ABSTRACT
Within an element, the finite-element method makes use of interpolation models for the displacement vector u(X, t) and temperature T(X, t) (and pressure p=−trace(τ)/3 in incompressible or near-incompressible materials):
u(X,t)=NT(X)γ(t), T(X,t)−T0=vT(X)θ(t), p=ξT(X)ψ(t), (18.1)
in which T0 is the temperature in the reference configuration, assumed constant. Here, N, v, and ξ are shape functions and γ, θ, and ψ are vectors of nodal values. Application of the strain-displacement relations and their thermal analogs furnishes
(18.2)
in which U is a 9×9 universal permutation tensor such that VEC(AT)=UVEC(A), and e=VEC(E) is the Lagrangian strain vector. Also, is the gradient operator referred to the deformed configuration. The matrix β and the vector βT are typically expressed in terms of isoparametric coordinates.
