Metal Foams: A Design Guide

and its deviatoric (i.e. shear) components Sij, giving
A constitutive model for metal foams 81
ij D Sij C mυij ⊲7.3⊳
where υij is the Kronecker delta symbol, and takes the value υij D 1ifi D j,
and υij D 0 otherwise. The von Mises effective stress, e, then becomes
2 3
e D 2SijSij ⊲7.4⊳
In similar fashion, the strain rate Pε is a symmetric 3 ð 3 tensor, with three
direct components ( Pε11, Pε22, Pε33) and three shear components ( Pε12, Pε23, Pε31).
The volumetric strain rate is defined by
Pεm Pε11 CPε22 CPε33 DPεkk ⊲7.5⊳
and the strain rate can be decomposed into its volumetric part Pεm and deviatoric
part Pε 0 ij according to
Pεij DPε 0 ij
C 1
3 υij Pεm
⊲7.6a⊳
The strain rate Pε can be written as the sum of an elastic strain rate Pε E and
a plastic strain rate Pε P . In an analogous manner to equation (7.6a), the plastic
strain rate can be decomposed into an deviatoric rate Pε P0 and a mean rate
, such that
Pε P m
Pε P kk
Pε P ij DPεP0 1
ij C 3υij Pε P m
⊲7.6b⊳
Now, for fully dense metallic solids, plastic flow occurs by slip with no
change of volume, and so the volumetric plastic strain rate Pε P m PεP kk equals
zero. Then, a useful scalar measure of the degree of plastic straining is the
effective strain rate Pεe, definedby
Pε 2 e
2
3 PεP0 ij PεP0 ij
⊲7.7⊳
where the factor of 2
3 has been introduced so that Pεe equals the uniaxial plastic
strain rate in a tension (or compression) test on an incompressible solid.
In conventional Prandtl–Reuss J2 flow theory, the yield criterion is written
8 e Y � 0 ⊲7.8⊳
and the plastic strain rate Pε P ij is normal to the yield surface 8 in stress space,
and is given by
Pε P ij DPεe
∂8
⊲7.9⊳
∂ ij