N ote
A Numerical Method for Creep Deformation of Solids
This note is concerned with extension of the procedure described by Andrews and Hancock [1] to time-dependent problems in which motion is slow enough that inertial forces are negligible. Time dependence of the solution may arise from stress relaxation of the material and from time-dependent boundary conditions. The procedure involves advancing through time in finite steps and iterating to achieve stress equilibrium at each step in time. It is discussed in terms of a Maxwellian viscoelastic model that can be generalized to nonlinear cases.
Stress components cri s are decomposed into stress deviators and pressure:
Pressure is uniquely determined by volume, but stress deviator components obey a stress relaxation law, which in differential form is
dsis = 21~ dei~ --sis dtlr
where e~s is strain deviator, /z is shear modulus, and -r is relaxation time. If ~ is constant this is linear Maxwellian viscoelasticity. In nonlinear cases ~" is a function of stress. To have a properly covariant description, ~-should be expressed as a function of stress invariants.
The finite difference equation used to advance stress in one zone from time step n to step n + 1 is derived as follows. Let (0~j)~+l/2 be the strain deviator rate, found from velocities, for that zone for advancing from time step n to a particular iteration at time step n + 1. A finite difference analog of the differential equation above is (Si3 n+l = (Sij) n -t-2tZ(ii3 n+l/z At -1/2[(s.