--- title: State-Based Peridynamic Formulation url: https://www.emergentmind.com/topics/state-based-peridynamic-formulation type: topic --- # State-Based Peridynamic Formulation State-based peridynamic formulation is a non-local continuum formulation in which the classical local balance laws involving spatial derivatives of the displacement field are replaced by integral balance laws over a finite horizon, so that each material point interacts with all other points in its family through vector-valued “states.” In this setting, the force in a bond may depend on the collective deformation of all bonds within the horizon rather than only on bond stretch, which removes the bond-based restriction on constitutive behavior and supports ordinary, non-ordinary, and correspondence constructions for elasticity, fracture, anisotropy, generalized continua, and coupled fields [1410.8655][2004.11477][2004.14223]. ## 1. Kinematic structure and state variables For a material point \(X\) in a reference body \(B_0\subset\mathbb R^3\), the horizon is written \[ H(X)=\{\,X'\mid \|X'-X\|\le \delta\}\,, \] with \(\delta\) the nonlocal interaction radius. State notation associates to each bond \(\xi=X'-X\) a quantity defined on the family of \(X\). The basic objects are the reference or relative-position state \(\underline{\mathbf X}\langle\xi\rangle=\xi\), the deformation state \(\underline{\mathbf Y}\langle\xi\rangle=y(X',t)-y(X,t)\), and the force state \(\underline{\mathbf T}\langle\xi\rangle\), which assigns a bond-force density to each bond [1710.06827][2004.14223]. In small-strain linearized formulations, the deformation state is often written in displacement form. Examples include the relative displacement state \[ U[X](\xi)=u(X')-u(X), \] the relative micro-rotation state \[ \Theta[X](\xi)=\theta(X')-\theta(X), \] and the averaged micro-rotation state \[ \bar\Theta[X](\xi)=\tfrac12\bigl(\theta(X')+\theta(X)\bigr), \] used in micropolar peridynamics to construct nonlocal approximations of micropolar strain and wryness [1410.8655]. In fracture-oriented linear peridynamic solid formulations, the scalar bond-strain is \[ S(y,x,t;u)=\frac{(u(y,t)-u(x,t))\cdot e_{y-x}}{|y-x|}, \qquad e_{y-x}=\frac{y-x}{|y-x|}, \] and the dilatation is defined by a weighted horizon integral that corresponds to the local divergence in the small-horizon limit [1903.00924]. Two nonlocal tensors recur throughout the literature. The first is the shape tensor, \[ K(X)=\int_{H(X)} \omega(|\xi|)\,\xi\otimes\xi\,dV_{X'}, \] whose inverse appears in correspondence mappings. The second is the nonlocal deformation gradient, \[ F(X)=\Bigl[\int_{H_\delta(X)}\omega(|\xi|)\,\underline Y\langle\xi\rangle\otimes\xi\,dV_{X'}\Bigr]\,K(X)^{-1}, \] or higher-order RK/GMLS variants of the same operator. Under linearized kinematics, \(F-I\approx \nabla u\) [1710.06827][2004.11477]. A basic distinction is between ordinary and non-ordinary state-based models. In ordinary state-based peridynamics, the force vector state is written \[ T[x,t]\{\xi\}=t[x,t]\{\xi\}\,M[x,t]\{\xi\}, \qquad M[x,t]\{\xi\}=\frac{Y[x,t]\{\xi\}}{|Y[x,t]\{\xi\}|}, \] so the pairwise force direction is aligned with the deformed bond. In non-ordinary theories, the traction on a bond is not constrained to lie parallel to the bond direction, which permits direct embedding of general anisotropic constitutive tensors through correspondence [2010.08043][1710.06827]. ## 2. Balance laws and constitutive correspondence The strong-form state-based peridynamic equation of motion has the generic structure \[ \rho(X)\,\ddot u(X,t)=\int_{H(X)} \Bigl\{ T[X,t](\xi)-T[X+\xi,t](-\xi) \Bigr\}\,dV_{X'}+b(X,t), \] or equivalent antisymmetric forms. This replaces the local divergence of stress by a nonlocal internal-force integral while retaining the usual body-force term \(b\) and inertia \(\rho \ddot u\) [1410.8655][1710.06827][2101.01434]. In correspondence models, constitutive structure is imported from classical continuum mechanics by first computing a nonlocal deformation measure and then evaluating a classical stress law. One common form is \[ \underline{\mathbf T} = \omega(\|\underline{\mathbf X}\|;\delta)\; \mathbf P(X)\,\mathbf F(X)^{-T}\,\underline{\mathbf X}, \] or, in the anisotropic small-strain setting, \[ P(x)=C:(F-I),\qquad \underline T[x,t]\langle\xi\rangle = \omega(|\xi|)\,P(x)^T\,B(x)\,\xi. \] Because the full fourth-order stiffness tensor \(C_{ijkl}\) enters directly, generally anisotropic materials can be represented without reducing them to orientation-dependent isotropic bonds [1710.06827][2004.11477]. Energy-based formulations express the constitutive law through Fréchet derivatives of a nonlocal internal energy. For micropolar peridynamics, if the internal energy density \(e(U,\Theta)\) depends on the tensorial deformation measures \(y\) and \(K\), then \[ T[X](\xi)=\frac{\partial e}{\partial U[X](\xi)}, \qquad M[X](\xi)=\frac{\partial e}{\partial \Theta[X](\xi)}. \] With Eringen’s linear elastic micropolar stored energy \[ w(y,K)=\tfrac12\bigl[\lambda (\operatorname{tr}y)^2+2\mu\,y:y+\alpha(\operatorname{tr}K)^2+\beta K:K+\gamma K\cdot K^T\bigr], \] the induced correspondence mapping becomes \[ T[X](\xi)=\omega(|\xi|)\,[\sigma(X)K^{-1}:\xi], \qquad M[X](\xi)=\omega(|\xi|)\,[\mu(X)K^{-1}:\xi], \] with classical micropolar stress and couple-stress tensors recovered pointwise [1410.8655]. Angular momentum requires special attention. In the micropolar extension, angular momentum is an independent balance law with a moment vector-state \(M[X,t](\xi)\), a micro-inertia \(J\), and a body-couple \(\ell\); the resulting formulation exactly enforces the integral angular-momentum law for any subregion. By contrast, the non-ordinary anisotropic correspondence model notes that angular momentum is not automatically conserved unless special symmetry conditions on the stiffness tensor are enforced [1410.8655][1710.06827]. A common misconception is therefore that every state-based formulation automatically satisfies moment balance in the same way; the literature distinguishes clearly between cases where antisymmetry suffices, cases requiring constitutive symmetry, and micropolar theories with explicit couple balance. ## 3. Local limit, asymptotic compatibility, and spatial consistency A central requirement is recovery of the local continuum model as the horizon shrinks. For smooth fields, Taylor expansion of \(u(x+\xi)\) inside the horizon shows that the nonlocal dilatation approaches \(\nabla\cdot u\) and the nonlocal internal-force operator approaches \(\nabla\cdot \sigma\). One formulation states that, for \(u\in C^4(\Omega)\), \[ \rho\,\ddot u=\nabla\cdot\sigma+b+O(\delta^2), \] and under suitable smoothness on \(\lambda,\mu\), the nonlocal solution converges strongly in \(L^2\) to the local solution at rate \(O(\delta^2)\) [2202.06578]. The linear peridynamic solid model is explicitly parameterized so that, as \(\delta\to 0\), the operator recovers the Navier equations \[ L_0u=-\nabla\cdot[\lambda(\operatorname{tr}E)I+2\mu E]=f, \qquad E=\tfrac12(\nabla u+\nabla u^T) \] [2101.01434]. Boundary truncation introduces the classical “surface effect.” In the LPS framework, the ball \(B_\delta(x)\) is truncated near \(\partial\Omega\), so the original nonlocal dilatation does not converge to \(\nabla\cdot u\). A corrected dilatation \[ \theta^{\mathrm{corr}}(x)=\frac{d}{m(\delta)} \int_{B_\delta(x)\cap\Omega} K(|y-x|)\,(y-x)\cdot M(x)\,[u(y)-u(x)]\,dy \] is introduced, with \[ M(x)= \Bigl[ \frac{d}{m(\delta)} \int_{B_\delta(x)\cap\Omega} K(|y-x|)\,(y-x)\otimes(y-x)\,dy \Bigr]^{-1}. \] If \(\partial\Omega\) is \(C^3\) and \(u\in C^2\), then for points within \(\delta\) of the boundary, \[ |\theta^{\mathrm{corr}}(x)-\nabla\cdot u(x)|=O(\delta^2), \] while away from the boundary the original dilatation is recovered [2101.01434]. Traction loading is another nontrivial issue. In the same framework, a consistent nonlocal traction prescription modifies the body-force term for points within \(\delta\) of \(\partial\Omega_N\), and a patch-test result states that for linear \(u\), the Neumann-modified operator reproduces the classical traction exactly if the truncated ball is symmetric and to \(O(\delta)\) otherwise [2101.01434]. A plausible implication is that asymptotic compatibility in state-based peridynamics depends as much on boundary treatment as on the interior constitutive map. Spatial discretization is commonly meshfree. Optimization-based quadrature constructs weights by solving a local constrained minimization problem that reproduces a prescribed polynomial space. For the LPS model, one formulation minimizes \(\sum_j \omega_{j,i}^2\) subject to exactness for kernels generated by \(P_5\), and shows interior truncation error \(O(\delta^2)\) for \(u\in C^3\), \(\theta\in C^2\) [2101.01434]. In heterogeneous brittle fracture, the same strategy is reported as asymptotically compatible provided \(h/\delta\) is bounded, with the discrete operator retaining the \(O(\delta^2)\) local-limit rate [2202.06578]. Rigorous convergence theory has been established for both finite element and finite difference discretizations of state-based fracture models. For linear continuous finite elements, the convergence rate is \[ C_t \Delta t + C_s h^2/\epsilon^2 \] in the mean square norm, with \(C_t\) and \(C_s\) independent of \(\Delta t\) and \(h\) [1903.00924]. In a Hölder-space finite difference analysis, the full-discrete error satisfies \[ E^k=O(\Delta t+h^\gamma/\epsilon^2), \] more precisely \[ \sup_k E^k \le \exp[T(1+L_3/\epsilon^2)]\,T\,[\,C_t\,\Delta t+C_s\,h^\gamma/\epsilon^2\,] \] for \(\gamma\in(0,1]\) [1805.00296]. ## 4. Fracture, damage, and evolving discontinuities One of the main motivations for state-based peridynamics is that fracture discontinuities are handled naturally because forces are transmitted through bond integrals rather than spatial derivatives. The anisotropic non-ordinary formulation states explicitly that crack initiation, propagation, branching, and coalescence can be modeled without special crack-tip enrichment or re-meshing [1710.06827]. In bond-breaking implementations, fracture is introduced by an irreversible degradation variable on each bond. In the asymptotically compatible LPS fracture formulation, the bond strain is \[ s_{ij}= \frac{|(u_j-u_i)+(x_j-x_i)|-|x_j-x_i|}{|x_j-x_i|}, \] and the bond \(ij\) is broken when \(s_{ij}>s_0\), where \(s_0\) is chosen so that the nonlocal fracture energy matches the material \(G_0\). A mask \(\gamma_{j,i}(t)\in\{0,1\}\) is then used to distinguish intact and broken quadrature weights, and “crack surfaces remain perfectly sharp by virtue of broken bonds” [2101.01434]. State-based fracture models also exist in energy-potential form. A class analyzed by Jha and Lipton uses a pairwise bond potential \(\mathcal W^\varepsilon\) based on a smooth convex–concave function \(f\) and a hydrostatic potential \(\mathcal V^\varepsilon\) based on \(g\), which may be quadratic or multi-well. The total force splits into a bond-based part \(\mathcal L_T^\varepsilon\) and a dilatational part \(\mathcal L_D^\varepsilon\), so fracture and cavitation can be associated with softening in either tensile strain or hydrostatic strain [1903.00924][1805.00296]. Composite and anisotropic damage criteria can be superposed on correspondence models. In the anisotropic laminate formulation of Hattori et al., the average Cauchy stress on a bond is rotated into local fiber coordinates and evaluated with the Tsai–Hill criterion; when \(\Psi\ge 1\), the bond is irreversibly broken, and a nodal damage index \[ \phi(x,t)=1-\frac{\int_{H_\delta(x)}\mu(\xi,t)\,dV_{x'}}{\int_{H_\delta(x)}dV_{x'}} \] summarizes local bond loss [1710.06827]. Randomly heterogeneous materials have also been treated within a stochastic state-based framework. In Fan et al., micromechanical parameters are modeled by a finite-dimensional random vector or by random variables obtained from truncating the Karhunen-Loève decomposition or principal component analysis, the random space is sampled by probabilistic collocation, and each deterministic sample is solved with optimization-based meshfree quadrature. The reported scheme sustains asymptotic compatibility spatially and achieves an algebraic or sub-exponential convergence rate in the random space as the number of collocation points grows; in glass-ceramics, averaged fracture toughness computed from crack initiation and growth simulations shows good consistency with experimental measurements [2202.06578]. ## 5. Specialized variants and domain-specific reductions State-based peridynamics has developed into a family of formulations rather than a single constitutive template. The following variants are explicitly reported in the cited literature. | Variant | Additional structure | Reported scope | |---|---|---| | Micropolar PD | micro-rotation, moment state, couple stress | size-dependent linear elasticity | | Shell PD | curved bonds, surface force and moment states | spherical/cylindrical shells, flat plates | | Flexoelectric PD | displacement, polarization, potential states | centrosymmetric dielectrics with strain-gradient coupling | | Non-ordinary anisotropic PD | full stiffness tensor in correspondence | generally anisotropic materials and composites | | OSB-PD elastoplasticity | volumetric/deviatoric state split, \(J_2\)-equivalent yield | large rotations in 2D | | XOSBPD | non-spherical horizons via Lagrange multipliers | arbitrary horizon shapes, non-uniform discretization | | Graphene OSBPD | coarse-grained MD-calibrated bond force law | monolayer deflection and perforation | Micropolar peridynamics augments each material point with an independent micro-rotation degree of freedom and introduces additional constitutive moduli \(\alpha,\beta,\gamma\). The combination \[ l^2=\gamma/\mu \] defines an intrinsic material length scale, so size-dependent stiffening emerges through the stress–couple-stress mapping while the horizon remains a numerical parameter. Homogenized one-dimensional Timoshenko-type beam reductions were derived for both the micropolar and standard non-polar variants, and representative examples show pronounced bending stiffening when \(h\sim l\) and reduced stress concentration around small holes when \(a