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Creep

This chapter describes FrontISTR's creep constitutive law. Selection and input details are in the Material Data chapter.

Creep Phenomenon and Strain Decomposition

Time-dependent deformation under constant stress is called creep. The viscoelastic behavior described in the previous chapter can be regarded as a linear form of creep; here we describe nonlinear creep.

The constitutive equation is typically formulated by adding a creep strain \(\varepsilon^c\) to the instantaneous strain. The creep strain rate \(\dot{\varepsilon}^c\) is given as a function of stress and total creep strain:

\[ \dot{\varepsilon}^c \equiv \frac{\partial \varepsilon^c}{\partial t} = \beta(\sigma, \varepsilon^c). \]

If the instantaneous strain is the elastic strain \(\varepsilon^e\), the total strain is

\[ \varepsilon = \varepsilon^e + \varepsilon^c, \qquad \varepsilon^e = c^{-1} : \sigma, \]

where \(c\) is the elastic stiffness tensor.

Time Integration and Stress Update

As in elastoplasticity, a numerical time-integration scheme is required. The creep constitutive equation is

\[ \sigma_{n+1} = c : (\varepsilon_{n+1} - \varepsilon_{n+1}^c), \]
\[ \varepsilon_{n+1}^c = \varepsilon_n^c + \Delta t \, \beta_{n+\theta}, \]

where

\[ \beta_{n+\theta} = (1 - \theta) \beta_n + \theta \beta_{n+1}. \]

The simplified residual is

\[ R_{n+1} = \varepsilon_{n+1} - c^{-1} : \sigma_{n+1} - \varepsilon_n^c - \Delta t \, \beta_{n+\theta} = \mathbf{0}. \]

In the Newton-Raphson iteration, with \(\sigma_{n+1} = \sigma_n\) as the initial guess and the FE strain increment given,

\[ R_{n+1}^{(k+1)} = \mathbf{0} = R_{n+1}^{(k)} - (c^{-1} + \Delta t \, c_{n+1}^c) \, d\sigma_{n+1}^{(k)}, \]

where

\[ c_{n+1}^c = \left.\frac{\partial \beta}{\partial \sigma}\right|_{n+\theta} = \theta \left.\frac{\partial \beta}{\partial \sigma}\right|_{n+1}. \]

Iterating until \(R = \mathbf{0}\), the stress \(\sigma_{n+1}\) and the consistent tangent

\[ c_{n+1}^* = (c^{-1} + \Delta t \, c_{n+1}^c)^{-1} \]

are obtained.

Norton Law

For the specific form of \(\beta\), FrontISTR adopts the Norton model in which the equivalent creep strain rate \(\dot{\varepsilon}^{cr}\) is a function of the Mises stress \(q\) and time \(t\):

\[ \dot{\varepsilon}^{cr} = A q^n t^m, \]

where \(A\), \(m\), and \(n\) are material constants.