A rate-dependent stochastic damage–plasticity model for quasi-brittle materials

Abstract

In this work, a rate-dependent model for the simulation of quasi-brittle materials experiencing damage and randomness is proposed. The bi-scalar plastic damage model is developed as the theoretical framework with the damage and the plasticity opening for further developments. The governing physical reason of the material rate-dependency under relatively low strain rates, which is defined as the Strain Delay Effect, is modeled by a differential system. Then the description of damage is established by further implementing the rate-dependent differential system into the random damage evolution. To reproduce the evolution of plasticity under a variety of stress conditions, a multi-variable phenomenological plastic model is proposed and the description of plasticity is then formulated. An explicit integration algorithm is developed to implement the proposed model in the structural simulation. The model results are validated by a series of numerical tests that cover a wide variety of stress conditions and loading rates. The proposed model and algorithm offer a package solution for the nonlinear dynamic structural simulations.

Appendix: Derivations of Stefan effect

In a viscous fluid, the stress $\sigma _{ij}$ is often decomposed into the pressure $p$ and the shear stress $\tau _{ij}$ as follows

$$\begin{aligned} \sigma _{ij}=\tau _{ij}-p\delta _{ij} \end{aligned}$$

(95)

where $\delta _{ij}$ is the Kronecker delta. The constitutive law for the incompressible Newtonian fluid reads

$$\begin{aligned} \tau _{ij}=\mu (v_{i,j}+v_{j,i}) \end{aligned}$$

(96)

where $\mu$ is the viscosity and $v_i$ is the velocity in the i-th direction. And the incompressibility reads

$$\begin{aligned} v_{k,k}=0 \end{aligned}$$

(97)

The equilibrium without body forces and inertia effects reads

$$\begin{aligned} \sigma _{ij,j}=0 \end{aligned}$$

(98)

Substituting Eq. (95) into Eq. (98) and using Eqs. (96) and (97), we have

$$\begin{aligned} \mu v_{i,jj}=p_{,i} \end{aligned}$$

(99)

Combining Eqs. (97) and (99) and expanding the tensor expressions into regular expressions, we have the following governing equation for the incompressible Newtonian inertia free flow.

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu \left( \frac{\partial ^2v_1}{\partial x_1^2}+\frac{\partial ^2v_1}{\partial x_2^2}+\frac{\partial ^2v_1}{\partial x^2_3}\right) =\frac{\partial p}{\partial x_1} \\ \mu \left( \frac{\partial ^2v_2}{\partial x_1^2}+\frac{\partial ^2v_2}{\partial x_2^2}+\frac{\partial ^2v_2}{\partial x^2_3}\right) =\frac{\partial p}{\partial x_2} \\ \mu \left( \frac{\partial ^2v_3}{\partial x_1^2}+\frac{\partial ^2v_3}{\partial x_2^2}+\frac{\partial ^2v_3}{\partial x^2_3}\right) =\frac{\partial p}{\partial x_3} \\ \frac{\partial v_1}{\partial x_1}+\frac{\partial v_2}{\partial x_2}+\frac{\partial v_3}{\partial x_3}=0 \end{array}\right. } \end{aligned}$$

(100)

The model problem to solve the Stefan effect is shown in Fig. 16. Two parallel, circular plates with radius $R$ that are separated by a Newtonian (incompressible) liquid with viscosity $\mu$ and thickness $h$. The next step is to obtain the expression of the force $F_v$ applied on the plates by solving Eqs. (100) with the boundary conditions shown in Fig. 16.

Fig. 16

Model problem of Stefan effect

As the model problem is axisymmetric, Eqs. (100) could be casted into the following form.

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu \left( \frac{\partial ^2 v_r}{\partial r^2}+\frac{1}{r}\frac{\partial v_r}{\partial r}+\frac{\partial ^2 v_r}{\partial z^2}-\frac{v_r}{r^2} \right) =\frac{\partial p}{\partial r}\\ \mu \left( \frac{\partial ^2 v_z}{\partial r^2}+\frac{1}{r}\frac{\partial v_z}{\partial r}+\frac{\partial ^2 v_z}{\partial z^2} \right) =\frac{\partial p}{\partial z}\\ \frac{\partial v_r}{\partial r}+\frac{v_r}{r}+\frac{\partial v_z}{\partial z}=0 \end{array}\right. } \end{aligned}$$

(101)

[34] further assumes

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial ^2 v_1}{\partial x_1^2} \ll \frac{\partial ^2 v_1}{\partial x_3^2},~\frac{\partial ^2 v_1}{\partial x_2^2} \ll \frac{\partial ^2 v_1}{\partial x_3^2}\\ \frac{\partial ^2 v_2}{\partial x_1^2} \ll \frac{\partial ^2 v_1}{\partial x_3^2},~\frac{\partial ^2 v_2}{\partial x_2^2} \ll \frac{\partial ^2 v_1}{\partial x_3^2}\\ \frac{\partial v_3}{\partial x_3}=0,~\frac{\partial p}{\partial x_3}=0 \end{array}\right. } \end{aligned}$$

(102)

Subtstuting Eq. (102) into Eq. (101) yields

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu \frac{\partial ^2 v_r}{\partial z^2}=\frac{d p}{d r}\\ \frac{\partial p}{\partial z}=0\\ \frac{\partial v_r}{\partial r}+\frac{v_r}{r}+\frac{d v_z}{d z}=0 \end{array}\right. } \end{aligned}$$

(103)

The first equation in Eqs. (103) suggests the solution of $v_r$ in the following form

$$\begin{aligned} v_r=\frac{z(z-h)}{2\mu }\frac{d p}{d r} \end{aligned}$$

(104)

Substituting Eq. (104) into the third equation of Eqs. (103), we obtain

$$\begin{aligned} \frac{d v_z}{d z}=-\frac{z(z-h)}{2\mu }\left( \frac{d^2 p}{d r^2}+\frac{1}{r}\frac{d p}{d r}\right) \end{aligned}$$

(105)

An integration from $0$ to $h$ with respect to $z$ gives

$$\begin{aligned} \frac{h^3}{12\mu }\left( \frac{d^2 p}{d r^2}+\frac{1}{r}\frac{d p}{d r}\right) =v_z|_{z=h}=\dot{h} \end{aligned}$$

(106)

The solution of Eq. (106) could be expressed as follows

$$\begin{aligned} p(r)=\frac{3\mu \dot{h}}{h^3}r^2+C_1\ln r+C_2 \end{aligned}$$

(107)

Considering the conditions that $p(R)=p_0$ and $p(0)$ is bounded, we further have

$$\begin{aligned} p(r)=p_0-\frac{3\mu \dot{h}}{h^3}(R^2-r^2) \end{aligned}$$

(108)

By integrating the additional pressure over the plate, we have the force induced by the viscosity

$$\begin{aligned} F_v=\int _0^R 2\pi r \big [p_0-p(r) \big ]dr=\frac{3\mu \pi R^4}{2h^3}\dot{h} \end{aligned}$$

(109)

Considering the following expressions of stress and strain

$$\begin{aligned} \sigma _v=\frac{F_v}{\pi R^2},~\dot{\epsilon }=\frac{\dot{h}}{h} \end{aligned}$$

(110)

Eq. (109) further reads

$$\begin{aligned} \sigma _v=A\dot{\epsilon },~A=\frac{\alpha _S\mu }{\gamma _h^2} \end{aligned}$$

(111)

where the shape coefficient $\alpha _S=\frac{3}{2}$ and the aspect ratio $\gamma _h=\frac{h}{R}$ for circular plates. Although derived considering the circular plate, Eqs. (111) offers the general form of viscous coefficients $A$. And it is also observed that the aspect ratio $\gamma _h$ plays a very important role for the rate-dependency between stress and strain rate besides the viscosity $\mu$.