Two-Layer Viscoplasticity

The material behavior is broken down into three parts: elastic, plastic, and viscous. Figure 1 shows a one-dimensional idealization of this material model, with the elastic-plastic and the elastic-viscous networks in parallel. The following subsections describe the elastic and the inelastic (plastic and viscous) behavior in detail.

Figure 1. One-dimensional idealization of the two-layer viscoplasticity model.

The calibration procedure is best explained in the context of the one-dimensional idealization of the material model. In the following discussion the viscous behavior is assumed to be governed by the Norton-Hoff rate law, which is given by

In the expression above the subscript V denotes quantities in the elastic-viscous network alone. This form of the rate law may be chosen, for example, by choosing a time-hardening power law for the viscous behavior and setting $m=0$. For this basic case there are six material parameters that need to be calibrated (Figure 1). These are the elastic properties of the two networks, $K_P$ and $K_V$; the initial yield stress ${\sigma }_y$; the hardening $H^{{}^{\prime }}$; and the Norton-Hoff rate parameters, A and n.

The experiment that needs to be performed is uniaxial tension under different constant strain rates. A static (effectively zero strain rate) uniaxial tension test determines the long-term modulus, $K_P$; the initial yield stress, ${\sigma }_y$; and the hardening, $H^{{}^{\prime }}$. The hardening is assumed to be linear for illustration purposes. The material model is not limited to linear hardening, and any general hardening behavior can be defined for the plasticity model. The instantaneous elastic modulus, $K=K_P+K_V$, can be measured by measuring the initial elastic response of the material under nonzero, relatively high, strain rates. Several such measurements at different applied strain rates can be compared until the instantaneous moduli does not change with a change in the applied strain rate. The difference between K and $K_P$ determines $K_V$.

To calibrate the parameters A and n, it is useful to recognize that the long-term (steady-state) behavior of the elastic-viscous network under a constant applied strain rate, ${\dot{\epsilon }}_o$, is a constant stress of magnitude ${\sigma }_V=A^{-\frac{1}{n}}\dot{\epsilon }_o{}^{\frac{1}{n}}$. Under the assumption that the hardening modulus is negligible compared to the elastic modulus ($K_P\gg H^{{}^{\prime }}$), the steady-state response of the overall material is given by

where $\sigma$ is the total stress for a given total strain $\epsilon$. To determine whether steady state has been reached, one can plot the quantity $\overline{\sigma }=\sigma -{\sigma }_y-H^{{}^{\prime }}\epsilon$ as a function of $\epsilon$ and note when it becomes a constant. The constant value of $\overline{\sigma }$ is equal to $A^{-\frac{1}{n}}\dot{\epsilon }_o{}^{\frac{1}{n}}$. By performing several tests at different values of the constant applied strain rate ${\dot{\epsilon }}_o$, it is possible to determine the constants A and n.

The material is active during all stress/displacement procedure types. In a static analysis step where the long-term response is requested (see Static Stress Analysis), only the elastic-plastic network will be active; the elastic-viscous network will not contribute in any manner. In particular, the stress in the viscous network will be zero during a long-term static response. If the creep effects are removed in a coupled temperature-displacement procedure or a soils consolidation procedure, the response of the elastic-viscous network will be assumed to be elastic only. During a linear perturbation step, only the elastic response of the networks is considered.

Some stress/displacement procedure types (coupled temperature-displacement, soils consolidation, and quasi-static) allow user control of the time integration accuracy of the viscous constitutive equations through a user-specified error tolerance. In other procedure types where no such direct control is currently available (static, dynamic), you must choose appropriate time increments. These time increments must be small compared to the typical relaxation time of the material.