Analysis and Numerical Approximation of an Integro-differential Equation Modeling Non-local Effects in Linear Elasticity

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Long-range interactions for linearly elastic media resulting in nonlinear dispersion relations are modeled by an initial-value problem for an integro-differential equation (IDE) that incorporates non-local effects. Interpreting this IDE as an evolutionary equation of second order, well-posedness in L ∞(ℝ) as well as jump relations are proved. Moreover, the construction of the micromodulus function from the dispersion relation is studied. A numerical approximation based upon quadrature is suggested and carried out for two examples, one involving jump discontinuities in the initial data corresponding to a Riemann-like problem

Analysis and numerical approximation of an integro-differential equation modelling non-local effects in linear elasticity

Long-range interactions for linearly elastic media resulting in nonlinear dispersion relations are modelled by an initial-value problem for an integro-differential equation (IDE) that incorporates non-local effects. Interpreting this IDE as an evolutionary equation of second order, well-posedness in L^{\infty}(\rz) as well as jump relations are proved. A numerical approximation based upon quadrature is suggested and carried out for two examples, one involving jump discontinuities in the initial data corresponding to a Riemann-like problem

Determination of Ballistic Limit for IM7/8552 Using Peridynamics

Significant testing is required to design and certify primary aircraft structure subject to High Energy Dynamic Impact (HEDI) events; current work under the NASA Advanced Composites Consortium (ACC) HEDI Project seeks to determine the state-of-the-art of dynamic fracture simulations for composite structures in these events. This paper discusses one of four Progressive Damage Analysis (PDA) methods selected for this project: peridynamics, through EMU implementation. A brief discussion of peridynamic theory is provided, followed by an outline of ballistic impact testing performed for model development and assessment. Detailed modeling approach and test-analysis correlation for a single open test case are presented, followed by the results of a series of blind predictions made prior to testing and test-analysis correlation performed with measured NASA test results. Specifically, we present simulation results for the ballistic limit (V50) of IM7/8552 composite panels ballistically tested with an impactor representative of a high-velocity fan-blade-out condition. In particular, force and displacement history and the damage state determined analytically are compared to measured results. Ultimately, peridynamics has the ability to predict damage patterns, impact force and deflections during a high energy dynamic impact event on composite panels of different layups using two different types of impactors. Blind predictions were promising and increased confidence in the model for impact simulation. There are open questions regarding the fidelity of the test fixture idealization in regards to stiffness and damping which will need to be addressed in future work

E.: Numerical simulation of the dynamics of a nonlocal, inhomogeneous, infinite bar

Abstract. In this paper, we develop an efficient numerical method based on Gauß-Hermite quadrature to calculate the one-dimensional dynamic response of a nonlocal, peridynamic bar composed of (inhomogeneous) linear material. The principal physical characteristic of the peridynamic theory is the presence of long-range forces leading to nonlinear dispersion relations while the principal mathematical characteristic is the presence of a stationary Barbashin operator in the integro-differential equation of motion. We calculate two examples corresponding to continuous and discontinuous, Riemann-like initial conditions. As the analytical solutions for these examples are known, they serve as validation problems for the proposed numerical scheme. Mathematical Subject Classification: 74H15, 65R20, 45K0

The peridynamic equation and its spatial discretisation

Different spatial discretisation methods for solving the peridynamic equation of motion are suggested. The methods proposed are tested for a linear microelastic material of infinite length in one spatial dimension. Moreover, the conservation of energy is studied for the continuous as well as discretised problem. First Published Online: 14 Oct 201

Analysis and Numerical Approximation of an Integrodifferential Equation Modeling Non-local Effects in Linear Elasticity

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Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot

This study presents an analysis approach based on a merger of the finite element method and the peridynamic theory. Its validity is established through qualitative and quantitative comparisons against the test results for a stiffened composite curved panel with a central slot under combined internal pressure and axial tension. The predicted initial and final failure loads, as well as the final failure modes, are in close agreement with the experimental observations. This approach demonstrates the capability of the PD approach to assess the durability of complex composite structures