304 16 Constitutive Relations
16.2 Elasticity
A property typical for a solid body is its elastic response to a small deformation. Of
course, ‘small’ is relative, and really small for brittle substances. On the other hand,
an elastic stick can be bent to a considerable amount. The elastic behavior is described
by a linear constitutive relation between the stress tensor and the deformation tensor.
The fourth rank elasticity tensor characterizes the elastic properties of specific solids.
16.2.1 Elastic Deformation of a Solid, Stress Tensor.
Letrbe the position vector to a volume element within a solid body. When this
solid is subjected to a small deformation, this volume element is displaced to the
positionr′=r+u(r). Now consider two neighboring points which are separated
bydrin the undeformed state. After the deformation, the difference vector between
these two points isdr′=dr+du. The difference in the displacement is duμ=
drν∇νuμ+...where the higher order terms, indicated by the dots, can be disregarded
for neighboring points. The distance squared, between these two points is dr^2 =
drμdrμin the undeformed state and
(dr′)^2 =drμ′drμ′=(drμ+drν∇νuμ)(drμ+drκ∇κuμ),in the deformed state. Thus one has
(dr′)^2 =(δμν+ 2 uνμ)drμdrν, uνμ=1
2
[
∇νuμ+∇μuν+(∇μuλ)(∇νuλ)]
,
(16.11)
whereuνμ=uμνis thedeformation tensor. Like any symmetric tensor,uνμcan be
diagonalized. With the principal values of the deformation tensor denoted byu(i),
i= 1 , 2 ,3, relation (16.11) is equivalent to
(dr′)^2 =(
1 + 2 u(^1 ))
dr 12 +(
1 + 2 u(^2 ))
dr 22 +(
1 + 2 u(^3 ))
dr^23 ,in the principal axes system. The relative deformation-induced change of the length,
along the principal directioni,is((dr)′i−dri)/dri=
√
1 + 2 u(i)− 1 ≈u(i). Accord-
ingly, the volume dV′in the deformed state is related to the original volume dVby
dV′=√
1 + 2 u(^1 )√
1 + 2 u(^2 )√
1 + 2 u(^3 )dV≈(
1 +u(^1 )+u(^2 )+u(^3 ))
dV.Thus the relative changeδV/dVof the volume
δV/dV=(dV′−dV)/dV=u(^1 )+u(^2 )+u(^3 )=uλλ, (16.12)