Rate-dependent metal plasticity (creep)

ProductsAbaqus/Standard

High-temperature “creep” in structures is one important class of examples of the application of such a material model. Because such problems generally involve relatively small amounts of inelastic straining (otherwise the structure is not a suitable design), the explicit, forward Euler method is often satisfactory as an integrator for the flow rule. This method is only conditionally stable, but the stability limit is usually sufficiently large compared to the time history of interest in such cases that the explicit method is very economical. Cormeau (1975) has developed formulae for the stability limit for most common cases of stress induced creep, and these results are used to monitor stability. For this explicit approach the integration is trivial. Combining the integrated flow rule

Δ ε^{p l} = {Δ t \frac{\partial G_{c r}}{\partial σ} |}_{t}

with the integrated strain rate decomposition and the (linear) elasticity gives

\begin{matrix} (1) & {σ |}_{t + Δ t} = D^{e l} : (ε_{t + Δ t} - {ε^{p l} |}_{t} - {Δ t \frac{\partial G_{c r}}{\partial σ} |}_{t}) . \end{matrix}

All of the terms on the right-hand side of this set of equations are known when the constitutive integration is done, so these equations define $σ_{t + Δ t}$ explicitly.

There also exist many problems involving rate-dependent plastic response in which the characteristic relaxation times for the material under the stress states to which it is subjected are very short compared to the time period of interest in the analysis, so the conditional stability of the explicit operator will only allow very short time increments. For such cases it can be more economical to use the backward Euler method because of its unconditional stability. Abaqus always uses the implicit method for high strain rate applications to avoid time increment restrictions being introduced by considerations of stability in the integration of the constitutive model. Abaqus will also use the implicit method in all geometrically nonlinear problems and in problems for which rate-independent plasticity is active simultaneously.

The backward Euler method is implicit; and because the plastic strain rate is usually a strong function of stress, some care must be taken to develop an effective algorithm to solve the nonlinear algebraic equations that result from the use of this operator. The problem has been posed formally in Integration of plasticity models. The main difficulty is to obtain a reasonable starting guess for $Δ ε^{p l}$ . For this we proceed as follows.

For simplicity, we consider rate-dependent behavior only and the particular form of flow rule defined by

{\dot{ε}}^{p l} = {\dot{\bar{ε}}}^{s w} \frac{1}{3} I + {\dot{\bar{ε}}}^{c r} n,

where ${\dot{\bar{ε}}}^{s w}$ is the “equivalent swelling strain rate,” ${\dot{\bar{ε}}}^{c r}$ is the “equivalent creep strain rate,” and $n$ is the gradient of the deviatoric stress potential,

n = \frac{\partial \tilde{q}}{\partial σ},

where $\tilde{q}$ is the Mises or Hill stress potential (defined in Stress potentials for anisotropic metal plasticity).

The “equivalent strain rates” are part of the stress potential for the plastic response and, therefore, are assumed to have evolution laws of the form

{\dot{\bar{ε}}}^{s w} = h^{s} (p, \tilde{q}, {\bar{ε}}^{s w}, {\bar{ε}}^{c r}, θ, \dots)

and

{\dot{\bar{ε}}}^{c r} = h^{c} (p, \tilde{q}, {\bar{ε}}^{s w}, {\bar{ε}}^{c r}, θ, \dots) .

Backward Euler integration of the flow equation gives

Δ ε^{p l} = Δ {\bar{ε}}^{s w} \frac{1}{3} I + Δ {\bar{ε}}^{c r} n,

where $n$ is understood to be evaluated at time $t + Δ t$ , and

\begin{matrix} (2) & Δ {\bar{ε}}^{s w} = Δ t h^{s} (p, \tilde{q}, {\bar{ε}}^{s w}, {\bar{ε}}^{c r}, θ, \dots) \end{matrix}

and

\begin{matrix} (3) & Δ {\bar{ε}}^{c r} = Δ t h^{c} (p, \tilde{q}, {\bar{ε}}^{s w}, {\bar{ε}}^{c r}, θ, \dots) . \end{matrix}

$Δ {\bar{ε}}^{c r}$ and $Δ {\bar{ε}}^{s w}$ are usually defined in user subroutine CREEP.

The solution to the algebraic problem is obtained by first finding reasonable initial guesses for $Δ {\bar{ε}}^{c r}$ and $Δ {\bar{ε}}^{s w}$ and then solving the full problem.

The Mises and Hill equivalent stress definitions ( $\tilde{q}$ ) both have the property that

n : σ = \tilde{q} .

We also have the simple relationship

I : σ = - 3 p .

The initial estimates for $Δ {\bar{ε}}^{c r}$ and $Δ {\bar{ε}}^{s w}$ are obtained by projecting the problem onto $n^{e l}$ and $I$ , where $n^{e l}$ is $\partial \tilde{q} / \partial σ$ defined at $σ^{e l}$ , the stress state that would arise at the end of the increment if there were no inelastic deformation during the increment. The projections are

\begin{matrix} (4) & \tilde{q} = {\tilde{q}}^{e l} - G Δ {\bar{ε}}^{c r} - B Δ {\bar{ε}}^{s w}, \end{matrix}

and

\begin{matrix} (5) & p = p^{e l} - B Δ {\bar{ε}}^{c r} - K Δ {\bar{ε}}^{s w}, \end{matrix}

where

{\tilde{q}}^{e l} = n : σ^{e l},

p^{e l} = - \frac{1}{3} I : σ^{e l},

G = n : D^{e l} : n,

K = \frac{1}{9} I : D^{e l} : I,

and

B = \frac{1}{3} n^{e l} : D^{e l} : I .

Equation 2 to Equation 5 are a set of nonlinear equations that can be solved for $Δ {\bar{ε}}^{c r}$ and $Δ {\bar{ε}}^{s w}$ . We solve these equations by Newton's method and then use this solution as the starting estimate for solving the complete problem. When the Mises stress potential is used and the problem is not plane stress, this starting estimate is the solution to the complete problem because the Mises stress potential is a circle in the deviatoric plane.

Rate-dependent metal plasticity (creep)

User subroutine CREEP