Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 20

the temperature. In an isotropic body, the thermal expansion is the same in all directions,
so that an unrestrained three-dimensional element experiences a uniform expansion but
no angular distortions. Thus, in the unrestrained isotropic body, the temperature change
leads to normal thermal strains while not producing shear strains, i.e.,

x y z yz xz xy

= = = α T, γ = γ = γ = 0 (1.45)

In Eqs. (1.43) and (1.44), these thermal strains lead to the terms

1
1
1
0
0
0

1
1
1
0
0
0

E α T

0 E 0

= α T − =− (1.46)

1 − 2 ν

1.3.3 Anisotropic Material
For the most general case of an elastic anisotropic body, all components in the matrix of
the constitutive law are non-zero, but still the symmetry is preserved [Timoshenko and
Goodier, 1970; Leipholz, 1968, p. 96]:

σ

a 11 a 12 a 13 a 14 a 15 a 16

x x

σ

a 22 a 23 a 24 a 25 a 26

y y

a 33 a 34 a 35 a 36 σ

z z

=

(1.47)

γ a 44 a 45 a 46 τ

xy
γ xz
γ yz

xy
τ xz
τ yz

Symmetric a 55 a 56

a 66

E − 1 = σ

σ

c 11 c 12 c 13 c 1 4 c 15 c 16

x x

σ c 22 c 23 c 2 4 c 25 c 26

y
σ z
τ xy
τ xz
τ yz

y
z
γ xy
γ xz
γ yz

c 33 c 34 c 35 c 36

= (1.48)

c 44 c 45 c 46

Symmetric c 55 c 56

c 66

E
This is, however, an extreme case for which it is very difficult to identify all of the coefficients.
There are many materials with a simpler structure, such as those found in rolled sheet
metals, wood, and honeycomb fabrications. If the properties of a material differ only in three
orthogonal directions it is called orthotropic , and nine independent parameters suffice to
describe the material. In Eqs. (1.47) and (1.48), the barred quantities are zero for orthotropic
(or isotropic) materials.

σ =

1.4 Equations of Equilibrium

The description of equilibrium at any point in a body is characterized by (local) differential
equations involving stresses and internal (volume or body) forces such as those generated by
gravity, acceleration, or magnetic fields. They can be derived for the two-dimensional case
by considering the configuration of Fig. 1.9. Over the infinitesimal distance dx , the change