arXiv subjects

Prashant K. Jha

Publications and source records attributed to Prashant K. Jha.

22 records · Page 2Linked to original sources

arXiv2018

Finite element approximation of nonlocal fracture models

We consider nonlocal nonlinear potentials and estimate the rate of convergence of time stepping schemes to the peridynamic equation of motion. We begin by establishing the existence of $H^2$ solutions over any finite time interval. Here spatial approximation by finite element interpolations are considered. The energy stability of the associated semi-discrete time stepping scheme is established and the approximation of strong and weak formulations of the evolution using FE interpolations of $H^2$ solutions are investigated. The strong and weak form of approximations are shown to converge to the actual solution in the mean square norm at the rate $C_tΔt +C_s h^2/ε^2$ where $h$ is the mesh size, $ε$ is the size of nonlocal interaction and $Δt$ is the time step. The constants $C_t$ and $C_s$ are independent of $Δt$, and $h$. In the absence of nonlinearity a CFL like condition for the energy stability of the central difference time discretization scheme is developed.

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arXiv2018

Numerical convergence of finite difference approximations for state based peridynamic fracture models

In this work, we study the finite difference approximation for a class of nonlocal fracture models. The nonlocal model is initially elastic but beyond a critical strain the material softens with increasing strain. This model is formulated as a state-based perydynamic model using two potentials: one associated with hydrostatic strain and the other associated with tensile strain. We show that the dynamic evolution is well-posed in the space of Hölder continuous functions $C^{0,γ}$ with Hölder exponent $γ\in (0,1]$. Here the length scale of nonlocality is $ε$, the size of time step is $Δt$ and the mesh size is $h$. The finite difference approximations are seen to converge to the Hölder solution at the rate $C_t Δt + C_s h^γ/ε^2$ where the constants $C_t$ and $C_s$ are independent of the discretization. The semi-discrete approximations are found to be stable with time. We present numerical simulations for crack propagation that computationally verify the theoretically predicted convergence rate. We also present numerical simulations for crack propagation in precracked samples subject to a bending load.

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arXiv2018

Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics

We quantify the numerical error and modeling error associated with replacing a nonlinear nonlocal bond-based peridynamic model with a local elasticity model or a linearized peridynamics model away from the fracture set. The nonlocal model treated here is characterized by a double well potential and is a smooth version of the peridynamic model introduced in n Silling (J Mech Phys Solids 48(1), 2000). The solutions of nonlinear peridynamics are shown to converge to the solution of linear elastodynamics at a rate linear with respect to the length scale $ε$ of non local interaction. This rate also holds for the convergence of solutions of the linearized peridynamic model to the solution of the local elastodynamic model. For local linear Lagrange interpolation the consistency error for the numerical approximation is found to depend on the ratio between mesh size $h$ and $ε$. More generally for local Lagrange interpolation of order $p\geq 1$ the consistency error is of order $h^p/ε$. A new stability theory for the time discretization is provided and an explicit generalization of the CFL condition on the time step and its relation to mesh size $h$ is given. Numerical simulations are provided illustrating the consistency error associated with the convergence of nonlinear and linearized peridynamics to linear elastodynamics.

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arXiv2018

Numerical analysis of nonlocal fracture models in Hölder space

In this work, we calculate the convergence rate of the finite difference approximation for a class of nonlocal fracture models. We consider two point force interactions characterized by a double well potential. We show the existence of a evolving displacement field in Hölder space with Hölder exponent $γ\in (0,1]$. The rate of convergence of the finite difference approximation depends on the factor $C_s h^γ/ε^2$ where $ε$ gives the length scale of nonlocal interaction, $h$ is the discretization length and $C_s$ is the maximum of Hölder norm of the solution and its second derivatives during the evolution. It is shown that the rate of convergence holds for both the forward Euler scheme as well as general single step implicit schemes. A stability result is established for the semi-discrete approximation. The Hölder continuous evolutions are seen to converge to a brittle fracture evolution in the limit of vanishing nonlocality.

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