Peridynamics Research Articles

A novel non-linear peridynamic model for quasi-brittle materials under cyclic loading

In the present paper, a novel non-linear peridynamic (PD) model is proposed to numerically simulate the fatigue behaviour of quasi-brittle materials. The model implements a non-linear degradation law for the bonds, function of the number of loading cycles. More precisely, a calibration function and a calibration parameter are included in such a law. The novel PD model is applied to simulate cyclic three-point bending experimental tests performed on concrete notched beams (available in the literature). The results are presented in terms of fatigue lifetime, Paris law parameters and crack path.

A high-order multi-time-step scheme for bond-based peridynamics

A high-order multi-time-step scheme (MTS) for the bond-based peridynamic (PD) model, an extension of classical continuous mechanics widely used for analyzing discontinuous problems like cracks, is proposed. The MTS scheme discretizes the spatial domain with a meshfree method and advances in time with a high-order Runge–Kutta method. To effectively handle discontinuities (cracks) that appear in a local subdomain in the solution, the scheme employs the Taylor expansion and Lagrange interpolation polynomials with a finer time step size, that is, coarse and fine time step sizes for smooth and discontinuous subdomains, respectively, to achieve accurate and efficient simulations. By eliminating unnecessary fine-scale resolution imposed on the entire domain, the MTS scheme outperforms the standard STS scheme for PD by significantly reducing computational costs, particularly for problems with discontinuous solutions, as demonstrated by comprehensive theoretical analysis and numerical experiments.

Finite deformation micropolar peridynamic theory: Variational consistency of wryness measure

This paper concerns developments in both classical and peridynamic (PD) finite deformation micropolar elasticity theory. The first part presents an alternative proof of a theorem that demonstrates variational consistency of the wryness measure in classical finite deformation micropolar elasticity which is one of the novel contributions of this work. The linearized version of the proposed wryness measure is compared with the wryness measure of small deformation micropolar elasticity. In the second part of this work, a finite deformation micropolar PD theory is presented. After proposing an additional integro-differential equation along with the standard PD equation of motion, the global balance of angular momentum is proved to be satisfied. The balance of virtual work is derived for the micropolar PD theory. Next, constitutive correspondence approach is used to relate PD force and moment states with their classical counterparts. The nonlocal versions of the strain and the proposed wryness measure are used in the constitutive correspondence. The constitutive equations of the classical micropolar theory are presented and how they can be incorporated in PD framework is discussed. We introduce a new bond breaking criterion for PD micropolar materials based on critical stretch and critical relative rotation. Quasi-static numerical simulations on deformations of plate with a hole and fracture of a double notched sheet are presented. Dynamic simulations concern wave propagation through a solid specimen and plate with a hole. Validation of the simulation results with the finite element solutions, boundary element solutions, and experimental observations demonstrate the potential of our approach.

A multiscale fracture model using peridynamic enrichment of finite elements within an adaptive partition of unity: Experimental validation

Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Here, we apply Peridynamic (PD) enrichment to propagate cracks in the PUM global–local enrichment scheme. We apply linear elasticity globally and PD over local zones where fractures occur. The elastic fields provide appropriate boundary data for local PD simulations on a subdomain containing the crack tip to grow the crack. Once the updated crack path is found the elastic field in the body and surrounding the crack is updated using the PUM basis with an elastic field enrichment near the crack. The subdomain for the PD simulation is chosen to include the current crack tip as well as features that influence crack growth. This paper is part II of this series and validates the combined PD/PUM simulator against the experimental results. The results of numerical simulation show that we attain good agreement between experiment and simulation with a local PD subdomain that is moving with the crack tip and adaptively chosen size.

Modelling ductile damage in metals and alloys through Weyl condition exploiting local gauge symmetries

Local translational and scaling symmetries in space–time is exploited for modelling ductile damage in metals and alloys over wide ranges of strain rate and temperature. The invariant energy density corresponding to the ductile deformation is constructed through the gauge invariant curvature tensor by imposing the Weyl like condition. In contrast, the energetics of the plastic deformation is brought in through the gauge compensating field emerged due to local translation and attempted to explore the geometric interpretation of certain internal variables often used in classical viscoplasticity models. Invariance of the energy density under the local action of translation and scaling is preserved through minimally replaced space–time gauge covariant operators. Minimal replacement introduces two non-trivial gauge compensating fields pertaining to local translation and scaling. These are used to describe ductile damage, including plastic flow and micro-crack evolution in the material. A space–time pseudo-Riemannian metric is used to lay out the kinematics in a finite-deformation setting. Recognizing the available insights in classical theories of viscoplasticity, we also establish a correspondence of the gauge compensating field due to spatial translation with Kröner’s multiplicative decomposition of the deformation gradient. Thermodynamically consistent coupling between viscoplasticity and ductile damage is ensured through an appropriate degradation function. Non-ordinary state-based (NOSB) peridynamics (PD) discretization of the model is used for numerical implementation. We conduct simulations of uniaxial deformation to validate the model against available experimental evidence and to assess its predictive features. The model’s viability is tested in reproducing a few experimentally known facts, viz., strain rate locking in the stress–strain response, whose origin is traced to a nonlinear microscopic inertia term arising out of the space–time translation symmetry. Finally, we solved 2D and axisymmetric deformation problems for qualitatively validating the model’s viability. NOSB peridynamics axisymmetric formulation in finite deformation setup is also presented.

Modeling via peridynamics for crack propagation in laminated glass under fire

Despite laminated glass is widely used in building construction, its susceptibility to cracking and collapsing during an accidental fire can intensify fire propagation, leading to substantial casualties and economic losses. In this work, we developed a meshfree framework for modeling thermomechanical cracking of laminated glass via peridynamic (PD) theory. First, to overcome the incomplete input experimental data for implementing thermal boundary conditions, we proposed both the uniform and non-uniform temperature field strategies. Second, PD interface model and the stochastic distribution of defects were implemented within this framework. Third, the challenging problem of crack path selection in thermally loaded borosilicate/steel bi-material beams was studied to validate the feasibility of the proposed model. It is demonstrated that the proposed framework can capture the temperature history and crack features of laminated glass under fire, showing great consistency with experimental results. Our findings show that with the decrease of the glass strength or the increase of the defect content, more islands form in the central region of the glazing, implying higher risks of glass fall-out in fire conditions. This model constitutes a promising tool for predicting the fire performance of laminated glass and contributes to designing safer engineering structures.

A neural network peridynamic method for modeling rubber-like materials

Peridynamic (PD) is a powerful tool for simulating the large deformation and failure process of many types of materials. However, its use in modeling rubber-like materials is limited due to the complex constitutive nature of the material, and low efficiency and numerical oscillation caused by PD. To address this issue, a neural network (NN) non-ordinary state-based peridynamics (NOSB PD) method is developed to model the large deformation and failure behavior of rubber-like materials. This method is free of the zero-energy modes, and can significantly improve the computational efficiency. Unlike the traditional NOSB PD method that formulates the force density vector based on the deformation gradient, this method uses a deep NN to map the bond related quantities to the force density vector. The accuracy and efficiency of the proposed method are demonstrated through a series of numerical examples. Additionally, this method can be applied to various hyperelastic materials for which analytical constitutive models exist.

Multi-GPU parallel acceleration scheme for meshfree peridynamic simulations

Peridynamics (PD) using integral equations has unique advantages in modeling structural damage evolution, but the expensive computational and memory costs associated with the nonlocal nature of PD severely limit its engineering applications. This study presents a CUDA-based multi-GPU parallel scheme for meshfree PD simulation, which is compatible with dual-horizon PD supporting multiscale spatial discretization. Efficient utilization of computing resources across all available GPUs is achieved by rationalizing data structures across multiple GPUs and using the CUDA memory model. Using atomic operations to avoid computational dual-bond forces further improves code performance. The classical Kalthoff-Winkler experiment and compact-tensile fatigue test were first carried out using 3 NVIDIA Tesla T4 GPUs. Two examples showed that the proposed scheme significantly accelerates the PD model and that the acceleration enhances approximately linearly with the number of GPUs. The speedup of the PD interaction force computation reaches up to 560.7x. Compared to double-precision floating-point values, the GPU performs much better in handling single-precision values. Finally, a PD simulation of rail head heat conduction under moving heat sources with nearly 76 million particles was realized, demonstrating the potential of the proposed scheme to support large-scale PD simulations.

Rockburst prediction for deep tunneling near fault based on the PD-BEM method

A multiscale model coupling the peridynamic (PD) and boundary element method (BEM) is proposed to analyze the rockburst in deep tunnels near fault. The entire rockburst processes of the rock masses around the opening are captured by PD while the field conditions and boundary are described by BEM, which can take full advantage of their salient features and model the rockburst problems efficiently. The proposed model is verified by comparing its predictions with those from the field observations in “11.28” rockburst at the Jinping II Hydropower Station in China. Parametric analysis considering the dip angle and location of the fault as well as the ground stress is performed to reveal the rockburst mechanism of deep tunnel near fault. Numerical results indicate that the rockburst intensity of rock tunnel is a function of the distance between the fault and tunnel, the dip angle of fault and ground stress. Different failure modes, including buckling failure, slip failure and local slabbing failure are found in the simulation, which are controlled by the fault near the tunnel. Moreover, a new index for rockburst is proposed to predict the tendency and intensity of rockburst associated with deep tunneling near a fault.

Peridynamic correspondence model with strain gradient elasticity for microstructure dependent size effects

This study presents a peridynamic (PD) correspondence model with strain gradient elasticity (SGE) to capture size effect on strength of nano- and micro-scale structures. The classical elasticity theory lacking a length scale parameter does not account for the microstructure-dependent size effects. Strain Gradient (SG) elasticity is an extension of the classical elasticity with a length scale parameter(s) which can be linked to the microstructure of the material. Peridynamic theory introduces damage into the constitutive relations in a natural way by allowing for interactions of a material point within its horizon which is referred to as the internal length parameter. The correspondence model of the PD theory permits the incorporation of the constitutive relationship of SG theory. Therefore, the present approach combines the effects of PD and SGE length scale parameters on the stiffness and strength of the material. The resulting equations present two length parameters: the horizon of a material point in PD theory and the characteristic length in SGE theory. The PD equation of motion with SGE (PDSG) is free of spatial derivatives and allows for the imposition of classical and nonclassical boundary conditions. The PDSG is first applied to investigate the deformation response of a one-dimensional nanoscale film subjected to a quasi-static tensile load and a dynamic load through an initial constant strain. The static response is compared with the analytical solution for varying ratio of SG length parameter to PD horizon. The dynamic response is compared with a computational solution while satisfying the nonclassical boundary conditions. Subsequently, the PDSG is applied to study the deformation response of a two-dimensional nanoscale film subjected to a quasi-static axial load and a specified tangential end displacement. These two cases are considered under special displacement constraints to compare with the available analytical solutions. Subsequently, a two-dimensional nanoscale film with or without an internal crack is subjected to uniform tensile loading. Finally, crack propagation is simulated under incremental displacement loading. The results capture the analytical predictions and the expected increase in stiffness with increasing ratio of SG length parameter to PD horizon.

Peridynamic modeling of seepage in multiscale fractured rigid porous media

AbstractThe numerical simulation of seepage in fractured porous media holds significant relevance for subsurface energy development. The presence of fractures at various scales profoundly influences the hydraulic properties of porous media during seepage. A peridynamic (PD)‐based frame is proposed for the seepage problem analysis of multiscale fractured porous media, in which micro‐fractures are implicitly represented like the porous media, macro‐fractures are explicitly represented. For fluid flow modeling, the peridynamic bonds are divided into three types: pore seepage bond, fracture seepage bond, and fluid exchange bond. The micro‐permeabilities of different bonds are derived by Darcy's law, Poiseuille's law, and fluid exchange model, respectively. The proposed peridynamic model can naturally simulate the seepage problem of porous rock with multiscale fractures in a unified framework. Then, four cases of pore seepage and fracture seepage are analyzed using the proposed model, fractured porous media seepage with micro‐fractures and macro‐fractures are respectively considered. The predicted results are compared with the available theoretical solutions, the close agreement shows that the wide validity and applicability of the proposed model in multiscale seepage problems.

A thermo-chemo-mechanically coupled peridynamics for investigating crack behavior in solids

In engineering applications, the phenomenon of cracking is often accompanied by a coupled multiphysics effect. Peridynamics (PD) is an effective approach for solving cracking problems, but currently, no general PD model accounts for the coupling of multiple physical fields. In this work, we develop a PD model of coupled deformation, heat conduction, species diffusion, and chemical reactions. First, we establish the equations for mass, linear momentum, and energy. Then we establish fully coupled constitutive laws that interpret the interactions between the various fields and formulate evolution equations that govern the flux of species and heat. These laws and equations are developed based on the inequality of energy dissipation and the principles of chemical kinetics. Species diffusion and chemical reactions are treated as separate processes to study their effects on the Helmholtz free energy density of solids and the subsequent formation and propagation of cracks. In addition, certain coupling coefficients are calibrated by equating the corresponding physical quantities in the PD model with those in continuum mechanics. Four specific cases are simulated to validate the model, including redistribution of vacancies in ceramics, hydrogen traps, and embrittlement in metals.

Peridynamic simulation of fatigue crack growth in porous materials

Pores are a key type of concerns in additive manufacturing technology. They can significantly affect materials’ fatigue resistance and fatigue crack growth in the structures. In this paper, an intermediately-homogenized (IH) peridynamic (PD) fatigue crack model is introduced for simulating the fatigue crack growth in porous materials. The effect of micro-scale heterogeneity on fatigue failure is preserved in the IH-PD model, where stochastically-generated pre-damage matches materials’ porosity. The PD J-integral, a key parameter used to connect the fatigue simulation and the experimental measurements, is calculated under different porosities, and the independence of PD J-integral on its integration path is verified. Then, the PD fatigue crack model combined with the IH-PD material model is applied to study the dependence of the crack path and growth rate on porosity in compact tension alloy samples. The normalized fatigue life between experimental measurements and numerical results is compared. The normalized fatigue life-porosity relationship calculated from the IH-PD fatigue model matches the average trend of the experimental measurements.

A stable formulation of correspondence‐based peridynamics with a computational structure of a method using nodal integration

SummaryIn this paper, we lay out a variational framework for correspondence‐based peridynamic (PD) formulations of solid mechanics. Using the framework, we address the numerical instabilities of the original version of correspondence‐based PD by developing a natural stabilization technique that avoids costly bond‐associated approaches and retains the structure of a method with nodal integration. Accuracy, robustness, and efficiency of the proposed naturally stabilized correspondence‐based PD are demonstrated on several computational test cases ranging from linear elastostatics to large deformation elasto‐plasticity. The computational methodology developed is particularly effective for handling materials undergoing nearly‐incompressible deformations.

Study on fatigue damage tolerance of rail steel materials using peridynamics

With railways transitioning into the maintenance era, this study proposes a method to ensure structural safety and control maintenance costs. We develop a Peridynamics (PD) fatigue crack growth prediction model to determine the safe crack threshold for rail steel material under cyclic loading, known as the fatigue damage tolerance size. By conducting fatigue tests, we obtain the PD parameters for the rail steel and rigorously validate the model's reliability. Using both the compact tensile (CT) specimen model and a transient wheel-rail contact model, we analyze factors influencing crack propagation and fatigue damage tolerance. The results reveal that higher cyclic loads and smaller stress ratios result in shorter fatigue life and smaller damage tolerance sizes. Additionally, larger initial crack angles, faster train speeds, and heavier axle loads reduce fatigue crack propagation life and damage tolerance sizes. Ultimately, based on damage tolerance analysis, it is recommended to limit crack propagation in U71Mn rail steel within a 3.8-millimeter safety range for reliable operation.

A 3D peridynamic model for fracture analysis of transversely isotropic solids

The classical continuum mechanics (CCM) has limitations in simulating crack initiation and propagation, and the peridynamic (PD) theory overcomes these shortcomings. This study proposed a 3D transversely isotropic PD model based on the micro-beam bond model. Firstly, the micromoduli of the micro-beam bond applied to the transversely isotropic 3D solid model are derived based on the elastic matrix relationship between the different coordinate systems. Secondly, the strain energy density expressions of arbitrary material points are expressed in both the PD theory and the CCM theory using strain and elastic matrix under the assumption of homogenization, respectively. Finally, by equating the strain energy density expressions under the two theories based on the conservation of energy, the constitutive model parameters of the 3D transversely isotropic PD model are solved. Numerical examples of crack propagation demonstrate the effectiveness of the proposed model in addressing dynamic 3D fracture problems.

Peridynamics for the fracture study on multi-layer graphene sheets

Understanding the fracture properties of graphene sheets is a crucial step towards their practical applications. However, due to the limitations of experimental operations and all-atom (AA) methods, investigating the fracture of large-sized nano graphene sheets remains a formidable challenge. Especially, study on the layer-by-layer fracture of multi-layer graphene sheets (MLGS) is nearly impossible. To overcome this challenge, a peridynamic (PD) model is proposed in this study, which comprises the intra-layer part and the inter-layer part. The proposed PD model is validated by comparing the fracture toughness and the fracture forms of MLGS with existing experiments. It is found that the uniaxial tensile stress-strain curve of pre-cracked MLGS is closely related to the number of graphene layers in MLGS. The fracture property of MLGS can be enhanced by increasing the number of graphene layers, reducing the pre-crack length and blunting the pre-crack tip. Notably, asynchronous crack propagation with independent path observed in MLGS is a unique mechanism for strengthening the fracture property, which is distinct from monolayer graphene sheet. In this work, the PD theory is extended for the first time to investigate the in-plane fracture of large-sized nano MLGS.

Ordinary state-based peridynamic model for out-of-plane deformation and damage analysis of composite laminates

An ordinary state-based peridynamic (PD) model for predicting out-of-plane mechanical behaviour of composite laminates has been proposed, in which three types of PD bonds are used to describe the reinforcement properties. The non-local strain energy density and peridynamic formulations are obtained based on the principle of virtual work by using Total Lagrange formulation, and the force density is reformulated by interpolation technique. Since the Mindlin plate is regarded as the research object of this PD model, the transverse shear deformation of laminates is considered. Also, the dilatation term is included in the force density vector, and flaws in Poisson's ratio of materials can be overcome. In the proposed PD, the critical strain energy density rather than the critical curvature is adopted as the failure criterion. The capability of the developed PD model was demonstrated by the bending examples of composite laminates with different fiber orientations, and damage analysis was further conducted to demonstrate the strong capability of proposed PD model in replicating the damage process of laminates. In addition, a single-layer material point model (SLMPM) can be implemented in the proposed PD algorithm, and the computational efficiency of numerical models will be greatly improved.

Non-local modelling of freezing and thawing of unsaturated soils

A large part of the earth’s surface is covered by seasonally or permanently frozen soils. Considering the negative impact of climate change, future development of such regions can be underpinned by mathematical methods for accurate analysis of heat and moisture transport in freezing and thawing soils. Reported in this paper is a novel non-local formulation of water and heat transport in unsaturated soils. The formulation uses bond-based peridynamics (PD) and consists of a set of integral–difference formulations of energy and mass conservation. Specific features of freezing/thawing soils are incorporated by a combination of van Genuthcen and Clausius–Clapeyron relations. Computational results are compared with four sets of laboratory experiments to demonstrate the efficiency of the developed approach. The model can be used to analyse the effect of water flow on heat transfer in soils during thawing of permafrost soils. Further, it can be applied in modelling climate change effects, and can be used for construction of coupled physically justified models of frost heave.

PERIDYNAMIC SIMULATION OF FRACTURING IN HETEROGENEOUS ROCK BASED ON X-RAY DIFFRACTION AND SCANNING ELECTRON MICROSCOPE TESTS

The mechanical properties and fracture behavior of rocks are significantly influenced by their microscopic characteristics. At present, there is a prevalent treatment of rocks as homogeneous materials or an oversimplified assumption of mesoscopic material properties following the Weibull distribution, often overlooking the impact of mineral composition and porosity. This study addresses these limitations by utilizing X-ray diffraction (XRD) and scanning electron microscope (SEM) tests to capture and characterize the microscopic features of rocks, including mineral composition and porosity. Subsequently, a mesoscopic peridynamic (PD) model is developed employing the Knuth-Durstenfeld shuffling algorithm to accurately reflect the real rock microstructure. The efficacy of this method is validated through experiments conducted on sandy mudstone and fine-grained sandstone. Moreover, a parametric analysis is performed, considering variations in porosities and mineral compositions. As porosity increases, numerous small cracks emerge laterally in the model, causing a notable decline in the rock's strength. Rock comprising a single mineral composition typically displays linear failure behavior. Conversely, a rock with a diverse array of minerals tends to exhibit non-linear failure behavior, indicating an increased level of heterogeneity within the material.