Selection of finite-element mesh parameters in modeling the growth of hydraulic fracturing cracks

The effect of the mesh geometry on the accuracy of solutions obtained by the finite-element method for problems of linear fracture mechanics is investigated. The guidelines have been formulated for constructing an optimum mesh for several routine problems involving elements with linear and quadratic approximation of displacements. The accuracy of finite-element solutions is estimated based on the degree of the difference between the calculated stress-intensity factor (SIF) and its value obtained analytically. In problems of hydrofracturing of oil-bearing formation, the pump-in pressure of injected water produces a distributed load on crack flanks as opposed to standard fracture mechanics problems that have analytical solutions, where a load is applied to the external boundaries of the computational region and the cracks themselves are kept free from stresses. Some model pressure profiles, as well as pressure profiles taken from real hydrodynamic computations, have been considered. Computer models of cracks with allowance for the pre-stressed state, fracture toughness, and elastic properties of materials are developed in the MSC.Marc 2012 finite-element analysis software. The Irwin force criterion is used as a criterion of brittle fracture and the SIFs are computed using the Cherepanov–Rice invariant J-integral. The process of crack propagation in a linearly elastic isotropic body is described in terms of the elastic energy release rate G and modeled using the VCCT (Virtual Crack Closure Technique) approach. It has been found that the solution accuracy is sensitive to the mesh configuration. Several parameters that are decisive in constructing effective finite-element meshes, namely, the minimum element size, the distance between mesh nodes in the vicinity of a crack tip, and the ratio of the height of an element to its length, have been established. It has been shown that a mesh that consists of only small elements does not improve the accuracy of the solution.     

Measurement of Residual Elastic Strains in a Titanium Alloy Using High Energy Synchrotron X-Ray Diffraction

Residual elastic strains in a bent bar of titanium alloy Ti-6Al-4V were measured using high energy diffraction on station 16.3 at SRS Daresbury. Using a single bounce Laue crystal monochromator, diffraction peaks were collected for reflections (00.2), (10.1), (10.2) and (11.0) from the hcp alpha phase of the titanium alloy. Reference values of the lattice spacing for each of the reflections were found from the diffraction pattern collected from a stress-free sampling volume. The residual elastic strain values calculated on the basis of each reflection were then computed and plotted as a function of position across the bent bar. The average macroscopic residual elastic strain was computed using an averaging procedure taking into account the multiplicity of each reflection. Energy dispersive white beam diffraction from the same bent bar was used to collect diffraction patterns over the range of lattice spacings between 0.8 and 2.2 Å. Detector calibration was carried out using the procedure described in Liu et al. (2005) and detailed interpretation of the energy dispersive profiles was carried out allowing the identification of average residual elastic strains in the two principal phases present in the titanium alloy considered, the α-Ti hcp and the β-Ti bcc phases. Peak-specific residual strain profiles computed on the basis of monochromatic measurements show significant differences reflecting the variation in the elastic and plastic properties with grain orientation, i.e., crystal anisotropy. Using the contrast between the elastic and plastic properties of different directions within the α-Ti hcp lattice, the difference between residual elastic strains measured for (00.2) and (11.0) reflections was plotted, as well as the ‘difference strain’ between (00.2) and (10.1) reflections. These profiles show a good qualitative correlation with the plastic strain profile introduced by inelastic bending that was computed from the analysis of Pawley refinement of the energy-dispersive diffraction measurements.     

玄武岩中裂隙分布形式对声波传播的影响

复杂岩体含有大量的裂隙,这些裂隙尺寸及其分布形式等对弹性波传播都有很大的影响.本文加工了含单个裂隙、双裂隙和三个裂隙的玄武岩岩样单元对其进行组合,进行了25kHz、 50kHz、 400kHz、 600kHz和1000kHz 等5种频率的声波测试.通过考虑垂直或平行波传播方向的裂隙长度,来探索裂隙分布形式和不同裂隙长度对弹性波传播的影响,研究玄武岩的频散效应和波的衰减.结果表明:裂隙方向与波传播方向夹角对弹性波传播有很大的影响.当裂隙方向与波传播方向垂直时,散射效应最大;而当裂隙方向与波传播方向平行时,影响最小.上述结果可为理论模型和数值分析提供依据.     

Experimental validation of multiscale modeling of indentation of suspended circular graphene membranes

In recent studies, the in-plane elastic properties of graphene have been computed via Density Functional Theory (DFT) and expressed in the form of a higher-order continuum elastic constitutive model. The studies predict that graphene exhibits an anisotropic and non-linear elastic response at high strains. However, one study predicts that the rupture mechanism of graphene at its intrinsic strength is due to elastic instability whereas another study predicts the rupture mechanism at its intrinsic strength is due to phonon instability. In the present paper, we use the higher-order continuum elastic constitutive model within the context of the finite element method to simulate a set of experiments of the indentation of circular freestanding monatomic graphene membranes. There is a close correspondence between the measured and predicted measured force vs. displacement responses of indented graphene, providing experimental validation for the constitutive response. Further, there is a close correspondence between the measured and predicted breaking force of graphene via the elastic instability mechanism. Thus, the results suggest that the elastic instability precipitates failure of pristine graphene at its intrinsic strength, and also provides further experimental validation of the constitutive response.     

Parameter identification for cracks in elastic plate structures based on remote strain fields

A method for the detection of cracks in plate structures is presented. In contrast to most of the common monitoring concepts taking advantage of the reflection of elastic waves at crack faces, the presented approach is based on the strain measured at different locations on the surface of the structure. This allows both the identification of crack position parameters, such as length, location and angles with respect to a reference coordinate system and the calculation of stress intensity factors (SIF). The solution of the direct problem is performed on the basis of the BFM (body force method). The inverse problem is solved applying the particle swarm optimization (PSO) algorithm. The BFM is based on the principle of linear superposition which allows the calculation of the strain field in a cracked body. The strain at an arbitrary point in the structure is replaced by the strain provided by body force doublets in the uncracked structure. The doublets as well as external loads are parameters which have to be determined solving the inverse problem by minimizing a fitness function, which is defined by a square sum of residuals between measured strain distributions and computed ones for an assumed crack. The PSO algorithm applied to the fitness function operates on the basis of a swarm of candidate solutions. Once knowing loading and crack parameters, the SIF can be determined.     

Shear band propagation from a crack tip subjected to Mode II shear wave loading

Results are reported for pressure–shear plate impact experiments in which pre-cracked 4340 steel plates are subjected to Mode II loading. Experiments show the propagation of a shear band ahead of the initial crack. Finite element simulations are used to interpret the results. Normal and transverse velocity–time profiles measured at the rear surface of the target can be simulated reasonably well using even an elastic model for the material response. A propagating shear band is obtained when the material is modeled as having reduced shearing resistance described by a thermo-viscoplastic power law, and complete loss of shearing resistance when the shear strain reaches a critical value. However, the predicted speed of propagation of the tip of the shear band is substantially less than required to explain the lengths of the bands observed in the experiments. Adjustments of parameters of the power-law model have little effect on the overall length of the band. Possible reasons for differences between predicted and measured shear band speeds are examined. Further reduction in the shearing resistance in the shear band appears to be essential for the simulated bands to be as long as those observed in the experiments.     

粘弹性夹心夹层结构卸载后的裂纹持续扩展现象研究

以聚氨酯弹性体钢夹层板为研究对象,对其进行了三点弯曲实验,对不同类型试件的实验结果进行了总结。实验发现,在加载过程中,在弯曲挠度超过一定程度时,夹层板结构的角点位置出现开裂,由于层间强度不足和夹心材料的粘弹性性质,卸载后裂纹会在一段时间内沿层间继续发生扩展,最后导致整个面板发生脱粘或夹心撕裂。针对这一现象,以弹性地基梁理论为基础,计算了上下面板的挠度差异,并使用层间粘结强度来分析试件的临界状态,获得了软夹芯试件和硬夹芯试件的临界破坏的裂纹长度。结果表明,使用层间粘结强度分析方法,获得的结果都能够描述聚氨酯弹性体钢夹层板的层间延迟破坏的特点。     

On the effects of characteristic lengths in bending and torsion on Mode III crack in couple stress elasticity

The problem of a stationary semi-infinite crack in an elastic solid with microstructures subject to remote classical K_III field is investigated in the present work. The material behavior is described by the indeterminate theory of couple stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the material as well as for the strong size effects arising at small scales. The stress and displacement fields turn out to be strongly influenced by the ratio between the characteristic lengths. Moreover, the symmetric stress field turns out to be finite at the crack tip, whereas the skew-symmetric stress field displays a strong singularity. Ahead of the crack tip within a zone smaller than the characteristic length in torsion, the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution, due to the relative rotation of the microstructural particles currently at the crack tip. The asymptotic fields dominate within this zone, which however has limited physical relevance and becomes vanishing small for a characteristic length in torsion of zero. In this limiting case the full-field solution recovers the classical K_III field with square-root stress singularity. Outside the zone where the total shear stress is negative, the full-field solution exhibits a bounded maximum for the total shear stress ahead of the crack tip, whose magnitude can be adopted as a measure of the critical stress level for crack advancing. The corresponding fracture criterion defines a critical stress intensity factor, which increases with the characteristic length in torsion. Moreover, the occurrence of a sharp crack profile denotes that the crack becomes stiffer with respect to the classical elastic response, thus revealing that the presence of microstructures may shield the crack tip from fracture.     

The mixed mode crack problem in an FGM layer bonded to a homogeneous half-plane

In this paper, the plane elasticity problem of an arbitrarily oriented crack in a FGM layer bonded to a homogeneous half-plane is considered. The problem is modeled by assuming that the elastic properties of the FGM layer are exponential functions of the thickness coordinate and are continuous at the interface of the FGM layer and the half-plane.The Fourier transform technique is used to reduce the problem to the solution of a system of Cauchy-type singular integral equations, which are solved numerically. The stress intensity factors are computed for various crack orientations, crack locations and material parameters. The results show that crack length, crack orientation and the non-homogeneity parameter of the strip material have significant effect on the fracture of the FGM layer.     

Mode mixity and nonlinear viscous effects on toughness of interfaces

This paper examines steady-state crack growth at interfaces between polymeric materials and hard substrates under quasi-static conditions. The polymeric material is taken to be an elastic nonlinear viscous solid while the substrate is treated as a rigid material. Void growth and coalescence in the rate-dependent fracture process zone is modeled by a nonlinear viscous porous strip of cell elements. In the first part of this paper, the polymeric background material surrounding the process zone is assumed to be purely elastic. Under fixed mode mixity, the computed interface toughness is found to be a monotonically increasing function of crack velocity; toughness also increases rapidly with higher rate sensitivity. This behavior can be explained in terms of voids growing in a strain-rate strengthened process zone. In the second part of the paper, the background material is also treated as an elastic nonlinear viscous solid. The competition between work of separation in the process zone and energy dissipation in the background material leads to a U-shaped toughness–crack velocity curve. Effects of mode mixity, initial porosity, rate sensitivity, as well as the initial yield strain on toughness are studied. The simulations produce trends that agree with interface toughness vs. crack velocity data reported in experimental studies for rubber toughened epoxy-paste adhesive and urethane acrylate adhesive.     

A new hybrid technique for in-plane characterization of orthotropic materials

In this paper we present a novel hybrid procedure for the in-plane mechanical characterization of orthotropic materials. The material identification reverse engineering problem is solved by combining speckle interferometry and numerical optimization. The rationale behind the entire process is the following: for any specimen to be characterized and which has been subjected to some loading condition, it is possible to express the difference between experimental data and analytical/numerical predictions by means of an error function ψ, which depends on the elastic constants of the material. The ψ error will decrease as the elastic constants come close to their target values. Here, we build the ψ function as the difference between the displacement field measured with speckle interferometry and its counterpart computed by means of finite element analysis. Since the ψ function is highly non-linear, it has to be optimized with a global optimization algorithm, which perform a random search in the elastic constants design space. The hybrid material identification process finally allows us to determine values of the elastic constants. In order to prove the feasibility of the present approach, we have determined the in-plane elastic properties of an eight-ply composite laminate (woven fiberglass-epoxy) used as a substrate for printed circuit boards. The results indicate that the procedure proposed in this paper was able to accurately characterize the material under investigation. Remarkably, the elastic constants found by the identification procedure were less than 0.7% different from their target values, while the residual error between the displacements measured by speckle interferometry and those computed at the end of the optimization process was less than 3%. L. Lamberti is an Assistant Professor, and C. Pappalettere (SEM Member and President of the Italian Society of Stress Analysis) is Professor of Mechanical Engineering and Experimental Mechanics, Politecnico di Bari, Dipartimento di Ingegneria Meccanica e Gestionale, Viale Japigia 182, 70126 Bari, Italy     

大变形条件下具有不同裂纹长度试件的断裂韧性J_C工程估算

本文首先通过有限元分析,在不同长度裂纹尺寸结构的临界破坏载荷下计算出临界断裂韧性值,然后从细观入手,在文献[1][2]等的工作基础上,通过理论推导得到了三维应力度参数R_σ与断裂韧性的关系。经过验证计算,证明该参数可用于描述结构尺寸效应的影响,能够在平面应变断裂韧性(参考值)的基础上,确定不同尺寸结构的临界断裂韧性J_C的值,适用于工程上的应用和推广。     

Limitations of growth regimes for cracks initiated in smooth fatigue specimens

The stages of the growth of small cracks initiating at natural flaws in smooth specimens subjected to fatigue loading are characterized and the dominant propagation mechanisms and corresponding fracture paths are described. Characteristic crack lengths are introduced to assess the transition between the regimes of microstructurally small cracks, physically small cracks and long cracks. A log Δσ-log a-diagram is used to derive estimates of these crack lengths. It is shown that simple formulate can be found which relate these characteristic crack lengths to mechanical and material parameters that can be measured using standard fracture mechanics specimens and fatigue tests.     

Identification of parameters of a plane elliptic crack in an isotropic linearly elastic body from the results of a single uniaxial tension test

The method earlier developed by one of the authors for identifying ellipsoidal defects is numerically tested for the applicability to the problem of identification of a degenerate ellipsoidal defect, i.e., an elliptic crack. The method is based on the reciprocity functional and the assumption that the displacements are measured in a uniaxial tension test of an isotropic linearly elastic body. Calculations show that the earlier developed method is also efficient for identification of an elliptic crack and its parameters (the center coordinates, the normal to the crack plane, and the directions and lengths of the semiaxes) can be determined with high accuracy. Some examples where the crack has a non-elliptic shape are also considered. It is discovered that, in many cases, the ellipsoids that were constructed by formulas reconstructing the ellipsoidal crack from the data on the external boundary of the body that correspond to a nonelliptic crack, approximate the actual defect with sufficient accuracy. The method stability was investigated with respect to noise in the initial data.     

Remarks on the energy release rate for an antiplane moving crack in couple stress elasticity

This paper is concerned with the steady-state propagation of an antiplane semi-infinite crack in couple stress elastic materials. A distributed loading applied at the crack faces and moving with the same velocity of the crack tip is considered, and the influence of the loading profile variations and microstructural effects on the dynamic energy release rate is investigated. The behavior of both energy release rate and maximum total shear stress when the crack tip speed approaches the critical speed (either that of the shear waves or that of the localized surface waves) is studied. The limit case corresponding to vanishing characteristic scale lengths is addressed both numerically and analytically by means of a comparison with classical elasticity results.     

Filament-breaking length—a measure of elasticity in extension

A technique has been developed to assess elasticity in extension of fluids which are weakly elastic. The technique is based on stretching a fluid sample held between two small closely spaced co-axial disks until the fluid bridge or filament breaks. The distance between the disks on breakup, ‘the breaking length’, is measured. When the fluid is elastic, this length is greater than that of an equivalent inelastic fluid. An inelastic baseline was established by measuring the breaking lengths of a wide variety of Newtonian fluids. Measurements with weakly elastic fluids reveal that the extra breaking length increases with polymer type and concentration and with the rate of stretching, the expected behavior for elastic liquids. The breaking lengths of model paper coatings were measured and found to correlate with the degree of misting in a roll coating machine, indicating that droplet formation is related to extensional elasticity.     

Stress intensity factors for an edge interface crack in a bonded semi-infinite plate for arbitrary material combination

Although a lot of interface crack problems were previously treated, few solutions are available under arbitrary crack lengths and material combinations. In this paper the stress intensity factors of an edge interface crack in a bonded strip are considered under tension with varying the crack length and material combinations systematically. Then, the limiting solutions are provided for an edge interface crack in a bonded semi-infinite plate under arbitrary material combinations. In order to calculate the stress intensity factors accurately, exact solutions in an infinite bonded plate are also considered to produce proportional singular stress fields in the analysis of FEM by superposing specific tensile and shear stresses at infinity. The details of this new numerical solution are described with clarifying the effect of the element size on the stress intensity factor. It is found that for the edge interface crack the normalized stress intensity factors are not always finite depending upon Dunders’ parameters. This behavior can be explained from the condition of the singular stress at the end of bonded strip. Convenient formulas are also given by fitting the computed results.     

Reduction of<Emphasis Type="Italic">K</Emphasis><Subscript>I</Subscript> and<Emphasis Type="Italic">K</Emphasis><Subscript>II</Subscript> by the shape-memory effect in a TiNi shape-memory fiber-reinforced epoxy matrix composite

Shape-memory TiNi fiber-reinforced/epoxy matrix composites have been fabricated, and the suppression of crack-tip stress intensity and the change in fracture toughness have been systematically investigated. Stress-strain data for these composite specimens with notches at various angles and different crack lengths in the transverse direction have been measured in tensile tests. The stress intensity factor at the crack tip is experimentally determined from photoelastic fringe patterns. The decreases inK values are attributed to the compressive stress field in the matrix induced when the pre-strains of the TiNi fiber contract to their initial length upon heating above the austenitic final temperature. We present the influences of the pre-strain of TiNi fibers and the compressive domain size between a crack tip and fiber on theK value.     

A continuum approach to the analysis of the stress field in a fiber reinforced composite with a transverse crack

The stress field in a periodically layered composite with an embedded crack oriented in the normal direction to the layering and subjected to a tensile far-field loading is obtained based on the continuum equations of elasticity. This geometry models the 2D problem of fiber reinforced materials with a transverse crack. The analysis is based on the combination of the representative cell method and the higher-order theory. The representative cell method is employed for the construction of Green’s functions for the displacements jumps along the crack line. The problem of the infinite domain is reduced, in conjunction with the discrete Fourier transform, to a finite domain (representative cell) on which the Born–von Karman type boundary conditions are applied. In the framework of the higher-order theory, the transformed elastic field is determined by a second-order expansion of the displacement vector in terms of local coordinates, in conjunction with the equilibrium equations and these boundary conditions. The accuracy of the proposed approach is verified by a comparison with the analytical solution for a crack embedded in a homogeneous plane.Results show the effects of crack lengths, fiber volume fractions, ratios of fiber to matrix Young’s moduli and matrix Poisson’s ratio on the resulting elastic field at various locations of interest. Comparisons with the predictions obtained from the shear lag theory are presented.     

Mode-III crack kinking under stress-wave loading

A horizontally polarized step-stress wave is incident on a semi-infinite crack in an elastic solid. At the instant that the crack tip is struck, the crack starts to propagate in the forward direction, but under an angle κπ with the plane of the original crack. In this paper a self-similar solution is obtained for the particle velocity of the diffracted cylindrical wave field. The use of Chaplygin's transformation reduces the problem to the solution of Laplace's equation in a semi-infinite strip containing a slit. The Schwarz-Christoffel transformation is employed to map the semi-infinite strip on a half-plane. An analytic function in the half-plane which satisfies appropriate conditions along the real axis, can subsequently be constructed. The Mode-III stress-intensity factor at the tip of the kinked crack has been computed for angles of incidence varying from normal to grazing incidence, for angles of crack kinking defined by -0.5?κ?0.5, and for arbitrary subsonic crack tip speeds.