Nonlocal models and two-scale homogenization for dynamics in heterogeneous elastic media

Project 1. Multiscale Analysis of Heterogeneous Media in the Peridynamic Formulation

The peridynamic formulation introduced in (Silling, 2000) is a non-local continuum theory for deformable bodies. Material particles interact through a pairwise force field that acts within a prescribed horizon. Interactions depend only on the difference in the displacement of material points and spatial derivatives in the displacement are avoided. This feature makes it an attractive model for the autonomous evolution of discontinuities in the displacement for problems that involve cracks, interfaces, and other defects.

In this project new tools are developed for the analysis of heterogeneous peridynamic media involving two distinct length scales over which different types of peridynamic forces interact. The setting treated here involves a long range peridynamic force law perturbed in space by an oscillating short range peridynamic force. The oscillating short range force represents the presence of heterogeneities. Here we treat particle or fiber reinforced composite media. It is also assumed that there is a sharp density variation associated with the heterogeneities. In this treatment we carry out the analysis in the small deformation setting. For this case the reference and deformed configurations are taken to be the same and both long and short range forces are given by linearizations of the peridynamic bond stretch model introduced in (Silling, 2000).

The relative length scale over which the short range forces interact is denoted by $\epsilon$ and points inside the domain containing the heterogeneous material are specified by $x$. Here we will suppose the heterogeneities are periodically dispersed on the length scale $\epsilon=\frac{1}{n}$ for some choice of $n=1,2,\ldots$ The deformation inside the medium is both a function of space and time $t$ and is written $u^{\epsilon}(x,t)$. The multi-scale analysis of the peridynamic formulation proceeds using the concept of two-scale convergence. The two-scale convergence originally introduced in the context of partial differential equations turns out to provide a natural setting for identifying both the coarse scale and fine scale dynamics inside peridynamic composites. The theory and application of the two-scale convergence is used to develop a novel two-scale peridynamic equation for the dynamics see Alali and Lipton, 2012). The two-scale formulation is described by introducing a rescaled or microscopic variable $y=x/\epsilon$. The solution of the two-scale dynamics is a deformation $u(x,y,t)$ that depends on both variables $x$ and $y$.

The rescaled solution $u(x,x/\epsilon,t)$ is shown to provide a strong approximation to the actual deformation $u^\epsilon(x,t)$ inside the peridynamic material see Alali and Lipton, 2012). This is is proved using an evolution law for the error $e^\epsilon(x,t)=u^\epsilon(x,t)-u(x,x/\epsilon,t)$. It is shown that $e^\epsilon(x,t)$ vanishes in the $L^p$ norm, with respect to the spatial variables, when the length scale of the oscillation tends to zero for all $p$ in the interval $1\leq p\leq\infty$. The advantage of using the two-scale dynamics as a computational model is that it has the potential to lower computational costs associated with the explicit peridynamic modeling of millions of heterogeneities.

It is important for the modeling to recover the dynamics that can be measured by strain gages or other macroscopic measuring devices. Typical measured quantities involve averages of the deformation $u^\epsilon(x,t)$ taken over a prescribed region $V$ with volume denoted by $|V|$. To this end we denote the unit period cell for the heterogeneities by $Y$ and project out the fluctuations by averaging over $y$ and write \begin{eqnarray} u^H(x,t)=\int_Y u(x,y,t)dy. \label{avgerage} \end{eqnarray} In (Alali and Lipton, ) it is shown that \begin{eqnarray} \lim_{\epsilon\rightarrow 0}\frac{1}{|V|}\int_V u^\epsilon(x,t)\,dx=\frac{1}{|V|}\int_V u^H(x,t)\,dx. \label{volavg} \end{eqnarray} In this way we see that the average deformation is characterized by $u^H(x,t)$ when the scale $\epsilon$ of the microstructure is small. We split the deformation into microscopic and macroscopic parts and write $u(x,y,t)=u^H(x,t)+r(x,y,t)$. The interplay between the microscopic and macroscopic dynamics is given by a coupled system of evolution equations for $u^H$ and $r$. Here $r(x,x/\epsilon,t)$ is recognized as the corrector for this problem. The equations show that forces generated by the homogenized deformation inside the medium are related to the homogenized deformation through a history dependent constitutive relation. The explicit form of the constitutive relation is presented in section four where we present a homogenized evolution equation for the coarse scale dynamics written exclusively in terms of $u^H$, see Alali and Lipton, 2012).

This work has appeared in B. Alali and R. Lipton. Multiscale Dynamics of Heterogeneous Media in the Peridynamic Formulation. Journal of Elasticity. 2012, Volume 106, Issue 1, pp 71-103.


This work is supported by: Boeing Contract # 207114, AFOSR Grant FA 9550-05-1008, and NSF Grant DMS-0406374

Project 2. Multiscale Analysis of an Abstract Evolution Equation with Applications to Nonlocal Models for Heterogeneous Media

Together with Qiang Du and Tadele Mengesha we develop an abstract framework for multiscale analysis and homogenization for abstract linear evolution equations associated with a bounded operator see (Du, Lipton, and Mengesha, 2016). It provides a natural analog to that for the time-dependent local PDE models with highly oscillatory coefficients. We illustrate the method applying it to a special case, namely, the time-dependent state-based peridynamic model. Our study builds on the analysis of the first project see (Alali and Lipton, 2012). We note that in all of these works, it is hypothesized that the direct nonlocal interaction involves a long range peridynamic force perturbed in space by an oscillating short range peridynamic force representing the heterogeneous environment. This is consistent with standard PDE based homogenization for oscillatory coefficients $A$ and is easily seen on writing the oscillatory coefficient as the perturbation of the identity $A= I +\delta A$, where $I$ is the identity matrix and $\delta A=I-A$ is the oscillatory part of the coefficient.

We extend and generalize the homogenization and multiscaling framework introduced in (Alali and Lipton, 2012) to handle abstract evolution equations. In this work we consider a family of evolution equations expressed in terms of a bounded oscillatory operator $\mathcal{P}_{\epsilon}$ given by \begin{equation}\label{abstract-td 01} {\bf u}^{\epsilon}_{t} (\bf x, t) %\frac{\partial^{2}{\bf u}^{\epsilon}}{\partial_{tt}} (\bf x, t) = \mathcal{P}_{\epsilon} {\bf u}^{\epsilon} (\bf x, t) + {\bf b}_{\epsilon}(\bf x, t), \,\, \text{for $\bf x\in \Omega$}, t>0, \end{equation} where for positive integers $k$ and $d$, $\Omega\subset \mathbb{R}^{d}$ is an open bounded domain, and $\mathcal{P}_{\epsilon}: L^{p}(\Omega;\mathbb{R}^{k}) \to L^{p}(\Omega;\mathbb{R}^{k})$ is a family of bounded linear operators parametrized by $\epsilon > 0$ with the uniform bound \begin{eqnarray}\label{uniform bound} \|\mathcal{P}_{\epsilon}\| \leq C, \hbox{ for every } \epsilon. \end{eqnarray} Here the heterogeneous medium is periodic and highly oscillatory and the parameter $\epsilon$ characterizes the length scale of periodicity. The operator norm for operators defined on the space $L^{p}(\Omega;\mathbb{R}^{k})$ with $ < p < \infty$ is denoted by $\|\cdot\|$. The Cauchy data for the problem can also be $\epsilon$ periodic and is specified by ${\bf u}^{\epsilon}(\bf x,0) ={\bf u}_0^\epsilon(\bf x)$.

We are interested in understanding the evolution of the solutions ${\bf u}^\epsilon$ of the Cauchy problem as $\epsilon\rightarrow 0$. In order to maximize generality we describe periodic oscillations associated with the sequence $\mathcal{P}_{\epsilon}$ implicitly and define the concept of two-scale convergence for operators $\mathcal{P}_{\epsilon}\stackrel{2}{\rightarrow} \mathcal{P}_{0}$. This notion is quite natural for the description of highly oscillatory periodic media and is motivated by the theory of two-scale convergence defined for functions.

In this project we use the the notion of two-scale operator convergence together with two-scale compactness to show that the limiting dynamics is captured by the unique two scale limit $\mathbf{u}(\mathbf{x},\mathbf{y},t)$ satisfying the unfolded evolution \begin{equation} \label{two_scale_limit_full_form-Omega-times-Y-Abridged} \renewcommand{\arraystretch}{1} \partial_{t} {\bf u}(\bf x, \bf y, t) =\mathcal{P}_{0} {\bf u}(\bf x, \bf y,t) + {\bf b}(\bf x, \bf y, t), \end{equation} supplemented with initial conditions given by the two scale limit of the Cauchy data. Here ${\bf b}(\bf x, \bf y, t)$ denotes the two-scale limit of the righthand side $\mathbf{b}_\epsilon$.

The unfolded evolution $\mathbf{u}(\mathbf{x},\mathbf{y},t)$ provides the necessary framework for recovering the homogenized dynamics and the strong multiscale approximation to the dynamics. The approach is general and delivers both homogenized dynamics as well as strong approximations. In this context the strong approximation is the analogue of corrector theory for PDE based homogenization. We illustrate the method applying it to time-dependent state-based peridynamic models.

References

1. B. Alali and R. Lipton. Multiscale Dynamics of Heterogeneous Media in the Peridynamic Formulation. Journal of Elasticity. 2012, Volume 106, Issue 1, pp 71-103.
2. Q. Du, R. Lipton, and T. Mengesha. Multiscale Analysis of an Abstract Evolution Equation with Applications to Nonlocal Models for Heterogeneous Media. ESAIM: M2AN, 50 Number 5, 2016, pp. 1425-1455.
3. S.A. Silling. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids, 48:175-209, 2000.