Nucleation and growth of domains near crack tips in single crystal ferroelectrics

The nucleation and growth of domains is investigated near a stationary crack tip in a single crystal of ferroelectric material. The phase-field approach, applying the material polarization as the order parameter, is used as the theoretical modeling framework and the finite element method is used for the numerical solution technique. The electromechanical form of the J-integral is appropriately modified to account for the polarization gradient energy terms, and analyzed to illustrate the amount of shielding, or lack thereof, due to domain switching at the crack tip. It is shown that the nucleation of domains near the crack tip due to applied electric field is affected by applied stress. However, the crack-tip energy release rate can change significantly between the instant of domain nucleation and the final equilibrium domain configuration. Implications of these results for ferroelectric single crystal fracture criteria are discussed.     

Mechanical and thermal transitions in morphotropic PZN-pT and PMN-PT single crystals and their implication for sound projectors

Isothermal compression experiments on multidomain [001] oriented and poled ferroelectric rhombohedral PZN-0.07PT and PMN-0.30PT single crystals revealed elastic instabilities corresponding to zero field ferroelectric-ferroelectric phase transition under mechanical compression. The application of an appropriate dc bias field doubled the stability range of the ferroelectric rhombohedral state under uniaxial compression for both crystals and maintained a linear elastic response. Young's modulus as derived from the quasistatic, zero field stress-strain linear response agreed well with that derived from small signal resonance for the ferroelectric rhombohedral FR state of both PZN-PT and PMN-PT. Elastic compliances s(E)33 as determined from high temperature resonance revealed a monotonically decreasing Young's modulus as a function of temperature in the ferroelectric rhombohedral state with a sudden stiffening near the ferroelectric rhombohedral (FR)-ferroelectric tetragonal (FT) transition. The reversible ferroelectric-ferroelectric transition of morphotropic PZN-PT and PMN-PT single crystals as accessed by mechanical compression is discussed in terms of strain calculations from Devonshire's theory, domain unfolding, and morphotropic phase boundary shift with mechanical stress. The mechanically-induced and thermally-induced ferroelectric-ferroelectric transition trajectories are discussed in terms of the Devonshire theory. Implications of these observations for sound projectors are discussed. A single crystal tonpilz projector fabricated into a 16-element array and a segmented cylinder transducer demonstrated the outstanding capabilities of single crystals to achieve compact, broadband, and high-source level projectors when compared to conventional lead zirconate-titanate PZT8 projectors.     

Constitutive law for ferroelectric and ferroelastic single crystals: a micromechanical approach

This paper examines the switching process occuring in ferroelectric and ferroelastic single crystals under electro-mechanical loadings. Ferroelectrics undergoing a cubic to tetragonal phase transition are considered. The single crystal energy has three origins: elastic, electric and the incompatibilities of the spontaneous strain and electric displacement fieds between domains. The stress and electric fields fluctuate and present jumps at the domain walls. As a consequence, they induce electro-elastic interaction energy. Thus, it involves dissipation that the present work aims to capture through a micromechanical approach.     

Numerical simulation of intergranular and transgranular crack propagation in ferroelectric polycrystals

We present a phase-field model to simulate intergranular and transgranular crack propagation in ferroelectric polycrystals. The proposed model couples three phase-fields describing (1) the polycrystalline structure, (2) the location of the cracks, and (3) the ferroelectric domain microstructure. Different polycrystalline microstructures are obtained from computer simulations of grain growth. Then, a phase-field model for fracture in ferroelectric single-crystals is extended to polycrystals by incorporating the differential fracture toughness of the bulk and the grain boundaries, and the different crystal orientations of the grains. Our simulation results show intergranular crack propagation in fine-grain microstructures, while transgranular crack propagation is observed in coarse grains. Crack deflection is shown as the main toughening mechanism in the fine-grain structure. Due to the ferroelectric domain switching mechanism, noticeable fracture toughness enhancement is also obtained for transgranular crack propagation. These observations agree with experiment.     

Two collinear unequal cracks in a poled piezoelectric plane: Mode I case solved by a new approach of real fundamental solutions

The purpose of the present work is to study the problem of two collinear unequal cracks in a piezoelectric plane under mode I electromechanical loadings via a new approach. For the first time, real fundamental solutions are derived for in-plane piezoelectric governing equations. The cracks are simulated by continuously distributed generalized dislocations and Cauchy singular integral equations are established from the solution of a generalized point dislocation. Both the theorectical derivation and numerical computations are validated by the exact solution in a special case. Parametric studies are conducted to reveal the effects of crack space, crack length, electric loading and remanent electric displacement on energy release rate. It is found that negative electric displacement loading can decrease both the total energy release rate (TERR) and the mechanical strain energy release rate (MSERR), implying that it has a shielding effect on cracks definitely. Positive electric displacement loading can enhance MSERR, but meanwhile it can enhance or reduce TERR depending on the magnitude of the electric loading factor. The effect of a remanent electric displacement along the poling direction is equivalent to that of a positive electric field loading and should be considered in engineering design.     

Experimental evidence of enhanced ferroelectricity in Ca doped BiFeO3

Calcium (Ca)-doped bismuth ferrite (BiFeO₃) thin films prepared by using the polymeric precursor method (PPM) were characterized by X-ray diffraction (XRD), field emission gun scanning electron microscopy (FEG-SEM), transmission electron microscopy (TEM), polarization and piezoelectric measurements. Structural studies by XRD and TEM reveal the co-existence of distorted rhombohedral and tetragonal phases in the highest doped BiFeO₃ where enhanced ferroelectric and piezoelectric properties are produced by internal strain. Resistive switching is observed in BFO and Ca-doped BFO which are affected by the barrier contact and work function of multiferroic materials and Pt electrodes. A high coercive field in the hysteresis loop is observed for the BiFeO₃ film. Piezoelectric properties are improved in the highest Ca-doped sample due to changes in the crystal structure of BFO for a primitive cubic perovskite lattice with four-fold symmetry and a large tetragonal distortion within the crystal domain. This observation introduces magnetoelectronics at room temperature by combining electronic conduction with electric and magnetic degrees of freedom which are already present in the multiferroic BiFeO₃.     

Strain localization at the crack tip in single crystal CT specimens under monotonous loading: 3D Finite Element analyses and application to nickel-base superalloys

Three-dimensional Finite Element simulations of mode I crack tip fields in Compact Tension specimens are presented for elastic ideally-plastic F.C.C. single crystals. The computations are carried out within the framework of classical continuum crystal plasticity for three crack orientations: (001)[110],(110)[001] and (001)[100]. The attention is drawn on the strong differences between the plastic strain field obtained at the free surface and in the mid-section of the specimens. The results are compared, on the one hand, to analytical solutions for stationary cracks in single crystals under plane strain conditions and, on the other hand, to experimental tests on a single crystal nickel-based superalloy at room temperature. For this material, both octahedral and cube slip must be taken into account. A good agreement between experimental observations and numerical results is found in the structure of the strain localization bands observed at the free surface of (110)[001] cracked specimens. In particular, the evidence of kink banding near the crack tip is provided, confirmed by EBSD orientation mapping. The measured values of local lattice rotation are in agreement with the Finite Element prediction.     

A localised experimental–numerical technique for determining mixed mode strain energy release rates

Composite structures are being used increasingly in the aerospace industry due to their superior specific stiffness and strength. One key issue associated with such structures is delamination, and how to effectively predict this. A new method which is derived from localised test displacement data is presented to determine the mixed mode strain energy release rates of layered structures with a pre-existing crack. Images taken during experimentation of the vicinity of the crack tip are analysed at low load and high load to determine the displacement changes across the load variation. These displacements are applied as boundary conditions to a simple local numerical model including a constraint at the crack tip. The forces and displacements at the crack tip are taken as output data and combined with the Virtual Crack Closure Technique to predict strain energy release rates. Initial validation of the localised experimental–numerical technique (LENT) shows that applying experimental data to a numerical model does give reasonable agreement thus far in the established trends, and hence LENT is promising for use in determining mixed mode strain energy release rates and mode mixity ratios.     

A localised experimental–numerical technique for determining mixed mode strain energy release rates

Composite structures are being used increasingly in the aerospace industry due to their superior specific stiffness and strength. One key issue associated with such structures is delamination, and how to effectively predict this. A new method which is derived from localised test displacement data is presented to determine the mixed mode strain energy release rates of layered structures with a pre-existing crack. Images taken during experimentation of the vicinity of the crack tip are analysed at low load and high load to determine the displacement changes across the load variation. These displacements are applied as boundary conditions to a simple local numerical model including a constraint at the crack tip. The forces and displacements at the crack tip are taken as output data and combined with the Virtual Crack Closure Technique to predict strain energy release rates. Initial validation of the localised experimental–numerical technique (LENT) shows that applying experimental data to a numerical model does give reasonable agreement thus far in the established trends, and hence LENT is promising for use in determining mixed mode strain energy release rates and mode mixity ratios.     

Physical interpretation of fracture-toughening mechanisms

The basic idea behind the toughening of materials by the introduction of energy-absorbing or dissipating artefacts is critically re-examined. It is shown that energy dissipation by plastic deformation or other dissipative processes at the tip of a growing crack does not contribute to increasing the effective surface energy or the crack resistance of the material. Erroneous interpretations of toughening by the presence of fibres or by phase transformations occuring at the tip of a growing crack are discussed. It is argued that all processes which dissipate energy at the crack tip produce crack shielding and that this effect must be an important contribution to toughening. It is concluded that most of the features and properties embodied in methods of toughening can be explained by shielding effects and that the increase in toughness is due to a reduction in the local value of the crack extension force, or its equivalent stress intensity factor, and not to an increase in energy dissipated.     

A treatment of interfacial cracks in the presence of friction

Frictional sliding on interface crack surfaces results in weak crack tip stress singularity and zero strain energy release rate. A fracture criterion based on finite extension strain energy release rate, is proposed to capture the intrinsic fracture toughness. The finite extension strain energy release rate is shown to represent the magnitude of the singular stress field. Numerical simulations of a center crack in a bimaterial infinite medium under remote shear as well as fiber pull-out and push-out in composite materials are presented to illustrate the frictional effect in both small and large scale contacts near the crack tip.     

基于扩展有限元的陶瓷复合材料多重增韧机制

为了提高陶瓷材料的断裂韧性和可靠度,改善材料抵御破坏的能力,将优化的多重增韧机制应用到氧化铝基陶瓷材料的开发中。相变增韧机制可以耗散部分能量,降低裂纹尖端处的应力集中程度,阻止或延缓裂纹扩展速率。当增强相分布较为合理、材料的致密度较高时,裂纹偏转与桥接增韧机制可以有效地削弱裂纹扩展动力,提高材料的断裂韧性。利用扩展有限元（X-FEM）手段讨论了裂纹扩展问题,为分析陶瓷复合材料的多重增韧机制提供了新思路。     

聚合物梯度材料黏弹性断裂的双控制参数

针对组分材料体积分数任意分布的聚合物功能梯度材料,研究其在蠕变加载条件下Ⅰ型裂纹应力强度因子（SIFs）和应变能释放率的时间相依特征。由Mori-Tanaka方法预测等效松弛模量,在Laplace变换域中采用梯度有限元法和虚拟裂纹闭合方法计算断裂参数,由数值逆变换得到物理空间的对应量。分析边裂纹平行于梯度方向的聚合物功能梯度板条,分别考虑均匀拉伸和三点弯曲蠕变加载。结果表明,聚合物梯度材料应变能释放率随时间增加,其增大的程度与黏弹性组分材料体积分数正相关;材料的非均匀黏弹性性质产生应力重新分布,导致裂纹尖端应力场强度随时间变化,当裂纹位于黏弹性材料含量较低的一边时,应力强度因子随时间增加,反之,随时间减小。而且,材料的应力强度因子与时间相依的变化范围和体积分数分布以及加载方式有关,当体积分数接近线性分布时,变化最明显,三点弯曲比均匀拉伸的变化大。SIFs随时间的延长增加或减小、加剧或减轻裂纹尖端部位的“衰坏”,表明黏弹性功能梯度裂纹体的延迟失稳需要联合采用应力强度因子与应变能释放率作为双控制参数。     

Simulating small crack growth behaviour using crystal plasticity theory and finite element analysis

Predictions of small crack growth under cyclic loading in aluminium alloy 7075 are performed using finite element analysis (FEA), and results are compared with published experimental data. A double‐slip crystal plasticity model is implemented within the analyses to enable the anisotropic nature of individual grains to be approximated. Small edge‐cracks in a single grain with a starting length of 6 μm are incrementally grown following a node‐release scheme. Crack‐tip opening displacements (CTOD) and crack opening stresses are calculated during the simulated crack growth, and da/dN against ΔK diagrams are computed. Interactions between the crack tip and a grain boundary are also considered. The computations are shown to accurately capture the magnitude and the variability normally observed in small crack fatigue data.     

Effects of remanent field on an elliptical flaw and a crack in a poled piezoelectric ceramic

Commonly used piezoelectric ceramics such as PZT and PLZT are polarized ferroelectric polycrystals. After poling, remanent strains and a remanent polarization exist in a ceramic material. Remanent field can affect the electroelastic field and consequently plays a critical role in fracture of poled ceramics. Based on a linear constitutive law, the electroelastic field and the energy release rate of an elliptical cavity (or a crack) in a poled piezoelectric are re-examined in this study by including the effects of remanent field. It is noted that the remanent field generally has a minor effect on the stress field and a pronounced effect on the electric field at the apex of the major axis of an elliptical flaw. When the permittivity of the cavity is small, the effect of remanent polarization is similar to that of a very strong electric field applied along the poling direction. However, for the case of a conducting flaw, the remanent field does not influence the electroelastic field and energy release rate. Energy release rate of a flaw in a poled ferroelectric ceramic with and without the remanent polarization is generally different.     

R-curve behaviour of PZT ceramics near the morphotropic phase boundary

The mechanical failure of PZT ceramics was characterized by measuring R-curves for compositions near and at the morphotropic phase boundary (MPB) where tetragonal and rhombohedral phases coexist in equal quantities. The R-curve behaviours (an increasing fracture toughness with crack extension) were identified by indentation-fracture testing and they were analysed to determine the key parameters. The fracture toughness of the PZT ceramics consisted of three different terms, representing particular microstructural processes in front of advancing cracks, that is, intrinsic cleavage, 90° domain switching and microcracking. Their relative contributions to an overall crack-extension resistance varied with the length of the advancing crack and, more importantly, with the compositions of the PZT. In the compositional range where the tetragonal phase was dominant, the R-curves were determined by domain switching and microcracking. However, the compositional dependency of the fracture toughness was due to the microcracking mechanism. On the other hand, in regions rich in rhombohedral phases, the R-curves were essentially determined by domain switching in the crack-tip area. The R-curves characterized by the domain-switching mechanism were insensitive to the compositional variation near the MPB. Our results also demonstrated that R-curve analysis could be used to probe further into the microstructural responses of materials in front of advancing cracks and to quantify them particularly in systems like PZT where several different toughening processes compete with each other.     

Stress intensity factor analyses of interface cracks between dissimilar anisotropic materials using the finite element method

Delamination along an interface between dissimilar materials is the primary cause of failure in microstructures like electronic packages, micro-electro-mechanical systems (MEMS), and so on. Fracture mechanics is a powerful tool for the evaluation of delamination. However, many materials used in microstructures such as composite materials and single crystals are anisotropic materials. Stress intensity factors of an interface crack between dissimilar anisotropic materials, which were proposed by Hwu, are useful for evaluating the reliability of microstructures. However, numerical methods that can analyze the stress intensity factors of an interface crack between anisotropic materials have not been developed. We propose herein a new numerical method for the analysis of an interface crack between dissimilar anisotropic materials. The stress intensity factors of an interface crack are based on the generalized plane strain condition. The energy release rate is obtained by the virtual crack extension method in conjunction with the finite element method for the generalized plane strain condition. The energy release rate is separated into individual modes of the stress intensity factors K_I, K_II, and K_III, using the principal of superposition. The target problem to be solved is superposed on the asymptotic solution of displacement in the vicinity of an interface crack tip, which is described using the Stroh formalism. Analyses of the stress intensity factors of center interface cracks between semi-infinite dissimilar anisotropic media subjected to concentrated self-balanced loads on the center of crack surfaces and to uniform loads are demonstrated. The present method accurately provides mode-separated stress intensity factors using relatively coarse meshes for the finite element method.     

Modelling ferroelastic domain switching at a stationary crack tip in a single crystal with account of transformation stresses due to domain reorientation

The present paper deals with ferroelastic domain switching around a stationary crack tip in a single crystal for potential applications in the context of crack toughening. The main focus is directed towards the effect of transformation stresses on the domain. The transformation stresses themselves are coupled to the extent and the shape of the domain. Therefore, they and the domain are unknown a-priori and have to be determined simultaneously. The model is applied to a crack in barium titanate and indicates a major effect of the transformation stresses on the observed needle-like domain shapes. Comparison with experimental observations shows a reasonable agreement although essential properties of barium titanate as ferroelectricity and piezoelectricity are neglected yet.     

Effect of compositional variations on electrical properties in phase switching (Pb,La)(Zr,Ti,Sn)O3 thin and thick films

The composition-dependent electrical properties in (Pb,La)(Zr,Ti,Sn)O₃ antiferroelectric-ferroelectric phase switching thin and thick films have been systematically studied and compared with bulk ceramics. The films were deposited on Pt-buffered silicon substrates by a sol-gel method. The results show that the dependence of low-field dielectric properties on compositions in the films is similar to that in bulk ceramics but the variation of high field properties (polarization or hysteresis loops) is quite different, which may be attributed to the special mechanical boundary condition of the films. While all the films with compositions in the antiferroelectric tetragonal region in the phase diagram demonstrate the existence of remanent polarization in the hysteresis loops, the films with zero remanent polarization can be obtained in the antiferroelectric orthorhombic region. This is because the films are under high tensile stress due to the thermal mismatch between the film and substrate, which tends to stabilize the ferroelectric phase and causes the retention of ferroelectric phase for the films in the antiferroelectric tetragonal region because of their relatively small free energy difference between the antiferroelectric phase and ferroelectric phase.     

Cyclic crystal plasticity analyses of stationary, microstructurally small surface cracks in ductile single phase polycrystals

ABSTRACT This paper explores the effects of microstructural heterogeneity on the cyclic crack tip opening and sliding displacements for stationary, microstructurally small transgranular surface cracks in a single phase metallic polycrystal using planar double slip crystal plasticity computations. Crack tip displacements are examined under plane strain conditions for stationary cracks of different lengths relative to grain size as a function of the applied nominal strain amplitude for tension-compression and cyclic shear. Nominal strain amplitudes range from well below to slightly above the nominal cyclic yield strength for each type of loading condition. Results indicate the complex nature of the crack tip sliding and opening displacements as functions of nominal strain amplitude and orientation of the nearest neighbour grains, the influence of the free surface in promoting the cyclic opening displacement even for cracks in the first surface grain, the rather restricted limits of applicability of linear elastic fracture mechanics, and very interesting crack tip plasticity effects which include crack tip displacement ratcheting or progressive accumulation, even for completely reversed, proportional applied loading. Results are compared for cases with and without crack face friction.