Page 181 - The Combined Finite-Discrete Element Method

164 DEFORMABILITY OF DISCRETE ELEMENTS

This is represented as a shape changing part:

1 2 1
˘
E d = (V /(| det F|) − I) = (B/(| det F|) − I) (4.178)
2 2
     
∂x c ∂x c ∂x c ∂y c
1
1   ∂x i ∂y i   ∂x i ∂x i  1 0 
     − 
2  (| det F|)  ∂y c ∂y c   ∂x c ∂y c  0 1 
∂x i ∂y i ∂y i ∂y i
and volume changing part

1 T 1 1/2 2 1 10
˘
E s = (V s V − I) = I (| det F|) − I = (| det F|− 1) (4.179)
s
2 2 2 01
From the matrix of the strain tensor, the matrix of the Cauchy stress tensor is obtained
using the constitutive law. For instance, for plane stress homogeneous isotropic material,
using the plane stress equations for Hooks law, is given by

E
σ 1 = (ε 1 + vε 2 ) (4.180)
2
(1 − v )
E
σ 2 = (vε 1 + ε 2 )
2
(1 − v )
The same constitutive law can be written as follows:

E 1 E 1
σ 1 = ε 1 − (ε 1 + ε 2 ) + (ε 1 + ε 2 ) (4.181)
(1 + v) 2 (1 − v) 2
E 1 E 1
σ 2 = ε 2 − (ε 1 + ε 2 ) + (ε 1 + ε 2 )
(1 + v) 2 (1 − v) 2

where therefore ‘volume’ change

1

(ε 1 + ε 2 ) (4.182)
2
is separated from the change in the shape of the material element. By analogy and using
the Green–St. Venant strain tensor, the following constitutive law is arrived at:

E E
˜
˜
T = E d + E s + 2µD (4.183)
(1 + v) (1 − v)
where the last term is due to the rate of deformation.