The peridynamic formulation for transient heat conduction

In bodies where discontinuities, like cracks, emerge and interact, the classical equations for heat and mass transfer are not well suited. We propose a peridynamic model for transient heat (or mass) transfer which is valid when the body undergoes damage or evolving cracks. We use a constructive approach to find the peridynamic formulation for heat transfer and test the numerical convergence to the classical solutions in the limit of the horizon (the nonlocal parameter) going to zero for several one-dimensional problems with different types of boundary conditions. We observe an interesting property of the peridynamic solution: when two m-convergence curves, corresponding to two different horizons, for the solution at a point and an instant intersect, the intersection point is also the exact classical (local) solution. The present formulation can be easily extended to higher dimensions and be coupled with the mechanical peridynamic description for thermomechanical analyses of fracturing bodies, or for heat and mass transfer in bodies with evolving material discontinuities.

[1]  Steven J. Plimpton,et al.  Implementing peridynamics within a molecular dynamics code , 2007, Comput. Phys. Commun..

[2]  A. C. Eringen,et al.  Nonlocal theory of wave propagation in thermoelastic plates , 1991 .

[3]  T. L. Warren,et al.  A non-ordinary state-based peridynamic method to model solid material deformation and fracture , 2009 .

[4]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[5]  S. Silling,et al.  Peridynamic modeling of membranes and fibers , 2004 .

[6]  P. Furmański.  A mixture theory for heat conduction in heterogeneous media , 1994 .

[7]  Jifeng Xu,et al.  Peridynamic Analysis of Impact Damage in Composite Laminates , 2008 .

[8]  S. Silling,et al.  A meshfree method based on the peridynamic model of solid mechanics , 2005 .

[9]  Huajian Gao,et al.  Modern topics and challenges in dynamic fracture , 2005 .

[10]  P. Furmański.  Effective macroscopic description for heat conduction in heterogeneous materials , 1992 .

[11]  Mahan,et al.  Nonlocal theory of thermal conductivity. , 1988, Physical review. B, Condensed matter.

[12]  I. Kunin,et al.  Elastic Media with Microstructure I: One-Dimensional Models , 1982 .

[13]  Isaac Shnaid,et al.  Thermodynamically consistent description of heat conduction with finite speed of heat propagation , 2003 .

[14]  Dominic G.B. Edelen,et al.  Irreversible thermodynamics of nonlocal systems , 1974 .

[15]  Milan Jirásek,et al.  Nonlocal integral formulations of plasticity and damage : Survey of progress , 2002 .

[16]  Huajian Gao,et al.  Taylor-based nonlocal theory of plasticity: numerical studies of the micro-indentation experiments and crack tip fields , 2001 .

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

[18]  Daniel Rittel,et al.  Experimental investigation of transient thermoplastic effects in dynamic fracture , 1998 .

[19]  George Z. Voyiadjis,et al.  Gradient plasticity theory with a variable length scale parameter , 2005 .

[20]  R. Lehoucq,et al.  Convergence of Peridynamics to Classical Elasticity Theory , 2008 .

[21]  A. Cemal Eringen,et al.  Theory of nonlocal thermoelasticity , 1974 .

[22]  Norman A. Fleck,et al.  A phenomenological theory for strain gradient effects in plasticity , 1993 .

[23]  M. Greenberg.  Advanced Engineering Mathematics , 1988 .

[24]  Christer Persson,et al.  In-situ ESEM study of thermo-mechanical fatigue crack propagation , 2008 .

[25]  J. Virmont,et al.  Nonlocal heat transport due to steep temperature gradients , 1983 .

[26]  Stewart Andrew Silling,et al.  Dynamic fracture modeling with a meshfree peridynamic code , 2003 .

[27]  A. Eringen,et al.  On nonlocal elasticity , 1972 .

[28]  F. Bobaru.  Influence of van der Waals forces on increasing the strength and toughness in dynamic fracture of nanofibre networks: a peridynamic approach , 2007 .

[29]  G. Voyiadjis,et al.  Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments , 2004 .

[30]  S. Silling,et al.  Peridynamics via finite element analysis , 2007 .

[31]  S. Silling,et al.  Convergence, adaptive refinement, and scaling in 1D peridynamics , 2009 .

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

[33]  Kaushik Bhattacharya,et al.  Kinetics of phase transformations in the peridynamic formulation of continuum mechanics , 2006 .

[34]  Miroslav Grmela,et al.  Finite-speed propagation of heat: a nonlocal and nonlinear approach , 1998 .

[35]  Anthony G. Evans,et al.  A critical assessment of theories of strain gradient plasticity , 2009 .

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

[37]  S. L. Sobolev,et al.  Equations of transfer in non-local media , 1994 .

[38]  S. Silling,et al.  Peridynamic States and Constitutive Modeling , 2007 .