Peridynamic Models for Random Media Found by Coarse Graining

Abstract

Using coarse graining, the upscaled mechanical properties of a solid with small scale heterogeneities are derived. The method maps internal forces at the small scale onto peridynamic bond forces in the coarse grained mesh. These upscaled bond forces are used to calibrate a peridynamic material model with position-dependent parameters. These parameters incorporate mesoscale variations in the statistics of the small scale system. The upscaled peridynamic model can have a much coarser discretization than the original small scale model, allowing larger scale simulations to be performed efficiently. The convergence properties of the method are investigated for representative random microstructures. A bond breakage criterion for the upscaled peridynamic material model is also demonstrated.

Appendix: Derivation of the NOSB Influence Function

In this Appendix it is shown that the form of the influence function $\omega$ can be obtained from the form of the CG smoothing function $\Omega$. It is convenient to use the continuum expression for the CG bond forces:

$$\begin{aligned} \textbf{f}(\textbf{q},\textbf{x})=\int \int \Omega (\textbf{X},\textbf{x})\Omega (\textbf{Q},\textbf{q})\textbf{F}(\textbf{Q},\textbf{X})\;{ {\text {d}} }\textbf{Q}\;{ {\text {d}} }\textbf{X} \end{aligned}$$

(104)

where the integrals are over the entire body. Assume that the lattice forces are short-range, that is, $d\ll R$. Also let $\Omega$ be radially symmetric, that is, $\Omega (\textbf{X},\textbf{x})$ depends only on $|\textbf{X}-\textbf{x}|$. Further assume that $\textbf{X}$ is in the interior of the body, that there are no external forces, and that the body and the deformation are both homogeneous. Define $\textbf{P}=\textbf{Q}-\textbf{X}$. Then (104) becomes

$$\begin{aligned} \textbf{f}(\textbf{q},\textbf{x})=\int \int \Omega (\textbf{X},\textbf{x})\Omega (\textbf{X}+\textbf{P},\textbf{q})\textbf{F}_0(\textbf{P})\;{ {\text {d}} }\textbf{P}\;{ {\text {d}} }\textbf{X} \end{aligned}$$

(105)

where $\textbf{F}_0$ is a function independent of $\textbf{X}$. Since the body is in equilibrium,

$$\begin{aligned} \int \textbf{F}_0(\textbf{P})\;{ {\text {d}} }\textbf{P}=\mathbf{{0}}. \end{aligned}$$

(106)

Since d is small, terms in a Taylor expansion for $\Omega$ near $\textbf{X}$ higher than first order can be neglected for present purposes. From (105),

$$\begin{aligned} \textbf{f}(\textbf{q},\textbf{x})=\int \int \Omega (\textbf{X},\textbf{x})\Big [ \Omega (\textbf{X},\textbf{q})+\textbf{P}\cdot \nabla \Omega (\textbf{X},\textbf{q}) \Big ] \textbf{F}_0(\textbf{P})\;{ {\text {d}} }\textbf{P}\;{ {\text {d}} }\textbf{X}. \end{aligned}$$

(107)

In view of (106),

$$\begin{aligned} \textbf{f}(\textbf{q},\textbf{x})=\varvec{\sigma }_0\int \Omega (\textbf{X},\textbf{x})\nabla \Omega (\textbf{X},\textbf{q})\;{ {\text {d}} }\textbf{X} \end{aligned}$$

(108)

where the stress tensor is given by

$$\begin{aligned} \varvec{\sigma }_0=\int \textbf{F}_0(\textbf{P})\otimes \textbf{P}\;{ {\text {d}} }\textbf{P}. \end{aligned}$$

(109)

Now recall the peridynamic bond stress for a corrospondence material with influence function $\omega$:

$$\begin{aligned} \textbf{f}(\textbf{q},\textbf{x})=\frac{1}{K_0}\varvec{\sigma }_0\omega (| {\varvec{\xi }} |) {\varvec{\xi }} , \qquad {\varvec{\xi }} =\textbf{q}-\textbf{x}. \end{aligned}$$

(110)

in which, under the present assumptions, the shape tensor is isotropic, that is $\textbf{K}=K_0\textbf{1}$ for some scalar $K_0$. Comparing (108) and (110), the conclusion is that

$$\begin{aligned} \omega (\textbf{q},\textbf{x})=\frac{1}{|\textbf{q}-\textbf{x}|}\left| \int \Omega (\textbf{X},\textbf{x})\nabla \Omega (\textbf{X},\textbf{q})\;{ {\text {d}} }\textbf{X}\right| \end{aligned}$$

(111)

where $K_0$ is omitted, since the normalization of $\omega$ is handled within the correspondence model. Under the present assumptions, $\omega$ depends only on $|\textbf{q}-\textbf{x}|$, and (111) therefore simplifies to

$$\begin{aligned} \omega (\xi )=\frac{1}{\xi }\left| \int _{{\mathcal {B}}}\Omega (\textbf{X},\mathbf{{0}})\nabla \Omega (\textbf{X},\xi \textbf{e})\;{ {\text {d}} }\textbf{X}\right| \end{aligned}$$

(112)

where $\textbf{e}$ is any fixed unit vector and, taking liberties with the notation, $\omega (|\textbf{q}-\textbf{x}|)$ replaces $\omega (\textbf{q},\textbf{x})$. ${\mathcal {B}}$ is a neighborhood of $\mathbf{{0}}$ with radius R. A typical curve for $\omega$ is shown in Fig. 20, in which tent-shaped smoothing functions are assumed.

Equation (112) provides the expression for the influence function in an NOSB peridynamic material model derived from the underlying smoothing functions that are used for coarse graining.

Fig. 20

Influence function for an NOSB material model as determined from the smoothing functions