Page 258 - Intro to Tensor Calculus

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Alternative Approach to Constitutive Equations

The constitutive equation defined by Hooke’s generalized law for isotropic materials can be approached
from another point of view. Consider the generalized Hooke’s law

σ ij = c ijkl e kl , i,j,k,l =1, 2, 3.

If we transform to a barred system of coordinates, we will have the new Hooke’s law

σ ij = c ijkl e kl , i,j,k,l =1, 2, 3.

For an isotropic material we require that
c ijkl = c ijkl .

Tensors whose components are the same in all coordinate systems are called isotropic tensors. We have
previously shown in Exercise 1.3, problem 18, that

c pqrs = λδ pq δ rs + µ(δ pr δ qs + δ ps δ qr )+ κ(δ pr δ qs − δ ps δ qr )

is an isotropic tensor when we consider affine type transformations. If we further require the symmetry
conditions found in equations (2.4.3) be satisfied, we find that κ = 0 and consequently the generalized
Hooke’s law must have the form

σ pq = c pqrs e rs =[λδ pq δ rs + µ(δ pr δ qs + δ ps δ qr )] e rs
σ pq = λδ pq e rr + µ(e pq + e qp ) (2.4.23)
or σ pq =2µe pq + λe rr δ pq ,

where e rr = e 11 + e 22 + e 33 = Θ is the dilatation. The constants λ and µ are called Lame’s constants.
Comparing the equation (2.4.22) with equation (2.4.23) we find that the constants λ and µ satisfy the
relations
E νE
µ = λ = . (2.4.24)
2(1 + ν) (1 + ν)(1 − 2ν)
In addition to the constants E, ν, µ, λ, it is sometimes convenient to introduce the constant k, called the bulk
modulus of elasticity, (Exercise 2.3, problem 23), defined by

E
k = . (2.4.25)
3(1 − 2ν)
The stress-strain constitutive equation (2.4.23) was derived using Cartesian tensors. To generalize the
i
equation (2.4.23) we consider a transformation from a Cartesian coordinate system y ,i =1, 2, 3 to a general
i
coordinate system x ,i =1, 2, 3. We employ the relations
i
∂y m ∂y m ij ∂x ∂x j
g = , g =
ij i j m m
∂x ∂x ∂y ∂y
and
i
i
i
∂y ∂y j ∂y ∂y j ∂x ∂x j
σ mn = σ ij m n , e mn = e ij m n , or e rq = e ij r q
∂x ∂x ∂x ∂x ∂y ∂y