On the one hand, solid mechanics can be modelled on the following basis. The equilibrium equation is:
дг <jjj + Pj = 0 (11.1)
Where Pj is a member of P, the vector of volume forces, ji;- is a member of j, the Cauchy stress tensor, and д represents the spatial partial derivative operator:
di = (11.2)
dXi
The stress tensor is obtained by the time integration of an (elastic, elasto-plastic or elasto-visco-plastic) constitutive equation (see Chapter 9, Section 9.4.2; Laloui, 2001; Coussy & Ulm, 2001):
Jij = fn(j, e, Z) (11.3)
where (jij is the stress rate, e is the strain rate and Z is a set of history parameters (state variables, like e. g. the preconsolidation stress). In the most classical case of elasto-plasticity, this equation reduces to:
jч = hi
where Eejkl is a member of the elasto-plastic constitutive (stress-strain) tensor, Eep. Most constitutive relationships for geomaterials are non-linear ones and not as previously introduced in Eq. 9.1 in a linear version.
When modelling a solid mechanics problem with the finite element method, the most commonly used formulation is based on displacements that make up the vector l or on actual coordinates that make up the vector x. If one considers only small strains and small displacements, the strain rate reduces to the well-known Cauchy’s strain rate:
Sij — 2 (djlj + djli) (11.5)
The time dimension is not addressed for solid mechanics problems, except when a viscous term is considered in the constitutive model. Generally, the time that appears in the time derivatives inEqs. 11.3, 11.4 and 11.5 is only aformal one.
