Plasticity and creep

Everything stated for a linear material applies without exception to the description of the properties of a nonlinear material. In addition, four further material quantities must be specified, whose meaning is explained in detail in the Reference Manual:

the yield stress $\sigma_Y~[\text{Pa}]$,
the kinematic hardening component $Q_Y~[\text{Pa}]$,
the effective creep strain rate ${\dot\varepsilon}_c~[1/\text{h}]$,
the dilatancy factor $\Phi~[1]$.

Note
For computations of problems where this generalized plasticity model is insufficient, specialized nonlinear materials can be used, which are described in the Reference Manual.

The complete form of the MP set for a nonlinear material is:

MP set number T 1 V $E$ $\alpha$ $\nu$ $\rho$ $\sigma_Y$ $Q_Y$ ${\dot\varepsilon}_c$ $\Phi$

In nonlinear problems, material properties can depend on variables using any of the available functional dependencies, see the Reference Manual. The following variables are most commonly used, see the Reference Manual:

the mean stress $\sigma_m$,
the Lode parameter $\mu$,
the temperature $T$,
the effective plastic strain $\varepsilon_p$,
the effective stress $\sigma_e$,
the effective creep strain $\varepsilon_c$.

Note
For an elastoplastic computation without creep, it is not necessary to specify ${\dot\varepsilon}_c$ (i.e., the MP set may contain only six quantities). However, if the plasticity model used requires the dilatancy factor $\Phi$, 0 must be formally written in the 7th position.

Note
For a standalone creep computation without plasticity, $\mathtt{KMOD}=0$ is specified in the name.iP file.

Note
The dilatancy factor $\Phi$ is specified only when a non-associated flow rule is used, that is, only in quite exceptional cases.

Example

Let us consider a material with elastic properties $E=2\cdot10^5~\text{MPa}$ and $\nu=0.3$. Let the yield stress be $\sigma_{Y0}=320~\text{MPa}$ and the tangent hardening modulus $E_T=2\cdot10^3~\text{MPa}$. We will describe the material using a von Mises model with linear kinematic hardening.

According to the Reference Manual, the dependence $\sigma_Y(\varepsilon_p)$ must be specified, which in our case will be linear, $\sigma_Y(\varepsilon_p)=\sigma_{Y0}+E_p\varepsilon_p$. We first calculate the plastic modulus $$E_p = \frac{EE_T}{E-E_T}=2.02\cdot10^3~\text{MPa}.$$

The kinematic hardening component $Q_Y$ must be such that the elastic range $\sigma_Y-Q_Y=\sigma_{Y0}$ remains constant. Therefore $Q_Y(\varepsilon_p)=E_p\varepsilon_p$. We describe the dependences $\sigma_Y(\varepsilon_p)$ and $Q_Y(\varepsilon_p)$ using a polynomial (see the Reference Manual), where the effective plastic strain has identification number 7 (see the Reference Manual).

The name.i2 file will contain:

⋮
MP 1 T -1 I 7 V 2.0e11 0 V 0 0 V 0.3 0 V 0 0
              V 3.2e8 2.02e9 V 0 2.02e9
⋮
AS 1 /M 1 /…
⋮

Example

We apply the generalized plasticity model so that it corresponds to the isotropic Drucker-Prager model.

The Drucker-Prager yield condition has the form $$\sigma_e = Y-3\beta\sigma_m,$$ where $Y$ is a material parameter and $\beta$ is a material constant. Let $Y_t$ denote the yield stress in tension and $Y_c$ the yield stress in compression. Then $$\beta = \frac{Y_c-Y_t}{Y_c+Y_t}, \quad Y_c>Y_t, \quad Y = \frac{2Y_cY_t}{Y_c+Y_t}.$$

Note that, in general, $Y = Y(\varepsilon_p,T)$. In that case, $$Y(\varepsilon_p,T) = Y_c(\varepsilon_p,T)\frac{2Y_t}{Y_c+Y_t}$$ is specified, whereby the model faithfully reproduces the $\sigma$–$\varepsilon$ curve in compression (but not in tension).

According to the Reference Manual, the dependence $\sigma_Y(\sigma_m,\varepsilon_p,T) = Y(\varepsilon_p,T) - 3\beta\sigma_m$ must be specified. The identification numbers for the independent variables $\sigma_m$, $\varepsilon_p$, and $T$ are 1, 7, and 5 (see the Reference Manual). In our case, the material hardens isotropically, so $Q_Y=0$. Within the MP set, the quantities $E$, $\alpha$, $\nu$, $\rho$, and $\sigma_Y$ are described as functions of variable numbers 1, 7, and 5.