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It does not contain any spatialderivatives. The specific force term is a functional depending on a vector-valued function, f , calledthe pairwise force function (PFF). The PFF gives the force per unit volume squared on a particle at xdue to a particle at x0. In conventional continuum-mechanics theory, this functional is the pergence ofthe stress tensor. All constitutive properties of a material are given by specifying the PFF.Figure 1 shows a sphere Sδ(x) of radius δ centered at the point x. It is convenient to assume thatthere is a distance δ such that the PFF function vanishes outside Sδ(x) for each point x in the domain ofanalysis. The quantity δ is called the horizon since a particle cannot “see” a force beyond its horizon. Thisassumption is natural for the types of materials that are used for extreme-loading analysis as discussedbelow. It is also reasonable for forces that die off sufficiently rapidly or become shielded by interveningparticles.The appropriate value of δ depends on the physical nature of the application. At the nanoscale, itis determined by the distance over which physical interactions between atoms or molecules occur. Atthe macroscale, it is somewhat arbitrary since, for any given value of δ, the parameters in the PFF canbe chosen to match the bulk-elastic properties of the material, as well as its most important fractureproperties [Silling and Askari 2005]. However, in numerical modeling, typically δ is chosen to be threetimes the grid spacing. Values much smaller than this value result in undesirable grid effects such ascracks growing along the rows or columns of the grid. Values much larger than this may result inexcessive wave dispersion and computational time.Several properties of materials follow from the form of the PFF in (1). First, the dependence of f onx − x0implies that the materials are homogeneous. Here, we will not consider extensions of peridynamictheory for nonhomogeneous materials. Second, the materials have no memory of their deformationhistory. Modeling materials with memory is a potential area for future research and development ofperidynamic theory. Perhaps the fractional calculus [Oldham and Spanier 1974] can be useful since thefractional calculus has been useful for rheology of materials with memory (see articles in [Hilfer 2000]).Consider the functional dependence of PFF shown in (1). It is convenient to express the PFF in termsof the new set of variables, ξ and η, whereξ = x0− x and η = u(x0, t)−u(x, t). (2)ξ is the relative position of particles at x and x0in the reference configuration and η is the difference indisplacements at these points.Figure 2 illustrates the relationship among the variables introduced thus far.In this figure and elsewhere, u0= u(x0, t). From this figure, η+ξ is the relative position of the particlesoriginally at x, x0in the deformed configuration.In terms of the new variables η and ξ , the PFF f becomes Newton’s laws not only lead to the fundamental equation of peridynamics but also imply anotherproperty of the PFF f . Newton’s third law states that the force on a particle at x due to a particle at x0must be the negative of the force on a particle at x0due to a particle at x. Therefore for (1) to satisfyNewton’s third law, f (η, ξ ) must satisfyf (−η,−ξ ) = − f (η, ξ ), for all η, ξ . (4)Thus, f (η, ξ ) is an odd function of (η, ξ ).The next property of the PFF follows from the requirement to conserve angular momentum in theabsence of external forces. If angular momentum were not conserved, then two particles initially at restwould move even in the absence of external forces. Thus, to ensure conservation of angular momentum,f (η, ξ ) must satisfy(η+ξ )× f (η, ξ ) = 0, for all η, ξ , (5)where “×” is the cross product. This expression implies that the force between any two particles mustbe parallel to their current relative position. Therefore, (5) implies that f (η, ξ ) must have the functionalformf (η, ξ ) = F(η, ξ )(η+ξ ), for all η, ξ , (6)where F is a scalar-valued function. Since f (η, ξ ) is an odd function of (η, ξ ), F(η, ξ ) must be an evenfunction of (η, ξ ).Gravity is important to determine the long-term consequences of impacts to structures. Before con-tinuing the discussion of material modeling in peridynamics, we note that gravity is included in (1) as abody-force density. The body-force density for gravity, bgravity is given bybgravity = ρ(x)g, (7)where g is the acceleration vector due to gravity with magnitude g = 9.814m/s2.In the previous discussion, we showed that Newton’s third law and the physical requirement to con-serve angular momentum in the absence of external forces imply that the PFF must be in the directionof the relative displacement in the deformed configuration as given by (6). Here, we develop the general form of the PFF for materials that are used in loading analyses. Although many of these results werefirst published in [Silling 2000], we provide this short summary and discussion for completeness and toprovide results that are used in Section 6 to model gases as peridynamic materials.All constitutive properties of a material are given by specifying the PFF. Previously, we showed thatthe dependence of the PFF f on ξ implies that the material is homogeneous. If a material is isotropic,then there is a further restriction on the functional dependence of f . Proper behavior of f under theorthogonal rotation group in three dimensions implies that F in (6) must be invariant under this group.Hence, the scalar function F depends only on the length of η, the angle between η and ξ , and the lengthof ξ . However, we find it more convenient to use the length of η+ξ rather than η. Therefore, for isotropicmaterials, the most general form for F isF(η, ξ ) = I (p, q, r ), where p = |η+ξ |, q = η • ξ , and r = |ξ |, (8)where I is some scalar-valued function and “•” is the dot product.A peridynamic material is said to be microelastic if and only if there exists a scalar-valued functionw(η, ξ ) such thatf (η, ξ ) = ∂w∂η(η, ξ ). (9)The function w is called the micropotential. It is important to realize that the derivatives in (9) are notthe spatial derivatives that are to be avoided by using peridynamic theory.The micropotential has units of energy per unit volume squared. It represents potential energy densityassociated with a bond. We may define a functional that for a displacement u is the local displacementenergy density,Wu(x, t) = 12ZZZ KRw(u0−u, x0− x)dV0. (10)Equation (10) is the energy density at the point x and time t associated with a displacement u of all theparticles in domain R. The factor of 1/2 is present since only half the energy is associated with eachendpoint of the bond.Integration of (10) over the body yields the total macroscopic energy functional. Silling [2000] showedthat the time rate of change of the kinetic energy plus the time rate of change of the total macroscopicenergy given by the integral of (10) over the body equals the work on the body done by external forceswhen the fundamental equation of peridynamics