Stress and strain field near tip of mode III growing crack in materials with creep behavior

Using a proposed constitutive relation for materials with creep behavior, the stress and strain distribution near the tip of a Mode III growing crack is examined. Asymptotic equations of the crack tip field are derived and solved numerically. The stresses remain finite at the crack tip. Obtained qualitatively is the crack tip velocity and the local autonomy of the near tip field solution is discussed.     

A ferroelectric and ferroelastic 3D hexahedral curvilinear finite element

An isoparametric 3D electromechanical hexahedral finite element integrating a 3D phenomenological ferroelectric and ferroelastic constitutive law for domain switching effects is proposed. The model presents two internal variables which are the ferroelectric polarization (related to the electric field) and the ferroelastic strain (related to the mechanical stress). An implicit integration technique of the constitutive equations based on the return-mapping algorithm is developed. The mechanical strain tensor and the electric field vector are expressed in a curvilinear coordinate system in order to handle the transverse isotropy behavior of ferroelectric ceramics. The hexahedral finite element is implemented into the commercial finite element code Abaqus® via the subroutine user element. Some linear (piezoelectric) and non linear (ferroelectric and ferroelastic) benchmarks are considered as validation tests.     

一种新的计算各向异性材料裂纹尖端应力强度因子杂交元法

利用有限元特征分析法研究了平面各向异性材料裂纹端部的奇性应力指数以及应力场和位移场的角分布函数,以此构造了一个新的裂纹尖端单元。文中利用该单元建立了研究裂纹尖端奇性场的杂交应力模型,并结合Hellinger-Reissner变分原理导出应力杂交元方程,建立了求解平面各向异性材料裂纹尖端问题的杂交元计算模型。与四节点单元相结合,由此提出了一种新的求解应力强度因子的杂交元法。最后给出了在平面应力和平面应变下求解裂纹尖端奇性场的算例。算例表明,本文所述方法不仅精度高,而且适应性强。     

Elastostatic crack tip behavior for a rubber-like material

Analyzed in this work is the elastostatic field near a crack tip in a rubber-like material. Asymptotic equations for a crack opened symmetrically about its plane are derived from assumed forms of the strain energy density and constitutive relation that applies to large and finite strain and remains valid even when the strain tends to infinity in the limit. Near field solutions are obtained in regions that decreases and increases in size as the crack tip is approached. Their singular character depends on the constitutive parameters and is evaluated numerically.     

Stress and energy density fields near a blunted crack front in power hardening material

Using Jaumann and Dienes rates of Euler stress in elastic-plastic constitutive equations of finite deformation, plane strain finite element analysis for a compact tension specimen with a blunted crack front is made. The Euler stress, Kirchhoff stress and volume strain energy density near a blunted crack tip are computed. Constitutive relations with different deformation rates affect the the near crack tip solution in a region within an order of magnitude of the crack opening displacement. The results differed from the corresponding solution of deformation plasticity (or nonlinear elasticity) with increasing deformation. They are smaller in a local region of about 2 to 10 times of the crack opening distance.The volume energy density near the crack tip is computed, the stationary values of which determine the locations of extensive yielding and possible sites of crack initiation. It remained nearly constant with increasing deformation. Such a character tends to support the volume energy density criterion as a means for quantifying the ductile fracture behavior of metals.     

The elastostatic plane strain mode I crack tip stress and displacement fields in a generalized linear neo-Hookean elastomer

A general asymptotic plane strain crack tip stress field is constructed for linear versions of neo-Hookean materials, which spans a wide variety of special cases including incompressible Mooney elastomers, the compressible Blatz–Ko elastomer, several cases of the Ogden constitutive law and a new result for a compressible linear neo-Hookean material. The nominal stress field has dominant terms that have a square root singularity with respect to the distance of material points from the crack tip in the undeformed reference configuration. At second order, there is a uniform tension parallel to the crack. The associated displacement field in plane strain at leading order has dependence proportional to the square root of the same coordinate. The relationship between the amplitude of the crack tip singularity (a stress intensity factor) and the plane strain energy release rate is outlined for the general linear material, with simplified relationships presented for notable special cases.     

Fully coupled, multi-axial, symmetric constitutive laws for polycrystalline ferroelectric ceramics

In this paper, a general form for multi-axial constitutive laws for ferroelectric ceramics is constructed. The foundation of the theory is an assumed form for the Helmholtz free energy of the material. Switching surfaces and associated flow rules are postulated in a modified stress and electric field space such that a positive dissipation rate during switching is guaranteed. The resulting tangent moduli relating increments of stress and electric field to increments of strain and electric displacement are symmetric since changes in the linear elastic, dielectric and piezoelectric properties of the material are included in the switching surface. Finally, parameters of the model are determined for two uncoupled cases, namely non-remanent straining ferroelectrics and purely ferroelastic switching, and then for the fully coupled ferroelectric case.     

Numerical simulation of domain switching in multilayer ferroelectric actuators

Micromechanical finite element methods are developed based on a nonlinear constitutive model of ferroelectric polycrystals. Electromechanical behaviors ahead of an internal electrode tip are numerically simulated in multilayer ferroelectric actuators. Around the electrode edge, the nonuniform electric field generates a concentration of stress due to the incompatible strain as well as spontaneous strain. The preferred domain switching enhances the concentration of residual stress and may cause the actuators to crack. An electrically permeable crack emanating from an internal electrode is analyzed. A large scale domain switching zone is found in the vicinity of crack tips. The larger the actuating strain and electric field are, the larger the switching zone will be. The size of switching zone even reaches the scale of crack length with increasing electromechanical loading.     

A finite element model of ferroelastic polycrystals

A finite element model of switching in polycrystalline ferroelastic ceramics is developed. It is assumed that a crystallite switches if the reduction in mechanically driven potential energy of the system exceeds a critical value per unit volume of switching material. Stress induced (i.e. ferroelastic) switching is a change of permanent strain in characteristic crystallographic directions. Martensitic twinning is one example, but the strain response of ferroelectric materials has the same characteristics. The model is suitable for representing ferroelastic systems such as shape memory alloys and as a preliminary model for ferroelectric/ferroelastic materials such as perovskite piezoelectrics. In the simulations, each crystallite is represented by a finite element and the crystallographic principal direction for each crystallite is assigned randomly. Different critical values for the energy barrier to switching are selected to simulate stress vs strain hysteresis loops of a ceramic lead lanthanum zirconate titanate (PLZT) at room temperature. The measured stress versus strain curves of polycrystalline ceramics designated PZT-A and PZT-B are also reproduced by the model.     

平面幂律塑性Ⅰ型裂纹尖端应力场全解

在航空航天、船舶、石油管道和核电等领域,服役结构或部件在长期极端条件下运行,不可避免地会产生裂纹,因此,为研究含裂纹结构的准静态断裂行为,必须了解裂纹尖端附近区域的应力应变场特点.对于幂律材料裂纹构元,研究平面应变和平面应力条件下Ⅰ型裂纹尖端应力场的解析分布.基于能量密度等效和量纲分析,推导了能量密度中值点代表性体积单元(representative volume element, RVE)的等效应力解析方程,并定义其为应力因子,进而针对有限平面应变和平面应力紧凑拉伸(compact tension, CT)试样和单边裂纹弯曲(single edge bend, SEB)试样,以应力因子作为应力特征量,并构造用于表征裂尖等效应力等值线的蝶翅轮廓式和扇贝轮廓式三角特殊函数,提出描述幂律塑性条件下平面I型裂纹尖端应力场的半解析模型.该半解析模型形式简单,对CT和SEB试样的裂尖应力场的预测结果与有限元分析的结果比较表明,两者之间均密切吻合,模型公式可直接用于预测Ⅰ型裂纹尖端应力分布,方便于断裂安全评价和理论发展.     

黏弹性功能梯度材料裂纹问题的有限元方法

针对组分材料体积含量任意分布的黏弹性功能梯度材料裂纹问题建立有限元分析途径. 通过Laplace变换,将黏弹性问题转化到象空间中求解,基于反映材料非均匀的梯度单元和裂纹尖端奇异特性的奇异单元计算象空间中的位移、应力和应变场,应用虚拟裂纹闭合方法得到应变能释放率,分别由应力和应变能释放率确定应力强度因子. 给出这些断裂参量在物理空间和象空间之间的对应关系,由数值逆变换求出其在物理空间的相应值. 文中分析两端均匀受拉的黏弹性边裂纹板条,首先针对松弛模量表示为空间函数和时间函数乘积的特殊梯度材料进行计算,结合对应原理验证方法的有效性. 然后分析组分材料体积含量具有任意梯度分布的情形,由Mori-Tanaka方法预测象空间中的等效松弛模量. 计算结果表明,蠕变加载条件下,应变能释放率随时间增加,其增大程度与黏弹性组分材料体积含量相关. 由于梯度材料的非均匀黏弹性性质,产生应力重新分布,导致应力强度因子随时间变化,其变化范围与组分材料的体积含量分布方式有关.     

Finite deformation analysis of crack tip fields in plastically compressible hardening–softening–hardening solids

Crack tip fields are calculated under plane strain small scale yielding conditions. The material is characterized by a finite strain elastic–viscoplastic constitutive relation with various hardening–softening–hardening hardness functions. Both plastically compressible and plastically incompressible solids are considered. Displacements corresponding to the isotropic linear elastic mode I crack field are prescribed on a remote boundary. The initial crack is taken to be a semi-circular notch and symmetry about the crack plane is imposed. Plastic compressibility is found to give an increased crack opening displacement for a given value of the applied loading. The plastic zone size and shape are found to depend on the plastic compressibility, but not much on whether material softening occurs near the crack tip.On the other hand, the near crack tip stress and deformation fields depend sensitively on whether or not material softening occurs. The combination of plastic compressibility and softening(or softening–hardening) has a particularly strong effect on the near crack tip stress and deformation fields.     

Effect of crack-tip shape on the near-tip field in glassy polymer

The physical nature of a crack tip is not absolutely sharp but blunt with finite curvature. In this paper, the effects of crack-tip shape on the stress and deformation fields ahead of blunted cracks in glassy polymers are numerically investigated under Mode I loading and small scale yielding conditions. An elastic–viscoplastic constitutive model accounting for the strain softening upon yield and then the subsequently strain hardening is adopted and two typical glassy polymers, one with strain hardening and the other with strain softening–rehardening are considered in analysis. It is shown that the profile of crack tip has obvious effect on the near-tip plastic field. The size of near-tip plastic zone reduces with the increase of curvature radius of crack tip, while the plastic strain rate and the stresses near crack tip enhance obviously for two typical polymers. Also, the plastic energy dissipation behavior near cracks with different curvatures is discussed for both materials.     

铁中Ⅰ型裂纹裂尖场与裂尖形变机制分析

应用有限元及连续介质力学计算了平面应交条件下体心立方铁中不同取向裂纹的裂尖应力场和弹性应变能密度分布,从宏微观相结合角度分析了裂尖场分布与裂尖微结构演化的相互关联.指出裂尖塑性变形的具体形式与裂尖滑移面上分切应力的大小密切相关,并从能量角度解释了裂尖相变产生的原因,最后结合相关分子动力学研究成果探讨了裂尖奇异性区域.     

NiTi形状记忆合金裂纹尖端热力耦合行为研究

本文对NiTi形状记忆合金I型裂纹尖端热力耦合行为进行了数值仿真分析和实验验证。建立了包含相变和热力耦合的本构模型,通过有限元计算得到了裂纹尖端附近的纵向应变、马氏体体积分数和温度场分布,依据马氏体相变情况对裂纹尖端有效应力强度因子进行了修正,揭示了加载速率对形状记忆合金裂纹尖端有效应力强度影子的影响规律。参数研究表明,随着加载频率的增加,裂纹尖端附近温度逐渐升高,马氏体相变区域逐渐缩小,有效应力强度因子呈下降趋势,形状记忆合金表现出增韧效应,有助于减缓裂纹扩展。本研究结果对于揭示热力耦合作用下超弹性形状记忆合金疲劳裂纹扩展规律具有重要参考意义。     

裂尖大应变细观断裂研究

本文用反映空穴形核成长的Gurson本构方程来描述裂尖区域材料在大应变情形下的力学特性，并进一步考虑了空穴演变对材料杨氏模量的影响。文中用上述本构方程分别结合弹塑性大应变有限元方法对平面应变I型裂纹问题作了计算，分析了裂尖应力分布、裂尖形状变化和裂尖空穴演变过程，并与用Prandtl-Reuss本构方程教育处的结果作了比较。     

Analyses on micro-fracture of crack tip under large deformation conditions

In the present paper, Gurson's constitutive equation, which takes into account the development of voids, is used to study the behaviour of the material in the region near crack tip. Furthermore, the effect of void development on Young's modulus, which was not considered by Gurson, is taken into consideration. The analyses on void development, on stress distribution near crack tip, and on the variance of COD for the plane strain mode I problem are carried out with the large elastic-plastic deformation finite element method. The results are compared with those estimated from the Prandtl-Reuss constitutive equation.     

A viscoplastic study of crack-tip deformation and crack growth in a nickel-based superalloy at elevated temperature

Viscoplastic crack-tip deformation behaviour in a nickel-based superalloy at elevated temperature has been studied for both stationary and growing cracks in a compact tension (CT) specimen using the finite element method. The material behaviour was described by a unified viscoplastic constitutive model with non-linear kinematic and isotropic hardening rules, and implemented in the finite element software ABAQUS via a user-defined material subroutine (UMAT). Finite element analyses for stationary cracks showed distinctive strain ratchetting behaviour near the crack tip at selected load ratios, leading to progressive accumulation of tensile strain normal to the crack-growth plane. Results also showed that low frequencies and superimposed hold periods at peak loads significantly enhanced strain accumulation at crack tip. Finite element simulation of crack growth was carried out under a constant ΔK-controlled loading condition, again ratchetting was observed ahead of the crack tip, similar to that for stationary cracks.A crack-growth criterion based on strain accumulation is proposed where a crack is assumed to grow when the accumulated strain ahead of the crack tip reaches a critical value over a characteristic distance. The criterion has been utilized in the prediction of crack-growth rates in a CT specimen at selected loading ranges, frequencies and dwell periods, and the predictions were compared with the experimental results.     

裂尖超钝化:实验、理论与数值模拟

本文报道了在改性聚丙烯裂尖平面应变段发生超钝化的实验发现。利用环向冷拉模型,可说明裂尖超钝化的形成取决于冷拉传播速率与加载率之比。裂尖超钝化可使材料韧性,尤其是冲击韧性的改善,达到一个新的水平。本文在超弹性-粘塑性各向异性损伤的有限变形本构模型下进行了详尽的数值计算。模拟结果成功地再现了裂尖超钝化的基本特征。     

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.