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Peridynamics is a formulation of
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such ...
that is oriented toward
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defo ...
s with discontinuities, especially fractures.


Purpose

The peridynamic theory is based on
integral equation In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n ...
s, in contrast with the classical theory of continuum mechanics, which is based on
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s. Since
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s do not exist on crack surfaces and other singularities, the classical equations of
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such ...
cannot be applied directly when such features are present in a
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defo ...
. 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 Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics ...
. For example, in peridynamics, there is no need for a separate crack growth law based on a
stress intensity factor In fracture mechanics, the stress intensity factor () is used to predict the stress state ("stress intensity") near the tip of a crack or notch caused by a remote load or residual stresses. It is a theoretical construct usually applied to a h ...
.


Definition and basic terminology

The basic equation of peridynamics is the following
equation of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
: :\rho(x)\ddot u(x,t)=\int_R f(u(x',t)-u(x,t),x'-x,x)dV_ + b(x,t) where x is a point in a body R, t is time, u is the
displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a s ...
field, and \rho is the mass density in the undeformed body. x' is a dummy variable of integration. The
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
valued
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
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
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
s of f are force per volume squared. The function f is called the "pairwise force function" and contains all the
constitutive Constitutive may refer to: * In physics, a constitutive equation is a relation between two physical quantities * In ecology, a constitutive defense is one that is always active, as opposed to an inducible defense * Constitutive theory of statehood ...
(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 Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
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 In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy ...
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.


Peridynamic states

{{POV check, section, date=November 2012 The theory described above assumes that each peridynamic bond responds independently of all the others. This is an oversimplification for most materials and leads to restrictions on the types of materials that can be modeled. In particular, this assumption implies that any isotropic linear elastic solid is restricted to a Poisson ratio of 1/4. To address this lack of generality, the idea of "peridynamic states" was introduced. This allows the force density in each bond to depend on the stretches in all the bonds connected to its endpoints, in addition to its own stretch. For example, the force in a bond could depend on the net volume changes at the endpoints. The effect of this volume change, relative to the effect of the bond stretch, determines the Poisson ratio. With peridynamic states, any material that can be modeled within the standard theory of
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such ...
can be modeled as a peridynamic material, while retaining the advantages of the peridynamic theory for fracture. One can find extended discussion of the integral form of equations of solid body mechanics and restrictions on the form of kernel in I.A.Kunin “Theory of elastic media with microstructure. Nonlocal theory of elasticity. 1975 (In Russian); I. A. Kunin, Elastic Media with Microstructure I. One-Dimensional Models (Springer, Berlin, 1982); I. A. Kunin, Elastic Media with Microstructure II. Three-Dimensional Models (Springer, Berlin, 1983)(In English).


See also

*
Fracture mechanics Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics ...
*
Continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such ...
* Movable cellular automaton


External links


Peridigm, an open-source computational peridynamics code

website on peridynamics

PeriDoX open-source repository for peridynamics and its documentation

Implementation of finite element and finite difference approximation of Nonlocal models
Continuum mechanics Fracture mechanics