On the coupling between macroscopic material degradation and interfiber bond fracture in an idealized fiber network

The numerical analysis performed here, using a finite element network model, provides a number of important results regarding the evolution of micro fractures in planar random fiber networks where the only active microscopic fracture mechanism is bond fracture. The fibers are randomly distributed in the network meaning that the network is considered having in-plane isotropic properties on the macroscopic scale. The network is loaded so that, in an average sense, homogenous macroscopic stress and strain fields are present.Several conclusions are drawn. It is found that the development of macroscopic material degradation follows an exponential two-parameter law, consisting of an onset parameter and a fracture rate parameter, justifying a previous theory derived by the authors. The fracture rate parameter is linearly related to the inverse of the bond density above a certain density limit (percolation) and increases with increasing slenderness ratio of the fibers when keeping the bond density at a constant level. The strain energies stored in interfiber bonds are exponentially distributed over the whole network. The numerical analysis reveals that there is a linear relation between the ratio of fractured and initial number of loaded bonds, and the network’s macroscopic material stiffness normalized with its pristine stiffness, confirming earlier findings based on experimental observations. At localization the analyzed theory looses its validity because the fracture process is no longer randomly distributed over the whole network. Localization coincides with location of peak load in force–displacement tensile tests.     

一种基于材料细观特征量的PBX拉伸强度理论模型

为给塑性黏结炸药(PBX)的力学强度设计提供支撑、探索材料细观特征量与材料强度之间的定量规律,应用微裂纹扩展区理论,将PBX炸药的单轴拉伸过程中力学响应特征的变化归结为扩展裂纹取向角度的增加,将扩展裂纹最大取向角与拉伸强度相关联,构建了基于材料细观特征量的拉伸强度理论模型,并采用不同温度的单轴拉伸实验验证了该理论模型的有效性。研究表明:该拉伸强度理论模型可以实现对PBX炸药拉伸强度与炸药微裂纹密度、颗粒/黏结剂界面性能以及颗粒/黏结剂体系的表观杨氏模量、泊松比等细观特征量之间关系的定量描述。     

Analysis of the strain field in the vicinity of a crack-tip in an in-plane isotropic paper material

Strains, computed by the finite element method, are evaluated and compared to an experimentally determined strain field. The analyzed low-density paper has been designed to ensure bond–breakage as the dominating damage mechanism and the paper material is approximately in-plane isotropic. An optical non-contact displacement measuring system has been used in fracture tests to determine the strain field in the crack-tip region of a pre-fabricated crack. Additionally, acoustic emission monitored tensile tests have been conducted to determine onset and evolution of damage processes and thereby enabling calibration of required constitutive parameters. The results suggest that the investigated paper material can tolerate significantly higher strains than what is predicted by a classic elastic–plastic J₂-flow theory. Immediately before onset of the final fracture (i.e., localization), the experimental measured normal strain in the near-tip region is around 60% higher than the computed strain when using exclusively an elastic–plastic theory for the corresponding load while the strain computed utilizing a non-local damage theory is of the same order of magnitude as the experimentally measured strain. Hence, it seems essential to include a non-local continuum theory to describe strains in the near-tip region quantitatively correct for paper materials. It is demonstrated that path independence of the well-known J-integral does not prevail for this class of material models. Only for the special situation of a homogenous damage field in the crack-tip region may the stress and strain fields be described by the well-known HRR-solutions.     

Tensile failure of two-dimensional quasi-brittle foams

Stress redistribution caused by damage onset and the subsequent local softening plays an important role in determining the ultimate tensile strength of a cellular structure. The formation of damage process zones with struts dissipating a finite amount of fracture energy will require the macroscopic stress to be increased in order to continue structural damage. The goal of this paper is to investigate the influence of the fracture energy of the solid on the tensile fracture strength and the strain to fracture in quasi-brittle two-dimensional foams using a microstructural model. We analyze the mesoscopic damage and failure mechanisms in uniaxial tension. Relative density, strut cross-sectional profile, solid’s fracture strain, and fracture energy are varied systematically. The effect of the specific fracture energy on the peak behavior has been shown to be captured by the ratio of the fracture energy to the stored elastic energy. We have also explored the net section strength variation in the presence of a central crack at two different fracture energies. Comparison is made between two structurally identical quasi-brittle and ductile strain hardening foams to identify the differences in the damage mechanisms.     

A multiscale model for the ductile fracture of crystalline materials

In this paper, a multiscale model that combines both macroscopic and microscopic analyses is presented for describing the ductile fracture process of crystalline materials. In the macroscopic fracture analysis, the recently developed strain gradient plasticity theory is used to describe the fracture toughness, the shielding effects of plastic deformation on the crack growth, and the crack tip field through the use of an elastic core model. The crack tip field resulting from the macroscopic analysis using the strain gradient plasticity theory displayes the 1/2 singularity of stress within the strain gradient dominated region. In the microscopic fracture analysis, the discrete dislocation theory is used to describe the shielding effects of discrete dislocations on the crack growth. The result of the macroscopic analysis near the crack tip, i.e. a new K-field, is taken as the boundary condition for the microscopic fracture analysis. The equilibrium locations of the discrete dislocations around the crack and the shielding effects of the discrete dislocations on the crack growth at the microscale are calculated. The macroscopic fracture analysis and the microscopic fracture analysis are connected based on the elastic core model. Through a comparison of the shielding effects from plastic deformation and the discrete dislocations, the elastic core size is determined.     

Discretized virtual internal bond model for nonlinear elasticity

The virtual internal bond (VIB) is a micro–macro constitutive model. Although this model is based on a postulated discrete microstructure, it ultimately returns to a continuum constitutive relation through a homogenization process. The homogenization process can reduce the internal degrees of freedom, but it omits the effect of the individual micro bond that may play an important role in the fracture process. The present research develops a discrete system to represent the nonlinear elasticity by discretizing the continuous VIB. This discrete system is composed of unit cells, which can adopt any geometry with any number of bonds. The system is characterized by the force–displacement, not the stress–strain constitutive relationship. The nonlinear properties of this discrete system are governed by the micro-bond potential. The micro bond properties are related to Young’s modulus of the material, the volume and the bond number of the unit cell. For a given material, the unit cell has a certain topological structure and configuration. A discussion of two specific cases (the 2D triangular and 3D tetrahedral unit cells) suggests that the discrete system converges with decreasing unit cell size. In the unstructured unit cell scheme, the discrete system can almost precisely represent the initial Young’s modulus and the Poisson ratio of a nonlinear continuum. A mixed fracture example demonstrates that the present method can efficiently simulate the fracture propagation. The present paper provides a theory for developing a lattice-type mechanical model for nonlinear elasticity and provides new method for the fracture simulation of a nonlinear elastic material.     

考虑应变率效应的混凝土随机损伤本构模型研究

混凝土材料组分复杂且具有随机分布的特点, 其受力力学行为不可避免地存在非线性和随机性. 同时, 在动力荷载作用下, 混凝土材料具有不可忽视的率敏感性. 为了综合反映混凝土受力力学行为中的非线性、随机性与率敏感性, 本文从对材料的纳-微观裂纹扩展分析入手, 引入速率过程理论描述纳观裂纹的扩展速率, 并研究了对应的能量耗散过程. 在此基础上通过裂纹层级模型将纳观分析推演到微观尺度, 建立了微观能量耗散的基本表达式. 进而与微-细观随机断裂模型相结合, 形成了混凝土纳-微-细观随机损伤本构模型. 同时, 基于速率相关势垒的分析, 揭示了动力强度的提高源自加载速率和原子键断裂速率的竞争机制. 据此, 假定微裂纹间相互作用与应变率比值的相关关系以建立微弹簧能量耗散速率与应变率的联系, 实现了从静力本构模型向动力本构模型的扩展. 数值算例表明, 建议模型能够同时反映混凝土材料力学行为中的非线性、随机性和率敏感性. 最后通过与相关试验结果的对比, 验证了建议模型的正确性.      

Elastic–plastic behavior of non-woven fibrous mats

Electrospinning is a novel method for creating non-woven polymer mats that have high surface area and high porosity. These attributes make them ideal candidates for multifunctional composites. Understanding the mechanical properties as a function of fiber properties and mat microstructure can aid in designing these composites. Further, a constitutive model which captures the membrane stress–strain behavior as a function of fiber properties and the geometry of the fibrous network would be a powerful design tool. Here, mats electrospun from amorphous polyamide are used as a model system. The elastic–plastic behavior of single fibers are obtained in tensile tests. Uniaxial monotonic and cyclic tensile tests are conducted on non-woven mats. The mat exhibits elastic–plastic stress–strain behavior. The transverse strain behavior provides important complementary data, showing a negligible initial Poisson's ratio followed by a transverse:axial strain ratio greater than ?1:1 after an axial strain of 0.02. A triangulated framework has been developed to emulate the fibrous network structure of the mat. The micromechanically based model incorporates the elastic–plastic behavior of single fibers into a macroscopic membrane model of the mat. This representative volume element based model is shown to capture the uniaxial elastic–plastic response of the mat under monotonic and cyclic loading. The initial modulus and yield stress of the mat are governed by the fiber properties, the network geometry, and the network density. The transverse strain behavior is linked to discrete deformation mechanisms of the fibrous mat structure including fiber alignment, fiber bending, and network consolidation. The model is further validated in comparison to experiments under different constrained axial loading conditions and found to capture the constraint effect on stiffness, yield, post-yield hardening, and post-yield transverse strain behavior. Due to the direct connection between microstructure and macroscopic behavior, this model should be extendable to other electrospun systems and other two dimensional random fibrous networks.     

Constitutive relations of strain, anisotropic degradation, and fracture of filled polymer materials in prevailing-tension processes with varying axis direction and relaxations

In the course of monotone uniaxial tension, filled polymer materials quasi-isotropic in the initial state experience increasing structure fractures (local adhesive separation and cohesive tearing) whose directions are mainly perpendicular to the tension axis. After complete unloading and relaxation, the fracture lips close, and weaker secondary bonds are formed between them. Taking into account the anisotropy of the above-described process of deterioration of the material structure and mechanical properties (degradation), we suggest to characterize the state of each elementary material fiber by its own values of the structure parameters (damage, fracture, and maximum strain), which can be calculated (according to the model equations of uniaxial tension in a constant direction) from the effective strain history of the fiber. It is determined as the product of the current values of two factors, namely, the strain intensity and the influence function, whose argument is the angle between the directions of the fiber under study and the maximum principal strain. The form of the influence function depends on the material and reflects the degree of anisotropy of the damage arising in it. As a model of uniaxial tension in a constant direction, we use the earlier-proposed version of the nonlinear endochronic theory of ageing viscoelastic materials, which, in addition, contains the secondary bond parameter (with its own equation). We show how the proposed constitutive relations permit one to describe the decrease in the resistance and the ultimate strain during the second axial tension compared with a similar tension from the initial state and to determine the dependence of these effects on the angle between the directions of the preliminary and repeated tensions.     

各向异性对平板裂纹扩展路径的影响研究

近场动力学理论克服了经典连续介质力学在模拟裂纹扩展方面的不足。本文基于微梁键构建了一般各向异性近场动力学平面模型,并探讨了各向异性对于裂纹扩展路径的影响。首先,建立一般各向异性的微梁键,并基于插值法建立了一般各向异性近场动力学平面模型的应变能密度表达式;然后,利用应变能密度互等原理求解出近场动力学平面本构模型的一般各向异性参数;最后,通过带初始切口的平板断裂实验研究了不同各向异性参数对于平板裂纹扩展路径的影响。     

Progressive failure constitutive model of fracture plane in geomaterial based on strain strength distribution

Progressive failure constitutive model of fracture plane in geomaterial based on strain strength distribution is proposed. The basic assumption is that strain strength of geomaterial comply with a certain distribution law in space. Failure of tensile fracture plane and shear fracture plane in representative volume element (RVE) with iso-strain are discussed, and generalized failure constitutive model of fracture plane in RVE is established considering combined effect of tension and shear. Fracture plane consists of elastic microplanes and fractured microplanes. Elastic microplanes are intact parts of the fracture plane, and fractured microplanes are the rest parts of the fracture plane whose strain have ever exceeded their strain strength. Interaction mode on elastic microplanes maintains linear elasticity, while on fractured microplanes it turns into contact and complies with Coulomb’s friction law. Intact factor and fracture factor are defined to describe damage state of the fracture plane which can be easily expressed with cumulative integration of distribution density function of strain strength. Strong nonlinear macroscopic behavior such as yielding and strain softening can be naturally obtained through statistical microstructural damage of fracture plane due to distribution of strain strength. Elastic–brittle fracture model and ideal elastoplastic model are special cases of this model when upper and lower limit of distribution interval are equal.     

Failure prediction in anisotropic sheet metals under forming operations with consideration of rotating principal stretch directions

An approximate macroscopic yield criterion for anisotropic porous sheet metals is adopted to develop a failure prediction methodology that can be used to investigate the failure of sheet metals under forming operations. Hill's quadratic anisotropic yield criterion is used to describe the matrix normal anisotropy and planar isotropy. The approximate macroscopic anisotropic yield criterion is a function of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions. The Marciniak–Kuczynski approach is employed here to predict failure/plastic localization by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands in a material element of interest. The effects of the anisotropy parameter R, the material/geometric inhomogeneities, and the potential surface curvature on failure/plastic localization are first investigated. Then, a non-proportional deformation history including relative rotation of principal stretch directions is identified in a critical element of a mild steel sheet under a fender forming operation given as a benchmark problem in the 1993 NUMISHEET conference. Based on the failure prediction methodology, the failure of the critical sheet element is investigated under the non-proportional deformation history. The results show that the gradual rotation of principal stretch directions lowers the failure strains of the critical element under the given non-proportional deformation history.     

非贯通裂隙岩体损伤演化率相关性及变形特征

含非贯通裂隙岩体是自然界中岩体的主要赋存形式，其裂隙几何特征对岩体的强度及变形均产生显著影响。应变率对岩体的损伤演化及黏滞效应也具有显著的率相关性。首先，运用模型元件的方法，将非贯通裂隙岩体动态破坏过程视为具复合损伤、静态弹性特性、动态黏滞特性的非均质点组成，对黏弹性响应的Maxwell体进行改进，将细观损伤体与裂隙损伤演化的宏观损伤体根据等效应变假设并联组成宏细观复合损伤体，构建综合考虑岩体宏细观缺陷的动态损伤模型；其次，基于断裂力学及应变能理论，对岩体宏观裂隙动态扩展的能量机制进行分析，综合考虑初始裂隙应变能、裂隙动态损伤演化过程应变能、裂隙闭合应变能，得到裂隙岩体宏观动态损伤变量计算公式；最后，将模型计算结果与实验结果进行比较，模型计算结果与实验结果吻合较好，证明了模型的合理性，同时利用模型讨论了裂隙倾角、应变率、岩石性质对岩体变形特征的影响规律。     

The General Mathematical Theory of Plasticity and the Il’yushin Postulates of Macroscopic Definability and Isotropy

The physical laws characterizing the relation between stresses and strains are considered and analyzed in the general modern theory of elastoplastic deformations and in its postulates of macroscopic definability and isotropy for initially isotropic continuous media. The fundamentals of this theory in continuum mechanics were developed by A.A. Il’yushin in the mid-twentieth century. His theory of small elastoplastic deformations under simple loading became a generalization of Hencky’s deformation theory of flow, whereas his theory of elastoplastic processes which are close to simple loading became a generalization of the Saint-Venant–Mises flow theory to the case of hardening media. In these theories, the concepts of simple arid complex loading processes arid the concept of directing form change tensors are introduced; the Bridgman law of volume elastic change and the universal Roche–Eichinger laws of a single hardening curve under simple loading are adopted; and the Odquist hardening for plastic deformations is generalized to the case of elastoplastic hardening media for the processes of almost simple loading without consideration of a specific history of deformations for the trajectories with small arid mean curvatures. In this paper we discuss the possibility of using the isotropy postulate to estimate the effect of forming parameters in the stress-strain state appeared due to the strain-induced anisotropy during the change of the internal structures of materials. We also discuss the possibility of representing the second-rank symmetric stress and strain tensors in the form of vectors in the linear coordinate six-dimensional Euclidean space. An identity principle is proposed for tensors and vectors.     

Discontinuous bifurcation analysis in fracture energy-based gradient plasticity for concrete

Conditions for discontinuous bifurcation in limit states of selective non-local thermodynamically consistent gradient theory for quasi-brittle materials like concrete are evaluated by means of both geometrical and analytical procedures. This constitutive formulation includes two internal lengths, one related to the strain gradient field that considers the degradation of the continuum in the vicinity of the considered material point. The other characteristic length takes into account the material degradation in the form of energy release in the cracks during failure process evolution.The variation from ductile to brittle failure in quasi-brittle materials is accomplished by means of the pressure dependent formulation of both characteristic lengths as described by Vrech and Etse (2009).In this paper the formulation of the localization ellipse for constitutive theories based on gradient plasticity and fracture energy plasticity is proposed as well as the explicit solutions for brittle failure conditions in the form of discontinuous bifurcation. The geometrical, analytical and numerical analysis of discontinuous bifurcation condition in this paper are comparatively evaluated in different stress states and loading conditions.The included results illustrate the capabilities of the thermodynamically consistent selective non-local gradient constitutive theory to reproduce the transition from ductile to brittle and localized failure modes in the low confinement regime of concrete and quasi-brittle materials.     

Free-energy density functions for nematic elastomers

The recently proposed neo-classical theory for nematic elastomers generalizes standard molecular-statistical Gaussian network theory to allow for anisotropic distributions of polymer chains. The resulting free-energy density models several of the novel properties of nematic elastomers. In particular, it predicts the ability of nematic elastomers to undergo large deformations with exactly zero force and energy cost—so called soft elasticity. Although some nematic elastomers have been shown to undergo deformations with unusually small applied forces, not all do so, and none deform with zero force. Further, as a zero force corresponds to infinitely many possible deformations in the neo-classical theory, this non-uniqueness leads to serious indeterminacies in numerical schemes. Here we suggest that the neo-classical free-energy density is incomplete and propose an alternative derivation that resolves these difficulties. In our approach, we use the molecular-statistical theory to identify appropriate variables. This yields the choice for the microstructural degrees of freedom as well as two independent strain tensors (the overall macroscopic strain plus a relative strain that indicates how the deformation of the elastomeric microstructure deviates from the macroscopic deformation). We then propose expressions for the free-energy density as a function of the three quantities and show how the material parameters can be measured by two simple tests. The neo-classical free-energy density can be viewed as a special case of our expressions in which the free-energy density is independent of the overall macroscopic strain, thus supporting our view that the neo-classical theory is incomplete.     

A 2-D lattice model for simulating the failure of paper

A new two-dimensional network model is proposed as a micromechanics model to simulate paper’s failure process due to sequentially breakages of fibers and/or bonds. Paper is approximated as a network composed of fibers any two of which link to each other by their intersecting point, namely so-called bond. Fibers distribute along three particular directions, leading to network’s macro-level isotropy. In the framework of finite element method, nodes correspond to fiber-to-fiber bonds, while elements are fiber segments between every two neighboring nodes and described by Timoshenko beam theory. Element breaks when its equivalent internal tensile stress reaches the tensile strength of fiber. Strength of nodes, i.e. fiber-to-fiber bonds is assumed to be dependant on shearing interaction between fibers, considering the dominant interaction is shearing in a plane problem. Numerical examples show the model’s capacity of reflecting basic failure characteristic in paper. Influences of fiber length and the ratio of fiber strength to bond strength are analyzed in detail.