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Peridynamics

Peridynamics is a nonlocal formulation of continuum mechanics that models material behavior through integral equations, enabling the simulation of discontinuities such as cracks and damage without relying on spatial derivatives or predefined fracture criteria.^[1] Developed by Stewart A. Silling at Sandia National Laboratories and published in 2000, it reformulates classical elasticity theory to incorporate long-range forces and interactions between material points separated by finite distances, addressing limitations in traditional local theories for handling singularities and multiscale phenomena.^[1]^[2] In peridynamics, each material point interacts with others within a spherical region called the horizon, typically on the order of millimeters to micrometers, through pairwise forces that depend on relative displacements; this nonlocal approach naturally captures wave dispersion and damage evolution across scales from atomic to structural levels.^[2] The theory encompasses two primary models: bond-based peridynamics, the original simpler variant where interactions are limited to axial forces along bonds between points, restricting material Poisson's ratios to fixed values (e.g., 1/4 in 3D); and state-based peridynamics, a more general extension introduced in 2007 that allows arbitrary force states, enabling full matching of classical elastic moduli and broader applicability to complex constitutive behaviors.^[1]^[3] Key advantages over classical continuum mechanics include the inherent ability to predict spontaneous crack initiation, propagation, and branching in brittle, ductile, and composite materials, as well as seamless integration of multiscale modeling without mesh dependencies or ad hoc failure rules.^[4]^[2] Applications span fracture mechanics in concrete and ceramics, fatigue cracking under cyclic loading, impact and explosive responses in structures, and emerging areas like additive manufacturing defects and fluid-structure interactions, with ongoing advancements in computational efficiency through adaptive methods and coupling with finite element analysis.^[4]^[5]^[6]

Introduction

Etymology

The term "peridynamics" was coined by Stewart Silling in his seminal 2000 paper introducing the theory as a reformulation of continuum mechanics to handle discontinuities and long-range forces.^[7] It derives from the Greek prefix peri-, meaning "around" or "near," and dynamis, meaning "force" or "power," reflecting the model's emphasis on pairwise force interactions between material points within a finite spatial neighborhood, or "horizon," surrounding each point. This nomenclature highlights the non-local nature of peridynamics, distinguishing it from classical local theories like continuum dynamics, where forces act instantaneously at a point without extending over a surrounding region.^[8]

History

Peridynamics draws inspiration from earlier non-local continuum theories, including 19th-century lattice models that discretized materials into interacting particles and mid-20th-century Cosserat theories, which incorporated rotational degrees of freedom and internal length scales to address limitations in classical elasticity.^[9]^[10] The theory was formally introduced in 2000 by Stewart Silling at Sandia National Laboratories, motivated by the need to simulate material failure processes, such as crack initiation and propagation, without the singularities inherent in partial differential equations of classical continuum mechanics.^[7] In his seminal paper, Silling proposed a reformulation of elasticity using integral equations that account for long-range forces between material points, enabling a unified treatment of continuous and discontinuous deformations.^[7] This initial bond-based peridynamic model treated interactions as pairwise forces between points within a finite horizon, avoiding the need for explicit crack tracking.^[7] Early development progressed with Silling's 2003 presentation of a meshfree numerical implementation for dynamic fracture modeling, which demonstrated the theory's capability to handle complex crack patterns without mesh dependency.^[11] This was followed by a 2005 journal publication co-authored with Ebrahim Askari, detailing a stable meshfree method for solving peridynamic equations of motion in solid mechanics problems.^[12] A key evolution occurred in 2007 when Silling and Askari introduced the state-based peridynamic formulation, which generalized the bond-based model by allowing forces to depend on the collective states of multiple bonds, enabling more realistic constitutive behaviors and broader material modeling.^[13] Further milestones included applications to dynamic brittle fracture, as explored in Silling's 2007 work on constitutive modeling that incorporated damage mechanics for fracture simulation.^[13] The 2010s saw significant growth in adoption, with the release of the open-source Peridigm code in 2012 by Sandia researchers, facilitating parallel simulations of multi-physics problems involving failure.^[14] Institutional contributions were pivotal, with Sandia National Laboratories leading foundational research and code development, the Air Force Research Laboratory funding projects on peridynamic modeling of composites and fatigue, and academic institutions like the University of Arizona—through Erdogan Madenci's group—and Texas A&M University advancing theoretical extensions and applications.^[15]^[16]^[17] Research has continued into the 2020s, with advancements in peridynamic modeling of additive manufacturing processes and impact loading in materials like concrete.^[18]

Purpose and Advantages

Peridynamics was developed primarily to address the limitations of classical continuum mechanics in modeling spontaneous fracture and damage initiation in materials, where partial differential equations fail due to singularities at discontinuities such as cracks.^[7] By reformulating the equations of motion using integral forms rather than spatial derivatives, peridynamics enables the natural emergence of cracks without requiring predefined crack paths or auxiliary tracking algorithms.^[7] This approach is particularly motivated by the need to simulate problems where the location and evolution of discontinuities are unknown in advance, allowing the same governing equations to apply uniformly across the domain, including on and off crack surfaces.^[7] Key advantages of peridynamics include its inherent ability to handle discontinuities without special mathematical treatments, ensuring mesh-independent simulations for crack propagation paths.^[4] In state-based formulations, the theory satisfies fundamental conservation laws for linear momentum and energy, providing a physically consistent framework for dynamic problems.^[19] It also facilitates seamless coupling with multiscale and multiphysics models, such as integrating atomic-scale interactions with continuum behavior or combining mechanical deformation with thermal effects in composites.^[4] Specific benefits manifest in its capacity to predict complex crack patterns, including branching and kinking, as demonstrated in dynamic fracture simulations that align closely with experimental observations.^[20] Peridynamics proves robust for high-strain-rate scenarios, such as ballistic impacts, where it captures rapid damage evolution and fragmentation without numerical instabilities associated with local theories.^[21] However, the non-local nature of interactions leads to higher computational costs compared to classical methods, necessitating efficient discretization strategies for practical applications.^[4]

Fundamentals

Definition and Basic Terminology

Peridynamics is a nonlocal formulation of continuum mechanics that reformulates the equations of motion using integral equations rather than partial differential equations, enabling the modeling of material behavior including discontinuities such as cracks without special treatment.^[1] In this framework, the force density at a material point \mathbf{x} depends on the collective deformation states of all points within a finite neighborhood, known as the horizon, allowing for long-range interactions across the material.^[1] The central equation of motion is given by

\rho \ddot{\mathbf{u}}(\mathbf{x}, t) = \int_{H_{\mathbf{x}}} \mathbf{f}(\mathbf{u}(\mathbf{y}, t) - \mathbf{u}(\mathbf{x}, t), \mathbf{y} - \mathbf{x}, \mathbf{x}, t) \, dV_{\mathbf{y}} + \mathbf{b}(\mathbf{x}, t),

where \rho is the material density, \ddot{\mathbf{u}} is the acceleration, H_{\mathbf{x}} is the horizon of \mathbf{x}, \mathbf{f} is the pairwise force function between points \mathbf{x} and \mathbf{y}, and \mathbf{b} is the body force density.^[1] This integro-differential equation avoids spatial derivatives, making it inherently meshfree and applicable to irregular geometries and failure scenarios.^[22] Key terminology in peridynamics includes the horizon H_{\mathbf{x}}, defined as the family of material points that interact with \mathbf{x}, typically a ball of radius \delta(\mathbf{x}) centered at \mathbf{x}.^[1] A bond refers to the pairwise interaction between two material points \mathbf{x} and \mathbf{y} within each other's horizons, governed by the force function \mathbf{f}.^[22] The reference configuration describes the initial, undeformed positions of material points, while the deformed configuration accounts for their displaced positions after applying displacements \mathbf{u}.^[22] Peridynamic points are the material points themselves, each associated with a small volume element in the continuum body.^[1] The force density in the equation of motion has units of force per unit volume, such as N/m³, ensuring dimensional consistency with the acceleration term scaled by density.^[22] The horizon radius \delta introduces a material length scale for nonlocal effects, typically chosen in numerical implementations to be 3 to 4 times the spacing between discretization points to balance accuracy and computational efficiency.^[22]

Comparison to Classical Continuum Mechanics

Peridynamics represents a nonlocal reformulation of continuum mechanics, in contrast to the local nature of classical theories. In classical continuum mechanics, the equation of motion is expressed as a partial differential equation (PDE) involving spatial derivatives, such as Navier's equation \rho \ddot{\mathbf{u}} = \nabla \cdot \boldsymbol{\sigma} + \mathbf{b}, where \rho is density, \ddot{\mathbf{u}} is acceleration, \boldsymbol{\sigma} is the stress tensor, and \mathbf{b} is body force; this formulation assumes that material response at a point depends only on the state at that point and its immediate infinitesimal neighborhood.^[1] Peridynamics, however, replaces these derivatives with an integral over a finite-volume neighborhood called the horizon, yielding an integro-differential equation of the form \rho \ddot{\mathbf{u}}(\mathbf{x},t) = \int_{H_{\mathbf{x}}} \mathbf{f}(\mathbf{u}(\mathbf{x}',t) - \mathbf{u}(\mathbf{x},t), \mathbf{x}' - \mathbf{x}, t) \, dV_{\mathbf{x}'} + \mathbf{b}(\mathbf{x},t), where \mathbf{f} is a pairwise force function and H_{\mathbf{x}} is the horizon of radius \delta around point \mathbf{x}.^[1] This nonlocal structure introduces an intrinsic length scale through the horizon parameter \delta, which governs the range of interactions and allows peridynamics to bridge continuum and atomistic scales, whereas classical mechanics is inherently scale-free in its local limit. A key distinction arises in handling discontinuities such as cracks. Classical theories encounter singularities in spatial derivatives at discontinuities, necessitating specialized techniques like extended finite element methods (XFEM) or cohesive zone models to track and propagate cracks, often requiring prior knowledge of crack paths.^[1] In peridynamics, discontinuities emerge naturally as the progressive failure of bonds within the horizon, without altering the governing equations or introducing auxiliary variables; damage is incorporated directly into the force function \mathbf{f}, enabling spontaneous crack initiation and growth in arbitrary directions.^[1] Both frameworks conserve linear momentum through their respective balance laws, but peridynamics ensures angular momentum conservation via the symmetry of pairwise forces across bonds, such that the force state produces no net moment about any point. Energy conservation in peridynamics is achieved through variational principles for elastic materials, where the force state derives from a strain energy density function, mirroring the potential energy formulation in classical elasticity while accommodating nonlocal interactions. Overall, the Volterra-type integro-differential equations of peridynamics contrast with the hyperbolic PDEs of classical mechanics, providing a unified description for both smooth deformations and fractures without reliance on differentiability.^[1]

Bond-Based Peridynamics

Core Principles

In the bond-based peridynamics model, the interaction between any two material points \mathbf{x} and \mathbf{y} is governed by a pairwise force that depends exclusively on the bond vector \boldsymbol{\xi} = \mathbf{y} - \mathbf{x} and the relative displacement \boldsymbol{\eta} = \mathbf{u}(\mathbf{y}) - \mathbf{u}(\mathbf{x}), expressed as \mathbf{f} = \mathbf{f}(\boldsymbol{\xi}, \boldsymbol{\eta}). This assumption simplifies the force description to a function of local bond deformation, enabling a nonlocal representation of material behavior without reliance on spatial derivatives.^[7] A fundamental symmetry in this model is the action-reaction principle, where the force exerted by \mathbf{y} on \mathbf{x} is equal and opposite to that exerted by \mathbf{x} on \mathbf{y}, mathematically \mathbf{f}(\boldsymbol{\xi}, \boldsymbol{\eta}; \mathbf{x}, \mathbf{y}) = -\mathbf{f}(-\boldsymbol{\xi}, -\boldsymbol{\eta}; \mathbf{y}, \mathbf{x}). This pairwise balance inherently conserves linear momentum across the continuum, as the net force on any isolated system of points sums to zero. Additionally, angular momentum is conserved because the forces are central, meaning they act collinearly along the bond vector \boldsymbol{\xi}; however, this central force assumption imposes a limitation in three-dimensional linear isotropic elastic materials, restricting the Poisson's ratio to exactly $1/4.^[7] The equation of motion for a material point \mathbf{x} specializes to

\rho \ddot{\mathbf{u}}(\mathbf{x}) = \int_{H_{\mathbf{x}}} \mathbf{f}(\boldsymbol{\xi}, \boldsymbol{\eta}) \, dV_{\mathbf{y}} + \mathbf{b}(\mathbf{x}),

where \rho is the material density, \ddot{\mathbf{u}} is the acceleration, \mathbf{b}(\mathbf{x}) is the body force density, and the integral is over the neighborhood H_{\mathbf{x}} of \mathbf{x}.^[7] Interactions in the model are confined to a finite domain known as the horizon H_{\mathbf{x}} = \{\mathbf{y} : |\mathbf{y} - \mathbf{x}| \leq \delta \}, where \delta > 0 is the horizon radius; beyond this distance, the force function \mathbf{f} vanishes, truncating the integro-differential operator and defining the extent of nonlocal effects. This truncation ensures computational tractability while capturing long-range interactions essential for modeling discontinuities.^[7]

Constitutive Models

In bond-based peridynamics, constitutive models describe the pairwise force interactions between material points within the horizon, enabling the representation of material behavior without relying on spatial derivatives. For hyperelastic materials, the force density \mathbf{f}(\boldsymbol{\eta}, \boldsymbol{\xi}) is derived from a scalar potential w(\boldsymbol{\eta}, \boldsymbol{\xi}) as \mathbf{f}(\boldsymbol{\eta}, \boldsymbol{\xi}) = \frac{\partial w}{\partial |\boldsymbol{\xi} + \boldsymbol{\eta}|} \frac{\boldsymbol{\xi} + \boldsymbol{\eta}}{|\boldsymbol{\xi} + \boldsymbol{\eta}|}, where \boldsymbol{\eta} is the relative displacement and \boldsymbol{\xi} is the reference relative position vector. This formulation ensures path-independent strain energy, as the work done by the force over any closed deformation path is zero, consistent with hyperelasticity principles.^[1] A simplified prototype for linear elastic behavior uses \mathbf{f} = c s \frac{\boldsymbol{\xi} + \boldsymbol{\eta}}{|\boldsymbol{\xi} + \boldsymbol{\eta}|}, where s = \frac{|\boldsymbol{\xi} + \boldsymbol{\eta}| - |\boldsymbol{\xi}|}{|\boldsymbol{\xi}|} is the bond stretch and c is a constant micro-modulus. This model assumes small deformations and isotropic response, with the force magnitude proportional to the relative extension of the bond. To align with classical linear elasticity, the micro-modulus c is calibrated by equating the peridynamic strain energy density to the classical expression for uniform deformation. In three dimensions, for instance, c = \frac{12 E}{\pi \delta^4} under the constraint of Poisson's ratio \nu = 1/4, where E is Young's modulus and \delta is the horizon radius; in two-dimensional plane strain (with fixed \nu = 1/4), c = \frac{9 E}{2 \pi \delta^3 (1 - \nu^2)}. These relations recover classical moduli for small stretches while fixing \nu to specific values inherent to the bond-based framework.^[1]^[23] Extensions to viscoelastic behavior in bond-based peridynamics incorporate time-dependence through convolution integrals, modifying the elastic force as \mathbf{f}(t) = \int_0^t G(t - \tau) \frac{\partial}{\partial \tau} \left[ c s(\tau) \frac{\boldsymbol{\xi} + \boldsymbol{\eta}(\tau)}{|\boldsymbol{\xi} + \boldsymbol{\eta}(\tau)|} \right] d\tau, where G(t) is the relaxation modulus. This approach captures hereditary effects but remains limited by the bond-based restriction on Poisson's ratio and inability to model full tensorial responses like shear decoupling. Such models are typically applied to quasi-static or dynamic problems in polymers or composites. Thermodynamic consistency is ensured by matching the peridynamic micropotential w to the classical strain energy density for infinitesimal deformations, yielding W_{PD} = \frac{1}{2} \int_{\mathcal{H}} w(\boldsymbol{\eta}, \boldsymbol{\xi}) dV_{\boldsymbol{\xi}} \approx \frac{1}{2} \boldsymbol{\varepsilon} : \mathbf{C} : \boldsymbol{\varepsilon} in the limit of vanishing horizon, where \boldsymbol{\varepsilon} is the strain tensor and \mathbf{C} is the classical stiffness tensor. This equivalence validates the model for undamaged, small-strain regimes across dimensions.^[24]

Micro-Modulus Functions

In bond-based peridynamics, the micro-modulus function c(|\xi|), where \xi is the relative position vector between interacting material points, weights the pairwise force interactions based on the distance between points and serves to characterize the material's nonlocal stiffness. For uniform isotropic materials, it is typically defined to be nonzero only within a fixed horizon distance \delta, beyond which interactions vanish, ensuring computational tractability while capturing long-range effects essential for modeling discontinuities.^[1] The most common form is the constant or cylindrical micro-modulus, expressed as c(s) = c_0 for s < \delta and c(s) = 0 otherwise, where s = |\xi| and c_0 is a material-specific constant. This uniform weighting assumes equal influence from all points within the horizon, simplifying the formulation and enabling exact reproduction of classical linear isotropic elasticity in three dimensions when properly calibrated.^[1]^[25] To achieve smoother spatial variation and reduce numerical artifacts near boundaries, a triangular or linear micro-modulus is employed, given by c(s) = c_0 \left(1 - \frac{s}{\delta}\right) for s < \delta and zero otherwise. This decaying profile mimics a conical distribution of interactions, promoting more gradual transitions in force densities and improving convergence in simulations of wave propagation or heterogeneous media.^[25] For applications requiring approximation of infinite-range nonlocal effects within a finite domain, the Gaussian or normal micro-modulus is used: c(s) = c_0 \exp\left(-\frac{s^2}{2\lambda^2}\right), where \lambda is a scale parameter often set proportional to \delta to enforce practical support. This exponentially decaying form enhances accuracy in capturing dispersive behaviors and is particularly suited for brittle fracture scenarios where sharp localization occurs.^[25] Higher-order polynomial forms, such as the quartic micro-modulus c(s) = c_0 \left(1 - \left(\frac{s}{\delta}\right)^2\right)^2 for s < \delta and zero otherwise, provide improved differentiability and higher-order accuracy in matching classical solutions. These are beneficial for refined modeling of stress concentrations and crack branching, as the smoother decay minimizes Gibbs-like oscillations in peridynamic responses.^[25] Calibration of these functions ensures equivalence to classical continuum mechanics by matching the strain energy density, typically through integrating the micro-modulus over the horizon to recover macroscopic elastic constants like the bulk modulus K. For instance, in three dimensions with a general radial form c(s), the condition involves \int_0^\delta c(s) s^3 \, ds = \frac{18 K}{\pi \delta^4}, which for the constant case yields c_0 = \frac{18 K}{\pi \delta^4} assuming a Poisson's ratio of $1/4. Similar integrals adjusted for the functional form are used for non-constant profiles to maintain consistency with shear and bulk moduli.^[1]

State-Based Peridynamics

Formulation Overview

State-based peridynamics generalizes the bond-based formulation by incorporating collective interactions among bonds, allowing the force on any given bond to depend on the deformation states of surrounding bonds rather than solely on pairwise interactions. This extension addresses limitations in the bond-based model, such as its restriction to central forces that fix the Poisson's ratio at 1/4 for isotropic three-dimensional materials. The core concept in state-based peridynamics is the force state, denoted as \mathbf{T}(\mathbf{x}, t)(\boldsymbol{\xi}), which at a point \mathbf{x} and time t assigns to each relative position vector (or bond) \boldsymbol{\xi} a force-density vector per unit volume in the reference configuration of the family of bonds emanating from \mathbf{x}. This force state encapsulates how the material at \mathbf{x} exerts forces on points within its horizon H_{\mathbf{x}}, a spherical neighborhood of radius \delta centered at \mathbf{x}. The equation of motion in state-based peridynamics is expressed as

\rho \ddot{\mathbf{u}}(\mathbf{x}, t) = \int_{H_{\mathbf{x}}} \left[ \mathbf{T}(\mathbf{x}, t)(\boldsymbol{\xi}) - \mathbf{T}(\mathbf{x} + \boldsymbol{\xi}, t)(-\boldsymbol{\xi}) \right] \, dV_{\mathbf{x}'} + \mathbf{b}(\mathbf{x}, t),

where \boldsymbol{\xi} = \mathbf{x}' - \mathbf{x}, \rho is the material density, \mathbf{u} is the displacement field, \mathbf{b} is the body force density, and the integral balances forces across the horizon to ensure action-reaction pairwise equilibrium. Force states are categorized as ordinary or non-ordinary. In ordinary state-based peridynamics, the force state \mathbf{T} for a bond depends exclusively on the stretch and orientation of that individual bond, akin to an extension of the bond-based approach but with greater flexibility in constitutive relations. Non-ordinary states, however, permit the force on a bond to arise from the collective influence of all bonds within the horizons of its endpoints, enabling accurate representation of full three-dimensional linear elasticity, including arbitrary Poisson ratios and shear behavior.^[26] For constitutive modeling, the force state \mathbf{T} can be decomposed into components such as a scalar force magnitude scaled by the bond direction divided by its length, plus dilatational and deviatoric (or rotational) contributions that capture volumetric and distortional responses, respectively. This decomposition facilitates the derivation of material models that align with classical elasticity while preserving the nonlocal nature of peridynamics. In the limit as the horizon size \delta approaches zero, the state-based peridynamic formulation converges to the classical continuum mechanics equations, where the nonlocal integral reduces to the divergence of the Cauchy stress tensor.

Advanced Features and Extensions

State-based peridynamics extends beyond the limitations of bond-based formulations by enabling the modeling of complex material behaviors through the use of force states that incorporate collective interactions within a material point's horizon.^[27] In non-ordinary state-based models, the Poisson's ratio \nu can deviate from the fixed value of $1/4 inherent to bond-based peridynamics, achieving arbitrary isotropic Poisson ratios through the dilatational force state.^[26] This is accomplished by defining the dilatational force state as

\mathbf{D}(\mathbf{x},t) = \int_{H_\mathbf{x}} s(\boldsymbol{\xi}',\boldsymbol{\eta}') \, dV_{\boldsymbol{\xi}'},

where s(\boldsymbol{\xi}',\boldsymbol{\eta}') represents the scalar force state derived from bond extensions and relative displacements, allowing the volumetric response to couple with shear deformations in a manner consistent with classical elasticity for any \nu.^[27] Plasticity and viscoplasticity in state-based peridynamics are modeled by evolving the reference configuration or introducing plastic stretches within the force states, enabling the simulation of irreversible deformations without singularities.^[28] For instance, in ordinary state-based plasticity models, the force state is decomposed into elastic and plastic components, with plastic flow governed by yield criteria such as von Mises, where the reference bond length updates based on accumulated plastic strain to enforce incompressibility conditions analogous to classical theories.^[29] Viscoplastic extensions incorporate time-dependent evolution laws for the plastic stretch tensor in the states, capturing rate-sensitive behaviors like creep in metals under high temperatures.^[30] Multiphysics couplings in state-based peridynamics integrate mechanical force states with additional fields, such as temperature or electric potential, to model coupled phenomena in advanced materials. In thermo-peridynamics, temperature-dependent constitutive relations modify the force states, where thermal expansion influences bond stretches and heat conduction is nonlocal, allowing simulation of thermal stresses and phase transitions.^[31] Similarly, electro-peridynamics for piezoelectric materials extends the state formulation to include electric force states derived from polarization and electric fields, enabling the prediction of electromechanical coupling effects like converse piezoelectricity in fracture scenarios.^[32] Anisotropy is incorporated into state-based peridynamics by defining direction-dependent force states that reflect material microstructure, such as in composites or crystals, where bond stiffness varies with orientation relative to fiber directions or lattice symmetries.^[33] This approach uses tensorial representations in the force states to capture orthotropic or transversely isotropic responses, ensuring wave propagation and deformation align with classical anisotropic continuum models for applications like fiber-reinforced polymers.^[34] The variational correspondence principle in state-based peridynamics establishes energy equivalence between nonlocal force states and classical constitutive models, facilitating the direct adoption of hyperelastic potentials or plasticity laws from local theories.^[35] By matching the peridynamic strain energy density to its classical counterpart through horizon integrals, this method ensures thermodynamic consistency and eliminates spurious modes, as demonstrated in formulations for nearly incompressible materials where the correspondence stabilizes numerical implementations.^[36] Recent advances as of 2025 include consistent ordinary state-based formulations for anisotropic materials in 2D and 3D, enabling more accurate modeling of orthotropic behaviors, and integrations with finite element software such as ANSYS for variable horizon analyses.^[37]^[38]

Damage and Fracture Modeling

Bond Failure Mechanisms

In peridynamics, damage is modeled through the irreversible failure of bonds between material points, which allows for the natural emergence of discontinuities such as cracks without requiring special treatment of crack surfaces. The primary mechanism for bond failure in bond-based peridynamics is the critical stretch criterion, where a bond breaks if its relative stretch s exceeds a critical value s_0. This stretch s represents the normalized relative displacement between two points connected by the bond, as defined in the constitutive models. The critical stretch s_0 is calibrated such that the energy required to break all bonds crossing a fracture surface matches the material's critical energy release rate G_0 from classical fracture mechanics.^[12] To quantify the extent of damage at a material point \mathbf{x}, a local damage variable \phi(\mathbf{x}, t) is defined as \phi(\mathbf{x}, t) = 1 - \frac{\int_{H_{\mathbf{x}}} \mu(|\boldsymbol{\xi}|, t) \, dV_{\mathbf{y}}}{\int_{H_{\mathbf{x}}} dV_{\mathbf{y}}}, where H_{\mathbf{x}} is the horizon of \mathbf{x}, \boldsymbol{\xi} is the bond vector, and \mu(|\boldsymbol{\xi}|, t) is a scalar indicator that equals 1 for intact bonds and 0 for broken bonds. This variable measures the fraction of failed bonds within the horizon, providing a scalar field that evolves as damage progresses and can be used to visualize crack paths. Bond failure is progressive and history-dependent, with broken bonds remaining failed permanently, leading to weakening or complete loss of load-carrying capacity in affected regions; this irreversibility ensures that damage accumulates over time without healing.^[12] In state-based peridynamics, bond failure is incorporated by zeroing the force state \mathbf{T} associated with failed bonds, which modifies the pairwise forces between points while preserving linear and angular momentum conservation. Unlike bond-based models, this approach allows for more general constitutive relations, including those with Poisson's ratio greater than 1/4, and failure occurs when the deformation state exceeds a critical threshold, effectively nullifying contributions from damaged interactions. The energy dissipation during fracture arises from the work performed to break bonds, which is designed to correspond to the classical J-integral, ensuring consistency with energy-based fracture criteria in linear elastic fracture mechanics.^[13]

Critical Parameters and Criteria

In peridynamics, the critical stretch s_0 serves as the primary parameter for initiating bond failure, marking the threshold beyond which a bond is considered broken and damage begins to accumulate. This parameter is calibrated to the material's fracture toughness by equating the peridynamic energy dissipation during bond breakage to the classical critical energy release rate G_0, ensuring physical consistency. For bond-based peridynamics with a linear micro-modulus in two dimensions, the critical stretch is derived as

s_0 = \sqrt{\frac{4 \pi G_0}{9 E \delta}},

where E is the Young's modulus and \delta is the horizon size; this relation arises from integrating the pairwise bond energies across the fracture plane to match the surface creation energy.^[23] The critical energy release rate G_0 in peridynamics is formulated to be independent of the horizon \delta, achieved through the cumulative integration of bond micro-potentials that cross the prospective fracture surface. Each bond's stored elastic energy, released upon failure, contributes to the total surface energy, with the formulation accounting for the symmetric sharing of energy between interacting material points to avoid double-counting. This approach ensures that the predicted fracture energy remains consistent as the model parameters vary, providing a robust link to experimental measures of toughness.^[39] Failure criteria in peridynamics extend beyond simple stretch thresholds to accommodate diverse material behaviors. Energy-based criteria evaluate the total dilatational or distortional work expended in a bond up to breakage, offering a more comprehensive measure of failure that aligns with thermodynamic principles. In dynamic scenarios, strain-rate dependent criteria modify s_0 proportionally to the bond stretch rate, capturing increased toughness under high-speed loading as observed in impact experiments. Probabilistic variants incorporate statistical distributions for bond strengths, enabling simulation of microstructural variability and scatter in failure outcomes without deterministic assumptions. Recent developments include state-based models incorporating plasticity, such as J₂ yield criteria, for ductile fracture and thermo-mechanical coupling for multiphysics problems.^[40]^[41]^[42]^[43]^[44] Under uniform uniaxial loading, the onset of damage in peridynamics occurs approximately when the applied stress reaches σ_c ≈ E s_0, where bonds aligned with the loading direction first reach the critical stretch.^[45] Validation studies demonstrate that peridynamic models accurately replicate linear elastic fracture mechanics (LEFM) predictions for brittle mode I and mode II fractures, including crack initiation stresses and propagation speeds under quasi-static and dynamic conditions. These validations often involve benchmark problems like edge-cracked plates, where peridynamics recovers stress intensity factors and mixed-mode crack paths without ad hoc remeshing. However, limitations arise in ductile fracture scenarios, where the basic model's assumption of instantaneous bond breakage overlooks plastic dissipation and strain hardening, necessitating advanced state-based extensions or hybrid formulations for accurate representation.^[23]

Numerical Implementation

Discretization Techniques

Peridynamics employs meshfree discretization to approximate the nonlocal integrals in the equation of motion, representing the domain as a collection of material points with associated volumes V_i. Each material point x_i interacts with neighboring points x_j within its horizon \delta, and the integral \int_{\mathcal{H}_{x_i}} f(u(x_j, t) - u(x_i, t), x_j - x_i) \, dV_{x_j} is approximated by the sum \sum_j f_{ij} V_j, where f_{ij} denotes the pairwise force density between points i and j. This approach avoids traditional meshes, enabling natural handling of discontinuities without remeshing.^[46] To mitigate the computational cost of evaluating interactions across all pairs, which scales as O(N^2) for N points, horizon binning precomputes neighbors by partitioning the domain into spatial bins larger than \delta. Points are assigned to bins based on their positions, and interactions are limited to points in the same or adjacent bins, reducing the search to O(N) operations. This technique, adapted from molecular dynamics neighbor lists, is essential for large-scale simulations. Quadrature rules for the spatial integrals typically rely on uniform grid collocation in regular domains, where material points are spaced evenly and each contributes its full volume V_j in the sum, providing first-order accuracy. For improved precision in irregular geometries or higher-order convergence, Gauss quadrature can be applied over subdomains within the horizon, using multiple integration points to evaluate the integrand accurately while avoiding singularities. Recent advances include mesh-free discretization of peridynamic shell structures coupled with isogeometric analysis for enhanced accuracy in complex geometries.^[47] Additionally, element-based peridynamics has been developed for efficient fracture analysis.^[48] These methods ensure consistent approximation of the nonlocal operator.^[46] Boundary conditions in peridynamics require corrections due to incomplete horizons near surfaces, where fewer neighbors lead to artificial stiffening. Surface corrections adjust force densities by scaling contributions from partial volumes or using volume constraints to match classical boundary tractions. Alternatively, fictitious layers of material points, typically one horizon thick, are added outside the domain to restore full interactions, with displacements prescribed to enforce conditions like fixed or loaded boundaries.^[49] For dynamic problems, time integration commonly uses the explicit central difference scheme, updating velocities and displacements via \dot{u}_i^{n+1/2} = \dot{u}_i^{n-1/2} + \frac{\Delta t}{\rho V_i} \sum_j f_{ij}^n V_j and u_i^{n+1} = u_i^n + \Delta t \dot{u}_i^{n+1/2}. Stability demands adherence to the CFL condition, \Delta t < \frac{\delta}{c_d}, where c_d is the dilatational wave speed, preventing numerical instability from wave propagation across the horizon in one step.^[46]^[50]

Computational Challenges and Solutions

Peridynamics simulations face significant computational hurdles primarily due to the nonlocal nature of the interactions, which require evaluating contributions from all material points within a horizon \delta for each point in the domain. This leads to a high computational cost scaling as \mathcal{O}\left( N \frac{\delta^3}{h^3} \right) per time step, where N is the total number of material points and h is the discretization spacing, making it substantially more expensive than local methods like finite element analysis for large-scale problems.^[51] To address this, adaptive horizon approaches dynamically adjust \delta based on local deformation or damage levels, reducing the number of interactions in low-strain regions while maintaining resolution where needed.^[52] Recent developments include adaptive partitioned reduced order models for efficient static fracture simulations and dual-variable-horizon peridynamics for modeling fracture in porous materials.^[53]^[54] Additionally, GPU acceleration has been integrated into peridynamics codes to parallelize the intensive neighbor searches and force computations, achieving speedups of orders of magnitude for three-dimensional simulations.^[55] Convergence in peridynamics is analyzed through two distinct regimes: m-convergence, which fixes the horizon \delta and refines the discretization by letting h \to 0, and \delta-convergence, which reduces the horizon \delta \to 0 with fixed h. These studies reveal that the overall error in approximating classical elasticity solutions is typically of order \mathcal{O}(\delta^2 + h^2), highlighting the need for balanced refinement to achieve accuracy without excessive cost. Numerical implementations must carefully select the ratio \delta / h (often around 3–4) to ensure stable m-convergence before pursuing \delta-convergence, as improper choices can lead to oscillations or divergence in dynamic problems. Horizon effects introduce inaccuracies near free surfaces and boundaries, where material points have incomplete interaction neighborhoods, resulting in erroneous force densities or energy calculations. Common remedies include partial volume integration, which weights contributions based on the actual overlapping volume within the domain, and enriched boundary techniques that virtually extend the horizon using mirror or extrapolation methods to restore full interactions.^[51] These corrections improve boundary fidelity without significantly increasing global computational overhead. Parallelization is essential for scaling peridynamics to practical engineering scales, employing domain decomposition where the simulation domain is partitioned across processors, augmented by ghost layers—overlapping regions that store boundary data for nonlocal neighbor searches. This approach ensures load balancing while minimizing communication costs in distributed-memory environments using protocols like MPI. Open-source frameworks such as Peridigm support these strategies, enabling massively parallel simulations on high-performance computing clusters for problems involving millions of points.^[56] In damage and fracture scenarios, abrupt bond breakage can generate ghost forces—spurious attractions across crack surfaces due to the nonlocal kernel—causing unphysical instabilities or mesh tangling. Mitigation strategies involve gradual softening of the micro-modulus over a transition zone or incorporating rate-dependent critical stretch criteria, which delay failure and smooth the damage evolution, thereby preserving numerical stability.^[51]

Applications and Developments

Engineering Applications

Peridynamics has been applied to fracture mechanics problems, particularly for simulating brittle crack growth in materials such as concrete and ceramics. In concrete, the method captures the influence of aggregate size and position on crack paths during failure, enabling prediction of irregular propagation without predefined crack orientations.^[57] For ceramics, bilinear peridynamic constitutive models effectively simulate crack nucleation and growth in brittle systems, accounting for quasi-brittle transitions.^[58] A notable example is the dynamic fracture of glass plates under impact, where peridynamic simulations predict crack initiation and branching patterns that align with experimental observations from high-speed imaging of thin glass slides.^[59] In impact and blast loading scenarios, peridynamics models spallation in metals by naturally simulating void nucleation, growth, and coalescence under high strain rates, as demonstrated in additively manufactured steel subjected to dynamic failure.^[60] For bird-strike on aircraft structures, Eulerian peridynamic formulations handle large deformations and fragmentation in composite leading edges, avoiding remeshing by inherently tracking bond breakage during the impact event.^[61] This capability is particularly advantageous for capturing multi-fragmentation in hypervelocity impacts without artificial crack tracking algorithms.^[62] Peridynamics addresses fatigue and corrosion by modeling progressive damage accumulation in composites through bond-based fatigue criteria that predict crack initiation and propagation under cyclic loading.^[63] In corrosion scenarios, the approach simulates pitting-induced failure in metallic structures, such as pipelines, by coupling electrochemical degradation with mechanical bond weakening, leading to subsurface damage evolution and eventual crack transition.^[64] These models integrate damage criteria to forecast lifetime reduction due to environmental exposure without relying on local singularity assumptions.^[65] In biomechanics, peridynamics simulates bone fracture by incorporating non-local interactions to model heterogeneous material responses under impact, capturing crack paths in cortical bone that match experimental fracture toughness data. For soft tissue tearing, the framework handles large deformations in hyperelastic materials like elastomers or hydrogels, predicting rupture in tearing scenarios through bond strain measures that account for non-local effects in fibrous composites.^[66] Validation of peridynamics in engineering applications includes benchmarks against Sandia fracture experiments from 2007, where simulations of dynamic perforation in concrete plates reproduced experimental load-displacement curves and failure modes.^[67] Additionally, peridynamic predictions of stress wave propagation agree with photoelasticity experiments, visualizing wave reflection and dispersion in brittle media without singularities.^[68]

Recent Advances

Since approximately 2020, peridynamics has seen significant advancements in multiscale modeling, particularly through couplings with finer-scale methods like molecular dynamics. Concurrent atomistic-peridynamic simulations have been enabled by handshaking approaches that ensure seamless information exchange across scales, as demonstrated in a 2021 framework for multiscale modeling with peridynamics.^[69] This method facilitates the upscaling of atomic interactions to continuum levels, improving predictions of fracture in heterogeneous materials without spatial discontinuities. More recent efforts include efficient peridynamic statistical multiscale methods for composite fracture analysis, which integrate microstructural variability for enhanced accuracy in 2023 simulations.^[70] Integration of machine learning has accelerated peridynamics by developing data-driven micro-moduli and surrogate models to reduce computational demands. In 2023, a peridynamics-based machine learning model was introduced for one- and two-dimensional structures, using linear regression to derive bond forces from displacement data, achieving faster predictions while preserving nonlocal behavior.^[71] Building on this, peridynamic neural operators emerged in 2024 as data-driven nonlocal constitutive models, trained on experimental datasets to capture complex material responses like nonlinear elasticity and damage, with applications in surrogate modeling that cut simulation times by orders of magnitude.^[72] These approaches enforce physical constraints, such as energy conservation, to ensure model reliability in high-fidelity predictions. Multiphysics extensions have expanded peridynamics to coupled phenomena, notably fluid-peridynamic models for hydraulic fracturing and thermo-mechanical formulations for additive manufacturing. A 2022 hybrid finite element-peridynamic simulation captured fluid-driven fracture propagation in saturated porous media, accounting for leak-off effects and demonstrating realistic crack branching under pressure. For additive manufacturing, a fully coupled thermo-mechanical peridynamic model in 2024 simulated residual stress evolution and defect formation in layered builds, revealing how thermal gradients induce porosity and warping in metals like Ti6Al4V. These extensions enable predictive analysis of multiphase flows and heat transfer in dynamic processes. Hybrid methods combining peridynamics with finite element methods (FEM) have addressed efficiency in large-scale domains through improved interfaces. A 2024 matrix-based implementation of hybrid FEM-peridynamic models incorporated GPU acceleration, reducing computation times for hydro-mechanical problems by up to 100 times while maintaining accuracy at crack tips. Reviews from the same year highlight adaptive coupling techniques that transition smoothly from local FEM regions to nonlocal peridynamic zones, optimizing simulations of brittle failure in engineering structures.^[43] Open-source progress has lowered barriers to peridynamics adoption, with updates to tools like Peridigm enhancing parallel performance. The 2023 release of Peridigm, a C++-based meshfree code, supports large-scale simulations on parallel computers, including multi-GPU schemes that achieve 100x speedups for 3D problems via CUDA implementations. Complementary efforts include the 2021 PeriPy package, an OpenCL-enabled Python toolkit for high-performance solid mechanics, and redesigns of Peridigm for SIMT accelerators in 2021, enabling accessible multiscale computations.^[73] Emerging applications leverage these advances in challenging domains, such as additive manufacturing defects and earthquake fault dynamics. In additive manufacturing, 2024 peridynamic frameworks modeled process-induced defects like microcracks in 3D-printed concrete, optimizing build parameters to minimize fatigue nucleation from porosity. For earthquake simulations, a 2024 fault activation-shearing-sliding peridynamic model explored dynamic rupture propagation, capturing off-fault damage and slip evolution in heterogeneous media to better predict seismic hazards. These applications underscore peridynamics' growing role in real-world multiphysics challenges. In 2025, further progress included an efficient explicit-implicit adaptive method for peridynamic simulations to handle deformation and damage more effectively,^[74] a remote stress criterion for brittle fracture within the peridynamics framework,^[75] and the implementation of bond-based peridynamics in Ansys LS-DYNA for enhanced structural analysis capabilities.^[76]

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