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(1) is satisfied.The most general form of the PFF f for a microelastic material isf (η, ξ ) = H(p, ξ )(η+ξ ) p = |η+ξ |, (11)where H is a scalar-valued function of (p, ξ ). In this case, there is a micropotential w(p, ξ ) such thatH(p, ξ ) = 1p∂w∂p(p, ξ ). (12)The form of f in (11) implies that the force between any two points depends only on the distancebetween them in the deformed configuration. Thus, we may interpret a microelastic material as a materialin which each two points is connected by a spring that, in general, may be nonlinear. If a microelasticmaterial is isotropic, then (8) implies that H and w only depend on p and r , the magnitude of ξ . The final specialization on the PFF that we will adopt is a further development of the notion thatparticles in microelastic materials may be considered to be connected by springs. We consider materialswith a PFF having its magnitude proportional to the stretch s, wheres = p −rr, p = |η+ξ |, and r = |ξ |. (13)Materials having magnitudes proportional to the stretch are called proportional materials. The mostgeneral form of the PFF for proportional, microelastic materials isf (η, ξ ) = g(s, r )p(η+ξ ), (14)where g(s, r ) is a piecewise linear function of s. The function g is called the bond force betweentwo particles for a microelastic, proportional peridynamic material. Figure 3 shows the bond forcedependence on bond stretch for such materials. This figure also shows the bond force for a microplasticmaterial. The behavior of peridynamic microelastic and microplastic materials differ only on unloading.A microelastic material unloads reversibly back to zero stretch, while a microplastic material that isstretched beyond the elastic limit will retain some stretch when unloaded.Figure 3 shows a linear dependence with nonzero slope in the elastic regime and constant bond forcewhen the yield strength magnitude is exceeded in tension or compression. This figure also indicatesbond failure at some value of bond stretch. Proportional peridynamic materials fail irreversibly whenthe stretch exceeds a value, sc, called the critical stretch. Not only does the critical stretch define failureof a material, but it also assures the existence of a horizon for proportional materials. A microelastic,proportional material behaves reversibly as long as the stretch does not exceed the critical stretch.Let Vd (x, t) denote the volume of the material that connected to x by bonds that have been brokenand V0(x) denote the volume of material initially connected to x. Then the damage D(x, t) is definedbyD(x, t) = Vd (x, t)/V0(x). (15)3. Numerical methodTo solve the fundamental peridynamics equation of motion, (1), the domain is discretized into a set ofnodes, {xi }, as depicted in Figure 4. Each node has a known volume in the reference configuration. Thenodes form a computational grid. where ρi = ρ(xi ), uni= u(xi , tn), bni= b(xi , tn), and Vp is the volume of node p. The sum is taken overall nodes within the horizon δ of xi , {x p : |x p −xi | < δ}.The acceleration term in (16) is approximated by an explicit central difference:uni= un+1i−2uni+un−1i(1t)2, (17)where 1t is the constant time step. The error in (17) is well known to be second order in time [Sillingand Askari 2005].Silling and Askari [2005] investigated the accuracy of (16) with grid spacing and stability for a lin-earized PFF, wheref (η, ξ ) = C(ξ )η, where C(ξ ) = ∂ f (0, ξ )∂η. (18)In (18), C(ξ ) is called the micromodulus. The micromodulus is a second order tensor or 3×3 matrix.They showed that if the micromodulus and displacement u are twice continuously differentiable, then(16) is second-order accurate in grid spacing. Otherwise, if there are discontinuities in C or u, (16) isfirst-order accurate in grid spacing. They also presented a stability criterion for the time step 1t , whichstates that the maximum time step is of order of the horizon. Hence, the maximum stable time step islimited by the horizon rather than the grid spacing. Alternative approaches to integrating the peridynamicequation of motion, along with some important mathematical properties of one-dimensional solutions,are discussed by Weckner and Abeyaratne [2005] and Emmrich and Weckner [2007].4. The EMU computer codeEMU is the first computer code that is based on the peridynamic theory of continuum mechanics. Itis written in Fortran 90 and executes on computers operating under Unix/Linux or Windows operatingsystems and on parallel computers using MPI (message passing interface).