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4 Stress–Strain Relations
4.2 Linear Elasticity—Generalized Hooke’s Law
There is a unique relationship between stress and strain defined by Hooke’s law,
which is independent of time and loading history. The law assumes that all the strain
changes resulting from stress changes are instantaneous and the system is completely
reversible and all the input energy is recovered in unloading.
In case of uniaxial loading, stress is related to strain as
σ x = Eε x
(4.1)
where E is known as “Young’s modulus” or “Modulus of Elasticity”.
Expression (4.1) is applicable within the linear elastic range and is called Hooke’s
law.
In general, each strain is dependent on each stress. For example, the strain ε x
written as a function of each stress is
ε x = C 11 σ x + C 12 σ y + C 13 σ z + C 14 τ xy + C 15 τ yz
+ C 16 τ xx + C 17 τ xz + C 18 τ zy + C 19 τ yx
(4.2)
Similarly, stresses can be expressed in terms of strains stating that at each point
in a material, each stress component is linearly related to all the strain components.
This is known as “Generalized Hooke’s Law”.
Hence,
σ x = D 11 ε x + D 12 ε y + D 13 ε z + D 14 γ xy + D 15 γ yz
+ D 16 γ zx + D 17 γ xz + D 18 γ zy + D 19 γ yx
(4.3)
For the most general case of three-dimensional state of stress, Eq. (4.3) can be
written as
σ i j
=
D i jkl
{ε kl }
(4.4)
where
D i jkl
Elasticity matrix
σ i j
Stress components
{ ε kl }
Strain components
Since both stress σ i j and strain ε i j are second-order tensors, it follows that D i jkl
is a fourth-order tensor, which consists of 3
4
= 81 material constants if symmetry is
not assumed. Therefore in matrix notation, the stress–strain relations would be