## Home Page of Prashant K. Jha

Currently, I am working with Dr. Robert Lipton as a Postdoc. Our project work involves numerical analysis of nonlocal models. I finished my Ph.D. in August 2016 and my adviser was Dr. Kaushik Dayal. My research interests lie in numerical methods, multiscale modeling, and computational mechanics. If you have any question, feel free to get in touch.

In this page you will find list of projects on which I am currently working on as well as list of journals and book chapters published or under review. This webpage also contains discussion on key research work in the Research Work tab. Site also has pdfs and slideshows of my talks.

## Short Bio

## Curriculum Vitae

## Ph.D. Thesis

## Work in Progress

List of projects on which I am currently working on.

In this article, we present robust implementation of finite element approximation for nonlocal fracture models. Numerical results are provided for linear continuous finite element approximation to verify the convergence rate. Special care is taken to compute the nonlocal force at a material point accurately and easy to implement method is presented to compute nonlocal forces accurately.

In this article, we present apriori error estimates for finite difference approximation of a state-based peridynamics models. The model is shown to be well-posed in the space of Hölder continuous functions for any finite size of horizon. We present numerical results for different types of problems to numerically verify the claims. We also consider a problem with crack and show that the problem converges numerically, as we refine the mesh, at a rate closer to the theoretical value.

In this work, we consider a state-based peridynamics model and study the finite element approximation. The state-based model is shown to be well-posed on Sobolev space \(H^2\). This work is based on our earlier work where we considered the bond-based peridynamics model and obtained an apriori rate of convergence due to finite element approximation.

In this work, we introduce a regularized state-based peridynamics model for free fracture propagation. We show that the model is well posed and solution exists in \(L^2\). In the limit of vanishing nonlocality, we obtain the \(\Gamma\)-limit of peridynamic energy.

In this work, we consider the electrical interactions in nanostructures like nanorods, nanohelix, and nanofilms. We find that in the limit the energy is short range. We show that the results in this work agrees with the work of Gioia and James on thin film magnetics.

## Showcase of research work

Let \(D\) be the material domain and \(d=1,2,3\) be the dimension. Time domain is \([0,T]\). Let \(\epsilon\) be the size of horizon which controls the nonlocal interaction of a material point with its neighboring material points. If \(u : D\to \mathbf{R}^d\) is the displacement and \(x,y \in D\) two material points, then we define linearized strain of the bond between \(y\) and \(x\) as $$ S(y,x;u) = \frac{u(y) - u(x)}{|y-x|} \cdot \frac{y-x}{|y-x|}. $$ Let \(-\nabla PD^\epsilon(u)(x)\) denote the peridynamics force at point \(x\) corresponding to displacement field \(u\). We can write down the equation of evolution of displacement field \(u: [0,T]\times D \to \mathbf{R}^d\) as follows $$ \rho \ddot{u}(t,x) = -\nabla PD^\epsilon(u(t))(x) + b(t,x) $$ where \(u(t)\) denotes the function over \(D\) at time \(t\), \(\rho\) is the density, and \(b\) is the body force. Equation is complemented by initial condition \(u(0,x) = u_0(x)\) and \(\dot{u}(0,x) = v_0(x)\). Boundary condition \(u=0\) is specified over nonlocal boundary \(D_c:= \{ x\in D: \text{dist}(x,\partial D) \leq \epsilon \}\).

We consider bond-based model proposed in **Lipton 2014** and **Lipton 2016**. Peridynamics force is of the form
$$
-\nabla PD^\epsilon(u)(x) = \frac{2}{|B_\epsilon(x)|} \int_{B_\epsilon(x)} \partial_S W^\epsilon(S(u), y-x) \frac{y-x}{|y-x|} dy,
$$
where \(B_\epsilon(x)\) is the ball of radius \(\epsilon\). Potential \(W^\epsilon(S,y-x)\) as a function of bond-strain \(S\) and bond vector \(y-x\) is given by
$$
W^\epsilon(S,y-x) = \frac{1}{\epsilon |y-x|} J^\epsilon(|y-x|) f(|y-x|S^2),
$$
where \(J^\epsilon\) is the influence function and \(f\) is the nonlinear potential function which is assumed to be smooth, concave, and positive. It also satisfies following properties at two extreme points
$$
\lim_{r\to 0^+} \frac{f(r)}{r} = f'(0), \quad \lim_{r\to \infty} f(r) = f_\infty < \infty.
$$
We assume influence function is defined by rescaling of function \(J\), i.e. \(J^\epsilon(r) = J(r/\epsilon)\) where \(0\leq J \leq M\) for all \(r< 1\) and \(J(r) = 0\) for \(r\geq 1\). We have
$$
\partial_S W^\epsilon(S,y-x) = \frac{2S}{\epsilon} J^\epsilon(|y-x|) f'(|y-x|S^2),
$$
and therefore peridynamics force is given by
$$
-\nabla PD^\epsilon(u)(x) = \frac{4}{\epsilon|B_\epsilon(x)|} \int_{B_\epsilon(x)} J^\epsilon(|y-x|) f'(|y-x|S(y,x;u)^2) S(y,x;u) \frac{y-x}{|y-x|} dy.
$$
For existence in regular spaces like Hölder space and Hilbert spaces, we also need to introduce another function \(\omega(x): D\cup D_c \to \mathbf{R}\) in potential \(W^\epsilon\). This function takes value \(1\) in the interior of domain \(D\) and smoothly decays from \(1\) to \(0\) as we approach boundary \(\partial D\) from interior. It takes value \(0\) outside \(D\). To keep presentation simple, we omit this fact and consider simpler form of peridynamic potential and force.

The model is nonlinear and is regularized version of prototypical micro-elastic bond material (PMB). See below where we show force response of a bond for given strain for the nonlinear model as well as PMB material.

We consider Hölder space for numerical analysis. This choice of space is suitable for two reasons: one, it admits solution with discontinuity, and two, for nonlocal models it makes the estimation of errors easier. We first show existence of solutions (**Jha and Lipton 2017a**) in Hölder space \(C^{0,\gamma}(D;\mathbf{R}^d)\), where \(D\) is the material domain, \(d\) is the dimension, \(\gamma\) is the Hölder exponent taking values in \((0,1]\).

We can write finite difference approximation as follows $$ \frac{\hat{u}^{k+1}_i - \hat{u}^k_i}{\Delta t} = \hat{v}^{k+1}_i $$ $$ \frac{\hat{v}^{k+1}_i - \hat{v}^k_i}{\Delta t} = -\nabla PD^{\epsilon}(\hat{u}^k_h)(x_i) + b^k_i , $$ where \(u^k_i, v^k_i\) is the approximate solution at time \(t^k = k\Delta t\) and \(x_i = ih \in D\). \(\Delta t\) is the size of time step and \(h\) is the mesh size. \(D_h = D\cap h\mathbf{Z}\) denotes the uniform discretization of domain. We denote piecewise constant extension of discrete set \(\{u^k_i\}_i\) as \(u^k_h\) and is given by $$ u^k_h(x) = \sum_{i, x_i \in D} u^k_i \chi_{U_i}(x), $$ where \(U_i\) is the unit cell associated to node \(i\) and \(chi_{A}\) is the characteristics function which returns value \(1\) for points inside set \(A\) and \(0\) otherwise.

We have following result which shows the convergence of finite difference approximation

In upcoming paper, we discuss the implementation of finite difference approximation of peridynamics model using HPX framework. We also provide the numerical results and show convergence rate of approximation. Before we present few results, let us discuss the calculation of convergence rate without the exact solutions. This has also been discussed in **Jha and Lipton 2017b**. Consider three mesh sizes \(\{h_1,h_2,h_3\}\) with \(r = h_1/h_2 = h_2/h_3 > 1\). Let \(u_{h_i}\) for \(i=1,2,3\) be the approximate solution corresponding to three mesh sizes and let \(u_{exact}\) be the exact solution (possibly not known). Assuming that error behaves like \(||u_h - u_{exact}||_{L^2} \leq C h^\alpha\) where \(\alpha\) is the rate of convergence, we have
$$
||u_{h_1} - u_{h_2}||_{L^2} \leq ||u_{h_1} - u_{exact}||_{L^2} + ||u_{exact} - u_{h_2}||_{L^2} \leq C(1+r^\alpha) h_2^\alpha.
$$
After taking logarithm of above, we find
$$
\log (||u_{h_1} - u_{h_2}||_{L^2}) \leq \bar{C} + \alpha \log (h_2),
$$
where \(\bar{C} = \log(C) + \log (1+ r^\alpha) \). Similar expression can be derived for \(||u_{h_2} - u_{h_3}||_{L^2}\), i.e.
$$
\log (||u_{h_2} - u_{h_3}||_{L^2}) \leq \bar{C} + \alpha \log (h_3),
$$
Eliminating \(\bar{C} \), we find that the rate of convergence \(\alpha\) is bounded below by
$$
\frac{\log(||u_{h_1} - u_{h_2}||_{L^2}) - \log(||u_{h_2} - u_{h_3}||_{L^2})}{\log(r)}.
$$
Therefore, using above formula, we can compute the lower bound on rate of convergence without any knowledge of exact solution.

We now present the convergence result for two different initial conditions. Consider square material domain \(D = [0,1]^2\). Fix size of horizon \(\epsilon = 0.05\). We consider three mesh sizes \(h = \epsilon/2, \epsilon/4, \epsilon/8\). Time domain is \([0,2.0]\) and size of time step is \(\Delta t = 10^{-5}\).

For nonlinear peridynamics force \(-\nabla PD^\epsilon(u)(x)\), see above for expression, we consider influence function \(J^\epsilon(r) = 1\) for \(r <\epsilon\) and zero otherwise. Nonlinear potential \(f\) is given by \(f(r) = C(1-\exp[-r\beta]) \) with \(C = 1, \beta = 1\). Boundary condition \(u=0\) is prescribed on nonlocal boundary \(D_c:= \{ x\in D: \text{dist}(x,\partial D) \leq \epsilon \}\). We prescribe initial condition: $$ u_0(x) = a\exp [-|x-x_c|^2/\beta], v_0(x) = 0, $$ where \(a, x_c \in \mathbf{R}^2\). For first problem we choose \(a = (0.001, 0), x_c = (0.25,0.25), \beta = 0.003\). For second problem we choose \(a = (0.0002, 0.0007), x_c = (0.25,0.25), \beta = 0.01\).

Below is the plot of lower bound on rate of convergence at different times up to time \(T = 2.0\). We compare the results from theorem and find that convergence rate is mostly above \(1\) for most of the time. Fluctuations are expected as this is dynamics result.

In **Jha and Lipton 2017b**, we analyze quadrature based finite element approximation in one dimension. We present numerical results to verify the convergence rate. Before we discuss the numerical approximation, we present the result which shows that in the limit of size of horizon going to zero, the solution of peridynamics equation converges to the solution of elastodynamics equation.

1. Convergence of peridynamics solution in the limit size of horizon going to zero

Recall that \(-\nabla PD^\epsilon(u)(x)\) denotes the peridynamic force. Let \((0,1)\) is the material domain. Let \(u^\epsilon\) denote the solution of peridynamic equation for given size of horizon \(\epsilon > 0\). \(u^\epsilon\) solves $$ \rho \ddot{u}^\epsilon(t,x) = -\nabla PD^\epsilon(u^\epsilon(t))(x) + b(t,x). $$ Let \(u\) denote the solution of following elastodynamics equation $$ \rho \ddot{u}(t,x) = \mathbb{C} u_{xx}(t,x) + b(t,x). $$ We specify same initial condition for \(u^\epsilon\) and \(u\), i.e. \(u^\epsilon(0,x) = u_0(x) = u(0,x)\) and \(\dot{u}^\epsilon(0,x) = v_0(x) = \dot{u}(0,x)\). For \(u^\epsilon\), boundary condition \(u^\epsilon(t,x) = 0\) is specified on nonlocal boundary \((-\epsilon, 0]\cup [1,1+\epsilon)\) and for \(u\) boundary condition \(u(t,x) = 0\) is specified at \(x = 0,1\). \(\mathbb{C}\) is related to the peridynamic force in following way $$ \mathbb{C} = \frac{1}{\epsilon^2} \int_{x-\epsilon}^{x+\epsilon} J^\epsilon(|y-x|) f'(0) |y-x| dy. $$ Using the fact that \(J^\epsilon(r)\) is defined by rescaling of function \(J\), i.e. \(J^\epsilon(r) = J(r/\epsilon)\), it can be easily shown that $$ \mathbb{C} = \int_{-1}^{1} J(|z|) f'(0) |z| dz. $$ Recall that \(J^\epsilon\) and function \(f\) are used to describe peridynamics force. Peridynamics force in dimension \(d=1\) is given by $$ -\nabla PD^\epsilon(u)(x) = \frac{2}{\epsilon^2} \int_{x-\epsilon}^{x+\epsilon} J^\epsilon(|y-x|) f'(|y-x|S(y,x;u)^2) S(y,x;u) dy. $$ In 1-d, definition of bond strain is simplified and is given by $$ S(y,x;u) =\frac{u(y) - u(x)}{|y-x|}. $$ We will see next that \(\mathbb{C} u_{xx}\) is the limit of \(-\nabla PD^\epsilon\) when \(\epsilon \to 0\) and when field \(u\) is sufficiently smooth. We have following results which show convergence of peridynamics solution \(u^\epsilon\) to elastodynamics solution \(u\).

2. Numerical scheme

Let \(h\) be the size of mesh, \(\Delta t\) be the size of time steps, \(D_h = D\cap h\mathbf{Z}\) and \(D_{c,h} = D_c \cap h\mathbf{Z}\) be the discretization of spatial domain, and \([0,T] \cap \Delta t\mathbf{Z}\) be the discretization of time domain. Let index set \(K= \{i\in \mathbf{Z}: ih \in D\}\) correspond to domain \(D\) and \(K_c= \{i\in \mathbf{Z}: ih \in D_c\}\) correspond to nonlocal boundary \(D_c = (-\epsilon, 0] \cap [1,1+\epsilon)\). We define interpolation operator \( I_h[\cdot]\) for a given function \(g: \overline{D\cup D_c} \to \mathbf{R}\) as follows $$ I_h[g(y)] = \sum_{i\in K \cup K_c} g(x_i) \phi_i(y), $$ where \(\phi_i(y)\) is the value of interpolation function corresponding to node \(i\) at point \(y\). We consider linear interpolation function. For simplification we let \(\epsilon = mh\) where \(m\geq 1\) is integer. Let \(hat{u}^\epsilon_h(t)\) is the extension of discrete set \(\{\hat{u}^\epsilon_i\}_{i\in K\cup K_c}\) and is given by $$ \hat{u}^\epsilon_h(t,x) = E[\{\hat{u}^\epsilon_i\}_{i\in K\cup K_c}] = \sum_{i\in K \cup K_c} \hat{u}^\epsilon_i \phi_i(x). $$

Let \(\hat{u}^{\epsilon,k}_{i}\) denote the approximate solution at time \(t^k = k\Delta t\) and \(x_i = ih\) and for given \(\epsilon >0\). It satisfies following central difference scheme $$ \rho\frac{\hat{u}^{\epsilon,k+1}_{i} - 2 \hat{u}^{\epsilon,k}_{i} + \hat{u}^{\epsilon,k-1}_{i}}{\Delta t^2} = -\nabla PD^\epsilon (\hat{u}^{\epsilon,k}_h) (x_i) + b(t^k,x_i), $$ where \( \hat{u}^{\epsilon,k}_h \) is the extension of discrete set \(\{\hat{u}^{\epsilon,k}_{i} \}_{i\in K\cup K_c} \). We let \( \rho = 1\) for simplicity.

Stability can be shown for linearized peridynamics. Linearized peridynamics force is obtained by setting nonlinear term \(f'(|y-x|S^2)\) equal to \(f'(0)\). Denoting linearized peridynamic force as \(-\nabla PD^\epsilon_l(u)\), we have $$ -\nabla PD^\epsilon_l(u)(x) = \frac{2f'(0)}{\epsilon^2} \int_{x-\epsilon}^{x+\epsilon} J^\epsilon(|y-x|) S(y,x;u) dy. $$ If \(\hat{u}^{\epsilon,k}_{i}\) satisfies following $$ \rho\frac{\hat{u}^{\epsilon,k+1}_{i} - 2 \hat{u}^{\epsilon,k}_{i} + \hat{u}^{\epsilon,k-1}_{i}}{\Delta t^2} = -\nabla PD^\epsilon_l (\hat{u}^{\epsilon,k}_h) (x_i) + b(t^k,x_i) $$ then we can write above equation in simpler matrix form. Let \(U_h^k = (\hat{u}^{\epsilon,k}_i)_{i\in K}\) then \(U_h^k\) satisfies $$ \frac{U^{k+1}_h - 2 U^k_h + U^{k-1}_h}{\Delta t^2} = AU^k_h + B^k $$ with \(A = (a_{ij})\) given by $$ a_{ij} = \bar{a}_{ij} \quad (\text{if }j\neq i) $$ and $$ a_{ij} = -\sum_{m\in K\cup K_c, m\neq i} \bar{a}_{ik} \quad (\text{if }j = i) $$ where $$ \bar{a}_{ij} = \frac{2f'(0)}{\epsilon^2} \int_{x_i - \epsilon}^{x_i + \epsilon} \frac{\phi_j(y) J^\epsilon(|y-x_i|)}{|y-x_i|} dy. $$ \(B^k = (b^k_i)\) is given by $$ b_i^k = b(t^k,x_i) + \sum_{j\in K_c, j\neq i} \bar{a}_{ij} \hat{u}^{\epsilon,k}_{j}. $$ In above, for \(j \in K_c\), since \(K_c\) corresponds to nonlocal boundary \(D_c\), the second term is due to nonzero boundary condition \(\hat{u}^{\epsilon,k}_j\) specified on \(D_c\).

3. Convergence of numerical scheme

We have following results which estimate the difference between exact peridynamics force and approximate peridynamics force. We also state result which show that approximate peridynamics force converges to elastodynamics force in the limit \(h\leq \epsilon \to 0\).

4. Stability of central difference approximation and CFL condition

For linearized peridynamics, we can show using the properties of matrix \(A\) the stability of central difference scheme and also obtain the explicit CFL condition depending on material parameters and mesh size. We have following important result.

5. Numerical results

Let domain is \(D=(0,1)\), size of horizon \(\epsilon = 0.01\), time domain \([0,0.5]\), and size of time step is \(\Delta t = 0.00001\). We consider nondimensional equation, see **Jha and Lipton 2017b**. Influence function is given by \(J(r) = 2 \alpha r(-r^2/\beta)/\sqrt(\pi) \) with \( \alpha = 1, \beta = 0.4\). Nonlinear function is of the form \(f(r) = C(1-\epx[-r\beta])\) with \(C = 1, \beta = 1/M\) where \(M := 2\int_0^1 J(r) rdr\).

Initial condition is of the form: \(u_0(x) = a\exp[-(0.25 - x)^2/\beta] + a\epx[-(0.75-x)^2/\beta] \) and \(v_0(x) = 0\). We have two Gaussian pulse centered at \(x=0.25, 0.75\) as initial condition on displacement. We choose \(a = 0.005, \beta = 0.00001\). Below is the plot of rate of convergence at different time steps. We have considered three different mesh sizes \(h = \epsilon/100, \epsilon/200, \epsilon/400\) to compute the rate of convergence using formula described in this page. We expect \(O(h)\) convergence i.e. slope of \(1\). The plots show that solutions converge at a rate higher than \(1\) for most of the time steps. Note final time is chosen such that it includes the wave reflection effect. Characteristics wave speed is \(1\) in nondimensional form and Gaussian pulses are at \(0.25\) distance from either boundary. Thus \(0.5\) unit of time is enough to include the wave reflection effect.

In second problem we consider higher final time \(T = 1.7\). Initial condition is of the form: \(u_0(x) = a\exp[-(0.5 - x)^2/\beta]\) and \(v_0(x) = 0\). We have a Gaussian pulse centered at \(x=0.5\) as initial condition on displacement. We choose \(a = 0.005, \beta = 0.00001\). We fix size of horizon to \(\epsilon = 0.1\) and consider three different mesh sizes \(h = \epsilon/10, \epsilon/100, \epsilon/1000\) to compute the rate of convergence. The plots show that solutions stay around value \(1\) which agrees with theoretical claims.

In **Jha and Lipton 2017c**, we analyze finite element approximation and prove rate of convergence of \(O(\Delta t + h^2/\epsilon^2)\). We first show the existence of solutions in \(H^2(D;\mathbf{R}^d) \cap L^\infty(D;\mathbf{R}^d)\) over any finite time interval \((-T, T)\).

1. Existence of solution in \(H^2(D;\mathbf{R}^d)\cap L^\infty(D;\mathbf{R}^d)\)

We prove Lipschitz bound on peridynamics force, see **[Theorem 3.1, Jha and Lipton 2017c]** in space \(H^2(D;\mathbf{R}^d)\cap L^\infty(D;\mathbf{R}^d)\). Using bound on force, we establish local existence of solutions. Following theorem shows that local existence theorem can be extended to finally show existence of solution over any finite time interval \((-T,T)\).

2. Finite element approximation

Let \(V_h\) be the approximation of \(H^2_0(D;\mathbf{R}^d)\) associated to the linear continuous interpolation function over triangulation \(\mathcal{T}_h\). Here \(h\) denotes the size of finite element mesh. We seek approximate solution \(u^k_h, v^k_h\) at time \(t^k\) in \(V_h\) such that following holds for all \(\tilde{u} \in V_h\) $$ \left( \frac{u^{k+1}_h - u^k_h}{\Delta t}, \tilde{u} \right) = (v^{k+1}_h, \tilde{u}), $$ $$ \left( \frac{v^{k+1}_h - v^k_h}{\Delta t}, \tilde{u} \right) = (-\nabla PD^\epsilon(u^k_h), \tilde{u} ) + (b^k_h, \tilde{u}). $$ If we combine above two equations, we get standard central difference equation $$ \left( \frac{u^{k+1}_h - 2 u^k_h + u^{k-1}_h}{\Delta t^2}, \tilde{u} \right) = (-\nabla PD^\epsilon(u^k_h), \tilde{u} ) + (b^k_h, \tilde{u}). $$ For \(k=0\), we have for all \(\tilde{u} \in V_h\) $$ \left( \frac{u^1_h - u^0_h}{\Delta t^2}, \tilde{u}\right) = \frac{1}{2}(-\nabla PD^\epsilon(u^0_h), \tilde{u})+ \dfrac{1}{\Delta t} (v^0_h, \tilde{u}) + \dfrac{1}{2}(b^0_h, \tilde{u}) $$ where \(u^0_h, v^0_h, b^k_h\) are pojection of initial condition \(u^0, v^0\) and body force \(b(t^k)\).

2.1 Stability of central difference approximation of linearized peridynamics

We define linearized peridynamics force \(-\nabla PD^\epsilon_l(u)\) by replacing nonlinear term \(f'(|y-x|S(u)^2)\) with \(f'(0)\). It is given by $$ -\nabla PD^\epsilon_l(u)(x) = \frac{4f'(0)}{\epsilon|B_\epsilon(x)|} \int_{B_\epsilon(x)} J^\epsilon(|y-x|) S(y,x;u) \frac{y-x}{|y-x|} dy. $$ Let discrete energy, for \(u^k_{l,h}\) at time step \(k\), is defined as $$ \mathcal{E}(u^k_{l,h}) := \frac{1}{2} \left[ ||\bar{\partial}^+_t u^k_{l,h}||^2 - \frac{\Delta t^2}{4} a^\epsilon_l(\bar{\partial}^+_t u^k_{l,h}, \bar{\partial}^+_t u^k_{l,h}) + a^\epsilon_l(\overline{u}^{k+1}_{l,h} , \overline{u}^{k+1}_{l,h}) \right], $$ where $$ \overline{u}^{k+1}_h := \frac{u^{k+1}_h + u^k_h}{2}, $$ and $$ \bar{\partial}_t^+ u^k_h := \frac{u^{k+1}_h - u^{k}_h}{\Delta t}. $$ We have following stability result for linearized peridynamics

2.2 Convergence of approximation

We have following result which shows order of \(\Delta t + h^2/\epsilon^2\) convergence.

In **Lipton et al 2017d**, we propose new damage model within the framework of peridynamics theory. Damage model is state-based and has history dependent damage function. Let \(u:[0,T]\times D\to \mathbf{R}^d\) be the displacement field. Bond strain \(S(y,x,t;u)\) is defined as (similar to previous definition)
$$
S(y,x,t;u) = \frac{u(t,y) - u(t,x)}{|y-x|} \cdot e_{y-x}
$$
where \(e_{y-x} = \frac{y-x}{|y-x|}\). Hydrostatic strain \(\theta(x,t;u)\) is defined as
$$
\theta(y,t;u) = \frac{1}{|B_\epsilon(x)|} \int_{D\cap B_\epsilon(x)} \omega^\epsilon (|y-x|) |y-x| S(y,x,t;u) dy,
$$
where \(\omega^\epsilon(r)\) is the influence function similar to \(J^\epsilon(r)\). In this model, the peridynamics force is due to two interactions: one due to bond-bond interaction within horizon, and two, due to hydrostatic strain. Let \(\mathcal{L}^T(u)(x,t)\) denote the peridynamics force at \(x\) due to bond-bond interaction and \(\mathcal{L}^D(u)(x,t)\) denote the force at \(x\) due to hydrostatic strain. They are given by
$$
\mathcal{L}^T(u)(x,t)=\frac{2}{|B_\epsilon(x)|}\int_{D\cap B_\epsilon(x)}\frac{J^\epsilon(|y-x|)}{\epsilon{|y-x|}}H^T(u)(y,x,t)\partial_S f(\sqrt{|y-x|}S(y,x,t;u))e_{y-x}\,dy,
$$
and
$$
\mathcal{L}^D(u)(x,t)=\frac{1}{|B_\epsilon(x)|}\int_{D\cap B_\epsilon(x)}\frac{\omega^\epsilon(|y-x|)}{\epsilon^2}\left[H^D(u)(y,t)\partial_\theta g(\theta(y,t;u))+ H^D(u)(x,t)\partial_\theta g(\theta(x,t;u))\right]e_{y-x}\,dy.
$$
We mainly focus on two special functions appearing in above definition of forces. Function \(H^T(u)(y,x,t)\) denotes the damage of bond between \(y,x\) at time \(t\). Function \(H^D(u)(x,t)\) denotes the damage of a material point \(x\) at time \(t\) due to evolution of hydrostatic strain \(\theta(x,t;u)\) at material point \(x\). \(H^T\) and \(H^D\) are function which depend on history of bond-strain between \(y,x\) and history of hydrostatic strain at \(x\) respectively. They are defined as
$$
H^T(u)(y,x,t)=h\left(\int_0^tj_S(S(y,x,\tau;u))\,d\tau\right)
$$
and
$$
H^D(u)(x,t)=h\left(\int_0^t j_\theta(\theta(x,\tau;u))\,d\tau\right).
$$
\(j_S\) and \(j_\theta\) are non-decreasing positive function. Physically what it means is that once the bond-strain exceeds some critical strain for which \(j_S > 0\), then the bond is damaged for all time to come. If bond-strain remains below critical strain after exceeding it once then damage would be fixed to the value it attained when strain exceeded critical strain. However if bond strain keep exceeding critical strain for sufficient amount of time then the bond-damage will increase and ultimately \(H^T(y,x,t;u) = 0\) meaning the bond between \(y,x\) has been broken for all time \(t' \geq t\). Similar interpretation can be drawn for damage function \(H^D\).

We present few numerical results which highlight various properties of a damage model. In what follows, we will plot total damage of a material point \(x\) due to bond-bond interaction. We will choose parameters such that for the range of strains, damage will only be due to bond-bond interaction, i.e. \(H^D\) will remain \(1\). Total damage \(\phi(x,t;u)\) of a material point can be defined as $$ \phi(x,t;u) = 1 - \frac{\int_{D\cap B_\epsilon(x)} H^T(u)(y,x,t) dy}{\int_{D\cap B_\epsilon(x)} dy}. $$ \(\phi(x,t;u)\) shows extent of damage experienced by material point \(x\) at time \(t\).

1. Shear loading

We consider generic values of various parameters in the model and apply shear loading to the material. The boundary condition is described in the image below

We plot the value of \(\phi(x,t;u)\) at all mesh points at final time step. Video of simulation is also attached.

2. Bending loading

We apply three-point and four-point bending load on beam of length \(1\) and width \(0.15\). Beam is supported at two points in bottom. In case of three-point loading, monotonically increasing force is applied at center mesh node on top. In case of four-point loading, the monotonically increasing force is applied at two points on top, see picture below. Following image shows the setup for four-point test.

We plot the value of \(\phi(x,t;u)\) at all mesh points at final time step for four-point test. Video of simulation is attached for both four-point and three-point test.

## Journal papers

Below are research publications which are either submitted to the journal or are published.

[2017-1] Numerical analysis of nonlocal fracture models in Hölder space. Prashant K. Jha and Robert Lipton. SIAM Journal on Numerical Analysis. 56(2), 906-941, DOI. Published on April 10, 2018, submitted on January 18, 2017. Print pdf. Link to preprint: arxiv_preprint.

[2017-2] Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics. Prashant K. Jha and Robert Lipton. International Journal for Numerical Methods in Engineering. 114(13), 1389-1410, DOI. Published on March 12, 2018, submitted on July 1, 2017. Print pdf. Link to preprint: arxiv_preprint.

[2017-3] Free damage propagation with memory. Robert Lipton, Eyad Said and Prashant K. Jha. Journal of Elasticity. DOI. Published on March 14, 2018, submitted on July 15, 2017. Print pdf.

[2017-4] Finite element approximation of nonlocal fracture models. Prashant K. Jha and Robert Lipton. Submitted for review in IMA Journal of Numerical Analysis on January 9, 2018. Link to preprint: arxiv_preprint.

[2018-1] Finite difference approximation for state based Peridynamic fracture models. Prashant K. Jha and Robert Lipton. Yet to submit to journal for a review. Link to preprint: arxiv_preprint.

[2018-2] Implementation of Peridynamics utilizing HPX -- the C++ standard library for parallelism and concurrency. Patrick Diehl, Prashant K. Jha, Hartmut Kaiser, Robert Lipton and Martin Levesque. Submitted for review on June 18, 2018. Link to preprint: arxiv_preprint.

## Book chapters

Below are the book chapters submitted for publication.

[2018-1] Well posed nonlinear nonlocal fracture models associated with double well potentials. Prashant K. Jha and Robert Lipton. Book: Handbook of Nonlocal Continuum Mechanics for Materials and Structures. In production.

[2018-2] Finite difference and finite element in nonlocal fracture modeling: A-priori convergence rates. Prashant K. Jha and Robert Lipton. Book: Handbook of Nonlocal Continuum Mechanics for Materials and Structures. In production.

[2018-3] Dynamic brittle fracture from non-local double well potentials: A state based model. Robert Lipton, Eyad Said and Prashant K. Jha. Book: Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Published March 2018. DOI. Print pdf.

[2018-4] Dynamic damage propagation with memory: A state based model. Robert Lipton, Eyad Said and Prashant K. Jha. Book: Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Published March 2018. DOI. Print pdf.