For solid mechanics problems, the constitutive law form (Eqs. 11.3 and 11.4) is an incremental one and differs from the ones for diffusion problems (Eq. 11.7). The knowledge of the stress tensor at any time implies that a time-integrated constitutive law is required. The stress tensor is a state variable that is stored and transmitted from step to step based on its final/initial value, and this value plays a key role in the numerical algorithm.
Then, in nearly all finite element codes devoted to modelling, equilibrium is expressed at the end of the time steps, following a fully implicit scheme (t = 1), and using the end of step stress tensor value.
However, integrating the stress history with enough accuracy is crucial for the numerical process stability and global accuracy. Integration of the first order differential equation (Eq. 11.4):
tB
ctb = cta + J Eeps dt (11.31)
tA
can be based on similar concepts as the one described in the preceding paragraph (the superscripts of ct here indicating the time at which ct is evaluated). Various time schemes based on different т values may be used for which similar reflections on stability and accuracy can be made.
When performing large time steps, obtaining enough accuracy can require the use of sub-stepping: within each global time step (as regulated by the global numerical convergence and accuracy problem) the stress integration is performed at each finite element integration point after division of the step into a number a sub-steps allowing higher accuracy and stability.
