Peridynamic Theory Research Articles

Modelling soil desiccation cracking by peridynamics

Understanding of desiccation-induced cracking in soil has improved over the last 20–30 years through experimental studies, but progress in predictive modelling of desiccation cracking has been limited. The heterogeneous structure of soils and the multi-physics nature of the phenomenon, involving emergence and propagation of discontinuities, make the mathematical description and analysis a challenging task. The authors present a non-local hydro-mechanical model for soil desiccation cracking capable of predicting crack initiation and growth. The model is based on peridynamics (PD) theory. Attempts to model soil desiccation cracking by PD are limited to a purely mechanical description of the process that involves calibration of the parameters. In contrast, the model presented in this paper describes soil desiccation cracking as a hydro-mechanical problem, where moisture flow and deformation are coupled. This allows for investigating and explaining the mechanisms controlling the initiation and propagation of discontinuities. The model is applied and tested against two sets of experimental data to explain the typical features of drying-induced cracking of clays. The validations use experimental parameters (Young's modulus, water retention characteristics) and avoid calibrations to test the accuracy of the model. The correlations between the shrinkage of soil clay, changes in displacement fields and crack growth are demonstrated. Crack initiation, propagation and ultimate crack patterns simulated by the model are found to be in very good agreement with experimental observations. The results show that the model can capture realistically key hydraulic, mechanical and geometry effects on clay desiccation cracking.

The fractal version of the bond‐based peridynamics method

AbstractThis article presents a new formulation of bond‐based peridynamic theory (PD) encompassing material texture heterogeneity through fractal geometry concepts and transforming all relevant variables from Euclidean space to fractal space. The heterogeneity incorporation is realized through the introduction of a linear transformation fractal coefficient (LTFC). By considering a mean value for all LTFCs of all spatial coordinates. For example, x, y, z, and using the relationship between fractal dimension and the Hurst exponent . We present a fractal version of the bond‐based PD method that the equation of motion is conformed to fractal space. Validation of the formulation is accomplished in two steps: first, the effect of irregular mass distribution is characterized by the fractal dimension of the fracture surface roughness through experimental results of a series of single‐edge notched tensile tests of a ductile material. In the next step, the proposed formulation is calibrated with the fractal dimension obtained experimentally. The results are compared with the classical bond‐based PD formulation and the finite element analysis results. The obtained results in this article indicate that the irregular mass distribution of material can be introduced into particle‐based numerical methods using an adequate fractal exponent. The fractal version provides a good approximation of the fracture behavior of materials.

Peridynamic modeling and simulations on concrete dynamic failure and penetration subjected to impact loadings

In this paper, we simulate the dynamic fracture and failure of three-dimensional concrete structures using the peridynamic theory. Considering the heterogeneities and mesoscopic characteristics of concrete material, the dynamic failure of concrete is investigated by employing an intermediately homogenized peridynamic (IH-PD) model, in which information from the composition at the mesoscale is linked to the macroscale fracture behavior, and cement mortar, aggregates, and interfacial transition zone (ITZ) are taken into account. Moreover, a plastic constitutive model under the framework of peridynamics, taking the effect of volume change and hydrostatic pressure on the concrete mechanical behavior into account, is formulated to depict the complex dynamic mechanical response of concrete under the high-velocity impact. Several typical numerical examples, including elastic wave propagation in a concrete plate, fracture in a notched concrete beam, and penetration of concrete targets subjected to high-velocity impacting are investigated. The numerical results are compared with corresponding experimental observations to validate the capabilities of the proposed model and approach. Further comparison with the fully-homogenized model shows the advantages of the present model on studying impact failure problems in concrete-like materials and structures.

Determination of material properties of a linearly elastic peridynamic material

We investigate the properties of an isotropic linear elastic peridynamic material in the context of a three-dimensional state-based peridynamic theory, which considers both length and relative angle changes, and is based on a free energy function proposed in previous work that contains four material constants. To this end, we consider a class of equilibrium problems in mechanics to show that, in interior points of the body where deformations are smooth, the corresponding solutions in classical linear elasticity are also equilibrium solutions in peridynamics. More generally, we show that the equations of equilibrium are satisfied even when two of the four peridynamic constants are arbitrary. Pure torsion of a cylindrical shaft and pure bending of a cylindrical beam are particular cases of this class of problems and are used together with a correspondence argument proposed elsewhere to determine these two constants in terms of the elasticity constants of an isotropic material from the classical linear elasticity. One of the constants has a singularity in the Poisson ratio, which needs further investigation. Two additional experiments concerning bending of cylindrical beam by terminal load and anti-plane shear of a hollow cylinder, which do not belong to the previous class of problems, are used to validate these results.

A non-local method in peridynamic theory for simulating elastic wave propagation in solids

In order to simulate the elastic wave propagation in unbounded solids, this paper introduces a non-local method to construct absorbing boundary conditions in the bond-based peridynamic theory. To construct non-local absorbing boundary conditions in the form of absorbing layers with increasing damping, we first re-derived the particle motion equilibrium equation of the bond-based peridynamic theory, based on the principle of virtual work and Lagrange's equation on the premise of considering material damping. We also give the expression for the inter-particle damping force density which satisfies the balance laws of linear momentum and angular momentum. This paper also provides a process and algorithm for realizing the non-local absorbing boundary conditions in the bond-based peridynamic theory, and adopts the Fortran language for computer program compilation. Finally, we simulate the elastic wave propagation in semi-unbounded and unbounded solids as numerical examples. The results show that such non-local absorbing boundary conditions can stably absorb and attenuate incident elastic waves. Since it is derived in the time domain, this method does not require frequency-domain operations, such as Fourier and Laplace transformations, nor does it require wave field splitting.

Peridynamic simulation on hydraulic fracture propagation in shale formation

The inhomogeneity of shale reservoirs, bedding properties and in-situ stress make crack propagation paths complex and diverse in hydraulic fracturing. However, the simulations on hydraulic fracture propagation in shale gas reservoirs still require an effective alternative numerical method. In this paper, a peridynamic (PD) fluid–solid coupling model was developed for the numerical simulation on the hydraulic fracturing process in bedding shale formation. Firstly, a modified bond-based peridynamic (BB-PD) theory was formulated by introducing normal bond, tangential bond and pressure bond. This removes the limitation of classical BB-PD theory on Poisson's ratio. Secondly, a fluid–solid coupling model was established by combining the modified BB-PD theory with pressure driven fluid transport in porous medium. Thirdly, a fracturing criterion for the modified BB-PD theory was derived from energy conservation. Fourthly, the PD fluid–solid coupling model was numerically discretized and verified by three-point bending tests on shale beam. Their accuracy was further checked through the Terzaghi's consolidation problem. Finally, numerical simulations were conducted to explore the generation of hydraulic fracturing induced fracture network in bedding shale formation under different dip angles and horizontal stress ratios. The simulation results showed that the horizontal stress ratio has a greater effect on fracturing pressure, and the dip angle plays a crucial role in the shape of fracture network. The increase of horizontal stress ratio leads to the increase of fracturing pressure.

A Review of Peridynamics (PD) Theory of Diffusion Based Problems

The study of heat conduction phenomena using peridynamic (PD) theory has a paramount significance on the development of computational heat transfer. This is because PD theory has got an interesting feature to deal with the inherent nonlocal nature of heat transfer processes. Since the revolutionary work on PD theory by Silling (2000), extensive investigations have been devoted to PD theory. This paper provides a survey on the recent developments of PD theory mainly focusing on diffusion based peridynamic (PD) formulation. Both the bond-based and state-based PD formulations are revisited, and numerical examples of two-dimensional problems are presented.

Direct numerical simulation of the scouring of a brittle streambed in a turbulent channel flow

The natural processes involved in the scouring of submerged sediments are crucially relevant in geomorphology along with environmental, fluvial, and oceanographic engineering. Despite their relevance, the phenomena involved are far from being completely understood, in particular for what concerns cohesive or stony substrates with brittle bulk mechanical properties. In this frame, we address the investigation of the mechanisms that govern the scouring and pattern formation on an initially flattened bed of homogenous and brittle material in a turbulent channel flow, employing direct numerical simulation. The problem is numerically tackled in the frame of peridynamic theory, which has intrinsic capabilities of reliably reproducing crack formation, coupled with the Navier–Stokes equations by the immersed boundary method. The numerical approach is reported in detail here and in the references, where extensive and fully coupled benchmarks are provided. The present paper focuses on the role of turbulence in promoting the brittle fragmentation of a solid, brittle streambed. A detailed characterization of the bedforms that originate on the brittle substrate is provided, alongside an analysis of the correlation between bed shape and the turbulent structures of the flow. We find that turbulent fluctuations locally increase the intensity of the wall-stresses producing localized damages. The accumulation of damage drives the scouring of the solid bed via a turbulence-driven fatigue mechanism. The formation, propagation, and coalescence of scouring structures are observed. In turn, these affect both the small- and large-scale structures of the turbulent flow, producing an enhancement of turbulence intensity and wall-stresses. At the small length scales, this phenomenology is put in relation to the formation of vortical cells that persist over the peaks of the channel bed. Similarly, large-scale irregularities are found to promote the formation of stationary turbulent stripes and large-scale vortices that enhance the widening and deepening of scour holes. As a result, we observe a quadratic increment of the volumetric erosion rate of the streambed, as well as a widening of the probability density of high-intensity wall stress on the channel bed.

A non-linear direct peridynamics plate theory

In this paper a direct non-local peridynamics theory for thin plates is developed. Peridynamic points are assumed to behave like rigid bodies with independent translation and finite rotation degrees of freedom. The non-local mechanical interaction between points is characterized by force and moment vectors. The balance equations including the linear momentum, the angular momentum and the energy are presented. Peridynamic deformation states of the plate are introduced including the actual bond vector (relative translation of two points within the bond) and the relative finite rotation tensor (actual relative orientation of two points in the bond). The corresponding power-conjugate bond force and bond moment states are derived. The framework to develop constitutive equations for the plate peridynamic states is addressed. Special cases of the theory including plates with zero drilling moments, membranes, soft interlayers as well as stiff plate layers with small relative rotations are considered.

Crack propagation with different radius local random damage based on peridynamic theory

Crack propagation with different radius local random damage based on peridynamic theory

A peridynamic approach for modeling of two dimensional functionally graded plates

This study investigates the crack initiation and its progression in two-dimensional functionally graded (FG) plates under dynamic and quasi-static loading conditions by using an ordinary state-based PeriDynamic (OSB PD) theory. Functionally gradient materials (FGMs) are a new class of advanced materials that provide a smooth transition among the layers of materials to meet the desired requirements in engineering structures. The effective material properties in the FG plate were functionally tailored in two directions by employing a rule of mixture. The present OSB PD theory is very suitable for the failure analysis of materials without the use of surface and volume correction factors. The robustness of the present approach on monitoring the mixed-mode fractures is established with the experimental results under various loading conditions. The PD predictions successfully captured the critical load levels and damage evolutions. Subsequently, numerical analyses were carried out to assess the influence of the one- and two-dimensional material variations, boundary, and loading conditions on the fracture behavior of the FG plates. The compositional gradient exponent played a major role on the reaction force levels and crack trajectories. It was observed that two-dimensional material variations paved the way for increasing the plate strength and fracture resistance.

Microstructure-based deformation and fracture modeling of particulate reinforced composites with ordinary state-based peridynamic theory

The aim of this study is to develop a microstructure-based peridynamic (PD) model to predict the elastic–plastic deformation and fracture mechanism of the particulate reinforced composites. The developed state-based PD model is validated against the previous experimental study on the aluminum matrix reinforced by Sic particles. Three representative volume elements (RVEs) are extracted from a scanning electron microscope image by image processing, and the particle morphology is investigated. Methods are proposed to assign boundary conditions and allocate material properties to matrix/particle interface regions and calculate the surface correction in the PD computational framework for the particulate composites. Two different loading paths are simulated (loading and loading–unloading-reloading). Distribution of residual stress and residual plastic strain are obtained after unloading. Before damage initiation, the stresses in the particles, especially at the sharp corners, are relatively higher than in the matrix. The particle, matrix, and interface damage are detected in the analysis. Damage initiation occurs in narrow particles and matrices sandwiched by particles. The initiated microcracks are propagated and coalesce to make the principal crack, and the final fracture occurs by the propagation of the principal crack. It is found that the proposed model can accurately predict the main deformation and damage behavior of particulate composites.

A vector form conjugated-shear bond-based peridynamic model for crack initiation and propagation in linear elastic solids

A vector form conjugated-shear bond-based peridynamic model is proposed to overcome the limitation of the fixed Poisson’s ratio based on the assumption of central force between material points in bond-based peridynamic theory. In the virtue of this proposed approach, the angle variation of a pair of conjugated-shear bonds is derived as a vector in terms of the displacements at three points located at the conjugated-shear bonds instead of the direct use of the angle, which improves the computational efficiency in both two and three dimensional conditions. Emphasis is placed on deriving the relationship between micro moduli in the proposed model and macro constants by equating peridynamic strain energy to the macroscopic strain energy. It is found that the micro moduli in this proposed approach can reduce to classical bond-based peridynamic parameters when the rotation effect of conjugated-shear bonds is not considered. A comparative study of the proposed approach and the FEM or analytical solutions on a square isotropic plate and a three-dimensional rectangular block under uniaxial tension shows that this proposed approach can well investigate the behavior of elastic solids under both two and three dimensional conditions. We test the numerical convergence behavior of the proposed model to classical solutions in the varying horizon/grid spacing ratios or horizon sizes. Moreover, the improvement of computational efficiency of the proposed model is verified by comparison of the CPU time computed from the scalar form conjugated bond-based peridynamic model. The capability of the proposed model in simulating dynamic brittle fracture problems is demonstrated by comparing the results obtained by the proposed model with the previous numerical results in the case of Kalthoff-Winkler experiment.

The boundary element method of peridynamics

AbstractThe peridynamic theory brings advantages in dealing with discontinuities, dynamic loading, and nonlocality. The integro‐differential formulation of peridynamics poses challenges to numerical solutions of complicated and practical problems. Some important issues attract much attention, such as the computation of infinite domains, the treatment of softening of boundaries due to an incomplete horizon, and time accumulation error in dynamic processes. In this work, we develop the boundary element method of peridynamics (PD‐BEM). The numerical examples demonstrate that the PD‐BEM exhibits several features. First, for nondestructive cases, the PD‐BEM can be one to two orders of magnitude faster than the meshless particle method of peridynamics (PD‐MPM) that directly discretizes the computational domains; second, it eliminates the time accumulation error, and thus conserves the total energy much better than the PD‐MPM; third, it does not exhibit spurious boundary softening phenomena. For destructive cases where new boundaries emerge during the loading process, we propose a coupling scheme where the PD‐MPM is applied to the cracked region and the PD‐BEM is applied to the uncracked region such that the time of computation can be significantly reduced.

Peridynamic analysis of fatigue crack growth in fillet welded joints

Fatigue assessment is one of the significant factors to be considered for estimating design life of structures and structural reliability during operation. Especially, for welded structures, high stress concentration, residual stresses, weld geometry and weld quality make the structure more vulnerable to fatigue failures. Therefore, approaches that are more effective are required for estimating fatigue performances of welded structures. Existing classical methods to predict the crack propagation under cyclic loadings have some difficulties in treating complicated patterns of crack growth. However, peridynamic theory has advantages on dealing with discontinuities. In this study, a peridynamic fatigue model, which is a bond damage model of remaining life, is validated by considering ASTM E647 standard compact tension test for the crack growth. After validating the PD model, the fatigue performance of the fillet welded joint is presented by predicting the fatigue crack growth under cyclic loading conditions.

Evaluation of stress intensity factors under thermal effect employing domain integral method and ordinary state based peridynamic theory

In this article, several thermoelastic benchmark cases are studied within the framework of ordinary state based peridynamic theory (OSPD). By using OSPD, the limitations of geometrical discontinuity in fracture analysis can be overcome. Meanwhile, double nodes can also be avoided during crack definition. A domain integral method with thermal effect is applied in calculating the thermal stress intensity factors (TSIFs). Meanwhile, peridynamic differential operators are utilized to rewrite the spatial derivatives in the domain integral. Numerical investigations of TSIFs in the single and mixed-mode crack scenarios are provided respectively, and verified by the reference solutions. Good agreements between OSPD and the reference solutions show high performance and capability of the proposed method in thermoelastic fracture analysis.

Shape design optimization of dynamic crack propagation using peridynamics

Crack generation tends to be prevented in most of design problems but taking advantages of cracks with some flaw acceptability could provide better design margin in practical problems. To develop a crack control strategy, we propose a shape design optimization method using an adjoint shape sensitivity based on a peridynamics theory that does not require crack propagation path a priori and thus is advantageous for the problems in which the discontinuity initiates, propagates, and interacts dynamically. The adjoint design sensitivity analysis (DSA) method based on the peridynamics is developed for dynamic crack propagations in brittle materials, including the successive branching of cracks. For a performance measure of stretch between the designated points at final time, the design sensitivity with respect to the design parameters is derived to be used in the shape design optimization. Numerical examples are demonstrated to verify the accuracy of the developed design sensitivity to show excellent agreement with finite differencing for all the shape design variables. Also, the proposed DSA method works very well in the shape design optimization applications of creating or avoiding cracks in a specified region.

Crack Growth Simulation of Functionally Graded Materials Based on Improved Bond-Based Peridynamic Model.

Functionally graded materials (FGMs) are widely used in the aerospace industry, especially for the thermal protection shields of aerospace vehicles. Studies show that the initiation and expansion of micro-cracks are important factors that adversely affect the service life of these shields. Based on the peridynamic theory of bonds, an improved peridynamic model is proposed in the present study for FGMs. In the proposed model, integral equivalence is applied to calculate the required material parameters. Obtained results reveal that this method can better reflect the gradient change of material properties.

Accurate artificial boundary conditions for semi-discretized one-dimensional peridynamics

The peridynamic theory reformulates the equations of continuum mechanics in terms of integro-differential equations instead of partial differential equations. In this paper, we consider the construction of artificial boundary conditions (ABCs) for semi-discretized peridynamics using Green functions. Especially, the Green functions that represent the response to the single wave source are used to construct the accu2rate boundary conditions. The recursive relationships between the Green functions are proposed, therefore the Green functions can be computed through a differential and integral system with high precision. The numerical results demonstrate the accuracy of the proposed ABCs. The proposed method can be applied to modelling of wave propagation for other non-local theories and high-dimensional cases.

Numerical study on dynamic icebreaking process of an icebreaker by ordinary state-based peridynamics and continuous contact detection algorithm

The existence of ice and its interaction with structure greatly threaten the navigation of the icebreaker. Due to the damage characteristics of ice, the traditional numerical ways based on mesh method are challenging to solve ice-structure interaction. The nonlocal particle methods have increasingly gained popularity as an essential tool for calculating large deformation problems. In the present work, a meshfree particle-based computational model is developed to investigate the icebreaking process and dynamic icebreaking resistance. The model employs ordinary state-based Peridynamics theory to establish the constitutive relation for ice which is modelled as an isotropic homogeneous elastic material. Furthermore, a continuous contact detection algorithm is proposed, which can be applied to detect the contact process between irregular structures and particles, to study the process of ice-ship interaction. Two dimensional (2D) three-points bending of ice beam and 3D ice sphere impact are studied numerically to validate the method specifically in modelling ice failure. Then, the investigation on the continuous-mode icebreaking process is carried out. The comparison of the present numerical results with existing experimental data demonstrated the validity and accuracy of the present numerical model for analysing the icebreaking process and predicting the icebreaking resistance.