On the crack-tip stress singularity of interfacial cracks in transversely isotropic piezoelectric bimaterials

In this paper, characteristics of the interface crack-tip stress and electric displacement fields in transversely isotropic piezoelectric bimaterials are studied. The authors have proven, within the framework of the generalized Stroh formalism for piezoelectric bimaterials, that there is no coexistence of the parameters (oscillating) and κ (non-oscillating) in the interface crack-tip generalized stress field for all transversely isotropic piezoelectric bimaterials. This leads to the classification of piezoelectric bimaterials into one group that exhibits the oscillating property in the interface crack-tip generalized stress field and the other that does not. Fifteen (15) pair-combinations of six (6) piezoelectric materials PZT-4, PZT-5H, PZT-6B, PZT-7A, P-7, and BaTiO₃, which are commonly used in practice, are numerically analyzed in this study, and the results backup the above theoretical conclusions. Moreover, the associated eigenvectors for such material systems (with either =0 or κ=0) are also obtained numerically, and the result show that there still exist four linear independent associate eigenvectors for each bimaterial.     

On fatigue crack growth in ductile materials by crack-tip blunting

One of the basic mechanisms for fatigue crack growth in ductile metals is that depending on crack-tip blunting under tensile loads and re-sharpening of the crack-tip during unloading. In a standard numerical analysis accounting for finite strains it is not possible to follow this process during many cycles, as severe mesh distortion at the crack-tip results from the huge geometry changes developing during the cyclic plastic straining. In the present numerical studies, based on an elastic-perfectly plastic material model, crack growth computations are continued up to 200 full cycles by using remeshing at several stages of the plastic deformation. Three different values of the load ratio R=K_min/K_max are considered. It is shown that the crack-tip opening displacement, CTOD, typically undergoes a transient behaviour, with no crack closure during many cycles, before a steady-state cycling with crack closure at the tip starts to gradually develop.     

Stress distribution about a slowly growing crack determined by photoelastic-coating method

An experimental stress-analysis technique using a birefringent coating is reported for determining the stress distribution about a slowly growing crack. The maximum error of the test method for a large strain gradient is found to be less than 10 percent. For a plate with an internal crack, the experimentally determined stress distribution compares favorably with two numerical solutions. Comparison of stresses about an internal or double-edge crack to those about a single-edge crack indicates that the isochromatics bend over to about 45 deg with the plane of the crack in the former and are inclined at about 60 deg in the latter. Also, the stresses for a single-crack tip vary as the inverse square root of the radius, while the stresses for a double-crack tip follow anr ^?1/4 law more closely.     

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。     

Experimental investigation of GH singularity field of the growing crack-tip for different ductile metals

The displacementsu _x ,u _y inx, y directions at growing crack tip of the specimens with double edge cracks for four different alumium alloys and two coppers are measured by using moire method and optical spatial filtering technique. From experimental displacement fields, the displacement singularity fields are obtained and compared with GH theoretical field. Unknown constantsA and _yo in theoretical solution are determined from experimental data. The theoretical singularity field thus compared is given for plane-stress, mode-I, strain-hardening materials. The error in both the experimental and theoretical evaluations is within ± 10%. The experiments show that there exists dominant singularity region around a growing crack-tip. In the experiments, the strain hardening indexn amounts from 3.158 to 14. The shape of this dominant region ranges from butterfly wing to oblate or circular shape. The size and shape of GH dominant region depend on the material property, the specimen geometry and loading type. Inside GH-field, there is a 3-D deformed damage zone, where no GH singularity exists. Very near to the crack-tip, there is a fracture process zone.The project is supported by the National Science Foundation of China.     

Analysis of mode-1 crack-tip stress patterns in orthotropic materials

Summary A classification scheme of normal opening mode crack-tip stress patterns for static cracks in orthotropic (composite) materials is presented. Direct application within the framework of coating stress analysis technique as applied to unidirectional or multidirectional glass fibre reinforced epoxies weakened by a sharp crack and subjected to uniaxial and biaxial stress fields is developed. The mutual influence of changing parameters such as ratio of Young's moduli, prestressing parallel to the crack etc on the geometrical configuration of the iso-stress-lines can easily be identified from orthotropic stress loop diagrams.

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Analyse von Mode-1 Riss-Spitzen-Isospannungs-Feldern in orthotropen Materialien

Übersicht Dieser Beitrag umfaßt ein Klassifikationsschema für statische mode-1 Rißspitzenisospannungs-felder in orthotropen Kompositmaterialien. Die direkte Anwendung auf durch Risse geschwächte ein- oder mehrfachorientierte glasfaserverstärkte Kunststoffe im Rahmen des spannungsoptischen Oberflächenschichtverfahrens ist gegeben. Der Wechselwirkungseinfluß variabler Parameter wie z. B. Elastizitäts-moduln, homogene Spannung parallel zum Riß usw. auf die geometrische Konfiguration der Isospannungslinien ist leicht aus einem eigens entwickelten orthotropen Spannungslinien-Diagramm abzulesen.

     

Asymptotic crack-tip fields for dynamic crack growth in compressive power-law hardening materials

The effect of material compressibility on the stress and strain fields for a mode-I crack propagating steadily in a power-law hardening material is investigated under plane strain conditions. The plastic deformation of materials is characterized by the J₂ flow theory within the framework of isotropic hardening and infinitesimal displacement gradient. The asymptotic solutions developed by the present authors [Zhu, X.K., Hwang K.C., 2002. Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials. International Journal of Fracture 115, 323–342] for incompressible hardening materials are extended in this work to the compressible hardening materials. The results show that all stresses, strains, and particle velocities in the asymptotic fields are fully continuous and bounded without elastic unloading near the dynamic crack tip. The stress field contains two free parameters σ_eq0 and s₃₃₀ that cannot be determined in the asymptotic analysis, and can be determined from the full-field solutions. For the given values of σ_eq0 and s₃₃₀, all field quantities around the crack tip are determined through numerical integration, and then the effects of the hardening exponent n, the Poisson ratio ν, and the crack growth speed M on the asymptotic fields are studied. The approximate behaviors of the proposed solutions are discussed in the limit of ν → 0.5 or n → ∞.     

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.     

Characterization of the post-necking strain hardening behavior using the virtual fields method

The present study aims at characterizing the post-necking strain hardening behavior of three sheet metals having different hardening behavior. Standard tensile tests were performed on sheet metal specimens up to fracture and heterogeneous logarithmic strain fields were obtained from a digital image correlation technique. Then, an appropriate elasto-plastic constitutive model was chosen. Von Mises yield criterion under plane stress and isotropic hardening law were considered to retrieve the relationship between stress and strain. The virtual fields method (VFM) was adopted as an inverse method to determine the constitutive parameters by calculating the stress fields from the heterogeneous strain fields. The results show that the choice of a hardening law which can describe the hardening behavior accurately is important to derive the true stress–strain curve. Finally, post-necking hardening behavior was successfully characterized up to the initial stage of localized necking using the VFM with Swift and modified Voce laws.     

Finite deformation analysis of crack-tip opening in elastic-plastic materials and implications for fracture

Analyses of the stress and strain fields around smoothly-blunting crack tips in both non-hardening and hardening elastic-plastic materials, under contained plane-strain yielding and subject to mode I opening loads, have been carried out by use of a finite element method suitably formulated to admit large geometry changes. The results include the crack-tip shape and near-tip deformation field, and the crack-tip opening displacement has been related to a parameter of the applied load, the J-integral. The hydrostatic stresses near the crack tip are limited due to the lack of constraint on the blunted tip, limiting achievable stress levels except in a very small region around the crack tip in power-law hardening materials. The J-integral is found to be path-independent except very close to the crack tip in the region affected by the blunted tip. Models for fracture are discussed in the light of these results including one based on the growth of voids. The rate of void-growth near the tip in hardening materials seems to be little different from the rate in non-hardening ones when measured in terms of crack-tip opening displacement, which leads to a prediction of higher toughness in hardening materials. It is suggested that improvement of this model would follow from better understanding of void-void and void-crack coalescence and void nucleation, and some criteria and models for these effects are discussed. The implications of the finite element results for fracture criteria based on critical stress or strain, or both, is discussed with respect to transition of fracture mode and the angle of initial crack-growth. Localization of flow is discussed as a possible fracture model and as a model for void-crack coalescence.     

Transient dynamic fracture analysis by an extended meshfree method with different crack-tip enrichments

Investigation of transient dynamic stress intensity factors (DSIFs) of two-dimensional fracture problems of isotropic solids and orthotropic composites by an extended meshfree method is described. We adopt the recently developed extended meshfree radial point interpolation method (X-RPIM), which combines either the standard branch functions or the new linear ramp function associated with Heaviside functions to capture crack-tip behaviors. It is the first time the linear ramp function integrating into meshfree X-RPIM has been presented in a dynamical fracture context. We are particularly interested in exploring insight into the behaviors of DSIFs under dynamic impact loadings (e.g., step, blast and sine loading types) using our meshfree method. For some of these problems numerical examples have been performed using the new ramp functions, and the obtained results of DSIFs have also been compared with those using the standard enrichment functions under which the two schemes have the same setting. In each case it is found that the numerical solutions delivered using the X-RPIM associated with the ramp enrichments are in good agreement with those with the standard functions. The paper first describes formulations and then provides verification of our developed approach through a series of numerical examples in transient dynamic fracture for both solids and orthotropic composites. Illustration of scattered elastic stress waves propagating in the cracked body is depicted to take an insight look at the behavior of dynamic response.     

Characterization of material mechanical properties using strain correlation method combined with virtual fields method

Acta Mechanica Sinica - The isotropic and anisotropic behaviors are considered as the important formats of the constitutive behaviors, and can also be called the global properties. To improve the...     

The application of Moire interferometry in the measurement of displacement field and strain field at notch-tip and crack-tip

This paper presents the application of Moire interferometry in measuring the displacement and strain field at notch-tip and crack-tip before and after crack propagation. The experiment is carried out using a three point bending beam with a notch. TheN _x andN _y fringe patterns representing displacement field, and the ΔN _x/Δx and ΔN _γ/Δγ fringe patterns representing the strain field are obtained. The sensitivity of the measured displacement is 0.417 μm per fringe order. The displacement and strain distribution along the sectionx=0 have been worked out according toN _x andN _γ fringe patterns. The project supported by Chinese Academy of Sciences and National Natural Science Foundation of China     

Logarithmic singularity of an elastic composite wedge subjected to out-of-the-plane extensional strain

When an elastic composite wedge is not under a plane strain deformation, an out-of-the-plane extensional strain exists. The singularity analysis for the stresses at the apex of the composite wedge reduces to a system of non-homogeneous linear equations. When the composite wedge consists of two anisotropic elastic materials, it is shown that the stresses have the (ln r) term for all combinations of wedge angles with few exceptions. The same is true when the materials are isotropic except that the (ln r) term may appear in the form of r(ln r) in the displacements only. For these isotropic composite wedges therefore the stresses are bounded, though not continuous, at the apex. However, there are isotropic composite wedges for which the stress singularity is logarithmic. Conditions are given for isotropic composite wedges for which the stresses are (a) uniform, (b) non-uniform but bounded and (c) logarithmic. Unlike the r^−λ singularity, the existence of the (ln r) term does not depend on the complete boundary conditions.     

Description of anisotropy of isotropic materials caused by plastic strain

Many materials are polycrystalline media or hardened mechanical mixtures of complicated structure. For small elastic strains, they behave as isotropic media. But under stresses exceeding the elastic limit, these materials exhibit effects related to strain anisotropy. The problem of quantitatively estimating this phenomenon still remains open. In the present paper, without any assumptions, we reduce the nonlinear Novozhilov equations to a form typical of the constitutive relations for orthotropic media. We obtain expressions for generalized nonlinear elasticity characteristics by choosing a specific expression for the strain potential. We derive working formulas for calculating the elasticity coefficients in the principal directions and of all coefficients of the transverse strains and shear moduli in the principal planes of elastic symmetry. We describe methods for determining them from the results of extension-compression tests. We also analyze several results of tests published on this subject. We show that the effect related to the anomalous behavior of the transverse strain coefficient, which is observed both under compression and under tension, can be explained completely if the fact that the plastic strain is accompanied with strain anisotropy effects is taken into account.     

Logarithm strain singularity at tip of mode I growing crack in elasticand perfectly-plastic material

Reanalyzed in detail is the stress and strain distribution near the tip of a Mode I steadily growing crack in an elastic and perfectly-plastic material. The crack tip region is divided into five angular sectors, one of which is singular in character and represents a rapid transition zone that becomes a line of strain discontinuity in the limit as crack tip is approached. It is shown for an incompressible material (ν=0.5) under plane strain that the local strain in all the angular sectors possesses the same logarithm singularity, i.e., In r where r is the radial distance measured from the crack tip. This result also prevails for the compressible material ( v < 0.5) and resolves a long standing controversy concerning the strain singularity in the sector just ahead of the crack tip.     

On the local nature of the strain field calculation method for measuring heterogeneous deformation of cellular materials

A strain field calculation method based on the optimal local deformation gradient technique has been developed to calculate the ‘local’ strain tensor of cellular materials using cell-based finite element models. The local nature and accuracy of this method may be strongly dependent on the cut-off radius, which is introduced to collect the effective nodes for determining the optimal local deformation gradient of a node. Two different schemes are first analyzed to determine the suitable cut-off radius by characterizing the heterogeneous deformation of Voronoi honeycombs under uniaxial compression and we suggest that in Scheme 1, the cut-off radius defined based on the reference configuration is about 1.5 times the average cell radius; in Scheme 2, the cut-off radius defined based on the current configuration is about 0.5 times the average cell radius. Then, Scheme 3, a combined scheme of the two former schemes, is further suggested. It is demonstrated that the optimal cut-off radius in Scheme 3 characterizes the local strain reasonable well whether the compression rate is low or high. Finally, the strain field calculation method with the optimal cut-off radius is applied to reveal the evolution of the heterogeneous deformation of two different configurations of double-layer cellular cladding under a linear decaying blast load. The 2D fields and the 1D distributions of local engineering strain are calculated. These results interpret the shock wave propagation mechanisms in both claddings and provide useful understanding in the design of a double-layer cellular cladding.     

On the interaction between the input signal and the singularity expansion method

For transient excitation, it is well known that the Singularity Expansion Method (SEM) is a late time technique that is not in general suitable for early time. Lossy dielectric slab studies, with bounded integrable finite-duration input signal, have shown that the SEM representation for early time requires an additional branch cut contribution and a modification of the SEM series for completeness. Other researchers indicated that the SEM representation is not complete for most cases of practical interest and that it may not converge for early time. In this paper, we present an approach, using the input signal, to obtain a convergent SEM series that is valid for early as well as for late time. In this approach the branch cut contribution is not needed and instead we have pole contributions form the input signal and a modified SEM series. In addition, two forms of SEM representations will be presented, the traditional form and the time varying coupling coefficient form. Furthermore, the interaction and the effect of the input signal choice on the SEM representation will be clarified. This will be done for a dielectric slab backed by a perfect conductor.