Stability of Nonlocal Dirichlet Integrals and Implications for Peridynamic Correspondence Material Modeling

Nonlocal gradient operators are basic elements of nonlocal vector calculus that play important roles in nonlocal modeling and analysis. In this work, we extend earlier analysis on nonlocal gradient operators. In particular, we study a nonlocal Dirichlet integral that is given by a quadratic energy functional based on nonlocal gradients. Our main finding, which differs from claims made in previous studies, is that the coercivity and stability of this nonlocal continuum energy functional may hold for some properly chosen nonlocal interaction kernels but may fail for some other ones. This can be significant for possible applications of nonlocal gradient operators in various nonlocal models. In particular, we discuss some important implications for the peridynamic correspondence material models.

[1]  Richard B. Lehoucq,et al.  Force flux and the peridynamic stress tensor , 2008 .

[2]  Fahad Almutairi,et al.  Nonlocal vector calculus , 2018 .

[3]  J. Morel,et al.  On image denoising methods , 2004 .

[4]  D. Z. Turner,et al.  A Nonlocal Strain Measure for Digital Image Correlation. , 2014, 1409.2586.

[5]  Qiang Du,et al.  Mathematical Models and Methods in Applied Sciences c ○ World Scientific Publishing Company Sandia National Labs SAND 2010-8353J A NONLOCAL VECTOR CALCULUS, NONLOCAL VOLUME-CONSTRAINED PROBLEMS, AND NONLOCAL BALANCE LAWS , 2022 .

[6]  R. Lehoucq,et al.  Peridynamic Theory of Solid Mechanics , 2010 .

[7]  Kun Zhou,et al.  Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints , 2012, SIAM Rev..

[8]  Eitan Tadmor,et al.  Critical thresholds in flocking hydrodynamics with non-local alignment , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[9]  Guy Gilboa,et al.  Nonlocal Operators with Applications to Image Processing , 2008, Multiscale Model. Simul..

[10]  R. Lehoucq,et al.  Peridynamics for multiscale materials modeling , 2008 .

[11]  Jean-Michel Morel,et al.  Image Denoising Methods. A New Nonlocal Principle , 2010, SIAM Rev..

[12]  Roger Fosdick,et al.  A constitutive model for a linearly elastic peridynamic body , 2014 .

[13]  S. Silling.  Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces , 2000 .

[14]  Qiang Du,et al.  On the variational limit of a class of nonlocal functionals related to peridynamics , 2015 .

[15]  N. SIAMJ.,et al.  ANALYSIS AND COMPARISON OF DIFFERENT APPROXIMATIONS TO NONLOCAL DIFFUSION AND LINEAR PERIDYNAMIC EQUATIONS∗ , 2013 .

[16]  Stanley Osher,et al.  Image Recovery via Nonlocal Operators , 2010, J. Sci. Comput..

[17]  Philippe H. Geubelle,et al.  Non-ordinary state-based peridynamic analysis of stationary crack problems , 2014 .

[18]  Kun Zhou,et al.  Mathematical and Numerical Analysis of Linear Peridynamic Models with Nonlocal Boundary Conditions , 2010, SIAM J. Numer. Anal..

[19]  Qiang Du,et al.  Asymptotically Compatible Schemes and Applications to Robust Discretization of Nonlocal Models , 2014, SIAM J. Numer. Anal..

[20]  Stewart Andrew Silling,et al.  Crack nucleation in a peridynamic solid , 2010 .

[21]  Ted Belytschko,et al.  Overview and applications of the reproducing Kernel Particle methods , 1996 .

[22]  S. Silling.  Stability of peridynamic correspondence material models and their particle discretizations , 2016 .

[23]  Olaf Weckner,et al.  The effect of long-range forces on the dynamics of a bar , 2005 .

[24]  Qiang Du,et al.  Analysis of a nonlocal-in-time parabolic equation , 2016 .

[25]  Qiang Du,et al.  The bond-based peridynamic system with Dirichlet-type volume constraint , 2014, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[26]  J. Bourgain,et al.  Another look at Sobolev spaces , 2001 .

[27]  Zdeněk P. Bažant,et al.  Wave Dispersion and Basic Concepts of Peridynamics Compared to Classical Nonlocal Damage Models , 2016 .

[28]  Ted Belytschko,et al.  A meshfree unification: reproducing kernel peridynamics , 2014, Computational Mechanics.

[29]  Florin Bobaru,et al.  The peridynamic formulation for transient heat conduction , 2010 .

[30]  Qiang Du,et al.  Nonlocal Constrained Value Problems for a Linear Peridynamic Navier Equation , 2014 .

[31]  J. Monaghan.  Smoothed particle hydrodynamics , 2005 .

[32]  Raul Radovitzky,et al.  An extended constitutive correspondence formulation of peridynamics based on nonlinear bond-strain measures , 2014 .

[33]  John T. Foster.  Constitutive Modeling in Peridynamics , 2016 .

[34]  Stewart Andrew Silling,et al.  Linearized Theory of Peridynamic States , 2010 .

[35]  Augusto C. Ponce,et al.  An estimate in the spirit of Poincaré's inequality , 2004 .

[36]  Qiang Du,et al.  ANALYSIS OF A SCALAR PERIDYNAMIC MODEL WITH A SIGN CHANGING KERNEL , 2013 .

[37]  Qiang Du,et al.  Robust a posteriori stress analysis for quadrature collocation approximations of nonlocal models via nonlocal gradients , 2016 .