Page 236 - Finite Element Modeling and Simulations with ANSYS Workbench

Three-Dimensional Elasticity 221

Or in a matrix form:

σ = E ε

7.2.2 Displacement
The displacement field can be described as

u xyz(, ,)  u 1 

u =  v xyz(, ,)    (7.4)

 =  u 2
 w xyz,)   u 
(,
   3 
7.2.3 Strain–Displacement Relation
Strain field is related to the displacement field as given below

∂u ∂v ∂w
ε x = , ε y = , ε z = ,
∂x ∂y ∂z
(7.5)
∂v ∂u ∂w ∂v ∂u ∂w
γ xy = + , γ yz = + , γ xz = +
∂x ∂y ∂y ∂z ∂ ∂z ∂x

These six equations can be written in the following index or tensor form:

1 ( ∂ i u ∂u )
ε ij = + j , ij = 12 3
,,
,
2 ∂ j x ∂ i x
Or simply,

1
ε ij = u ( ij u+ ji ) (Tensor notation)
,
,
2
7.2.4 Equilibrium Equations
The stresses and body force vector f at each point satisfy the following three equilibrium
equations for elastostatic problems:

∂σ x + ∂τ xy + ∂τ xz + x f = 0,
∂x ∂y ∂z

∂τ yx + ∂σ y + ∂τ yz + y f = 0, (7.6)
∂x ∂y ∂z
τ
∂τ zx + ∂τ zy + ∂σ z + z f = 0
∂x ∂y ∂z