Page 32 - Using ANSYS for Finite Element Analysis A Tutorial for Engineers

IntroductIon to FInIte element AnAlysIs • 19
1.2.2.2 strain–displacement relationship

Strains can be calculated by differentiating displacement functions.
Differentiation of a function is possible only if it is continuous. There-
fore, the strain–displacement relations are also known as compatibility
equations and are given as follows:

u ∂ u ∂ v ∂
e = x ∂ g = y ∂ + x ∂
zy
x
v ∂ ∂ u ∂ w
e = y ∂ g = z ∂ + x ∂
y
xz
∂ w ∂ w v ∂
e = z ∂ g = y ∂ + ∂ ∂z
z
yz
In matrix form:

 ∂ 
 x ∂ 0 0 
 
 ∂ 
e  x   0 0 
   y ∂ 
 e y   ∂ 
 e   0 0  u 
 
 z  =  z ∂  v
g  xy   ∂ ∂   
   y ∂ x ∂ 0   w
  
 g xz   
 g   ∂ ∂ ∂ 
 yz   ∂z 0 ∂x 
 
 0 ∂ ∂ 
  ∂z ∂  y 

1.2.2.3 stress-strain relationships

For linear elastic materials, the stress–strain relations come from the gen-
eralized Hooke’s law. For isotropic materials, the two material properties
are Young’s modulus (or modulus of elasticity) E and Poisson’s ratio ν.
For a three-dimensional case, the state of stress at any point in relation to
the state of strain as follows: