Peridynamic Model Research Articles

Multi-scale peridynamic investigation of chloride penetration in concrete under drying-wetting cycles

A multi-scale peridynamic (PD) model was developed to simulate chloride penetrating process in concrete under drying-wetting cycles. The meso‑structure characteristics of concrete and advection-diffusion were considered simultaneously. The governing equations of chloride penetrating during drying-wetting cycles was originally established under the framework of PD theory. Subsequently, the corresponding boundary conditions for wetting and drying process were constructed. The accumulative effect of chloride under drying-wetting cycles was taken into account with the superposition method. Numerical examples demonstrate that the multi-scale PD model is robust and efficient in capturing the variation of chloride concentration and relative humidity. The results show that with the approach to the exposed surface, the differentials of relative humidity rise significantly. The chloride concentration is affected by positions of reference points and relative humidity. Namely, it increases first and then decrease or remain constant during every drying-wetting cycle. Notably, the maximum chloride concentration occurs in a region adjacent to exposed surface rather than on the exposed surface, which coincides well with the experimental phenomenon reported in the literature.

An efficient peridynamics-based statistical multiscale method for fracture in composite structures

This paper proposes a peridynamics-based statistical multiscale (PSM) framework to simulate the macroscopic structure fracture with high efficiency. The heterogeneities of composites, including the shape, spatial distribution and volume fraction of particles, are characterized within the representative volume elements (RVEs), and their impact on structure failure are extracted as two types of peridynamic parameters, namely, statistical critical stretch and equivalent micromodulus. At the microscale level, a bond-based peridynamic (BPD) model with energy-based micromodulus correction technique is introduced to simulate the fracture in RVEs, and then the computational model of statistical critical stretch is established through micromechanical analysis. Moreover, based on the statistical homogenization approach, the computational model of effective elastic tensor is also established. Then, the equivalent micromodulus can be derived from the effective elastic tensor, according to the energy density equivalence between classical continuum mechanics (CCM) and BPD models. At the macroscale level, a macroscale BPD model with the statistical critical stretch and the equivalent micromodulus is constructed to simulate the fracture in the macroscopic homogenized structures. The algorithm framework of the PSM method is also described. Two- and three-dimensional numerical examples illustrate the validity, accuracy and efficiency of the proposed method.

A peridynamic model for advection–reaction–diffusion problems

In this paper, a bond-based peridynamic (PD) advection–reaction–diffusion model is formulated to simulate processes involving chemical reactions, flows, and diffusion. In the formulation, the reaction rate, represented by a body force-like local term, depends on the local concentrations of the reactants, and the dependence varies according to the reaction type. Both linear and nonlinear dependencies are considered. A system of coupled PD equations for each substance is solved numerically. We verify and validate the PD advection–reaction–diffusion model by comparing results with those from a corresponding classical model and from experiments in both 1D and 2D, for a bimolecular reaction (A+B→AB) in a flowing aqueous environment, similar to purifying polluted water flowing in a channel, via chemical reactions. The model is also used to simulate bimolecular advection–reaction–diffusion in heterogeneous media with impermeable inclusions, and to simulate polymer degradation through hydrolysis and diffusion, similar to how biodegradable polymer implants dissolve.

The essential work of fracture in peridynamics

In this work, the essential work of fracture (EWF) method is introduced for a peridynamic (PD) material model to characterize fracture toughness of ductile materials. First, an analytical derivation for the path-independence of the PD J-integral is provided. Thereafter, the classical J-integral and PD J-integral are computed on a number of analytical crack problems, for subsequent investigation on how it performs under large scale yielding of thin sheets. To represent a highly nonlinear elastic behavior, a new adaptive bond stiffness calibration and a modified bond-damage model with gradual softening are proposed. The model is employed for two different materials: a lower-ductility bainitic-martensitic steel and a higher-ductility bainitic steel. Up to the start of the softening phase, the PD model recovers the experimentally obtained stress–strain response of both materials. Due to the high failure sensitivity on the presence of defects for the lower-ductility material, the PD model could not recover the experimentally obtained EWF values. For the higher-ductility bainitic material, the PD model was able to match very well the experimentally obtained EWF values. Moreover, the J-integral value obtained from the PD model, at the absolute maximum specimen load, matched the corresponding EWF value.

An improved peridynamic model for quasi-static and dynamic fracture and failure of reinforced concrete

In this paper, an improved peridynamic model is established to characterize the bond-slip behavior between ribbed steel rebar and concrete. An interfacial force term is introduced into the description of interaction between rebar and concrete considering the chemical bonding force, the friction, as well as the mechanical bite force, and a size-dependent non-uniform discretization algorithm for rebar and concrete is developped to reduce the calculation cost in numerical simulations. An equivalent modeling approach for ribs is presented as well, which can accurately reflect the interaction between ribs and concrete. Validation of the proposed model and approach is established through modeling pull-out problem of ribbed steel rebar, in which the crack initiation and propagation during the pull-out process can be accurately captured. Moreover, the dynamic fracture and impact failure of reinforced concrete are numerically modeled and analyzed. The numerical predictions are consistent with available experimental results, in terms of fracture characteristics of concrete, deformation of rebars, as well as the force–displacement curves.

Towards a unified nonlocal, peridynamics framework for the coarse-graining of molecular dynamics data with fractures

Molecular dynamics (MD) has served as a powerful tool for designing materials with reduced reliance on laboratory testing. However, the use of MD directly to treat the deformation and failure of materials at the mesoscale is still largely beyond reach. In this work, we propose a learning framework to extract a peridynamics model as a mesoscale continuum surrogate from MD simulated material fracture data sets. Firstly, we develop a novel coarse-graining method, to automatically handle the material fracture and its corresponding discontinuities in the MD displacement data sets. Inspired by the weighted essentially non-oscillatory (WENO) scheme, the key idea lies at an adaptive procedure to automatically choose the locally smoothest stencil, then reconstruct the coarse-grained material displacement field as the piecewise smooth solutions containing discontinuities. Then, based on the coarse-grained MD data, a two-phase optimization-based learning approach is proposed to infer the optimal peridynamics model with damage criterion. In the first phase, we identify the optimal nonlocal kernel function from the data sets without material damage to capture the material stiffness properties. Then, in the second phase, the material damage criterion is learnt as a smoothed step function from the data with fractures. As a result, a peridynamics surrogate is obtained. As a continuum model, our peridynamics surrogate model can be employed in further prediction tasks with different grid resolutions from training, and hence allows for substantial reductions in computational cost compared with MD. We illustrate the efficacy of the proposed approach with several numerical tests for the dynamic crack propagation problem in a single-layer graphene. Our tests show that the proposed data-driven model is robust and generalizable, in the sense that it is capable of modeling the initialization and growth of fractures under discretization and loading settings that are different from the ones used during training.

A peridynamic model based on generalized thermoelastic theory in a plate with oblique cracks

In this paper, a peridynamic model based on generalized thermoelastic theory is derived by irreversible thermodynamics. This model avoids the spatial derivative and can be easily applied to generalized thermoelastic problems with discontinuities. The transient temperature response is captured using the non-Fourier CattaneoVernotte (C-V) model. The interaction integral is employed to calculate the transient thermal stress intensity factors (TSIFs). Numerical examples are presented for analyzing the effects of crack orientation angle and multi-crack distribution on transient thermoelastic response. And the crack propagation behavior under non-Fourier thermal shock is further discussed in detail. It is found that with the increase of the angle between inner crack and heating boundary, the sensitivity of mode-I TSIFs to the non-Fourier effect increases, while the sensitivity of mode-II TSIFs to the non-Fourier effect decreases. This peridynamic model may be useful for understanding the crack behaviors of thermal protection materials under thermal shock.

Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity

Abstract We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz’ fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola’s identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient.

A bond-level energy-based peridynamics for mixed-mode fracture in rocks

This paper deals with the numerical simulation of initiation and propagation of mixed mode fracture in rocks. A bond-level energy-based peridynamic model is developed by introducing a dilatation function to capture both volumetric and deviatoric deformations. We proceed to define the nonlocal stresses to obtain the isotropic and deviatoric forces commensurate with the deformations. We then come up with a new failure model to link computational peridynamics with some phenomenological failure criteria, which is highly relevant for both brittle and quasi-brittle rocks. Finally, several numerical examples of mixed-mode fractures are presented to show the performance of our model.

Peridynamic modeling of void nucleation and growth in metal lines due to electromigration in a finite element framework

This study presents a peridynamic model to predict void formation and growth in metal lines due to electromigration. The nonlocal form of the nonlinear governing equations and boundary conditions for electromigration are derived by using the peridynamic differential operator (PDDO). The equation of motion includes the effect of diffusion strain due to electromigration. Coupled governing equations of atomic concentration and deformation are solved in the Ansys framework with native elements through implicit algorithms. The numerical results show that the current approach can accurately predict the distribution of the deformation, atomic concentration, and hydrostatic stress. Also, the void growth predictions agree well with the experimental predictions.

Accelerating convergence of crack propagation simulation in peridynamic models via high-order temporal discretization

This work proposes a high-order temporal discretization for reformulations of both bond-based and state-based peridynamic models modeling, e.g. crack propagation. It is discovered from various experiments that for such discontinuous problems, commonly-used second-order methods are less stable and more computationally consuming than the high-order method, which indicates that the high-order temporal discretization not only improves the accuracy and the efficiency for conventional smooth problems, but also exhibits a promising behavior for some non-smooth models such as peridynamics.

The unified nonlocal peridynamics-based phase-field damage theory

The peridynamic correspondence model is a promising and attractive candidate for the modeling of localized failure in solids, in that it can incorporate many local classical damage constitutive models through introducing a nonlocal averaged deformation gradient. However, the zero-energy mode and the limited bond breaking criterion greatly restrict the potential applications of this correspondence model. To address these two problems, a unified nonlocal peridynamics-based phase-field damage theory is proposed within the framework of thermodynamics. Firstly, the unified correspondence principle for the displacement field is developed on the basis of the energy compensation method in such a way that a new peridynamic deformation gradient, shape matrix and force state are derived and redefined. And then, the proposed principle is applied to derive the nonlocal phase-field damage constitutive model that the general nonlocal phase-field flux, phase-field flow state and phase-field internal force are defined. Moreover, we propose a mixed variational derivative method to obtain the coupled equilibrium governing equation and present the general linearization approach of the proposed peridynamics-based phase-field model (PD-PFM), where the double states for the coupled displacement and phase fields are derived in detail. It will be found that PD-PFM can not only resolve the zero-energy mode existed in the conventional peridynamic correspondence model, but also provide a rational criterion for the breakage of the peridynamic bond. Some representative numerical examples including mixed mode fracture of single-material media and interface fracture of ceramic coating systems are presented for the validation of PD-PFM. The satisfactory results show both quantitative and qualitative agreement with the available experiment.

An adaptive coupling approach of local and non-local micromechanics

This study proposes a novel method for predicting the microcrack propagation in composites based on coupling the local and non-local micromechanics. The special feature of this method is that it can take full advantages of both the continuum micromechanics as a local model and peridynamic micromechanics as a non-local model to achieve composite fracture simulation with a higher level of accuracy and efficiency. Based on the energy equivalence, we first establish the equivalent continuum micromechanics model with equivalent stiffness operators through peridynamic micromechanics model. These two models are then coupled into a closed equation system, and a transition region is introduced to achieve a smooth transition between them. A composite strength-induced adaptive algorithm is introduced to solve the unified model. Numerical examples for particle reinforced composites are considered to show the accuracy and performance of the present method. The micromechanics-based coupling method has the potential to efficiently simulate the microcrack propagation in various complex composite materials.

Peridynamic modeling of nonlinear surface acoustic waves propagating in orthotropic materials

Due to the elastic nonlinearity of the material, high-amplitude surface acoustic waves undergo nonlinear evolution during propagation and may lead to material failure. To enable the acoustical quantification of material nonlinearity and strength, a comprehensive understanding of this nonlinear evolution is necessary. This paper presents a novel ordinary state-based nonlinear peridynamic model for the analysis of the nonlinear propagation of surface acoustic waves and brittle fracture in anisotropic elastic media. The relationship between seven peridynamic constants and second- and third-order elastic constants is established. The capability of the developed peridynamic model has been demonstrated by predicting surface strain profiles of surface acoustic waves after propagating in the silicon (111) plane and the 〈112¯〉 direction. On this basis, the nonlinear wave-induced spatially localized dynamic fracture is also studied. The numerical results reproduce the main features of nonlinear surface acoustic waves and fracture observed in experiments.

A bond-based peridynamics modeling of polymeric material fracture under finite deformation

We have developed a bond-based peridynamics model to study the fracture of polymer networks under finite deformation. Our approach is based on the Kuhn–Grün model of randomly jointed polymers with extensible chains, which allows the rupture of polymers through the chain scission mechanism. We introduced a novel mesoscale potential function that intrinsically incorporates an Arruda–Boyce type model into a nonlocal continuum framework with the purpose of modeling the mechanical response of general polymeric materials and the fracture of polymers. We have demonstrated through a series of numerical examples, both two-dimensional and three-dimensional, that the proposed model is robust and efficient in modeling crack growth in polymeric materials. The model is carefully validated with experimental data and numerical results obtained on the basis of existing continuum mechanics polymer models. We have shown that the proposed peridynamic model is accurate and stable in various configurations and loading modes, such as mode I and shear crack opening. In comparison to existing continuum polymer models and fracture modeling methods, our present technique has the advantages of being theoretically simple, rigorous, and computationally fast. The numerical results show that the nonlocal polymer model can easily capture the fracture process in polymeric materials without complex ad hoc modeling parameters under finite deformation, and it has the potential to become a convenient simulation tool for modeling the polymer failure process in the design and optimization of polymeric materials.

Multi-adaptive spatial discretization of bond-based peridynamics

Peridynamic (PD) models are commonly implemented by exploiting a particle-based method referred to as standard scheme. Compared to numerical methods based on classical theories (e.g., the finite element method), PD models using the meshfree standard scheme are typically computationally more expensive mainly for two reasons. First, the nonlocal nature of PD requires advanced quadrature schemes. Second, non-uniform discretizations of the standard scheme are inaccurate and thus typically avoided. Hence, very fine uniform discretizations are applied in the whole domain even in cases where a fine resolution is per se required only in a small part of it (e.g., close to discontinuities and interfaces). In the present study, a new framework is devised to enhance the computational performance of PD models substantially. It applies the standard scheme only to localized regions where discontinuities and interfaces emerge, and a less demanding quadrature scheme to the rest of the domain. Moreover, it uses a multi-grid approach with a fine grid spacing only in critical regions. Because these regions are identified dynamically over time, our framework is referred to as multi-adaptive. The performance of the proposed approach is examined by means of two real-world problems, the Kalthoff–Winkler experiment and the bio-degradation of a magnesium-based bone implant screw. It is demonstrated that our novel framework can vastly reduce the computational cost (for given accuracy requirements) compared to a simple application of the standard scheme.

Experimental and numerical investigation of mixed mode fracture of high‐performance grouting materials based on peridynamics

AbstractThis paper investigates the fracture behavior of high‐performance grouting materials in the grouted connection section of marine structures, where they are subjected to complex stress states. This study utilizes a combination of experimental and numerical simulation methods to establish a reliable numerical simulation technique for the fracture process of high‐performance grouting materials. The mixed mode fracture behavior is analyzed using six different types of specimens, and the strain contour is analyzed using the Digital Image Correlation technique. An extended peridynamics model is proposed for the numerical simulation, which adopts a fracture criterion based on strain energy density. The accuracy of the model is verified qualitatively and quantitatively, and the simulation results are consistent with the experiments. Overall, this study provides insights into the fracture behavior of high‐performance grouting materials in complex stress states and presents a reliable numerical simulation technique for the fracture process.

A Modified Bond-Associated Non-Ordinary State-Based Peridynamic Model for Impact Problems of Quasi-Brittle Materials.

In this work, we have developed a novel bond-associated non-ordinary state-based peridynamic (BA-NOSB PD) model for the numerical modeling and prediction of the impact response and fracture damage of quasi-brittle materials. First, the improved Johnson-Holmquist (JH2) constitutive relationship is implemented in the framework of BA-NOSB PD theory to describe the nonlinear material response, which also helps to eliminate the zero-energy mode. Afterwards, the volumetric strain in the equation of state is redefined by the introduction of the bond-associated deformation gradient, which can effectively improve the stability and accuracy of the material model. Then, a new general bond-breaking criterion is proposed in the BA-NOSB PD model, which is capable of covering various failure modes of quasi-brittle materials, including the tensile-shear failure that is not commonly considered in the literature. Subsequently, a practical bond-breaking strategy and its computational implementation are presented and discussed by means of energy convergence. Finally, the proposed model is verified by two benchmark numerical examples and demonstrated by the numerical simulation of edge-on impact and normal impact experiments on ceramics. The comparison between our results and references shows good capability and stability for impact problems of quasi-brittle materials. Numerical oscillations and unphysical deformation modes are effectively eliminated, showing strong robustness and bright prospects for relevant applications.

A comparison study on numerical analysis for concrete dynamic failure using intermediately homogenized peridynamic model and meso-scale peridynamic model

In this paper, a three-dimensional meso-scale model of concrete is proposed within the peridynamic framework, containing cement mortar, aggregates, and ITZ (interfacial transition zone). The meso-structures characteristics are considered for numerical modeling of concrete, and ITZ is modeled by cross-interface bonds instead of actual modeling. Several typical numerical tests are performed to validate the capability of the meso-scale model. Meanwhile, a comparison study between the intermediately homogenized model and meso-scale model is conducted, and the difference in modeling and analysis for concrete dynamic failure among these two models is illustrated and discussed. There is no remarkable discrepancy in the concrete fracture pattern under dynamic loadings. Nevertheless, the meso-scale model has a better performance than the intermediately homogenized model for impact perforation tests of concrete.

An incremental bond‐based peridynamic method for elastoplastic problems

AbstractThe fracture problem of elastoplastic materials is an insistent challenge. It is difficult to implement the fracture simulations of elastoplastic materials in peridynamics (PD), especially in the bond‐based peridynamic (BB–PD) framework. The core issue is how to introduce non‐linear constitutive relationships into BB–PD model. To address this issue, an incremental form of BB–PD model was originally proposed. In the proposed model, a rapid strain identification technique was established. Based on this technique, the constitutive relationship is related to the deformation and the incremental bond force equation is formulated in a unified form. In addition, a fracture criterion controlled by energy was derived to implement plastic fracture simulations. Several numerical examples, including five elastoplastic deformation analyses and three quasi‐static and dynamic fractures, were implemented to verify the feasibility and efficiency of the proposed model. The results show that the proposed model can accurately analyze the deformation of elastoplastic problems, especially plastic deformation. The numerical examples also demonstrate the fracture prediction capability of the proposed model.