A coupled thermo-mechanical peridynamic model for fracture behavior of granite subjected to heating and water-cooling processes

Thermal damage and thermal fracture of rocks are two important indicators in geothermal mining projects.This paper investigates the effects of heating and water-cooling on granite specimens at various temperatures.The laboratory uniaxial compression experiments were also conducted.Then,a coupled thermo-mechanical ordinary state-based peridynamic(OSB-PD)model and corresponding numerical scheme were developed to simulate the damage of rocks after the heating and cooling processes,and the change of crack evolution process was predicted.The results demonstrate that elevated heating temperatures exacerbate the thermal damage to the specimens,resulting in a decrease in peak strength and an increase in ductility of granite.The escalating occurrence of thermal-induced cracks significantly affects the crack evolution process during the loading phase.The numerical results accurately reproduce the damage and fracture characteristics of the granite under different final heating temperatures(FHTs),which are consistent with the test results in terms of strength,crack evolution process,and failure mode.

A Coupled Thermomechanical Crack Propagation Behavior of Brittle Materials by Peridynamic Differential Operator

This study proposes a comprehensive,coupled thermomechanical model that replaces local spatial derivatives in classical differential thermomechanical equations with nonlocal integral forms derived from the peridynamic differential operator(PDDO),eliminating the need for calibration procedures.The model employs a multi-rate explicit time integration scheme to handle varying time scales in multi-physics systems.Through simulations conducted on granite and ceramic materials,this model demonstrates its effectiveness.It successfully simulates thermal damage behavior in granite arising from incompatible mineral expansion and accurately calculates thermal crack propagation in ceramic slabs during quenching.To account for material heterogeneity,the model utilizes the Shuffle algorithm andWeibull distribution,yielding results that align with numerical simulations and experimental observations.This coupled thermomechanical model shows great promise for analyzing intricate thermomechanical phenomena in brittle materials.

Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

Transient heat conduction problems widely exist in engineering.In previous work on the peridynamic differential operator(PDDO)method for solving such problems,both time and spatial derivatives were discretized using the PDDO method,resulting in increased complexity and programming difficulty.In this work,the forward difference formula,the backward difference formula,and the centered difference formula are used to discretize the time derivative,while the PDDO method is used to discretize the spatial derivative.Three new schemes for solving transient heat conduction equations have been developed,namely,the forward-in-time and PDDO in space(FT-PDDO)scheme,the backward-in-time and PDDO in space(BT-PDDO)scheme,and the central-in-time and PDDO in space(CT-PDDO)scheme.The stability and convergence of these schemes are analyzed using the Fourier method and Taylor’s theorem.Results show that the FT-PDDO scheme is conditionally stable,whereas the BT-PDDO and CT-PDDO schemes are unconditionally stable.The stability conditions for the FT-PDDO scheme are less stringent than those of the explicit finite element method and explicit finite difference method.The convergence rate in space for these three methods is two.These constructed schemes are applied to solve one-dimensional and two-dimensional transient heat conduction problems.The accuracy and validity of the schemes are verified by comparison with analytical solutions.

Sub-Homogeneous Peridynamic Model for Fracture and Failure Analysis of Roadway Surrounding Rock

The surrounding rock of roadways exhibits intricate characteristics of discontinuity and heterogeneity.To address these complexities,this study employs non-local Peridynamics(PD)theory and reconstructs the kernel function to represent accurately the spatial decline of long-range force.Additionally,modifications to the traditional bondbased PD model are made.By considering the micro-structure of coal-rock materials within a uniform discrete model,heterogeneity characterized by bond random pre-breaking is introduced.This approach facilitates the proposal of a novel model capable of handling the random distribution characteristics of material heterogeneity,rendering the PD model suitable for analyzing the deformation and failure of heterogeneous layered coal-rock mass structures.The established numerical model and simulation method,termed the sub-homogeneous PD model,not only incorporates the support effect but also captures accurately the random heterogeneous micro-structure of roadway surrounding rock.The simulation results obtained using this model show good agreement with field measurements from the Fucun coal mine,effectively validating the model’s capability in accurately reproducing the deformation and failure mode of surrounding rock under bolt-supported(anchor cable).The proposed subhomogeneous PD model presents a valuable and effective simulation tool for studying the deformation and failure of roadway surrounding rock in coal mines,offering new insights and potential advancements.

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

An Elastoplastic Fracture Model Based on Bond-Based Peridynamics

Fracture in ductile materials often occurs in conjunction with plastic deformation.However,in the bond-based peridynamic(BB-PD)theory,the classic mechanical stress is not defined inherently.This makes it difficult to describe plasticity directly using the classical plastic theory.To address the above issue,a unified bond-based peridynamics model was proposed as an effective tool to solve elastoplastic fracture problems.Compared to the existing models,the proposed model directly describes the elastoplastic theory at the bond level without the need for additional calculation means.The results obtained in the context of this model are shown to be consistent with FEM results in regard to force-displacement curves,displacement fields,stress fields,and plastic deformation regions.The model exhibits good capability of capturing crack propagation in ductile material failure problems.

The Boundary Element Method for Ordinary State-Based Peridynamics

The peridynamics(PD),as a promising nonlocal continuum mechanics theory,shines in solving discontinuous problems.Up to now,various numerical methods,such as the peridynamic mesh-free particlemethod(PD-MPM),peridynamic finite element method(PD-FEM),and peridynamic boundary element method(PD-BEM),have been proposed.PD-BEM,in particular,outperforms other methods by eliminating spurious boundary softening,efficiently handling infinite problems,and ensuring high computational accuracy.However,the existing PD-BEM is constructed exclusively for bond-based peridynamics(BBPD)with fixed Poisson’s ratio,limiting its applicability to crack propagation problems and scenarios involving infinite or semi-infinite problems.In this paper,we address these limitations by introducing the boundary element method(BEM)for ordinary state-based peridynamics(OSPD-BEM).Additionally,we present a crack propagationmodel embeddedwithin the framework ofOSPD-BEM to simulate crack propagations.To validate the effectiveness of OSPD-BEM,we conduct four numerical examples:deformation under uniaxial loading,crack initiation in a double-notched specimen,wedge-splitting test,and threepoint bending test.The results demonstrate the accuracy and efficiency of OSPD-BEM,highlighting its capability to successfully eliminate spurious boundary softening phenomena under varying Poisson’s ratios.Moreover,OSPDBEMsignificantly reduces computational time and exhibits greater consistencywith experimental results compared to PD-MPM.

ABAQUS and ANSYS Implementations of the Peridynamics-Based Finite Element Method(PeriFEM)for Brittle Fractures

In this study,we propose the first unified implementation strategy for peridynamics in commercial finite element method(FEM)software packages based on their application programming interface using the peridynamics-based finite element method(PeriFEM).Using ANSYS and ABAQUS as examples,we present the numerical results and implementation details of PeriFEM in commercial FEM software.PeriFEM is a reformulation of the traditional FEM for solving peridynamic equations numerically.It is considered that the non-local features of peridynamics yet possesses the same computational framework as the traditional FEM.Therefore,this implementation benefits from the consistent computational frameworks of both PeriFEM and the traditional FEM.An implicit algorithm is used for both ANSYS and ABAQUS;however,different convergence criteria are adopted owing to their unique features.In ANSYS,APDL enables users to conveniently obtain broken-bond information from UPFs;thus,the convergence criterion is chosen as no new broken bond.In ABAQUS,obtaining broken-bond information is not convenient for users;thus,the default convergence criterion is used in ABAQUS.The codes integrated into ANSYS and ABAQUS are both verified through benchmark examples,and the computational convergence and costs are compared.The results show that,for some specific examples,ABAQUS is more efficient,whereas the convergence criterion adopted in ANSYS is more robust.Finally,3D examples are presented to demonstrate the ability of the proposed approach to deal with complex engineering problems.

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 optimizationbased 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.

Peridynamic Shell Model Based on Micro-Beam Bond

Peridynamics(PD)is a non-localmechanics theory that overcomes the limitations of classical continuummechanics(CCM)in predicting the initiation and propagation of cracks.However,the calculation efficiency of PDmodels is generally lower than that of the traditional finite elementmethod(FEM).Structural idealization can greatly improve the calculation efficiency of PD models for complex structures.This study presents a PD shell model based on the micro-beam bond via the homogenization assumption.First,the deformations of each endpoint of themicro-beam bond are calculated through the interpolation method.Second,the micro-potential energy of the axial,torsional,and bending deformations of the bond can be established from the deformations of endpoints.Finally,the micro moduli of the shellmodel can be obtained via the equivalence principle of strain energy density(SED).In addition,a new fracture criterion based on the SED of the micro-beam bond is adopted for crack simulation.Numerical examples of crack propagation are provided,and the results demonstrate the effectiveness of the proposed PD shell model.

Study on Crack Propagation Parameters of Tunnel Lining Structure Based on Peridynamics

The numerical simulation results utilizing the Peridynamics(PD)method reveal that the initial crack and crack propagation of the tunnel concrete lining structure agree with the experimental data compared to the Japanese prototype lining test.The load structure model takes into account the cracking process and distribution of the lining segment under the influence of local bias pressure and lining thickness.In addition,the influence of preset cracks and lining section formon the crack propagation of the concrete lining model is studied.This study evaluates the stability and sustainability of tunnel structure by the Peridynamics method,which provides a reference for the analysis of the causes of lining cracks,and also lays a foundation for the prevention,reinforcement and repair of tunnel lining cracks.

The Coupled Thermo-Chemo-Mechanical Peridynamics for ZrB_(2) Ceramics Ablation Behavior

The ablation of ultra-high-temperature ceramics(UTHCs)is a complex physicochemical process including mechanical behavior,temperature effect,and chemical reactions.In order to realize the structural optimization and functional design of ultra-high temperature ceramics,a coupled thermo-chemo-mechanical bond-based peridynamics(PD)model is proposed based on the ZrB_(2) ceramics oxidation kinetics model and coupled thermomechanical bond-based peridynamics.Compared with the traditional coupled thermo-mechanical model,the proposedmodel considers the influenceof chemical reactionprocessonthe ablation resistanceof ceramicmaterials.In order to verify the reliability of the proposed model,the thermo-mechanical coupling model,damage model and oxidation kinetic model are established respectively to investigate the applicability of the proposedmodel proposed in dealing with thermo-mechanical coupling,crack propagation,and chemical reaction,and the results show that the model is reliable.Finally,the coupled thermo-mechanical model and coupled thermo-chemo-mechanical model are used to simulate the crack propagation process of the plate under the thermal shock load,and the results show that the oxide layer plays a good role in preventing heat transfer and protecting the internal materials.Based on the PD fully coupled thermo-mechanical model,this paper innovatively introduces the oxidation kinetic model to analyze the influence of parameter changes caused by oxide layer growth and chemical growth strain on the thermal protection ability of ceramics.The proposed model provides an effective simulation technology for the structural design of UTHCs.

Peridynamic Study on Fracture Mode and Crack Propagation Path of a Plate with Multiple Cracks Subjected to Uniaxial Tension

How to simulate fracture mode and crack propagation path in a plate with multiple cracks is an attractive but difficult issue in fracture mechanics.Peridynamics is a recently developed nonlocal continuum formulation that can spontaneously predict the crack nucleation,branch and propagation in materials and structures through a meshfree discrete technique.In this paper,the peridynamic motion equation with boundary traction is improved by simplifying the boundary transfer functions.We calculate the critical cracking load and the fracture angles of the plate with multiple cracks under uniaxial tension.The results are consistent with those predicted by classical fracture mechanics.The fracture mode and crack propagation path are also determined.The calculation shows that the brittle fracture process of the plate with multiple cracks can be conveniently and correctly simulated by the peridynamic motion equation with boundary conditions.

A Peridynamic Approach for the Evaluation of Metal Ablation under High Temperature

In this paper,the evaluations of metal ablation processes under high temperature,i.e.,the Al plate ablated by a laser and a heat carrier and the reactor pressure vessel ablated by a core melt,are studied by a novel peridynamic method.Above all,the peridynamic formulation for the heat conduction problem is obtained by Taylor’s expansion technique.Then,a simple and efficient moving boundary model in the peridynamic framework is proposed to handle the variable geometries,in which the ablated states of material points are described by an additional scalar field.Next,due to the automatic non-interpenetration properties of peridynamic method,a contact algorithm is established to determine the contact relationship between the ablated system and the additional heat carrier.In addition,the corresponding computational procedure is listed in detail.Finally,several numerical examples are carried out and the results verify the validity and accuracy of the present method.

二维冰-水-结构物作用的近场动力学模拟

为探讨冰、水、结构物之间的相互作用关系,采用近场动力学方法建立二维冰-水-圆柱结构物的运动计算模型,采用近场动力学微分算子模拟海水流动,采用常规态型近场动力学模拟海冰的损伤行为以及海冰与结构物之间的作用。进一步研究不同圆柱结构物移动速度和直径对冰载荷的影响,将数值计算结果与试验结果进行对比以验证方法的可行性,为研究冰、水、结构物相互作用提供理论支撑。

倾斜临空面下隧道掘进机滚刀贯入破岩研究

在临空面辅助隧道掘进机(tunnel boring machine,简称TBM)滚刀破岩的过程中,当滚刀在垂直临空面附近完成破岩后,垂直临空面将变为倾斜临空面。为了研究倾斜临空面对滚刀贯入破岩的影响,利用近场动力学方法建立了三维模型,考虑了临空面倾斜程度以及滚刀临空距离的变化,开展了一系列数值研究。模拟结果表明,在倾斜临空面辅助TBM滚刀贯入破岩工况下,岩石存在两种破碎模式,一种为内部裂纹扩展模式,另一种为内部压碎区扩大模式。在滚刀临空距离固定为40 mm的情况下,随着临空面倾角θ余切值cotθ的增加,岩石破碎逐渐从内部裂纹扩展模式转变为内部压碎区扩大模式,滚刀的法向峰值力与破岩消耗能量也逐渐增大,而且内部压碎区扩大模式的破岩效率比内部裂纹扩展模式低。随着滚刀临空距离的增加,滚刀法向峰值力、破岩消耗能量与破岩体积逐渐增加,同时滚刀的破岩效率也得到了提升。

基于近场动力学的岩石水力压裂数值模拟及裂隙网络定量分析

基于常规态型近场动力学方法实现了岩石裂隙扩展演化,采用实时追踪新生裂隙面并施加表面压力的方法来模拟压裂液与裂隙表面的相互作用,建立了岩石水力压裂的数值计算模型。根据数字图像处理技术将Zhang-Suen细化算法应用于水力裂隙网络的骨架提取,并采用统计方法计算裂隙网络形态参数,提出了水力裂隙网络量化计算方法。最后考虑加载速率、地应力条件及弹性模量等因素影响,研究了水力裂隙扩展过程及裂隙网络形态参数的演化。研究结果表明:当加载速率较小时,主裂隙朝着较大地应力的方向扩展,裂隙分支并不明显;增加加载速率会增大裂隙均宽和裂隙密度,能够促进裂隙的张开程度及裂隙数量,增强裂隙网络的复杂程度并提高其渗透能力。当水平向地应力和竖向地应力相同时,主裂隙呈十字交叉分布;随着竖向地应力的增大,水平裂隙受到抑制作用,主裂隙沿着竖向扩展,而裂隙总长和裂隙密度则呈增加的趋势。岩体弹性模量的增加会减少细小裂隙分支的扩展,使裂隙网络形态趋于简单化。

基于近场动力学方法的玻璃板冲击破坏数值模拟研究

基于近场动力学方法,考虑刚体冲击作用下材料损伤发生断裂破坏影响,建立了刚球冲击玻璃板的三维数值计算模型,并结合Kalthoff-Winkler试验,验证了计算模型的有效性。最后,探究了刚球初始冲击速度、刚球直径和玻璃板厚度等因素,对玻璃板的冲击破坏结构形态演化以及裂纹扩展机制的影响。研究结果表明,当初始冲击速度较小时,玻璃板裂纹表现为辐射状,随着冲击速度的增加,环向裂纹出现并逐渐形成复杂的裂纹网络。同时,玻璃板的裂纹长度和裂纹占比均随初始冲击速度的增加而增大,且近似呈线性关系。随着刚球直径的增加,其与玻璃板的接触范围扩大,致使玻璃板的损伤程度加剧,且更易形成结构复杂的裂纹网络。此外,增加玻璃板的厚度会延长冲击持续时间,使得玻璃板承受更大的冲击能,从而导致板厚较小时局部结构损伤严重,但整体影响范围较小;而板厚较大时结构损伤的影响范围扩大,局部损伤则有所减弱。

随机微裂纹对岩石宏观裂纹扩展的影响规律研究

岩石微裂纹控制着岩石的宏观力学特性和破裂行为,其分布具有不确定性。现有关于岩石裂纹扩展的研究侧重于确定性裂纹扩展规律研究,而随机微裂纹的影响研究不足。该文基于键型近场动力学(Peridynamics,PD)理论,引入随机分布函数预设微裂纹的几何、位置随机参数,对预设有宏观裂纹和随机微裂纹的岩石构件拉伸破坏过程进行大规模蒙特卡洛模拟,提出了物质点断裂失效概率因子(PFP)来描述物质点的破坏概率,揭示了随机微裂纹对岩石拉伸宏观裂纹扩展的影响规律。通过分析两个经典算例,分别与FEM和XFEM对比,其结果有较强的一致性。另外,该文引入了含宏观裂纹与微裂纹的岩石构件,根据微裂纹的位置、角度和长度随机分布,首次绘制不同情况下随机裂纹的PFP云图,获得了各参数对裂纹扩展的影响规律。该文首创了物质点断裂失效概率因子,定量描述微裂纹对宏观裂纹扩展的影响规律,为预测含随机微裂纹的岩石裂纹扩展研究提供了一种新思路新方法。

基于近场动力学的含损伤缺陷Q345钢裂纹扩展分析

船舶与海洋工程结构中由于材料本身缺陷或服役过程中产生的细小裂纹,使结构不可避免地产生裂纹损伤,造成应力集中而破坏。为了研究含裂纹缺陷低合金钢构件的裂纹扩展行为,以应用广泛的Q345钢为研究对象,基于近场动力学(Peridynamics,PD)理论提出一种改进的双参数PD方法。首先考虑PD力的非局部长程力特性,推导对应的键刚度系数;其次,根据含裂纹缺陷Q345钢拉伸破坏的线性和非线性的力学行为,构建双参数PD本构力函数的基本形式。通过研究不同双裂纹间距、长度和角度的Q345钢裂纹扩展行为,与试验结果对比验证本文改进的双参数PD方法的准确性,并且给出交变荷载情形下疲劳裂纹扩展的计算方法。结果可为船舶钢构件的优化设计和断裂预防提供参考。