State-Based Peridynamic Formulation

• State-based peridynamic formulation is a non-local continuum model that replaces classical spatial derivatives with integral balance laws to capture complex material behavior.
• It supports both ordinary and non-ordinary approaches, enabling accurate modeling of elasticity, fracture, anisotropy, and coupled fields through vector-valued states.
• The framework ensures asymptotic compatibility with local theories using horizon-based states and advanced quadrature techniques for precise spatial discretization.

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 (Chowdhury et al., 2014, Behzadinasab et al., 2020, Javili et al., 2020).

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 (Hattori et al., 2017, Javili et al., 2020).

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

$B_0\subset\mathbb R^3$0

and the averaged micro-rotation state

$B_0\subset\mathbb R^3$1

used in micropolar peridynamics to construct nonlocal approximations of micropolar strain and wryness (Chowdhury et al., 2014). In fracture-oriented linear peridynamic solid formulations, the scalar bond-strain is

$B_0\subset\mathbb R^3$2

and the dilatation is defined by a weighted horizon integral that corresponds to the local divergence in the small-horizon limit (Jha et al., 2019).

Two nonlocal tensors recur throughout the literature. The first is the shape tensor,

$B_0\subset\mathbb R^3$3

whose inverse appears in correspondence mappings. The second is the nonlocal deformation gradient,

$B_0\subset\mathbb R^3$4

or higher-order RK/GMLS variants of the same operator. Under linearized kinematics, $B_0\subset\mathbb R^3$5 (Hattori et al., 2017, Behzadinasab et al., 2020).

A basic distinction is between ordinary and non-ordinary state-based models. In ordinary state-based peridynamics, the force vector state is written

$B_0\subset\mathbb R^3$6

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 (Mousavi et al., 2020, Hattori et al., 2017).

2. Balance laws and constitutive correspondence

The strong-form state-based peridynamic equation of motion has the generic structure

$B_0\subset\mathbb R^3$7

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_0\subset\mathbb R^3$8 and inertia $B_0\subset\mathbb R^3$9 (Chowdhury et al., 2014, Hattori et al., 2017, Yu et al., 2021).

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

$H(X)=\{\,X'\mid \|X'-X\|\le \delta\}\,,$0

or, in the anisotropic small-strain setting,

$H(X)=\{\,X'\mid \|X'-X\|\le \delta\}\,,$1

Because the full fourth-order stiffness tensor $H(X)=\{\,X'\mid \|X'-X\|\le \delta\}\,,$2 enters directly, generally anisotropic materials can be represented without reducing them to orientation-dependent isotropic bonds (Hattori et al., 2017, Behzadinasab et al., 2020).

Energy-based formulations express the constitutive law through Fréchet derivatives of a nonlocal internal energy. For micropolar peridynamics, if the internal energy density $H(X)=\{\,X'\mid \|X'-X\|\le \delta\}\,,$3 depends on the tensorial deformation measures $H(X)=\{\,X'\mid \|X'-X\|\le \delta\}\,,$4 and $H(X)=\{\,X'\mid \|X'-X\|\le \delta\}\,,$5, then

$H(X)=\{\,X'\mid \|X'-X\|\le \delta\}\,,$6

With Eringen’s linear elastic micropolar stored energy

$H(X)=\{\,X'\mid \|X'-X\|\le \delta\}\,,$7

the induced correspondence mapping becomes

$H(X)=\{\,X'\mid \|X'-X\|\le \delta\}\,,$8

with classical micropolar stress and couple-stress tensors recovered pointwise (Chowdhury et al., 2014).

Angular momentum requires special attention. In the micropolar extension, angular momentum is an independent balance law with a moment vector-state $H(X)=\{\,X'\mid \|X'-X\|\le \delta\}\,,$9, a micro-inertia $\delta$0, and a body-couple $\delta$1; 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 (Chowdhury et al., 2014, Hattori et al., 2017). 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 $\delta$2 inside the horizon shows that the nonlocal dilatation approaches $\delta$3 and the nonlocal internal-force operator approaches $\delta$4. One formulation states that, for $\delta$5,

$\delta$6

and under suitable smoothness on $\delta$7, the nonlocal solution converges strongly in $\delta$8 to the local solution at rate $\delta$9 (Fan et al., 2022). The linear peridynamic solid model is explicitly parameterized so that, as $\xi=X'-X$0, the operator recovers the Navier equations

$\xi=X'-X$1

(Yu et al., 2021).

Boundary truncation introduces the classical “surface effect.” In the LPS framework, the ball $\xi=X'-X$2 is truncated near $\xi=X'-X$3, so the original nonlocal dilatation does not converge to $\xi=X'-X$4. A corrected dilatation

$\xi=X'-X$5

is introduced, with

$\xi=X'-X$6

If $\xi=X'-X$7 is $\xi=X'-X$8 and $\xi=X'-X$9, then for points within $X$0 of the boundary,

$X$1

while away from the boundary the original dilatation is recovered (Yu et al., 2021).

Traction loading is another nontrivial issue. In the same framework, a consistent nonlocal traction prescription modifies the body-force term for points within $X$2 of $X$3, and a patch-test result states that for linear $X$4, the Neumann-modified operator reproduces the classical traction exactly if the truncated ball is symmetric and to $X$5 otherwise (Yu et al., 2021). 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 $X$6 subject to exactness for kernels generated by $X$7, and shows interior truncation error $X$8 for $X$9, $\underline{\mathbf X}\langle\xi\rangle=\xi$0 (Yu et al., 2021). In heterogeneous brittle fracture, the same strategy is reported as asymptotically compatible provided $\underline{\mathbf X}\langle\xi\rangle=\xi$1 is bounded, with the discrete operator retaining the $\underline{\mathbf X}\langle\xi\rangle=\xi$2 local-limit rate (Fan et al., 2022).

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

$\underline{\mathbf X}\langle\xi\rangle=\xi$3

in the mean square norm, with $\underline{\mathbf X}\langle\xi\rangle=\xi$4 and $\underline{\mathbf X}\langle\xi\rangle=\xi$5 independent of $\underline{\mathbf X}\langle\xi\rangle=\xi$6 and $\underline{\mathbf X}\langle\xi\rangle=\xi$7 (Jha et al., 2019). In a Hölder-space finite difference analysis, the full-discrete error satisfies

$\underline{\mathbf X}\langle\xi\rangle=\xi$8

more precisely

$\underline{\mathbf X}\langle\xi\rangle=\xi$9

for $$\underline{\mathbf Y}\langle\xi\rangle=y(X',t)-y(X,t)$$0 (Jha et al., 2018).

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 (Hattori et al., 2017).

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

$$\underline{\mathbf Y}\langle\xi\rangle=y(X',t)-y(X,t)$$1

and the bond $$\underline{\mathbf Y}\langle\xi\rangle=y(X',t)-y(X,t)$$2 is broken when $$\underline{\mathbf Y}\langle\xi\rangle=y(X',t)-y(X,t)$$3, where $$\underline{\mathbf Y}\langle\xi\rangle=y(X',t)-y(X,t)$$4 is chosen so that the nonlocal fracture energy matches the material $$\underline{\mathbf Y}\langle\xi\rangle=y(X',t)-y(X,t)$$5. A mask $$\underline{\mathbf Y}\langle\xi\rangle=y(X',t)-y(X,t)$$6 is then used to distinguish intact and broken quadrature weights, and “crack surfaces remain perfectly sharp by virtue of broken bonds” (Yu et al., 2021).

State-based fracture models also exist in energy-potential form. A class analyzed by Jha and Lipton uses a pairwise bond potential $$\underline{\mathbf Y}\langle\xi\rangle=y(X',t)-y(X,t)$$7 based on a smooth convex–concave function $$\underline{\mathbf Y}\langle\xi\rangle=y(X',t)-y(X,t)$$8 and a hydrostatic potential $$\underline{\mathbf Y}\langle\xi\rangle=y(X',t)-y(X,t)$$9 based on $\underline{\mathbf T}\langle\xi\rangle$0, which may be quadratic or multi-well. The total force splits into a bond-based part $\underline{\mathbf T}\langle\xi\rangle$1 and a dilatational part $\underline{\mathbf T}\langle\xi\rangle$2, so fracture and cavitation can be associated with softening in either tensile strain or hydrostatic strain (Jha et al., 2019, Jha et al., 2018).

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 $\underline{\mathbf T}\langle\xi\rangle$3, the bond is irreversibly broken, and a nodal damage index

$\underline{\mathbf T}\langle\xi\rangle$4

summarizes local bond loss (Hattori et al., 2017).

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 (Fan et al., 2022).

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, $\underline{\mathbf T}\langle\xi\rangle$5-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 $\underline{\mathbf T}\langle\xi\rangle$6. The combination

$\underline{\mathbf T}\langle\xi\rangle$7

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 $\underline{\mathbf T}\langle\xi\rangle$8 and reduced stress concentration around small holes when $\underline{\mathbf T}\langle\xi\rangle$9 (Chowdhury et al., 2014).

For shells, a surface-based state formulation replaces straight bonds by geodesic curved bonds on the shell mid-surface, derives surface force and moment states by through-thickness integration of the three-dimensional force state, and introduces nonlocal approximations of surface strain and curvature through a surface shape tensor $U[X](\xi)=u(X')-u(X),$0. The resulting balances of linear and angular momentum recover the global three-dimensional balances for the full shell, and constitutive correspondence reproduces classical linear elastic shell resultants through $U[X](\xi)=u(X')-u(X),$1, $U[X](\xi)=u(X')-u(X),$2, and thickness $U[X](\xi)=u(X')-u(X),$3 exactly as in the local shell theory (Chowdhury et al., 2015).

Flexoelectric state-based peridynamics introduces three relative states,

$U[X](\xi)=u(X')-u(X),$4

and derives coupled nonlocal equilibrium equations from Hamilton’s principle. In this formulation, the mechanical force state depends on polarization, the electrical force state depends on displacement, and the global balance of linear momentum and electric charge follows from antisymmetry of the states. For Gaussian-type horizon kernels, Roy and Roy derive analytic Fourier-space and real-space Green’s functions for an infinite three-dimensional body under point mechanical and electrical loads (Roy et al., 2016).

The non-ordinary anisotropic formulation embeds the full fourth-order anisotropic stiffness tensor through a nonlocal deformation gradient and a first Piola–Kirchhoff stress, then combines this constitutive map with a composite damage criterion. Verification reported in dynamic fracture shows agreement of dynamic mode-I and mode-II stress-intensity factors with a converged $U[X](\xi)=u(X')-u(X),$5 FEM quadrilateral solution to within a few percent, and crack paths agree with published XFEM/FEM results and laboratory observations for unidirectional HTA/6376 laminate (Hattori et al., 2017).

Ordinary state-based elastoplasticity consistent with $U[X](\xi)=u(X')-u(X),$6 plasticity has been constructed in 2D by decomposing extension and force states into hydrostatic and distortional parts, defining two rate-independent yield functions equivalent to $U[X](\xi)=u(X')-u(X),$7 plasticity, and using an associated flow rule together with a per-bond return-mapping algorithm. The formulation is objective, works for large rotations, and was verified against Abaqus in a dog-bone tension test and a cantilever example with $U[X](\xi)=u(X')-u(X),$8 rotations (Mousavi et al., 2020).

The extended ordinary state-based model for non-spherical horizons uses Lagrange multipliers to guarantee that non-local dilatation and non-local strain-energy density equal local dilatation and local strain-energy density, respectively. It removes the spherical-horizon restriction, does not need volume and surface correction, allows non-uniform discretization with various horizon sizes, and recovers classical OSBPD exactly when the horizon is spherical (Liu et al., 2022).

At smaller scales, an ordinary state-based model for single-layer graphene was calibrated from coarse-grained molecular dynamics. The calibrated bond-force law depends on bond stretch and normalized bond length, the horizon can be rescaled for multiscale computation, and the resulting model was compared with experimental data for AFM deflection and perforation of a graphene monolayer (Silling et al., 2021).

6. Stability issues, boundary effects, and methodological controversies

Several recurrent numerical and conceptual issues define the current state of the field. One concerns the correspondence formulation itself. The higher-order RK-PD and GMLS-PD operators improve gradient accuracy, but one study states explicitly that improved quadrature alone does not suffice to handle correspondence-modeling instability issues. Standard correspondence models admit zero-energy modes, and the bond-associative remedy replaces the single-point stress on each bond by a bond-level stress $U[X](\xi)=u(X')-u(X),$9, where the bond-level deformation gradient contains a nonhomogeneous correction

$B_0\subset\mathbb R^3$00

The reported outcome is robust second-order convergence for smooth fields with BA-RK-PD and BA-GMLS-PD and first-order convergence for problems involving field discontinuities such as curvilinear free surfaces (Behzadinasab et al., 2020, Behzadinasab et al., 2020).

Wave propagation exposes the same instability in a different form. In one-dimensional dispersion studies, standard RK-PD and GMLS-PD exhibit severe dispersion and spurious zero-frequency modes, whereas BA-RK-PD and BA-GMLS-PD are far less dispersive. The same framework introduces a strong-form imposition of natural boundary conditions by separating a kinematic family $B_0\subset\mathbb R^3$01 from a stress-divergence family $B_0\subset\mathbb R^3$02, so that stress boundary conditions enter only in the stress summation while exact reproduction for linear $B_0\subset\mathbb R^3$03 is maintained in the kinematic update (Behzadinasab et al., 2020).

A second issue is the surface effect near geometry boundaries. In the improved non-ordinary state-based elastoplastic framework for geomaterials, the approximate deformation gradient $B_0\subset\mathbb R^3$04 is shown to be second-order accurate in the interior but only first-order accurate within a horizon radius $B_0\subset\mathbb R^3$05 from the surface; moreover, residual stresses persist within a larger range of $B_0\subset\mathbb R^3$06. To mitigate this, a divergence formulation of the non-local differential operator is applied in the surface band, together with a traction boundary condition consistent with the divergence of stress. The same work introduces a loading balance correction algorithm so that no node yields before the entire domain is in equilibrium, thereby addressing fictitious yielding during explicit quasi-static integration (Li et al., 19 Mar 2025).

A third issue is horizon geometry. Classical ordinary state-based models rely on spherical horizons and specially chosen influence functions so that non-local dilatation matches the trace of local strain and non-local strain-energy density equals the classical strain-energy density under homogeneous deformation. XOSBPD replaces this restriction by local $B_0\subset\mathbb R^3$07 and $B_0\subset\mathbb R^3$08 Lagrange-multiplier solves for hydrostatic and deviatoric generalized influence functions. This suggests that horizon shape is not merely a discretization choice but part of the constitutive consistency problem (Liu et al., 2022).

A final misconception concerns the relation between state-based peridynamics and other nonlocal formulations. Continuum-kinematics-inspired peridynamics was introduced precisely because ordinary state-based PD admits a continuum of Poisson ratios but its kinematics deviate from exact continuum mechanics and can exhibit spurious zero-energy modes; CPD distinguishes one-, two-, and three-neighbour interactions, and the latter two are stated to be fundamentally different from state-based interactions (Javili et al., 2020). The comparison clarifies that “state-based” names a specific nonlocal mechanics architecture rather than all meshfree nonlocal continuum models.

Taken together, the literature presents the state-based peridynamic formulation as a broad constitutive and computational framework whose defining elements are horizon-based states, nonlocal balance laws, and correspondence or energy mappings to local physics. Its mature forms now include asymptotically compatible quadrature, corrected dilatation, strong-form traction enforcement, bond-associative stabilization, generalized continua, electromechanical coupling, stochastic heterogeneity, and elastoplasticity, while the main open challenges continue to cluster around stability, boundary accuracy, and constitutive consistency near evolving discontinuities (Yu et al., 2021, Behzadinasab et al., 2020, Li et al., 19 Mar 2025).