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# Peridynamics

Peridynamics is a formulation of continuum mechanics that is oriented toward deformations with discontinuities, especially fractures.

The peridynamic theory is based on integral equations, in contrast with the classical theory of continuum mechanics, which is based on partial differential equations. Since partial derivatives do not exist on crack surfaces and other singularities, the classical equations of continuum mechanics cannot be applied directly when such features are present in a deformation. The integral equations of the peridynamic theory can be applied directly, because they do not require partial derivatives.

The ability to apply the same equations directly at all points in a mathematical model of a deforming structure helps the peridynamic approach avoid the need for the special techniques of fracture mechanics. For example, in peridynamics, there is no need for a separate crack growth law based on a stress intensity factor.

The basic equation of peridynamics is the following equation of motion:

$\rho(x)\ddot u(x,t)=\int_R f(u(x',t)-u(x,t),x'-x,x)dV_{x'} + b(x,t)$

where x is a point in a body R, t is time, u is the displacement vector field, and ρ is the mass density in the undeformed body. x' is a dummy variable of integration.

The vector valued function f is the force density that x' exerts on x. This force density depends on the relative displacement and relative position vectors between x' and x. The dimensions of f are force per volume squared. The function f is called the "pairwise force function" and contains all the constitutive (material-dependent) properties. It describes how the internal forces depend on the deformation.

The interaction between any x and x' is called a "bond." The physical mechanism in this interaction need not be specified. It is usually assumed that f vanishes whenever x' is outside a neighborhood of x (in the undeformed configuration) called the horizon.

The term "peridynamic," an adjective, was proposed in the year 2000 and comes from the prefix peri, which means all around, near, or surrounding; and the root dyna, which means force or power. The term "peridynamics," a noun, is a shortened form of the phrase peridynamic model of solid mechanics.

## Pairwise force functions

Using the abbreviated notation u = u(x,t) and u' = u(x',t) Newton's third law places the following restriction on f:

$\displaystyle f(u-u', x-x', x') = -f(u'-u, x'-x, x)$

for any x,x',u,u'. This equation states that the force density vector that x exerts on x' equals minus the force density vector that x' exerts on x. Balance of angular momentum requires that f be parallel to the vector connecting the deformed position of x to the deformed position of x':

$\displaystyle ((x'+u')-(x+u))\times f(u'-u, x'-x, x)=0$.

A pairwise force function is specified by a graph of | f | versus bond elongation e, defined by

$\displaystyle e=|(x'+u')-(x+u)|-|x'-x|$.

A schematic of a pairwise force function for the bond connecting two typical points is shown in the following figure:

## Damage

Damage is incorporated in the pairwise force function by allowing bonds to break when their elongation exceeds some prescribed value. After a bond breaks, it no longer sustains any force, and the endpoints are effectively disconnected from each other. When a bond breaks, the force it was carrying is redistributed to other bonds that have not yet broken. This increased load makes it more likely that these other bonds will break. The process of bond breakage and load redistribution, leading to further breakage, is how cracks grow in the peridynamic model.

## Bibliography

♦ S. A. Silling, "Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces," Journal of the Mechanics and Physics of Solids, Vol. 48 (2000) 175-209. DOI: 10.1016/S0022-5096(99)00029-0

♦ S. A. Silling, M. Zimmermann, and R. Abeyaratne, "Deformation of a Peridynamic Bar," Journal of Elasticity, Vol. 73 (2003) 173-190. DOI: 10.1023/B:ELAS.0000029931.03844.4f

♦ S. A. Silling and F. Bobaru, "Peridynamic Modeling of Membranes and Fibers," International Journal of Non-Linear Mechanics, Vol. 40 (2005) 395-409. DOI:10.1016/j.ijnonlinmec.2004.08.004

♦ O. Weckner and R. Abeyaratne, "The Effect of Long-Range Forces on the Dynamics of a Bar," Journal of the Mechanics and Physics of Solids, Vol. 53 (2005) 705-728. DOI: 10.1016/j.jmps.2004.08.006

♦ S. A. Silling and E. Askari, "A Meshfree Method Based on the Peridynamic Model of Solid Mechanics," Computers and Structures, Vol. 83 (2005) 1526-1535. DOI:10.1016/j.compstruc.2004.11.026

♦ K. Dayal and K. Bhattacharya, "Kinetics of Phase Transformations in the Peridynamic Formulation of Continuum Mechanics," Journal of the Mechanics and Physics of Solids, Vol. 54 (2006) 1811-1842. DOI: I0.1016/j.jmps.2006.04.001

♦ W. Gerstle, N. Sau, and S. Silling, "Peridynamic Modeling of Concrete Structures," Nuclear Engineering and Design, Vol. 237 (2007) 1250-1258. DOI: 10.1016/j.nucengdes.2006.10.002

♦ S. A. Silling, M. Epton, O. Weckner, J. Xu and E. Askari, "Peridynamic States and Constitutive Modeling," Journal of Elasticity, Vol. 88 (2007) 151-184. DOI: 10.1007/s10659-007-9125-1

♦ F. Bobaru, "Influence of van der Waals forces on increasing the strength and toughness in dynamic fracture of nanofibre networks: a peridynamic approach," Modelling and Simulation in Materials Science and Engineering, Vol. 15 (2007) 397-417. DOI: 10.1088/0965-0393/15/5/002