε23 = 2ε32 = ε4



2ε12 = 2ε21 = ε6



(6.1-7b)



where the factor of two on the shear strain components is introduced in

keeping with Eq 4.7-14. From these definitions, Hooke’s law is now written



σα = Cαβ εβ or



␴ = Cε

ε



(6.1-8)



with Greek subscripts having a range of six. In matrix form Eq 6.1-8 appears

as

σ 1  C11

σ  C

 2   21

σ 3  C31

 =

σ 4  C41

σ 5  C51

  

σ 6  C61

  



C12

C22

C32

C42

C52



C13

C23

C33

C43

C53



C14

C24

C34

C44

C54



C15

C25

C35

C45

C55



C62



C63



C64



C65



C16  ε1 

C26  ε 2 

 

C36  ε 3 

 

C46  ε 4 

C56  ε 5 

 

C66  ε 6 

 



(6.1-9)



We point out that the array of the 36 constants Cαβ does not constitute a tensor.

In view of our assumption to neglect thermal effects at this point, the

energy balance Eq 5.7-13 is reduced to the form

˙

u=



1

σ D

ρ ij ij



(6.1-10a)



which for small-deformation theory, by Eq 4.10-18, becomes

˙

u=



1

˙

σ ε

ρ ij ij



(6.1-10b)



The internal energy u in these equations is purely mechanical and is called

the strain energy (per unit mass). Recall now that, by the continuity equation

in Lagrangian form, ρo = ρJ and also that to the first order of approximation



∂ ui 

∂ ui

J = det F = det δ iA +

 ≈ 1+ ∂ X

∂ XA



A



(6.1-11)



Therefore, from our assumption of small displacement gradients, namely

∂ui/∂XA << 1, we may take J ≈ 1 in the continuity equation to give ρ = ρ 0 , a

constant in Eqs 6.1-10.

For elastic behavior under the assumptions we have imposed, the strain

energy is a function of the strain components only, and we write

u = u(εij)

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(6.1-12)



so that

˙

u=



∂u

˙

ε

∂ε ij ij



(6.1-13)



and by a direct comparison with Eq 6.1-10b we obtain



∂u

1

σ =

ρ ij ∂ ε ij



(6.1-14)



The strain energy density, W (strain energy per unit volume) is defined by

W = ρ0u



(6.1-15)



and since ρ = ρ 0 , a constant, under the assumptions we have made, it follows

from Eq 6.1-14 that



σ ij = ρ



∂ u ∂W

=

∂ ε ij ∂ ε ij



(6.1-16)



It is worthwhile noting at this point that elastic behavior is sometimes

defined on the basis of the existence of a strain energy function from which

the stresses may be determined by the differentiation in Eq 6.1-16. A material

defined in this way is called a hyperelastic material. The stress is still a unique

function of strain so that this energy approach is compatible with our earlier

definition of elastic behavior. Thus, in keeping with our basic restriction to

infinitesimal deformations, we shall develop the linearized form of Eq 6.1-16.

Expanding W about the origin, we have



( )



W ε ij = W (0) +



∂ W ( 0)

1 ∂ 2 W ( 0)

ε ij +

ε ε +L

∂ ε ij

2 ∂ ε ij∂ε km ij km



(6.1-17)



and, from Eq 6.1-16,



σ ij =



∂ W ∂ W ( 0) ∂ 2 W ( 0)

=

+

ε +L

∂ ε ij

∂ ε ij

∂ ε ij∂ε km km



(6.1-18)



It is customary to assume that there are no residual stresses in the unstrained

state of the material so that σij = 0 when εij = 0. Thus, by retaining only the

linear term of the above expansion, we may express the linear elastic constitutive equation as



σ ij =

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∂ 2 W ( 0)

ε = Cijkmε km

∂ ε ij∂ ε km km



(6.1-19)



based on the strain energy function. This equation appears to be identical

to Eq 6.1-3, but there is one very important difference between the two —

not only do we have the symmetries expressed by Eq 6.1-4, but now we also

have

Cijkm = Ckmij



(6.1-20)



due to the fact that



∂ 2 W ( 0) ∂ 2 W ( 0)

=

∂ ε ij∂ ε km ∂ ε km∂ ε ij

Thus, the existence of a strain energy function reduces the number of distinct

components of Cijkm from 36 to 21. Further reductions for special types of

elastic behavior are obtained from material symmetry properties in the next

section. Note that by substituting Eq 6.1-19 into Eq 6.1-17 and assuming a

linear stress-strain relation, we may now write



( )



W ε ij =



1

1

C ε ε = σ ε

2 ijkm ij km 2 ij ij



(6.1-21a)



which in the notation of Eq 6.1-8 becomes

W (εα ) =



1

1

C ε ε = σ ε

2 αβ α β 2 α α



(6.1-21b)



and by the symmetry condition Cαβ = Cβα we have only 21 distinct constants

out of the 36 possible.



6.2



Hooke’s Law for Isotropic Media, Elastic Constants



If the behavior of a material is elastic under a given set of circumstances, it

is customarily spoken of as an elastic material when discussing that situation

even though under a different set of circumstances its behavior may not be

elastic. Furthermore, if a body’s elastic properties as described by the coefficients Cijkm are the same in every set of reference axes at any point for a

given situation, we call it an isotropic elastic material. For such materials, the

constitutive equation has only two elastic constants. A material that is not

isotropic is called anisotropic; we shall define some of these based upon the

degree of elastic symmetry each possesses.



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Isotropy requires the elastic tensor C of Eq 6.1-3 to be a fourth-order isotropic

tensor. In general, an isotropic tensor is defined as one whose components are

unchanged by any orthogonal transformation from one set of Cartesian axes

to another. Zero-order tensors of any order — and all zeroth-order tensors

(scalars) — are isotropic, but there are no first-order isotropic tensors (vectors).

The unit tensor I, having Kronecker deltas as components, and any scalar

multiple of I are the only second-order isotropic tensors (see Problem 6.5). The

only nontrivial third-order isotropic tensor is the permutation symbol. The most

general fourth-order isotropic tensor may be shown to have a form in terms of

Kronecker deltas which we now introduce as the prototype for C, namely,

Cijkm = λδijδkm + µ(δikδjm + δimδjk) + β(δikδjm – δimδjk)



(6.2-1)



where λ, µ, and β are scalars. But by Eq 6.1-4, Cijkm = Cjikm = Cijmk. This implies

that β must be zero for the stated symmetries since by interchanging i and

j in the expression



β(δikδjm – δimδjk) = β(δjkδim – δjmδik)

we see that β = –β and, consequently, β = 0. Therefore, inserting the reduced

Eq 6.2-1 into Eq 6.1-3, we have



σij = (λδijδkm + µδikδjm + µδimδjk)εkm

But by the substitution property of δij, this reduces to



σij = λδijεkk + 2µεij



(6.2-2)



which is Hooke’s law for isotropic elastic behavior. As mentioned earlier, we see

that for isotropic elastic behavior the 21 constants of the generalized law

have been reduced to two, λ and µ, known as the Lamé constants. Note that

for an isotropic elastic material Cijkl = Cklij; that is, an isotropic elastic material

is necessarily hyperelastic.



Example 6.2-1

Show that for an isotropic linear elastic solid the principal axes of the stress

and strain tensors coincide, and develop an expression for the relationship

among their principal values.



Solution



ˆ

Let n( ) (q = 1, 2, 3) be unit normals in the principal directions of εij, and

associated with these normals the corresponding principal values are ε(q)

where (q = 1, 2, 3). From Eq 6.2-2 we form the dot products

q



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