Approaches to dynamic fracture modelling at finite deformations

The focus of the present paper is on the finite element modelling of dynamic fracture based on the concept of locally enriched element shape functions in the vicinity of the crack, in line with the eXtended Finite Element Method (X-FEM). For this purpose, the proper governing equations for the case of a propagating crack within a hyperelastic material is established, including the definition of the concept of material motion which kinematically describes the progression of the crack. Furthermore, two different approaches to describe the material degradation and separation are proposed. The first approach, denoted the material crack driving force model, is based on the concept of material (or configurational) forces associated with the material motion. The basic motivation is that, in this context, a driving force is identified at the crack tip, which points in the direction of maximum energy release upon crack propagation. An additional interesting feature of this force is that the projection in the crack propagation direction corresponds to the energy released for such a propagation, whereby an intuitive criterion for crack propagation based on the direction and magnitude of this force is proposed. The second approach is based on the classical cohesive zone concept, extended to include rate effects to capture experimentally observed phenomena such as growing process zones during propagation as well as limited crack propagation speeds well below the theoretical limit. Both models are investigated and compared in a couple of numerical examples in the latter part of the paper, showing both the predictive capabilities as well as some limitations of the two approaches. It has also been shown that, for a specific set of parameters, the two models can reproduce (almost) the same response.     

CRACK INTERACTING WITH AN INDIVIDUAL INCLUSION BY THE FRACTURE CRITERION OF CONFIGURATIONAL FORCE

基于构型力概念提出一种可判断裂纹起裂以及裂纹扩展方向的新断裂准则.该准则假设当构型合力值达到一个临界值时裂纹开始扩展,而裂纹扩展的方向则为构型合力的矢量方向.基于此断裂准则,本文开发构型力的有限元计算方法,实现对裂纹扩展的数值模拟,并着重对工程中常见的含孔洞/夹杂结构的裂纹扩展问题展开研究.研究结果表明,基于构型力的裂纹扩展准则可以很好地预测裂纹与孔/夹杂的干涉作用,其数值模拟结果与实验结果相符,从而验证了该裂纹扩展模拟方法的有效性.通过对裂纹和夹杂（圆孔、软夹杂、硬夹杂）干涉问题的数值模拟表明,裂纹前端夹杂对裂纹的扩展具有重要影响.裂纹的扩展方向与裂纹和夹杂的相对位置、以及夹杂类型密切相关.软夹杂和圆孔会吸引裂纹向其扩展,而硬夹杂会排斥裂纹扩展,裂纹在扩展过程中会绕开硬夹杂.当裂纹与夹杂夹角较小时,夹杂对裂纹扩展的影响作用明显,当夹角较大时,夹杂对裂纹扩展的影响较小;特别当裂纹与夹杂夹角为45°时,软夹杂和圆孔可能会抑制裂纹的扩展,使裂纹扩展发生止裂.研究结果有助于认清含孔洞/夹杂结构中的裂纹扩展或止裂问题,对于工程中的断裂问题具有重要指导意义.     

Study on the mechanisms and quantitative law of mode I supersonic crack propagation

Continuum mechanics predicts that the propagation speed of non-equilibrium information in solids is limited by the longitudinal wave speed, so is crack propagation. However, solids are essentially discrete systems. In this paper, via theoretical analysis and numerical simulations, it is demonstrated in a straightforward way that non-equilibrium disturbance (e.g. force, displacement, energy, and so on) can propagate at a supersonic speed in discrete systems, although the magnitude of the disturbance attenuates very quickly. In dynamic fracture, a cascade of atomic-bond breaking events provides an amplification mechanism to counterbalance the attenuation of the disturbance. Therefore, supersonic crack propagation can be realized in a domino way. Another key factor for supersonic crack propagation is to ensure sufficient energy flowing into the crack tip. Since most energy can only be transferred at a speed limited by the longitudinal wave speed, the conditions for the occurrence of supersonic crack propagation are not easily met in most situations, unless there is high pre-stored energy along the crack path or continuous energy supply from the loading concomitantly moving with the crack tip. A quantitative relation between supersonic crack propagation speed and material properties and parameters is given, which implies that knowing all the classical macroscopic quantities is not enough in determining the supersonic crack propagation speed, and the microstructure does play a role. Moreover, it is interesting to note that fracture toughness affects the crack propagation speed in the subsonic regime, but not in the supersonic regime, because the deformation/stress is uniform in front of a supersonic crack where strength criterion dominates.     

An energy-consistent fracture model for ferroelectrics

The fracture behavior of ferroelectrics has been intensively studied in recent decades, though currently a widely accepted fracture mechanism is still lacking. In this work, enlightened by previous experimental observations that crack propagation in ferroelectrics is always accompa-nied by domain switching, we propose a micromechanical model in which both crack propagation and domain switch-ing are controlled by energy-based criteria. Both electric energy and mechanical energy can induce domain switching, while only mechanical energy can drive crack propagation. Furthermore, constrained domain switching is considered in this model, leading to the gradient domain switching zone near the crack tip. Analysis results show that stress-induced ferroelastic switching always has a toughening effect as the mechanical energy release rate serves as the driving force for both fracture and domain switching. In compari-son, the electric-field-induced switching may have either a toughening or detoughening effect. The proposed model can qualitatively agree with the existing experimental results.     

A theoretical investigation on the vibrational characteristics and torsional dynamic response of circumferentially cracked turbo-generator shafts

Turbo-generator shafts are often subjected to complex dynamic torsional loadings, resulting in generation and propagation of circumferential cracks. Mode III fatigue crack growth generally results in a fracture surface consisting of peaks and valleys, resembling a factory roof. The fracture surface roughness depends on the material microstructure, the material yield strength, and the applied cyclic torque amplitude. This crack pattern can severely affect the vibration characteristics of the shafts. The accurate evaluation of the torsional dynamic response of the turbo-generator shafts entails considering the local sources of energy loss in the crack vicinity. The two most common sources of the energy loss are the local energy loss due to the plasticity at the crack tip and frictional energy loss due to interaction of mutual crack surfaces. A theoretical procedure for evaluating the values of the system loss factors corresponding to these sources of energy loss is presented. Furthermore, the local flexibility is obtained by evaluating the resistance of the cracked section of the shaft to the rotational displacement. The shaft material is assumed to be elastic perfectly plastic. The effects of the applied Mode III stress intensity factor and the crack surface pattern parameters on the energy loss due to the friction and the energy loss due to the plasticity at the crack tip are investigated. The results show that depending on the amplitude of the applied Mode III stress intensity factor, one of these energy losses may dominate the total energy loss in the circumferentially cracked shaft. The results further indicate that the torsional dynamic response of the turbo-generator shaft is significantly affected by considering these two sources of the local energy loss.     

Hydraulic fracturing of a saturated porous medium-I: General theory

This investigation deals with the problem of steady state hydraulic fracture in an infinite isotropic fluid-saturated elastic porous medium induced by a uniform pressure applied to the crack surfaces. A quasi-static approach is employed in the study. A boundary value problem is formulated and then analyzed by means of the Fourier transform associated with the Wiener-Hopf technique. Stress intensity factor and potential energy release rate are found by asymptotic analysis and the superposition principle as functions of the speed of crack propagation. The material breakdown process at the crack tip is discussed based on Dugdale's model. Finally, a brief discussion of the effect of pressure drop on the hydraulic fracture process and the decrease in crack speed during crack extension is included.     

Hydraulic fracturing of a saturated porous medium-II: Special cases

This investigation deals with the problem of steady state hydraulic fracture in an infinite isotropic fluid-saturated elastic porous medium induced by a uniform pressure applied to the crack surfaces. A quasi-static approach is employed in the study. A boundary value problem is formulated and then analyzed by means of the Fourier transform associated with the Wiener-Hopf technique. Stress intensity factor and potential energy release rate are found by asymptotic analysis and the superposition principle as functions of the speed of crack propagation. The material breakdown process at the crack tip is discussed based on Dugdale's model. Finally, a brief discussion of the effect of pressure drop on the hydraulic fracture process and the decrease in crack speed during crack extension is included.     

Energy release rate of small cracks in hyperelastic materials

The energy release rate of a small crack in an infinite hyperelastic medium, and subjected to large strain multiaxial loading conditions, is derived by considering the balance of configurational stresses acting on two planes: one cutting the center of the crack face, and the other at an infinite distance in front of the crack tip. The analysis establishes that the energy release rate of a small crack is always proportional to the size of the crack, irrespective of the loading conditions and the crack orientation. The balance of configurational stresses is illustrated for several benchmark cases including simple extension, pure shear and equibiaxial extension, and for perpendicular and inclined cracks.     

Analysis of dynamic growth of a tensile crack in an elastic-plastic material

The influence of inertia on the stress and deformation fields near the tip of a crack growing in an elastic-plastic material is studied. The material is characterized by the von Mises yield criterion and J₂ flow theory of plasticity. The crack grows steadily under plane strain conditions in the tensile opening mode. Features of the stress and deformation state at points near the moving crack tip are described for elastic-perfectly plastic response and for several crack propagation speeds. It is found that inertia has a significant effect on the elastic-plastic response of material particles near the crack tip, and that elastic unloading may occur behind the crack tip for higher speeds. The relationship between the applied crack driving force, represented by a remote stress intensity factor, and the crack tip speed is examined on the basis of a critical crack tip opening angle growth criterion. The calculated result is compared with dynamic fracture toughness versus crack speed data for a 4340 steel.     

基于有限断裂法和比例边界有限元法的裂纹扩展模拟

基于有限断裂法和比例边界有限元法提出了一种裂缝开裂过程模拟的数值模型。采用基于有限断裂法的混合断裂准则作为起裂及扩展的判断标准,当最大环向应力和能量释放率同时达到其临界值时,裂缝扩展。结合多边形比例边界有限元法,可以半解析地求解裂尖区域附近的应力场和位移场,在裂尖附近无需富集即可获得高精度的解。计算能量释放率时,只需将裂尖多边形内的裂尖位置局部调整,无需改变整体网格的分布,网格重剖分的工作量降至最少。裂缝扩展步长通过混合断裂准则确定,避免了人为假设的随意性,并可以实现裂缝变步长扩展的模拟,更符合实际情况。通过对四点剪切梁的复合型裂缝扩展过程的模拟,对本文模型进行了验证,并应用于重力坝模型的裂缝扩展模拟,计算结果表明,本文提出的模型简单易行且精度较高。     

梁在纯弯曲力矩作用下的动态塑性断裂

本文采用刚塑性分析方法,研究了带裂纹有限长韧性材料梁在纯弯曲力矩作用下的动态塑性断裂过程,本文考虑了由于断裂引起的附加轴向力对断裂过程的影响,并在分析中考虑了应变率对断裂过程的影响。文章给出了断裂过程中断裂截面上弯曲力矩和轴向力随时间的变化规律,以及裂纹长度,裂纹扩展速度和加速度随时间的变化规律。     

基于DIC的爆炸加载下脆性材料裂纹扩展规律的试验研究

通过数字图像相关法（DIC）,应用PMMA对爆炸加载条件下脆性材料的裂纹扩展规律进行了试验研究。基于对称性试验模型,实现了裂纹尖端位置和应变场信息的同步记录。以此为基础,通过对比分析获知,主应变场应变值最大点不能作为裂纹尖端的判断依据。并以动态裂纹扩展速度为参量,应用断裂动力学和最小二乘牛顿迭代法,计算出了考虑惯性效应的Ⅰ-Ⅱ混合型裂纹的应力强度因子:K_Ⅰ和K_Ⅱ值会随着裂纹扩展方向改变而发生突变;K_Ⅰ最大值为2.63 MPa·m1/2,最小值为0.89 MPa·m1/2;其整体变化趋势表明,爆炸加载条件下脆性材料裂纹扩展随能量积聚和释放呈循环阶梯式递减发展。     

Energy release rates in rubber during dynamic crack propagation

The theoretical understanding of the fracture mechanics of rubber is not as well developed as for other engineering materials, such as metals. The present study is intended to further the understanding of the dissipative processes that take place in rubber in the vicinity of a propagating crack tip. This dissipation contributes significantly to the total fracture toughness of the rubber and is therefore of great interest from a fracture mechanics point of view. To study this, a computational framework for analysing high-speed crack growth in a biaxially stretched rubber under plane stress is therefore formulated. The main purpose is to investigate the energy release rates required for crack propagation under different modes of biaxial stretching. The results show, that inertia comes into play when the crack speed exceeds about 50 m/s. The total work of fracture by far exceeds the surface energy consumed at the very crack tip, and the difference must be attributed to dissipative damage processes in the vicinity of the crack tip. The size of this damage/dissipation zone is expected to be a few millimetres.     

基于构型力内变量的界面损伤问题研究

本文基于材料构型力的基本理论和损伤力学中含有内变量的热力学框架,提出了新的损伤变量定义方式,为研究界面损伤问题提供了一种新思路。首先,基于双相弹性体的能量分析,给出界面材料构型力表达式,通过构型力的离散化方法,实现了其在有限元中的数值计算。其次,定义构型力为界面损伤内变量,进而提出一种新的损伤演化模型,并采用刚度劣化的方法,对该界面损伤模型进行数值实现。最后,通过对复合材料界面损伤问题（有裂纹或无裂纹）进行数值模拟,分析了其界面损伤发展趋势,探讨了此模型的合理性和优越性。基于构型力内变量的界面损伤模型,可为研究复合材料的界面损伤失效问题提供一种普适性的方法。     

Sharp-crack limit of a phase-field model for brittle fracture

A matched asymptotic analysis is used to establish the correspondence between an appropriately scaled version of the governing equations of a phase-field model for fracture and the equations of the two-dimensional sharp-crack theory of Gurtin and Podio-Guidugli (1996) that arise on assuming that the bulk constitutive behavior is nonlinearly elastic, requiring that surface energy provides the only factor limiting crack propagation, and assuming that the fracture kinetics are isotropic. Consistent with the prominence of the configurational momentum balance at the crack tip in the latter theory, the approach capitalizes on the configurational momentum balance that arises naturally in the context of the phase-field model. The model developed and utilized here incorporates irreversibility of the phase-field evolution. This is achieved by introducing a suitable constraint and by carefully heeding the influence of that constraint on the kinetics underlying microstructural changes associated with fracture. The analysis is predicated on the assumption that the phase-field variable takes values in the closed interval between zero and unity.     

Angle cracks in two bonded functionally graded piezoelectric material under anti-plane shear

Consider two bonded functionally graded piezoelectric material (FGPM) with finite height. Each material contains an arbitrary oriented crack. The material properties are assumed in exponential forms in the direction normal to the interface. The crack surface condition is assumed to be electrically impermeable or permeable. Using the Fourier transform technique, the problem can be reduced to a system of singular integral equations, which are then solved numerically by applying the Gauss-Chebyshev integration formula to obtain the stress intensity factors at the crack tips. Numerical calculations are carried out to obtain the energy density factor S and the energy release rate G. In impermeable case, the energy release rate has been shown to be negative as the electric loads are applied. The positive definite characteristic of the energy density factor makes it possible for predicting the fracture behavior of the cracked structure. The influences of the non-homogeneous parameters and crack orientation on the energy density factors at the crack tips are discussed in detail. The results show that the energy density factor at the crack tip will be increased when the crack tip is located within the softer material.     

Configurational forces in crystal plasticity: an analysis of the influence of grain boundaries on crack driving forces

The theory of configurational forces is applied to standard dissipative materials. As a computational example, the method is used to analyze a crystal plastic material law, which assumes flow in the basal and prismatic slip systems of a hexagonally close packed crystal. The material data mimic titanium. Numerical examples include the driving force on a crack interacting with a grain boundary. A simple algorithm is proposed to treat straight crack growth. It is shown how the configurational force at the crack tip is influenced by the presence of the grain boundary.     

Quasicontinuum simulation of single crystal nano-plate with a mixed-mode crack

The quasicontinuum (QC) method is employed to simulate a nickel single crystal nano-plate with a mixed-mode crack. Atomic stresses near the crack tip are fitted according to the elastoplastic fracture mechanics equations. It is found that the atomic stress fields neighboring the crack tip are also singular and controlled by the atomic stress intensity factors. And then the critical energy release rates for brittle and ductile fracture are computed and compared in order to predict crack propagation or dislocation emission. Four possible slip directions at the crack tip are pointed out. Finally, the slip direction around the crack tip is determined by the shear stress and it is well consistent with the atomic pictures from the QC simulation.     

On the fracture toughness of ferroelastic materials

The toughness enhancement due to domain switching near a steadily growing crack in a ferroelastic material is analyzed. The constitutive response of the material is taken to be characteristic of a polycrystalline sample assembled from randomly oriented tetragonal single crystal grains. The constitutive law accounts for the strain saturation, asymmetry in tension versus compression, Bauschinger effects, reverse switching, and strain reorientation that can occur in these materials due to the non-proportional loading that arises near a propagating crack. Crack growth is assumed to proceed at a critical level of the crack tip energy release rate. Detailed finite element calculations are carried out to determine the stress and strain fields near the growing tip, and the ratio of the far field applied energy release rate to the crack tip energy release rate. The results of the finite element calculations are then compared to analytical models that assume the linear isotropic K-field solution holds for either the near tip stress or strain field. Ultimately, the model is able to account for the experimentally observed toughness enhancement in ferroelastic ceramics.     

Two-temperature continuum thermomechanics of deforming amorphous solids

There is an ever-growing need for predictive models for the elasto-viscoplastic deformation of solids. Our goal in this paper is to incorporate recently developed out-of-equilibrium statistical concepts into a thermodynamically consistent, finite-deformation, continuum framework for deforming amorphous solids. The basic premise is that the configurational degrees of freedom of the material – the part of the internal energy/entropy that corresponds to mechanically stable microscopic configurations – are characterized by a configurational temperature that might differ from that of the vibrational degrees of freedom, which equilibrate rapidly with an external heat bath. This results in an approximate internal energy decomposition into weakly interacting configurational and vibrational subsystems, which exchange energy following a Fourier-like law, leading to a thermomechanical framework permitting two well-defined temperatures. In this framework, internal variables, that carry information about the state of the material equilibrate with the configurational subsystem, are explicitly associated with energy and entropy of their own, and couple to a viscoplastic flow rule. The coefficients that determine the rate of flow of entropy and heat between different internal systems are proposed to explicitly depend on the rate of irreversible deformation. As an application of this framework, we discuss two constitutive models for the response of glassy materials, a simple phenomenological model and a model related to the concept of Shear-Transformation-Zones as the basis for internal variables. The models account for several salient features of glassy deformation phenomenology. Directions for future investigation are briefly discussed.