Finite fracture mechanics analysis of crack onset at a stress concentration in a UD glass/epoxy composite in off-axis tension

The presence of stress concentrations at holes and notches is known to reduce the strength of composite materials. Due to complexity of the damage processes at a stress raiser in a composite, different modeling approaches have been developed, ranging from empirical point and average stress criteria to involved damage mechanics or cohesive zone-based models of failure. Finite fracture mechanics approach with a coupled stress and energy failure criterion, recently developed and applied mainly to cracking in homogeneous isotropic materials, allows predicting the appearance and propagation of a crack using material strength and toughness characteristics obtained from independent tests. The present study concerns application of the finite fracture mechanics to the analysis of cracking at a notch in a UD glass/epoxy composite subjected to tensile off-axis loading. Based on UD composite strength and intralaminar toughness characterized by separate tests, finite fracture mechanics analysis provided conservative estimates of crack onset stress at the notch.     

Modellierungskonzept für die Versagensbewertung von Laserschweißverbindungen

Failure assessment of laser weldments based on numerical modelling Classical fracture mechanics based assessments are no more sufficient to provide realistic predictions of the deformation and failure behaviour of welded structures. This situation can be improved by numerical modelling based on damage mechanics. A new concept will be provided, which is based on a cohesive model for crack growth simulation. The determination of the relevant material parameters is also considered where testing is combined with numerical simulation. For a laser weld joint, the gradient of the material properties has to be properly characterized. With miniature sized specimens, the material properties can be discretized by homogeneous layers. A new method, based on the digital image technique, has been introduced to determine the stress‐strain curves also in the large strain region due to necking. Test results on small bend bars containing a thin laser weld and a crack like defect in the centre show different crack path developments resulting from a competitive fracture situation. Mainly shear fracture mode occurs, in some cases also a pure normal fracture mode or a combination of both were observed. The concept presented is able to consider the crack development, if all occuring fracture modes are included in the analysis. However, a complete simulation of an extensive crack extension through a heterogeneous structure has not yet been verified.     

A finite element approach to anisotropic damage of ductile materials in large deformations Part I: an anisotropic ductile elastic-plastic-damage model

During past decades, many material models using the continuum damage mechanics (CDM) approach have been proposed successfully in the small deformation regime to describe inelastic behaviors and fracturing phenomena of a material. For ductile materials, large deformation takes place at the level of damage appearance. Damage is anisotropic in nature. In this paper, the ductile damage at finite deformations is modeled as an anisotropic tensor quantity. Then, a fourth-order symmetric stress correction tensor is proposed for computationally efficient and easy implementation in the finite element formulations. Consequently, an explicit form of the fourth-order constitutive equations of the proposed elastic-plastic-damage model is derived. Both isotropic and kinematic hardening effects are included in the formulation. The new constitutive model can predict not only the elastic-plastic behaviors, but also the sequential variations of ductile materials. An evaluation of the constitutive and damage evolution equations is presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

A comparative study of continuum damage models for crack propagation under gross yielding

In this paper, the problem of fracture initiation in an aluminum alloy thin plate containing a central crack is examined by employing several phenomenological continuum damage mechanics models. These models differ mainly in the selection of the kind of tensorial property the damage variable assumes, the nature of the equivalence postulate between damaged and pseudo undamaged material states, and the way damage evolution laws are formulated. Two formulations of damage effect tensor based on the engineering notation and the normative notation of stress and strain, respectively, are compared. In addition, the hypothesis of strain equivalence is compared to that of stress working equivalence. The error in the assumption of isotropic damage development in the crack tip process zone is also checked against that of anisotropic damage. In the numerical algorithm, both updated Lagrangian formulation and small displacement formulation of material non-linearity only are adopted and compared. The influence of non-proportionality in stress histories present in the crack tip region is accounted for by introducing a dynamic coordinate system of principal damage such that the principal direction of damage rotates in accordance with that of the loading. The calculated fracture initiation loads are finally compared with those determined experimentally.     

An Approach to Estimate Interface Shear Stress of Ceramic Matrix Composites from Hysteresis Loops

An approach to estimate interface shear stress of ceramic matrix composites during fatigue loading has been developed in this paper. By adopting a shear-lag model which includes the matrix shear deformation in the bonded region and friction in the debonded region, the matrix crack space and interface debonding length are obtained by matrix statistical cracking model and fracture mechanics interface debonding criterion. Based on the damage mechanisms of fiber sliding relative to matrix in the interface debonded region upon unloading and subsequent reloading, the unloading counter slip length and reloading new slip length are determined by the fracture mechanics method. The hysteresis loops of four different cases have been derived. The hysteresis loss energy for the strain energy lost per volume during corresponding cycle is formulated in terms of interface shear stress. By comparing the experimental hysteresis loss energy with computational values, the interface shear stress corresponding to different cycles can then be derived. The theoretical results have been compared with experimental data of three different ceramic composites.     

Generalized elastic damage theory and its application to composite plate

Based on continuum mechanics, a generalized damage theory for elastic material which can be used for anisotropic composite is presented in this paper. This theory for anisotropic elastic material has been proposed here from the stress-strain relation of the actual damaged material. Introducing a fourth order damage operator that may be formed by a symmetrical second order damage factor tensor, the constitutive equation of the damaged material has been set up. The expressions of components of both the stress tensor and the strain tensor of the damaged material and their first order invariants have been also derived. The application of this theory to the 2-dimensional composite laminate, including the technique estimating the components of the damage factor tensor and the damage variable tensor and also the practical measure technique of the damage in the whole process, have been explained in detail. Finally, the changes of the anisotropic elastic properties and the actual stress state of damaged material have been discussed and some interesting results have been obtained in this paper.     

Failure of plain concrete beam at impact load: 3D finite element analysis

In the paper, the results of numerical failure analysis of plain concrete beams loaded by impact three-point bending load are presented and discussed. The theoretical framework for the numerical analysis is continuum mechanics and irreversible thermodynamics. The spatial discretization is performed by the finite element method using update Lagrange formulation. Green–Lagrange stain tensor is used as a strain measure. To account for cracking and damage of concrete, the beam is modeled by the rate sensitive microplane model with the use of the so-called co-rotational stress tensor. Damage and cracking phenomena are modeled within the concept of smeared cracks. To assure objectivity of the analysis with respect to the size of the finite elements, crack band method is used. The contact-impact analysis is based on the mechanical interaction between two bodies—concrete beam (master) and dropping hammer (slave) falling on the mid span of the beam. The contact constrains are satisfied by Lagrange multiplier method, which is adapted for the explicit time integration scheme. To investigate the influence of loading rate on the failure mode of the beam parametric study is carried out. The numerical results are evaluated, discussed and compared with test results known from the literature. It is shown that the beam resistance and failure mode strongly depend on loading rate. For lower loading rates beam fails in bending (mode-I fracture). However, with increasing loading rate there is a transition of the failure mechanism from bending to shear. The results are in good agreement with theoretical and experimental results known from the literature.     

Deformation gradient tensor decomposition for representing matrix cracks in fiber-reinforced materials

A new method is presented for the representation of matrix cracks in continuum damage mechanics (CDM) models for fiber-reinforced materials. The method is based on the additive decomposition of the deformation gradient tensor into ‘crack’ and ‘bulk material’ components, analogous to the additive strain decomposition of the smeared-crack approach. The potential improvements to the accuracy of CDM models that utilize the presented method are demonstrated for a single element subjected to simple shear deformation and for a unidirectional open-hole tension specimen. The presented method avoids load transfer across matrix cracks and eliminates the prediction of spurious secondary failure modes that occurs when conventional strain-based CDM models are used in geometrically nonlinear finite element analyses involving large shear deformations.     

Computational modelling of static indentation-induced damage in glass

The present paper is focused on the numerical simulation of a glass plate subjected to static indentation by a spherical indenter. For this purpose, a combined approach of continuum damage mechanics (CDM) and fracture mechanics is performed. Results provided by an axisymmetric finite element model were compared with analytical solutions. A CDM based constitutive model with an anisotropic damage tensor was selected and implemented into a finite element code to study the damage of glass. The numerical results were analysed through the framework of the stress and damage distribution. Various regions with critical damage values were therefore predicted in good agreement with the experimental observations in the literature. In these regions, the directions of crack propagation, including both cracks initiating on the surface as well as in the bulk, were predicted using the strain energy density factor. Predicted directions were found in good agreement with those experimentally obtained in the literature results.     

复合固体推进剂黏弹性微裂纹损伤本构模型

为建立复合固体推进剂的损伤本构模型,在介观尺度上视其为微裂纹损伤,选取微裂纹密度为损伤内变量。在Abdel-Tawab本构方程的基础上,基于微裂纹均匀化理论,推导了损伤映射张量的一般形式。该张量通常具有非完全对称性,其物理意义是将真实应力空间中各向异性材料的多轴加载映射为等效应力空间中各向同性材料的更为复杂的多轴加载。其次,基于黏弹性动态裂纹扩展模型和裂纹扩展阻力曲线的概念,建立了损伤内变量的演化方程。该演化方程仅含4个物理意义明确的细观参数,并且参数的取值规律与宏观应力曲线的变化规律相一致。数值结果表明,建立的模型能够有效反映材料损伤的应变率、温度依赖性及各向异性特征,并且具有一定的蠕变损伤预测能力。     

A continuum damage mechanics approach to crack tip shielding in brittle solids

This paper focuses firstly on the development of a comprehensive anisotropic theory of continuum damage mechanics for brittle solids suffering progressive deterioration. The basic concept of damage parameterization is re-examined and a new set of damage variables introduced yielding a new damage effect tensor through which the effective stress and strain tensors are defined. Constitutive equations of the damaged material are established incorporating a new hypothesis on equivalence between damaged and undamaged responses of the material. The model is completed by introduction of a general damage characteristic tensor which accounts for the experimentally observed fact that the rate of damage growth depends nonlinearly on applied external loads. The established damage model is next applied to investigate the crack-tip shielding effect due to anisotropic microcracking. The ratio of near-tip to remote stress intensity factors is obtained in closed form. A moderate but definite effect of anisotropy of microcracking is observed. The case of isotropic damage is found to be the least effective in screening remote external loads and is in accord with the results obtained by other researchers using different approaches.     

高应变率加载下玻璃板碎片尺寸的理论模型

基于材料损伤理论和损伤断裂机理,分析了普通平板玻璃在高应变率下的损伤破碎规律。根据玻璃碎片破坏损伤应变能和断裂表面能之间的平衡关系,推导了不同应变率下预测玻璃碎片大小的计算公式。该模型能够直接、定量地给出玻璃板中由于裂缝间的贯通形成的平均碎片大小,同时能够求出玻璃碎片的数量。对于该模型,讨论了玻璃板尺寸及临界应变等因素对玻璃碎片大小的影响。     

Interfaces between fatigue,creep, and fracture

A discussion is presented of the fatigue process as one of initiating a crack and propagating it to failure, and a formula is presented for estimating the effective point of crack initiation. This formula is speculatively applied to three interface problems in fatigue: (a) Life of a quasi-brittle material which can sustain only a relatively small crack before failure takes place according to the laws of fracture mechanics. An example is presented to illustrate the procedure and to indicate the probable validity of the approach. (b) Estimation of the fatigue characteristics at high temperatures within the creep range of materials. By assuming that intercrystalline cracking has the effect of by-passing much of the crack initiation process, the number of cycles to failure becomes related more importantly to the crack propagation period. A numerical procedure for estimating life in terms of applied strain range is described, and its validity investigated by application to a number of materials for which data have been presented in the literature. (c) Application of a linear damage rule individually to crack initiation and crack propagation. It is possible to predict the effect of order of application of loads in a two-step cumulative fatigue test. The method is checked by using literature data.     

Damage Constitutive Equations and its Application to Fiber Reinforced Composites under Transverse Impact

Two continuous field variables, called as continuity tensor and damage variable tensor, are used to describe the anisotropic responses of an elastic-brittle material under transverse impact load. Based on the continuum damage mechanics, anisotropic damage constitutive equations in both full and incremental forms are proposed here. The expressions of effective elastic module tensor, damage variable tensor and damage propagation force tensor are further derived, and the methods for determining the tensors are explained in detail. An example of strain and damage response of a fiber reinforced composite laminated plate under transverse impact load is employed to demonstrate the application of this theory. In the example, the damage variable coupled with geometric large deformation of laminated plate is also considered. The calculating results illustrate the influence of damage on strain field in the impacted laminated plate.     

Characterization of microscopic damage in composite laminates and real-time monitoring by embedded optical fiber sensors

First, a methodology for observation and modeling of microscopic damage evolution in quasi-isotropic composite laminates is presented. Based on the damage observation using both an optical microscope and a soft X-ray radiography, a damage mechanics analysis is conducted to formulate the stiffness change due to transverse cracking. Then, both energy and stress criteria are combined to provide a valid procedure to predict the transverse crack evolution. The theoretical prediction is found to agree well with the experimental results for the transverse crack density as a function of strain as well as stress–strain curves. Then, another methodology is introduced using two kinds of embedded optical fiber sensors to detect and monitor the transverse crack evolution in composite laminates. One is plastic optical fibers (POF), where the loss in optical power is generated by local deformation of POF due to transverse cracking. The other is fiber Bragg grating (FBG) sensors, where the local strain distribution within the FBG gage length due to transverse cracking alters the power spectrum of the light reflected from the FBG sensors. Embedded optical fiber sensors are found to be a powerful method to detect and monitor the transverse crack evolution in composite laminates.     

电测法确定低强混凝土裂缝起裂和等效裂缝长度

该文设计了3种低强度混凝土三点弯曲切口梁,测试研究其双K断裂参数。试验中采用标距为5 mm和10 mm的应变片以半桥连接方式测量预制裂缝的起裂荷载,比较其工作性,发现应变片测量起裂荷载具有强度敏感性,短标距应变片更适宜测量本试验用低强度混凝土的裂缝起裂。试验中另沿韧带方向布置4组半桥应变片,根据各测点处拉应变回缩时的荷载与裂缝口张开位移,计算裂缝发展至测点高度时刻的等效裂缝长度,并与此时的实际缝长比较,结果表明双K断裂模型在预测裂缝长度方面具有较好的适用性。     

Computational fracture analysis of concrete gravity dams by crack-embedded elements––toward an engineering evaluation of seismic safety

From a point of view to utilize fracture mechanics of concrete for the solution of engineering problems, the problems in crack modeling, computational algorithm, and damping implementation are discussed in conjunction with safety assessment of concrete dam against large earthquakes. It is shown that the formulation for the crack-embedded element has an analogy with that of computational plasticity. This analogy enables us to utilize the return-mapping algorithm well established in computational plasticity for the dynamic analysis of crack growth in concrete. The ways of implementing damping to avoid diffused cracking are presented and simple examples of numerical analysis are shown to demonstrate the effect of damping and the performance of appropriate damping implementation for cracked elements.     

Fracture behavior of nano-layered coatings under tension

The fracture of brittle/ductile multilayers composed of equal thicknesses of Si and Ag layers evaporated on a thick substrate is studied with the aid of a four-point bending apparatus. The system variables include individual layer thickness (2.5 to 30 nm), total film thickness (0.5 to 3.5 μm) and substrate material (polycarbonate, aluminum alloy and hard steel). The fracture is characterized by transverse cracks that proliferate with load. The crack initiation strain ε_i is virtually independent of total film thickness and substrate material while increasing with decreasing layer thickness h, to a good approximation as ε_i ~ 1/h^1/2. At higher strains, film debonding and buckling are evident.The fracture conditions are determined with the aid of a 2D finite element analysis incorporating the inelastic response of the interlayer. A fracture scenario consisting of tunnel cracking in the brittle layers followed by cracking in the interlayers is shown to be capable of predicting the observed increase in crack initiation strain with decreasing layer thickness. To realize this benefit the interlayer must be compliant and tough to force tunnel cracking in the brittle layers. The explicit relation for the crack initiation strain obtained from the analysis can be used to assess fracture toughness and improve damage tolerance in nanoscale layered structures.