基于结构化变形驱动的非局部宏-微观损伤模型的真Ⅱ型裂纹模拟

Ⅱ型载荷作用下裂纹变形模式也为Ⅱ型的破坏问题称为真Ⅱ型破坏.准确定量地把握真Ⅱ型破坏的全过程是具有挑战性的问题.本文采用结构化变形驱动的非局部宏-微观损伤模型对真Ⅱ型破坏问题进行了模拟.根据结构化变形理论将点偶的非局部应变分解为弹性应变与结构化应变两部分,进而利用Cauchy-Born准则与结构化应变计算点偶的结构化正伸长量.在本文中,结构化应变取为非局部应变的偏量部分.当点偶的结构化正伸长量超过临界伸长量时,微细观损伤开始在点偶层次发展.将微细观损伤在作用域中进行加权求和得到拓扑损伤,并通过能量退化函数将其嵌入到连续介质-损伤力学框架中进行数值求解.进一步地,本文采用Gauss-Lobatto积分格式计算点偶的非局部应变,将积分点数目降低到4个,显著降低了前处理和非线性分析的计算成本.通过对Ⅱ型加载下裂尖应变场的分析揭示了采用偏应变作为结构化应变的原因.对两个典型真Ⅱ型破坏问题的模拟结果表明,本文方法不仅可以把握Ⅱ型加载下的真Ⅱ型裂纹扩展模式,同时可以定量刻画加载过程中的载荷-变形曲线,且不具有网格敏感性.最后指出了需要进一步研究的问题.     

Mode I and mode II crack tip asymptotic fields with strain gradient effects

The strain gradient effect becomes significant when the size of fracture process zone around a crack tip is comparable to the intrinsic material lengthl, typically of the order of microns. Using the new strain gradient deformation theory given by Chen and Wang, the asymptotic fields near a crack tip in an elastic-plastic material with strain gradient effects are investigated. It is established that the dominant strain field is irrotational. For mode I plane stress crack tip asymptotic field, the stress asymptotic field and the couple stress asymptotic field can not exist simultaneously. In the stress dominated asymptotic field, the angular distributions of stresses are consistent with the classical plane stress HRR field; In the couple stress dominated asymptotic field, the angular distributions of couple stresses are consistent with that obtained by Huang et al. For mode II plane stress and plane strain crack tip asymptotic fields, only the stress-dominated asymptotic fields exist. The couple stress asymptotic field is less singular than the stress asymptotic fields. The stress asymptotic fields are the same as mode II plane stress and plane strain HRR fields, respectively. The increase in stresses is not observed in strain gradient plasticity for mode I and mode II, because the present theory is based only on the rotational gradient of deformation and the crack tip asymptotic fields are irrotational and dominated by the stretching gradient. The project supported by the National Natural Science Foundation of China (19704100), National Natural Science Foundation of Chinese Academy of Sciences (KJ951-1-20), CAS K.C. Wong Post-doctoral Research Award Fund and Post-doctoral Science Fund of China     

I型定常扩展裂纹尖端的弹黏塑性场

考虑材料在扩展裂纹尖端的黏性效应，假设黏性系数与塑性应变率的幂次成反比，对幂硬化材料中平面应变扩展裂纹尖端场进行了弹黏塑性渐近分析，得到了不含间断的连续解，并讨论了I型裂纹数值解的性质随各参数的变化规律. 分析表明应力和应变均具有幂奇异性，并且只有在线性硬化时，尖端场的弹、黏、塑性才可以合理匹配. 对于I型裂纹，裂尖场不含弹性卸载区. 当裂纹扩展速度趋于零时，动态解趋于准静态解，表明准静态解是动态解的特殊形式；如果进一步考虑硬化系数为零的极限情况，便可退化为Hui和Riedel的非线性黏弹性解.     

Intersonic crack growth under time-dependent loading

The problem investigated in this paper is a mode II crack extending at a uniform intersonic speed in an otherwise unbounded elastic solid subjected to time dependent crack-face tractions. The fundamental solution for this problem is obtained analytically, which is then used to construct the general solution for an intersonic crack subjected to arbitrary time-dependent loading. For time-independent loading, this solution reduces to Huang and Gao’s [J. Appl. Mech 68 (2001) 169] fundamental solution. We have also studied two crack-face loadings that are of interest for engineering applications.     

Dynamic crack tip fields and dynamic crack propagation characteristics of anisotropic material

Based on mechanics of anisotropic material, the dynamic crack propagation problem of I/II mixed mode crack in an infinite anisotropic body is investigated. Expressions of dynamic stress intensity factors for modes I and II crack are obtained. Components of dynamic stress and dynamic displacements around the crack tip are derived. The strain energy density theory is used to predict the dynamic crack extension angle. The critical strain energy density is determined by the strength parameters of anisotropic materials. The obtained dynamic crack tip fields are unified and applicable to the analysis of the crack tip fields of anisotropic material, orthotropic material and isotropic material under dynamic or static load. The obtained results show Crack propagation characteristics are represented by the mechanical properties of anisotropic material, i.e., crack propagation velocity M and fiber direction α. In particular, the fiber direction α and the crack propagation velocity M give greater influence on the variations of the stress fields and displacement fields. Fracture angle is found to depend not only on the crack propagation but also on the anisotropic character of the material.     

Linear Elastic Solutions for Slotted Plates

Two-dimensional problems of finite-length blunted cracks cut into infinite plates subject to remote tractions are solved using complex variable theory. The slot geometry is composed of two flat surfaces connected by rounded ends. This special geometrical shape was derived by Riabouchinsky in the study of two-dimensional ideal fluid flow around parallel plates. The simpler antiplane slotted plate problem is addressed initially for this geometry. From this exact solution, the equivalent of a Westergaard stress potential is found and applied to the two other principal modes of fracture, which are plane elasticity problems. For a plate subject to uniform radial tension at infinity, an analytical solution is obtained that will reduce to the familiar mode I singular crack solution as the separation between the parallel faces of the slot becomes zero. For finite-width mode I slots, the rounded ends have tensile tractions which terminate at the adjoining flat surfaces of the slot, which remain traction-free. In this respect, the finite-width mode I slot problem resembles a Barenblatt cohesive zone model of a plane crack or a Dugdale plastic strip model of a plane crack, although the tractions will vary in magnitude along the slot ends rather than remaining uniform as in the former type of crack problems. Similarly, in the case of the finite-width mode II slot problem, the rounded ends of the slot have shear tractions, while the flat surfaces remain load-free. A distinguishing feature of the mode II slot solution over the mode I slot problem is that the maximum in-plane shear stress is constant along the rounded ends of the slot. Because of this, those particular regions of the boundary can represent incipient plastic yield based on either the Mises or Tresca yield condition under plane strain loading conditions. In this way, the problem resembles the plastic strip models of Dugdale, Cherepanov, Bilby-Cottrell-Swinden, and others. Notably, the mode III slot problem also has a constant maximum shear stress along the curved portions of the slot, while the entire slot boundary remains traction-free, unlike the mode II slot problem. Consequently, the mode III slot problem represents both a generalization of the standard mode III crack problem geometry, while simultaneously satisfying the boundary conditions of a plastic strip model.     

Perfect elastic-viscoplastic field at mode I dynamic propagating crack-tip

The viscosity of material is considered at propagating crack-tip. Under the assumption that the artificial viscosity coefficient is in inverse proportion to power law of the plastic strain rate, an elastic-viscoplastic asymptotic analysis is carried out for moving crack-tip fields in power-hardening materials under plane-strain condition. A continuous solution is obtained containing no discontinuities. The variations of numerical solution are discussed for mode I crack according to each parameter. It is shown that stress and strain both possess exponential singularity. The elasticity, plasticity and viscosity of material at crack-tip only can be matched reasonably under linear-hardening condition. And the tip field contains no elastic unloading zone for mode I crack. It approaches the limiting case, crack-tip is under ultra-viscose situation and energy accumulates, crack-tip begins to propagate under different compression situations.     

Dynamic crack growth under Rayleigh wave

A nonuniform crack growth problem is considered for a homogeneous isotropic elastic medium subjected to the action of remote oscillatory and static loads. In the case of a plane problem, the former results in Rayleigh waves propagating toward the crack tip. For the antiplane problem the shear waves play a similar role. Under the considered conditions the crack cannot move uniformly, and if the static prestress is not sufficiently high, the crack moves interruptedly. For fracture modes I and II the established, crack speed periodic regimes are examined. For mode III a complete transient solution is derived with the periodic regime as an asymptote. Examples of the crack motion are presented. The crack speed time-period and the time-averaged crack speeds are found. The ratio of the fracture energy to the energy carried by the Rayleigh wave is derived. An issue concerning two equivalent forms of the general solution is discussed.     

一种基于SHTB的Ⅱ型动态断裂实验技术

冲击剪切载荷作用下动态断裂韧性的测定是材料力学性能和断裂行为研究中重要组成部分.为了测定材料的Ⅱ型动态断裂韧性,许多学者采用不同的试样与实验方法进行了实验,但限于实验条件,裂纹断裂模式往往是I+Ⅱ复合型,而不是纯Ⅱ型,因而不能准确测得材料的Ⅱ型动态断裂韧性.鉴于此,本文基于分离式霍普金森拉杆(split Hopkinson tension bar,SHTB)实验技术,提出一种改进的紧凑拉伸剪切(modified compact tension shear,MCTS)试样,通过夹具对MCTS试样施加约束,从而保证试样按照纯Ⅱ型模式断裂.采用实验-数值方法对MCTS试样动态加载过程进行分析,将实验测得的波形输入有限元软件ANSYS-LSDYNA,得到了裂纹尖端应力强度因子-时间曲线,并与紧凑拉伸剪切(compact tension shear,CTS)试样进行了对比.同时采用数字图像相关法进行了实验,验证了有限元分析结果.结果表明,MCTS试样在整个加载过程中K_I K_Ⅱ,裂纹没有张开;而CTS试样在同样的加载过程中K_IK_Ⅱ,出现裂纹张开现象.这说明MCTS试样能够准确地测定材料的Ⅱ型动态断裂韧性,为材料动态力学测试提供了一种有效的实验技术.     

Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity

The present study aims at determining the elastic stress and displacement fields around the tips of a finite-length crack in a microstructured solid under remotely applied plane-strain loading (mode I and II cases). The material microstructure is modeled through the Toupin-Mindlin generalized continuum theory of dipolar gradient elasticity. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain tensor (as in classical elasticity) and the gradient of the strain tensor (additional term). A simple but yet rigorous version of the theory is employed here by considering an isotropic linear expression of the elastic strain-energy density that involves only three material constants (the two Lamé constants and the so-called gradient coefficient). First, a near-tip asymptotic solution is obtained by the Knein-Williams technique. Then, we attack the complete boundary value problem in an effort to obtain a full-field solution. Hypersingular integral equations with a cubic singularity are formulated with the aid of the Fourier transform. These equations are solved by analytical considerations on Hadamard finite-part integrals and a numerical treatment. The results show significant departure from the predictions of standard fracture mechanics. In view of these results, it seems that the classical theory of elasticity is inadequate to analyze crack problems in microstructured materials. Indeed, the present results indicate that the stress distribution ahead of the crack tip exhibits a local maximum that is bounded. Therefore, this maximum value may serve as a measure of the critical stress level at which further advancement of the crack may occur. Also, in the vicinity of the crack tip, the crack-face displacement closes more smoothly as compared to the standard result and the strain field is bounded. Finally, the J-integral (energy release rate) in gradient elasticity was evaluated. A decrease of its value is noticed in comparison with the classical theory. This shows that the gradient theory predicts a strengthening effect since a reduction of crack driving force takes place as the material microstructure becomes more pronounced.     

刚度微分法计算压电材料平面断裂问题

把计算应变能释放率的刚度微分法推广到压电材料平面断裂问题．在此基础上，利用压电材料平面断裂问题的有限元数值解作为真实场，用Ｓｏｓａ的平面问题裂端渐近解作为辅助场，由推广的交互Ｍ积分法求得了应力强度因子ＫＩ，ＫＩＩ和电位移强度因子ＫＩＶ．算例表明，计算结果与理论解符合得很好     

General solutions of plane problem for power function curved cracks

A new exact and universal conformal mapping is proposed. Using Muskhelishvili's complex potential method, the plane elasticity problem of power function curved cracks is investigated with an arbitrary power of a natural number, and the general solutions of the stress intensity factors (SIFs) for mode I and mode II at the crack tip are obtained under the remotely uniform tensile loads. The present results can be reduced to the well-known solutions when the power of the function takes different natural numbers. Numerical examples are conducted to reveal the effects of the coefficient, the power, and the projected length along the x-axis of the power function curved crack on the SIFs for mode I and mode II.     

广义平均形状改变能密度断裂准则及应用

基于形状改变能为材料起裂扩展控制参量的物理事实及裂纹尖端断裂控制区能量平均概念,提出了计及裂纹尖端应力场特征级数展开奇异项和常数项的广义平均形状改变能密度(GADSED)准则,建立了Ⅰ-Ⅱ-Ⅲ混合型裂纹断裂判据计算式,为工程结构疲劳断裂评估提供了新选择.基于提出的GADSED准则,系统分析了T应力对裂纹断裂判据的影响,结果表明:当参数|Bα|数值相同时:负值T应力的Ⅰ-Ⅱ混合型裂纹更易起裂扩展,正值T应力的K_Ⅰf值高于负值T应力的K_Ⅰf值,正负T应力的K_Ⅱf值相同;T应力在区间0Ⅰf值(3%以内),T应力降低了K_Ⅱf值.根据GADSED准则完成了双轴疲劳载荷平板表面Ⅰ型裂纹扩展寿命预报,结果表明:基于GADSED准则的裂纹扩展寿命预报值约为传统方法预报值的40.7%,可为工程实际评估提供借鉴.     

The stress intensity factor history for an advancing crack in a transversely isotropic solid under 3-D loading

The mode I extension of a half plane crack in a transversely isotropic solid under 3-D loading is analyzed. Firstly, the fundamental problem that the crack is subjected to a pair of unit point loads on its faces is considered. Transform methods are used to reduce the boundary value problem to a single integral equation that can be solved by the Wiener–Hopf technique. The Cagniard–de Hoop method is employed to invert the transforms. An exact expression is derived for the mode I stress intensity factor as a function of time and position along the crack edge. Based on the fundamental solution, the stress intensity factor history due to general loading is then obtained. Some features of the solutions are discussed through numerical results.     

正交异性材料平面裂纹尖端应力场

根据正交各向异性材料力学性能确定出了用应力函数表示的弹性力学基本方程,利用坐标变换和复变函数方法求解了正交异性材料平面裂纹体的应力边值问题。借鉴一般断裂力学解法构造了I型和II型裂纹问题的应力函数,推导出了正交各向异性板裂纹尖端区的奇异应力场。通过数值计算说明了裂纹尖端应力表达式的正确性,验证了裂尖前沿应力变化规律,即σx与材料特征参数h2成正比,而σy和τxy不随材料特性变化。     

T应力对光弹性条纹影响的理论分析

以裂尖弹性应力场的多参数模型为基础,研究I型、II型以及I/II混合型裂纹参数对光弹性条纹的影响. T应力的存在和符号影响着等色线条纹环的半径大小和旋转方向,对于纯I型或II型裂纹而言,条纹环的旋转角度只与T应力有关;而对于I/II混合型裂纹,条纹环旋转角度与K_{\rm I}, K_{\rm II}和T应力有关. T应力的存在使得I型裂纹在裂尖±π/3方向上出现2个各向同性点(T应力为正时),使得II型裂纹在裂尖后的裂纹面上出现1个各向同性点. 对于I/II混合型裂纹而言,当T应力为正时等倾线出现距裂尖半径不等的3个各向同性点;反之, T应力为负时在裂尖后只存在1个各向同性点,这些各向同性点分别与I型和II型裂纹情况具有相同的规律.      

Specification of crack opening and surface sliding in orthotropic viscoelastic material

Modelling of crack opening and surface sliding in an orthotropic viscoelastic material is made by introducing two coefficients: one for the surface displacement and surface friction. The material possesses orthotropy in two dimensions and viscoelastic property consisting of a Kelvin element in series with a spring. The method of Laplace transform is applied to obtain a closed form solution to the problem. Explicit expressions of Mode I and II stress intensity factors are computed together with crack surface opening. Trade-off between the Mode I and II stress intensity factors depends on the nature of material orthotropy.     

Perfect elastic-viscoplastic field at mode Ⅰ dynamic propagating crack-tip

The viscosity of material is considered at propagating crack-tip. Under the assumption that the artificial viscosity coefficient is in inverse proportion to power law of the plastic strain rate, an elastic-viscoplastic asymptotic analysis is carried out for moving crack-tip fields in power-hardening materials under plane-strain condition. A continuous solution is obtained containing no discontinuities. The variations of numerical solution are discussed for mode Ⅰ crack according to each parameter. It is shown that stress and strain both possess exponential singularity. The elasticity, plasticity and viscosity of material at crack-tip only can be matched reasonably under linear-hardening condition. And the tip field contains no elastic unloading zone for mode Ⅰ crack. It approaches the limiting case, crack-tip is under ultra-viscose situation and energy accumulates, crack-tip begins to propagate under different compression situations.     

Asymptotic fields in steady crack growth with linear strain-hardening

The asymptotic stress and velocity fields of a crack propagating steadily and quasi-statically into an elastic-plastic material are presented. The material is characterized by J₂-flow theory with linear strain- hardening. The possibility of reloading on the crack flanks is taken into account. The cases of anti-plane strain (mode III), plane strain (modes I and II), and plane stress (modes I and II) are considered. Numerical results are given for the strength of the singularity and for the distribution of the stress and velocity fields in the plastic loading, elastic unloading and plastic reloading regions, as functions of the strain-hardening parameter. An attempt is made to make a connection with the perfectly-plastic solutions in the limit of vanishing strain-hardening.     

VISCOPLASTIC SOLUTION TO FIELD AT STEADILY PROPAGATING CRACK TIP IN LINEAR- HARDENING MATERIALS

An elastic-viscoplastic constitutive model was adopted to analyze asymptotically the tip-field of moving crack in linear-hardening materials under plane strain condition. Under the assumption that the artificial viscosity coefficient was in inverse proportion to power law of the rate of effective plastic strain, it is obtained that stress and strain both possess power law singularity and the singularity exponent is uniquely determined by the power law exponent of the rate of effective plastic strain. Variations of zoning structure according to each material parameter were discussed by means of numerical computation for the tip-field of mode Ⅱ dynamic propagating crack, which show that the structure of crack tip field is dominated by hardening coefficient rather than viscosity coefficient. The secondary plastic zone can be ignored for weak hardening materials while the secondary plastic zone and the secondary elastic zone both have important influence on crack tip field for strong hardening materials. The dynamic solution approaches to the corresponding quasi-static solution when the crack moving speed goes to zero, and further approaches to the HR (Hui-Riedel) solution when the hardening coefficient is equal to zero.