Energy partitioning for a crack under remote shear and compression

The true nature and characteristics of crack growth mechanisms in geologic materials have not been adequately described and are poorly understood. The process by which deformation energy is converted to slipping and growing cracks under compressive stresses is complex and difficult to measure. A hybrid technique employing moiré interferometry as an experimental boundary condition to a finite element method (FEM) was employed for through-cracked polycarbonate plates under remote shear and compression. Cohesive end zone and dislocation slip models are used to approximate experimentally observed displacement characteristics. Shear-driven linear elastic fracture mechanics displacement predictions are shown to be inadequate for initial displacement progression. Moiré displacement fields of relative crack face slip reveal a near tip cohesive zone. The pre-slip moiré-FEM stress fields reveal that the maximum crack tip tensile stress occurs at approximately 45 degrees and further infers cohesive zone presence. A J integral formulation uses moiré displacement data and accounts for stored energy along the crack before and after shear driven crack face slip. These energy-partitioning results track the transfer of stored energy along the crack face to the crack tip until the entire crack is actively slipping. These laboratory-scale experiments capture basic mechanical behavior and simulate thousands of years of large-scale geologic feature displacement history in just a few hours.     

基于虚拟裂缝模型的混凝土断裂过程区研究

利用虚拟裂缝模型对混凝土断裂过程区进行了研究.以无限大板中心拉伸裂缝模型为例,将过程区裂缝张开位移采用多项式级数形式表示,求得了断裂过程区上的位移分布和粘聚力分布.进而分析了材料参数对断裂过程区上的位移、粘聚力、断裂过程区长度以及峰值外荷载的影响.结果表明:断裂过程区上的位移和粘聚力均为非线性分布.断裂过程区长度随骨料最大粒径增大而逐渐增大,随抗压强度增大而逐渐减小.峰值外荷载随骨料最大粒径和抗压强度增大均逐渐增大.     

Fracture toughness of transverse cracks in graphite/epoxy laminates at cryogenic conditions

Stress singularity of a transverse crack normal to ply-interface in a composite laminate is investigated using analytical and finite element methods. Four-point bending tests were performed on single-notch bend specimens of graphite/epoxy laminates containing a transverse crack perpendicular to the ply-interface. The experimentally determined fracture loads were applied to the finite element model to estimate the fracture toughness. The procedures were repeated for specimens under cryogenic conditions. Although the fracture loads varied with specimen thickness, the critical stress intensity factor was constant for all the specimens indicating that the measured fracture toughness can be used to predict delamination initiation from transverse cracks. For a given crack length and laminate configuration, the fracture load at cryogenic temperature was significantly lower. The results indicate that fracture toughness does not change significantly at cryogenic temperatures, but the thermal stresses play a major role in fracture and initiation of delaminations from transverse cracks.     

Observation of fracture process zone by laser speckle technique and governing mechanism in fracture of concrete

Fracture tests with a wedge-loading device are carried out on mortar and concrete specimens so as to have stable crack growth. Using laser speckle technique the length of macrocrack and the distribution of crack opening displacement are measured. Results are compared with those obtained by the boundary element method (BEM) analysis for a Dugdale-Barenblatt-type model of a fracture process zone. The governing mechanism in fracture of concrete and the mechanism which is represented by the model are discussed with special attention to the microcracking zone. It is deduced that a Dugdale-Barenblatt-type model does not represent the microcracking zone, thus implying that the microcracking zone and the bridging zone correspond to the pre-peak nonlinear part of the stress-strain curve in a uniaxial tension test and the post-peak tension-softening curve, respectively. It is concluded that the effect of microcracking on the maximum load is less significant than that of bridging. Possible models which include the effect of microcracking in addition to that of bridging are proposed.     

Delamination buckling: A finite element study with realistic delamination shapes,multiple delaminations and fibre fracture cracks

This paper presents a finite element model of a carbon fibre composite laminate with multiple delaminations of realistic shape and including fibre fracture cracks loaded under compression. The modelling technique is initially applied on circular and elliptical delaminations of single ply sublaminates, which are compared with existing analytical solutions. The techniques are then applied to models with multiple delaminations of realistic shape and their behaviour in buckling and post-buckling is captured. An inverse method is used to determine the stiffness reduction caused by the damage, and shows significant stiffness reduction caused by peanut shaped delaminations. When fibre fracture cracks are added, their contribution to further stiffness reduction is minimal but they have some significant effects on the buckling shapes.     

Estimation of width of fracture process zone in spruce wood by radial tensile test

Fracture behavior of spruce wood under radial tension was analyzed using nonlinear fracture mechanics as wood is classified as a quasi-brittle material. Stress-strain relationship with a strain-softening branch was obtained by digital image analysis and stress redistribution process, and the energy release rate of serial end-matched specimens was measured by performing a single-edge-notched tension test.The width of the fracture process zone (FPZ) was estimated by comparing two kinds of fracture energies. One was the dispersion energy per unit area to model strain localizations using a discontinuum model of damage theory, by integrating the stress-strain function with the strain-softening branch. The other was the energy release rate to determine crack growth. From this analysis, we determined that the width of the FPZ ranged from 0.3 mm to 0.5 mm in the radial direction. However, for a few specimens, the approximate stress-strain function could not be fitted into the stress-strain relationship obtained by the image analysis; it was observed that the fracture planes of these specimens tended to be more or less inclined.     

A distributed dislocation stress analysis for crazes and plastic zones at crack tips

A distributed dislocation method is developed to obtain analytically the applied stress as well as the surface stress profile along narrow plastic zones at the tip of a crack in a homogeneous tensile stress field. Replacing the plastic zone by a continuous array of mathematical dislocations, the stress field solution of this mixed boundary value problem (the displacement profile of the plastic zone is fixed while the tensile stresses are zero across the crack) can be solved. A computer program based on this stress field solution has been constructed and tested using the analytical results of the Dugdale model. The method is then applied to determining the surface stress profiles of crazes and plane-stress plastic deformation zones grown from electron microprobe cracks in polystyrene and polycarbonate respectively. The necessary craze and zone surface displacement profiles are determined by quantitative analysis of transmission electron micrographs. The surface stress profiles, which show small stress concentrations at the craze or zone tip falling to an approximately constant value which is maintained to the crack tip, are compared with those previously computed using an approximate Fourier transform method involving estimation of the displacement profile in the crack. The agreement between the approximate method and the exact distributed dislocation method is satisfactory.     

Modeling thermal and stress behavior of the fuel–clad interface in monolithic fuel mini-plates

A fuel development and qualification program is in process with the objective of qualifying very high density monolithic low enriched uranium–molybdenum fuel for high-performance research reactors. The monolithic fuel foil creates differences in the mechanical and structural characteristics of the fuel plate because of the planar interface created by the fuel foil and cladding. An initial finite element analysis model has been developed to investigate worst-case scenarios for the basic monolithic fuel plate structure using typical mini-plate irradiation conditions in the Advanced Test Reactor. Initial analysis shows that the stress normal to the fuel–clad interface dominates during irradiation and that the presence of small, rounded delaminations at the interface is not of great concern. However, larger and/or fuel–clad delaminations with sharp corners can create areas of concern, as maximum principal cladding stress, strain, displacement, and peak fuel temperature are all significantly increased. Furthermore, stresses resulting from temperature gradients that cause the plate to bow or buckle in an unconstrained fuel plate configuration is greatly enhanced in a constrained fuel plate configuration. The sensitivities of the model and input parameters are discussed, along with some overlap of initial experimental observations using as-fabricated plate characterization and post-irradiation examination.     

Damage prediction in composite sandwich panels subjected to low-velocity impact

The paper illustrates the application of a finite element tool for simulating the structural and damage response of foam-based sandwich composites subjected to low-velocity impact. Onset and growth of typical damage modes occurring in the composite skins, such as fibre fracture, matrix cracking and delaminations, were simulated by the use of three-dimensional damage models (for intralaminar damage) and interfacial cohesive laws (for interlaminar damage). The nonlinear behaviour of the foam core was simulated by a crushable foam plasticity model. The FE results were compared with experimental data acquired by impact testing on sandwich panels consisting of carbon/epoxy facesheets bonded to a PVC foam. Good agreement was obtained between predictions and experiments in terms of force histories, force–displacement curves and dissipated energy. The proposed model was also capable of simulating correctly nature and size of impact damage, and of capturing the key features of individual delaminations at different depth locations.     

A cohesive zone model for predicting delamination suppression in z-pinned laminates

This paper presents a cohesive zone model based finite element analysis of delamination resistance of z-pin reinforced double cantilever beam (DCB). The main difference between this and existing cohesive zone models is that each z-pin bridging force is governed by a traction-separation law derived from a meso-mechanical model of the pin pullout process, which is independent of the fracture toughness of unreinforced laminate. Therefore, two different traction-separation laws are used: one representing the toughness of unreinforced laminate and the other the enhanced delamination toughness owing to the pin bridging action. This approach can account for the large scale bridging effect and avoid using concentrated pin forces, thus removing the mesh dependency and permitting more accurate analysis solution. Computations were performed using a simplified unit strip model. Predicted delamination growth and load vs. displacement relation are in excellent agreement with the prediction by a complete model, and both models are in good agreement with test measured load vs. displacement relation. For a pinned DCB specimen, the unit strip model can reduce the computing time by 85%.     

SINGULAR p-VERSION FINITE ELEMENTS FOR STRESS INTENSITY FACTOR COMPUTATIONS

The finite element analysis of linear elastic fracture mechanics problems is complicated by the presence of the singular and finite non-singular stress distributions in the crack tip region. The availability of a constant stress term in addition to the singular term in the standard h-version singular finite elements is insufficient to model the finite nonsingular stress zone. A p-version singular finite element capable of modelling the higher-order non-singular stress terms in addition to the singular term and the constant term is presented. The formulation for the displacement substitution technique for computing the stress intensity factors using singular p-version triangular finite elements is developed. Unlike the standard h-version formulation, the stress intensity factors computed using the p-version displacement substitution technique do not depend on the specific arrangement and length of the quarter point elements, and require simple mesh designs as well as fewer number of degrees of freedom. Numerical studies comparing the convergence of the stress intensity factors computed by the p-version method against other available alternatives such as the h-version method and the contour integral method are presented to demonstrate the effectiveness of the present developments. © 1997 by John Wiley & Sons, Ltd.     

The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K _IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K ^H _IC for a single hydride platelet at the crack tip has been also estimated.     

Mechanisms and modeling of low cycle fatigue crack propagation in a pressure vessel steel Q345

The fatigue process near crack is governed by highly concentrated strain and stress in the crack tip region. Based on the theory of elastic–plastic fracture mechanics, we explore the cyclic J-integral as breakthrough point, an analytical model is presented in this paper to determine the CTOD for cracked component subjected to cyclic axial in-plane loading. A simple fracture mechanism based model for fatigue crack growth assumes a linear correlation between the cyclic crack tip opening displacement (ΔCTOD) and the crack growth rate (da/dN). In order to validate the model and to calibrate the model parameters, the low cycle fatigue crack propagation experiment was carried out for CT specimen made of Q345 steel. The effects of stress ratio and crack closure on fatigue crack growth were investigated by elastic–plastic finite element stress–strain analysis of a cracked component. A good comparison has been found between predictions and experimental results, which shows that the crack opening displacement is able to characterize the crack tip state at large scale yielding constant amplitude fatigue crack growth.     

In-plane impact behavior of honeycomb structures filled with linearly arranged inclusions

Honeycomb structures filled with linearly arranged inclusions were analyzed with a finite element method (FEM) to study how the arrangement of rigid inclusions affects the in-plane impact behavior of honeycomb structures. Each model was divided into several cell regions by inclusion lines. The analysis revealed the effect of inclusion lines on the mean stress of the cell region, maximum displacement of the cell region, and the order of deformed cell regions. Maximum displacement of the cell region was proportional to the width of the cell region, and mean stress of the cell region decreased as the width of the cell region increased. Approximate equations for the maximum displacement and mean stress of the cell region were derived. The approximations accounted for the deformation process of the honeycomb models with inclusion lines and revealed the dependence of the order of the deformed cell region on the mean stress of the regions. The validity of the approximate equations was confirmed by comparing them with experimental results. It was found that the approximate equations enabled us to design the in-plane impact behavior of honeycomb structures filled with linearly arranged inclusions.     

Higher order Cauchy–Born rule based multiscale cohesive zone model and prediction of fracture toughness of silicon thin films

In this work, we extend the multiscale cohesive zone model (MCZM) (Zeng and Li in Comput Methods Appl Mech Eng 199:547–556, 2010), in which interatomic potential is embedded into constitutive relation to express cohesive law in fracture process zone, to include the hierarchical Cauchy–Born rule in the process zone and to simulate three dimensional fracture in silicon thin films. The model has been applied to simulate fracture stress and fracture toughness of single-crystal silicon thin film by using the Tersoff potential. In this study, a new approach has been developed to capture inhomogeneous deformation inside the cohesive zone. For this purpose, we introduce higher order Cauchy–Born rules to construct constitutive relations for corresponding higher order process zone elements, and we introduce a sigmoidal function supported bubble mode in finite element shape function of those higher order cohesive zone elements to capture the nonlinear inhomogeneous deformation inside the cohesive zone elements. Benchmark tests with simple 3D models have confirmed that the present method can predict the fracture toughness of silicon thin films. Interestingly, this is accomplished without increasing of computational cost, because the present model does not require quadratic elements to represent heterogeneous deformation, which is the inherent weakness of the previous MCZM model. Quantitative comparisons with experimental results are performed by computing crack propagation in non-notched and initially notched silicon thin films, and it is found that our model can reproduce essential material properties, such as Young’s modulus, fracture stress, and fracture toughness of single-crystal silicon thin films.     

A 3D anisotropic elastoplastic‐damage model using discontinuous displacement fields

This paper is concerned with the development of constitutive equations for finite element formulations based on discontinuous displacement fields. For this purpose, an elastoplastic continuum model (stress–strain relation) as well as an anisotropic damage model (stress–strain relation) are projected onto a surface leading to traction separation laws. The coupling of both continuum models and, subsequently, the derivation of the corresponding constitutive interface law are described in detail. For a simple calibration of the proposed model, the fracture energy resulting from the coupled elastoplastic‐damage traction separation law is computed. By this, the softening evolution is linearly dependent on the fracture energy. The second part of the present paper deals with the numerical implementation. Based on a local and incompatible additive split of the displacement field into a continuous and a discontinuous part, the parameters specifying the jump of the displacement field are condensed out at the material level without employing the standard static condensation technique. To reduce locking effects, a rotating localization zone formulation is applied. The applicability and the performance of the proposed numerical implementation is investigated by means of a re‐analysis of a two‐dimensional L‐shaped slab as well as by means of a three‐dimensional ultimate load analysis of a steel anchor embedded in a concrete block. Copyright © 2004 John Wiley & Sons, Ltd.     

Combined continuum damage‐embedded discontinuity model for explicit dynamic fracture analyses of quasi‐brittle materials

In this paper, a novel constitutive model combining continuum damage with embedded discontinuity is developed for explicit dynamic analyses of quasi‐brittle failure phenomena. The model is capable of describing the rate‐dependent behavior in dynamics and the three phases in failure of quasi‐brittle materials. The first phase is always linear elastic, followed by the second phase corresponding to fracture‐process zone creation, represented with rate‐dependent continuum damage with isotropic hardening formulated by utilizing consistency approach. The third and final phase, involving nonlinear softening, is formulated by using an embedded displacement discontinuity model with constant displacement jumps both in normal and tangential directions. The proposed model is capable of describing the rate‐dependent ductile to brittle transition typical of cohesive materials (e.g., rocks and ice). The model is implemented in the finite element setting by using the CST elements. The displacement jump vector is solved for implicitly at the local (finite element) level along with a viscoplastic return mapping algorithm, whereas the global equations of motion are solved with explicit time‐stepping scheme. The model performance is illustrated by several numerical simulations, including both material point and structural tests. The final validation example concerns the dynamic Brazilian disc test on rock material under plane stress assumption. Copyright © 2014 John Wiley & Sons, Ltd.     

The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K _IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K ^H _IC for a single hydride platelet at the crack tip has been also estimated.     

A different view of the assumed stress hybrid method

A somewhat unconventional, yet simple, derivation of the assumed stress hybrid formulation is presented in which its relationship with displacement and stress-based finite element models is demonstrated. This gives insight into the hybrid method and shows that it represents a least energy fit of an equilibrium stress field to a displacement model strain field. By this means the attributes of both fields may be exploited in extending the analysis scope of hybrid models.     

用能量方法研究混凝土断裂过程区的力学性能

准脆性混凝土自由裂缝前缘断裂过程区的发展与其非线性断裂特征及尺寸效应现象密切相关。它的物理力学行为的量化分析对理解混凝土断裂破坏机理和建立适用于混凝土结构裂缝稳定分析和安全评估断裂准则尤为重要,一直是混凝土断裂力学研究的核心问题。该文依据Hillerborg给出的断裂能定义,给出了计算单位长度断裂过程区发展能量耗散的通用表达式。以三点弯曲梁为例,采用非线性软化本构关系,进一步给出了计算此平均能量耗散的具体步骤及对应的公式。在根据实测的三点弯曲梁的断裂能回归拟合了特征裂缝张开位移w0后,计算了每个试件整个断裂全过程中不同荷载时刻断裂过程区耗能的平均值。结果表明:随着裂缝扩展,断裂过程区能量耗散的变化和试件尺寸无关,可描述断裂过程区混凝土材料的力学性能。