Page 170 - The Combined Finite-Discrete Element Method

CONSTITUTIVE LAW 153

1 1
σ 1 = E(ε 3 − ε s ) + Eε s (4.112)
(1 + ν) (1 − 2ν)
Unlike Lam´ e constants, the above formulation completely separates volumetric strains in a
sense that the first part does not produce any change in the volume of the material element,
while the second part does not produce any change in the shape of the material element.
To write the above constitutive law in terms of a Green–St. Venants strain tensor,
homogeneous deformation is expressed as a composition of rotation g, followed by the
shape changing stretch s d , followed by the volume changing stretch s s :

f(p) = s s ◦s d ◦g (4.113)

The deformation gradient for this deformation is given by
F = V s V d R where R =∇g; V d =∇s d ; V s =∇s s ; (4.114)

It is worth mentioning that by definition,
| det F|= det V s ; and det V d = 1 (4.115)

The left Green–St. Venant strain tensor is therefore given by
1 T 1 T
˜
E = (FF − I) = [(V s V d R)(V s V d R) − I] (4.116)
2 2
1 T T 1 T T T
= [(V s V d R)R (V s V d ) − I] = [(V s V d RR V V − I]
s
d
2 2
1 T T 1 T 2/3
= [(V s V d V V − I] = [(V d V (| det F|) − I]
s
d
d
2 2
The last term in the above equation comes from the fact that the volumetric stretch carries
all volume change with it, and can therefore be written as three identical successive
stretches in any three mutually orthogonal directions:
#
3
V s = I | det F| (4.117)
Thus
T
V s V = I (| det F|) 2/3 (4.118)
s
The Green–St. Venant strain tensor due to the shape changing stretch is therefore given by

1 T 1
FF T
˜
E d = (V d V − I) = − I (4.119)
d
2 2 (| det F|) 2/3
while the Green–St. Venant strain tensor due to volume changing stretch is given by
1 T 1 2/3
(| det F|) 2/3 − 1
˜
E s = (V s V − I) = (I(| det F|) − I) = I (4.120)
s
2 2 2