# Peridynamic theory of solid mechanics.

## Abstract

The peridynamic theory of mechanics attempts to unite the mathematical modeling of continuous media, cracks, and particles within a single framework. It does this by replacing the partial differential equations of the classical theory of solid mechanics with integral or integro-differential equations. These equations are based on a model of internal forces within a body in which material points interact with each other directly over finite distances. The classical theory of solid mechanics is based on the assumption of a continuous distribution of mass within a body. It further assumes that all internal forces are contact forces that act across zero distance. The mathematical description of a solid that follows from these assumptions relies on partial differential equations that additionally assume sufficient smoothness of the deformation for the PDEs to make sense in either their strong or weak forms. The classical theory has been demonstrated to provide a good approximation to the response of real materials down to small length scales, particularly in single crystals, provided these assumptions are met. Nevertheless, technology increasingly involves the design and fabrication of devices at smaller and smaller length scales, even interatomic dimensions. Therefore, it is worthwhile to investigate whether the classical theory canmore »

- Authors:

- Publication Date:

- Research Org.:
- Sandia National Laboratories

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 984146

- Report Number(s):
- SAND2010-1233J

TRN: US201015%%850

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article

- Journal Name:
- Proposed for publication in Advances in Applied Mechanics.

- Additional Journal Information:
- Journal Name: Proposed for publication in Advances in Applied Mechanics.

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 36 MATERIALS SCIENCE; APPROXIMATIONS; ATOMS; DEFORMATION; DESIGN; DIMENSIONS; DISTRIBUTION; FABRICATION; INTEGRO-DIFFERENTIAL EQUATIONS; MONOCRYSTALS; PARTIAL DIFFERENTIAL EQUATIONS; ROUGHNESS; SIMULATION

### Citation Formats

```
Lehoucq, Richard B., and Silling, Stewart Andrew.
```*Peridynamic theory of solid mechanics.*. United States: N. p., 2010.
Web.

```
Lehoucq, Richard B., & Silling, Stewart Andrew.
```*Peridynamic theory of solid mechanics.*. United States.

```
Lehoucq, Richard B., and Silling, Stewart Andrew. Fri .
"Peridynamic theory of solid mechanics.". United States.
```

```
@article{osti_984146,
```

title = {Peridynamic theory of solid mechanics.},

author = {Lehoucq, Richard B. and Silling, Stewart Andrew},

abstractNote = {The peridynamic theory of mechanics attempts to unite the mathematical modeling of continuous media, cracks, and particles within a single framework. It does this by replacing the partial differential equations of the classical theory of solid mechanics with integral or integro-differential equations. These equations are based on a model of internal forces within a body in which material points interact with each other directly over finite distances. The classical theory of solid mechanics is based on the assumption of a continuous distribution of mass within a body. It further assumes that all internal forces are contact forces that act across zero distance. The mathematical description of a solid that follows from these assumptions relies on partial differential equations that additionally assume sufficient smoothness of the deformation for the PDEs to make sense in either their strong or weak forms. The classical theory has been demonstrated to provide a good approximation to the response of real materials down to small length scales, particularly in single crystals, provided these assumptions are met. Nevertheless, technology increasingly involves the design and fabrication of devices at smaller and smaller length scales, even interatomic dimensions. Therefore, it is worthwhile to investigate whether the classical theory can be extended to permit relaxed assumptions of continuity, to include the modeling of discrete particles such as atoms, and to allow the explicit modeling of nonlocal forces that are known to strongly influence the behavior of real materials.},

doi = {},

journal = {Proposed for publication in Advances in Applied Mechanics.},

number = ,

volume = ,

place = {United States},

year = {2010},

month = {1}

}