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Creep

This chapter describes the creep constitutive law adopted by FrontISTR. For details on selection and input specifications, see the functions section 03_material.

Creep Phenomenon and Additive Strain Decomposition

Time-dependent displacement under a constant-stress condition is a phenomenon called "creep".

The viscoelastic behavior described above can also be regarded as a type of linear creep phenomenon. Here, several nonlinear creep behaviors are described. A common approach is to formulate the constitutive relation by adding this behavior to the strain that occurs instantaneously, and to define the strain accumulated while a given constant load continues as the creep strain \(\varepsilon^c\). A constitutive relation including creep is usually expressed using the creep strain rate \(\dot{\varepsilon}^c\), defined as a function of stress and the total creep strain.

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

If the strain that occurs instantaneously is the elastic strain \(\varepsilon^e\), the total strain is expressed as the following sum including the creep strain.

\[ \varepsilon = \varepsilon^e + \varepsilon^c \]

where

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

(\(c\) is the elasticity tensor).

Norton Law

As a specific creep constitutive law, FrontISTR adopts the following Norton model. In this constitutive law, the equivalent creep strain rate \(\dot{\varepsilon}^{cr}\) is expressed as a function of the von Mises stress \(q\) and time \(t\) as follows.

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

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

Time Integration and Stress Update

As with plastic materials, a numerical time-integration method must be specified for a constitutive law that exhibits creep. The constitutive relation when creep is considered 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 \(\beta_{n+\theta}\) is

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

The creep strain increment \(\Delta \varepsilon^c\) is then defined by the following simplified nonlinear equation:

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

which is set to zero.

In the Newton-Raphson iteration, taking \(\sigma_{n+1} = \sigma_n\) as the initial stress and using the strain increment obtained from the finite element method, the iterative solution and increment are given by

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

where

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

with When iterating until the residual \(R\) becomes \(\mathbf{0}\), the stress \(\sigma_{n+1}\) and tangent modulus

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

are used.