Development of an approach to constitutive modelling of concrete: Isotropic damage coupled with plasticity

The paper presents an approach to constitutive modelling of concrete using damage mechanics and plasticity theory. The thermodynamic formulation, and parameter identification of a non-local coupled damage-plasticity model are presented in this study. The particular focus is the calibration of model parameters. It is shown that both the local parameters and the parameters governing the non-local interaction can be determined from experimental data reliably and consistently. A novel procedure is developed for parameter identification, using the separation of total dissipation energy into additive parts corresponding to different dissipation mechanisms. The relationship between the local and non-local parameters is also addressed, helping to obtain model responses consistent with the fracture energy of the material. The application of the model and the calibration procedure proposed in this study to the numerical failure analysis of concrete structures is illustrated through a series of real structural tests, showing both the performance of the model and the consistency of the proposed calibration procedure.     

On non-local and non-homogeneous elastic continua

A thermodynamic framework endowed with the concept of non-locality residual is adopted to derive non-local models of integral-type for non-homogeneous linear elastic materials. Two expressions of the free energy are considered: the former yields a one-component non-local stress, the latter leads to a two-component local–non-local stress since the stress is expressed as the sum of the classical local stress and of a non-local component identically vanishing in the case of constant strains. The attenuation effects are accounted for by a symmetric space weight function which guarantees the constant strain requirement as well as the dual constant stress condition everywhere in the body. The non-local and non-homogeneous elastic structural boundary-value problem under quasi-static loads is addressed in a geometrically linear range. The complete set of variational formulations for the structural problem is then provided in a unitary framework. The solution uniqueness of the non-local structural model is proved and the non-local FEM is addressed starting from the non-local counterpart of the total potential energy. Numerical applications are provided with reference to a non-homogeneous bar in tension using the Fredholm integral equation and the non-local FEM. The solutions show no pathological features such as numerical instability and mesh sensitivity for degraded bar conditions.     

准静态变形破坏的近场动力学分析

基于近场动力学理论,提出新的更能反映非局部长程力特性的物质点间作用力函数,并通过在物质点运动方程中引入局部阻尼、构造分级加载算法和系统失衡判断准则,实现了采用统一的近场动力学模型和算法进行从准静态变形、裂纹萌生和扩展直至结构破坏全过程的连续模拟和准确定量计算。     

基于近场动力学的薄板弯曲大变形及断裂分析

薄板结构仅在较小的荷载下就能产生较大的位移、旋转,甚至引发结构产生裂纹并扩展,进而发生结构整体断裂,因此,建立薄板结构在大变形过程中的裂纹扩展及断裂仿真模型,具有重要的工程实际意义.文章建立了用于薄板结构几何大变形和断裂分析的近场动力学(PD)和连续介质力学(CCM)耦合模型.首先基于冯·卡门假设,采用更新的拉格朗日法得到薄板在几何大变形增量步下的虚应变能密度增量公式,并利用虚功原理和均质化假设求出几何大变形微梁键的本构模型参数;接着分别建立几何大变形薄板PD模型与CCM模型的虚应变能密度增量,并建立了薄板几何大变形PD-CCM耦合模型;最后模拟了薄板结构在横向变形作用下的渐进断裂过程,得到与实验结果高度一致的仿真结果,验证了所提出的几何非线性PD-CCM耦合模型的精度.结果表明:本文所提出的薄板PD-CCM耦合模型具有简单高效,无需考虑材料参数限制和边界效应的特点,可以很好地用于预测薄板结构在几何大变形过程中的局部损伤和结构断裂,有利于薄板结构的断裂安全评价和理论发展.     

非常规态型近场动力学热黏塑性模型及其应用

在非常规态型近场动力学(non-ordinary state-based peridynamics, NOSB-PD) 理论框架下构建了考虑应变率效应、塑性硬化、热软化效应和材料断裂特征的非局部三维热黏塑性固体本构模型以及相应的非局部空间积分型数值算法, 并应用于金属类材料和构件在冲击载荷作用等工况下的高应变率热黏塑性变形与破坏分析. 通过对经典含初始裂纹Kalthoff-Winkler板冲击试验进行三维近场动力学模拟, 可得到裂纹的起裂角度、扩展路径、扩展速度以及裂纹扩展过程中靶板等效应力和温度分布, 所得结果与已有试验结果和其他数值方法结果吻合较好. 在此基础上, 应用该模型分析了不同冲击速度作用下金属靶板的变形与裂纹扩展过程, 结果表明: 该模型能较好地模拟不同冲击速度(应变率)情况下靶板的变形与破坏全过程. 随着冲击速度变化, 初始裂纹的起裂时间、扩展方向和扩展速度呈一定规律变化. 冲击速度越低, 起裂时间越晚(直至冲击速度低于某值时初始裂纹不扩展), 裂纹扩展速度峰值越低, 冲击过程中靶板温度峰值越低, 完全扩展所需时间越长.      

Non-local theory solution for in-plane shear of through crack

A non-local theory of elasticity is applied to obtain the plane strain stress and displacement field for a through crack under in-plane shear by using Schmidt's method. Unlike the classical elasticity solution, a lattice parameter enters into the problem that make the stresses finite at crack tip. Both the angular variations of the circumferential stress and strain energy density function are examined to associate their stationary value with locations of possible fracture initiation. The former criterion predicted a crack initiation angle of 54° from the plane of shear for the non-local solution as compared with about 75° for the classical elasticity solution. The latter criterion based on energy density yields a crack initiation angle of 80° for a Poisson's ratio of 0.28. This is much closer to the value that is predicted by the classical crack tips solution of elasticity.     

A new subregion mesh method for the investigation of the elastic-plastic impact in flexible multibody systems

Impact processes between flexible bodies often lead to local stress concentration and wave propagation of high frequency. Therefore, the modeling of flexible multi-body systems involving impact should consider the local plastic deformation and the strict requirements of the spa-tial discretization. Owing to the nonlinearity of the stiffness matrix, the reduction of the element number is extremely important. For the contact-impact problem, since different regions have different requirements regarding the element size, a new subregion mesh method is proposed to reduce the number of the unnecessary elements. A dynamic model for flexible multibody systems with elastic-plastic contact impact is established based on a floating frame of reference formulation and complete Lagrange incremental nonlin-ear finite-element method to investigate the effect of the elastic-plastic deformation as well as spatial discretization. Experiments on the impact between two bodies are carried out to validate the correctness of the elastic-plastic model. The proposed formulation is applied to a slider-crank system with elastic-plastic impact.     

A directed continuum model of micro- and nano-scale thin films

As is well known, classical continuum theories cease to adequately model a material’s behavior as long-range loads or interactions begin to have a greater effect on the overall behavior of the material, i.e., as the material no longer conforms to the locality requirements of classical continuum theories. It is suggested that certain structures to be analyzed in this work, e.g., columnar thin films, are influenced by non-local phenomena. Directed continuum theories, which are often used to capture non-local behavior in the context of a continuum theory, will therefore be used. The analysis in this work begins by establishing the kinematics relationships for a discrete model based on the physical structure of a columnar thin film. The strain energy density of the system is calculated and used to formulate a directed continuum theory, in particular a micromorphic theory, involving deformations of the film substrate and deformations of the columnar structure. The resulting boundary value problem is solved analytically to obtain the deformation of the film in response to applied end displacements.     

混凝土单轴受拉的非局部本构模型

混凝土受拉本构行为存在很强的局部软化现象,使得单轴受拉试验无法给出应力-应变关系,而只能给出应力-位移关系。本文根据内变量理论和等效应变假设建立了基于真实应变的混凝土单轴受力本构方程,并根据Weibull分布可以描述混凝土等脆性材料断裂过程的试验现象,建立了关于弹性应变的损伤演化规律。然后,通过假设平均应变与真实弹性应变的函数关系,在应力-平均应变的本构关系中采用平均弹性应变以描述其非局部行为,而在材料的损伤演化规律中采用真实弹性应变以描述其局部行为,由此建立了单轴受拉荷载条件下的非局部本构模型。最后,对一个单调受拉试验和一个反复受拉试验的仿真结果表明所提出的非局部本构模型可以准确地模拟试验结果。     

Non-local continuum mechanics and fractional calculus

The present work introduces fractional calculus into the continuum mechanics area describing non-local constitutive relations. Considering a one-dimensional body and assuming total stored energy depending not only upon the local strain but also upon a fractional derivative of the stain, an elastic model with non-local stress–strain behavior is introduced. Fractional calculus provides a natural framework for describing non-local constitutive relations and requires no assumptions for the interval of non-local influence. Furthermore, the proposed method works in finite intervals contrary to the existing theories requiring infinite domains.     

Effect of elastic target material on Taylor model

A theoretical model is proposed for a semi-infinite elastic bar struck axially by a flat-ended cylindrical projectile. The model is actually developed from the classical Taylor model except that the target is a semi-infinite bar with elastic behavior being considered. Particular attention is paid to the influence due to elastic wave propagation in the target bar on the energy partitioning between the projectile and target, which may result in the final length of the cylinder significantly different from the predictions of the classical Taylor model. The theoretical model is verified by numerical simulations, and the effects of several key non-dimensional parameters on the residual deformation and energy dissipation are discussed in detail. It is shown that the elastic effect of the target bar plays an important role in the prediction of plastic deformation of the cylindrical projectile.     

Strain gradient crystal plasticity analysis of a single crystal containing a cylindrical void

The effects of void size and hardening in a hexagonal close-packed single crystal containing a cylindrical void loaded by a far-field equibiaxial tensile stress under plane strain conditions are studied. The crystal has three in-plane slip systems oriented at the angle 60° with respect to one another. Finite element simulations are performed using a strain gradient crystal plasticity formulation with an intrinsic length scale parameter in a non-local strain gradient constitutive framework. For a vanishing length scale parameter the non-local formulation reduces to a local crystal plasticity formulation. The stress and deformation fields obtained with a local non-hardening constitutive formulation are compared to those obtained from a local hardening formulation and to those from a non-local formulation. Compared to the case of the non-hardening local constitutive formulation, it is shown that a local theory with hardening has only minor effects on the deformation field around the void, whereas a significant difference is obtained with the non-local constitutive relation. Finally, it is shown that the applied stress state required to activate plastic deformation at the void is up to three times higher for smaller void sizes than for larger void sizes in the non-local material.     

Fracture energy based constitutive models for tensile fracture of metal powder compacts

Diametral compression test or the Brazilian disc test is commonly used to characterise the tensile strength of brittle materials. A general fracture model based on energy assumptions is proposed for simulation of the discrete and localised tensile fracturing process in metal powder. The characteristics of the tensile fracture development of the central crack in diametral tested specimen is numerically studied. The softening rate of the model is obtained from the corresponding rate of the dissipated energy. Finite element simulations of the diametral compression test are performed with the proposed tensile fracture model used in conjunction with a Cap model for the deformation of the powder material. The results agree reasonably with experiments.     

非连续变形计算力学模型的粘弹性分析方法Ⅰ:基本理论

基于粘弹性广义有限单元和接触力元，发展了适用于多体相互作用系统非连续变形分析的粘弹性数值分析方法，通过虚功原理，给出了其分区参变量最小势能原理，从而阐明了其理论基础。粘弹性广义有限单元的本构关系可由粘弹性退化为弹性或刚性，因此本文所提出的方法可对由刚体、弹性体和粘弹性体所构成的复杂多体系统在外荷载作用下的力学行为进行数值模拟，同时能够比本文精确地直接得到多体之间的接触应力。     

基于颗粒离散元法的连接键应变软化模型及其应用

基于颗粒间的有限接触假设,提出了可表述颗粒间力、力矩传递的连接键模型. 为了表征连接键的塑性、损伤及断裂过程,在连接键中引入了考虑应变软化效应的Mohr-Coulomb 准则及最大拉应力准则. 单一连接键的单向拉伸测试及直剪测试表明了上述连接键应变软化模型的计算精度. 研究了颗粒体系的宏观应变能与颗粒平均配位数的对应关系. 通过计算发现,对于二维颗粒体系,当平均配位数为5 时,颗粒体系的宏观应变能与相同参数下连续介质方法(如有限元等) 计算获得的应变能基本一致. 利用上述连接键应变软化模型对岩石的单轴压缩过程进行了模拟,计算结果表明:岩石单轴压缩的应力应变曲线经历了线性上升段、非线性上升段、非线性下降段及缓变段等4 个阶段,并给出了上述4 个阶段与岩石内部损伤破裂状态的内在联系. 计算结果还表明,随着断裂应变的增大,岩石的破裂模式逐渐由拉剪复合型破裂向单一压剪型破裂转化;随着断裂应变的增大,峰值应力及达到峰值应力时的应变均逐渐增大,但峰值时的破裂度及终态时的破裂度将逐渐减小.      

SiC粒子非均布引起铝基复合材料的损伤分析

 根据电镜断面考察结果，以Gurson模型为本构方程的有限单元 法对包体模型及三维非均匀模型进行了详细分析. 为了评价应力-应变 关系及损伤的主要因素，考虑了基体中SiC粒子的体积率和径比的非均 匀分布. 其结果表明，用这种非均匀模型能很好地仿真铝基体在大量塑 性变形之后所发生的韧窝破坏过程. SiC粒子体积率、径比及其位置的 非均匀性，对局部和整体损伤过程与应力-应变关系的影响相当大. 当 Sic粒子径比为1.0，并在基体中均匀分布时，断裂应变会大幅度增大.     

On the overall elastic moduli of composites with spherical coated fillers

A micromechanical approach is presented to estimate the overall linear elastic moduli of three phase composites consisting of two phase coated spherical particles randomly dispersed in a homogeneous isotropic matrix. The theoretical method is based on Eshelby’s equivalent inclusion method and its recent extension by Shodja and Sarvestani [J. Appl. Mech. 68 (2001) 3] to evaluate the local field variables in case of double (multi) inhomogeneities. Using Tanaka–Mori theorem [J. Elasticity 2 (1972) 199] and a decomposition of Green’s function integral equation, the pair-wise average phase values of stress and strain in two interacting coated particles are estimated. Following Ju and Chen [Acta Mech. 103 (1994) 103; Acta Mech. 103 (1994) 123] the ensemble phase volume average of stress and strain fields can be evaluated within a representative volume element containing a finite number of coated particles. Comparisons with classical bounds are presented to illustrate the accuracy of the proposed method.     

Modelling of dynamic ductile fracture and application to the simulation of plate impact tests on tantalum

A model of dynamic damage by void nucleation and growth is proposed for elastic-viscoplastic materials sustaining intense loading. The model is dedicated to ductile materials for which fracture is caused by microvoiding. The material contains potential nucleation sites where microvoids are generated when the local pressure overcomes the nucleation pressure. A probability density function is adopted to describe the fluctuation of the nucleation pressure within the material. The void growth is described by using a hollow sphere model where micro-inertia effects are accounted for. The matrix weakening due to void growth is also included.The model has been first tested under uniaxial deformation. When the strain rate is assumed constant, the pressure inside the material has nearly a linear response up to a maximum. An analytical expression for the maximum pressure is proposed.Finite element simulations of plate impact tests have been carried out and compared to experiments on tantalum. From simulations based on the proposed model, an increase of the spall strength is observed with higher shock intensities. Therefore, the relationship between the velocity pullback and spall strength usually assumed in the literature (based on the acoustic approach) seems to be inadequate. Velocity profiles are simulated for different flyer thicknesses and different impact velocities with close agreement with experiments.