Towards a catastrophe theory for the mechanics of plasticity and fracture

The catastrophe theory has established that the conservative systems have only a few ways to lose their stability. We sketch a similar classification for a class of dissipative systems. Although our scope is limited to the case of two dissipative variables, we find six elementary catastrophes. We present examples from discrete plasticity, crack and friction mechanics.     

Nonlinear elastic mechanics of the ball-loaded blister test

The nonlinear elastic mechanics of spherically capped shaft or ball-loaded blister tests is presented. In the test model, a thin film is attached to a substrate with a circular hole running through the thickness of the substrate. A central load is applied to the film through the hole by a spherically capped shaft or a ball with a finite radius. The deformed blister is divided into two parts: a circular region in contact with the sphere of the cap or ball and an outer noncontact annulus. The Reissner’s plate theory is employed to describe the deformation of the contact part and the von Kármán plate theory for the noncontact annulus. A constitutive equation of coupled linear springs is obtained to quantify the effect of the substrate deformation on the blister deflection. For small deflection, the analytical solution of load-deflection is derived. For large deflection, an iteration approach is adopted to predict numerically the load-deflection curve. Finite-element analysis is conducted to verify the analytical and numerical solutions. The influence of the substrate deformation, residual stress, radius of the spherical cap or ball and the friction between the film and ball on the load-deflection relation is investigated.     

Limitations of elastic definitions in Al-Si-Mg cast alloys with enhanced plasticity: linear elastic fracture mechanics versus elastic-plastic fracture mechanics

Linear elastic fracture mechanics describes the fracture behavior of materials and components that respond elastically under loading. This approach is valuable and accurate for the continuum analysis of crack growth in brittle and high strength materials; however it introduces increasing inaccuracies for low-strength/high-ductility alloys (particularly low-carbon steels and light metal alloys). In the case of ductile alloys, different degrees of plastic deformation precede and accompany crack initiation and propagation, and a non-linear ductile fracture mechanics approach better characterizes the fatigue and fracture behavior under elastic-plastic conditions.To delineate plasticity effects in upper Region II and Region III of crack growth an analysis comparing linear elastic stress intensity factor ranges (ΔK_el) with crack tip plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) is presented. To compute plasticity corrected stress intensity factor ranges (ΔK_pl), a new relationship for plastic zone size determination was developed taking into account effects of plane-strain and plane-stress conditions (“combo plastic zone”). In addition, for the upper part of the fatigue crack growth curve, elastic-plastic (cyclic J based) stress intensity factor ranges (ΔK_J) were computed from load-displacement records and compared to plasticity corrected stress intensity factor ranges (ΔK_pl). A new cyclic J analysis was designed to compute elastic-plastic stress intensity factor ranges (ΔK_J) by determining cumulative plastic damage from load-displacement records captured in load-control (K-control) fatigue crack growth tests. The cyclic J analysis provides the true fatigue crack growth behavior of the material. A methodology to evaluate the lower and upper bound fracture toughness of the material (J_IC and J_max) directly from fatigue crack growth test data (ΔK_FT(JIC) and ΔK_FT(Jmax)) was developed and validated using static fracture toughness test results. The value of ΔK_FT(JIC) (and implicitly J_IC) is determined by comparing the plasticity corrected elastic fatigue crack growth curve with the elastic-plastic fatigue crack growth curve. A most relevant finding is that plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) are in remarkably good agreement with cyclic J analysis results (ΔK_J), and provide accurate plasticity corrections up to a ΔK corresponding to J_IC (i.e. ΔK_FT(JIC)). Towards the end of the fatigue crack growth test (above ΔK_FT(JIC)) when plasticity is accompanied by significant tearing, the cyclic J analysis provides a more accurate way to capture the true behavior of the material and determine ΔK_FT(Jmax). A procedure to decouple and partition plasticity and tearing effects on crack growth rates is given.Three cast Al-Si-Mg alloys with different levels of ductility, provided by different Si contents and heat treatments (T61 and T4) are evaluated, and the effects of crack tip plasticity on fatigue crack growth are assessed. Fatigue crack growth tests were conducted at a constant stress ratio, R = 0.1, using compact tension specimens.     

Solutions of the second elastic-plastic fracture mechanics parameter in test specimens

Extensive finite element analyses have been conducted to obtain solutions of the A-term, which is the second parameter in three-term elastic-plastic asymptotic expansion, for test specimens. Three mode I crack plane-strain test specimens, i.e. single edge cracked plate (SECP), center cracked plate (CCP) and double edge cracked plate (DECP) were studied. The crack geometries analyzed included shallow to deep cracks. Solutions of A-term were obtained for material following the Ramberg-Osgood power law with hardening exponent of n = 3, 4, 5, 7 and 10. Remote tension loading was applied which covers from small-scale to large-scale yielding. Based on the finite element results, empirical equations to predict the A-terms under small-scale yielding (SSY) to large-scale yielding conditions were developed. In addition, by using the relationships between A and other commonly used second fracture parameters such as Q factor and A₂-term, the present solutions can be used to calculate parameters A₂ and Q as well. The results presented in the paper are suitable to calculate the second elastic-plastic fracture parameters for test specimens for a wide range of crack geometries, material strain hardening behaviors and loading conditions.     

A fracture mechanics analysis of the double cleavage drilled compression test specimen

The “double cleavage drilled compression” specimen is used for the determination of crack growth rates under mode-I and general mixed-mode loading conditions. In this paper, a fracture mechanics analysis is presented for the DCDC specimen, using a finite-element analysis to determine stresses, strains and surface displacements. Weight functions are derived, and the consequence of load misalignment is demonstrated. Finally, side-surface displacements at the crack terminating region are studied using a 3-dimensional, finite-element computation in combination with results from the literature that are based on an ‘‘asymptotic analysis”. Calculated side-surface displacements in the vicinity of the crack tip are found to agree with published values obtained by others via measurements using the atomic force microscope.     

Use of a plasticity condition for porous bodies to solve certain fracture mechanics problems

An analysis is made of the effect of porosity on the fracture of a tensioned specimen. The problem for a plane stress state and plane strain is solved using a plasticity condition for porous bodies. Relations are obtained for the dependence of the dilatation rate in the neck of the specimen and the normal and tangential components of the stresses on the porosity of the material. The effect of porosity on the size of the plastic zone at the crack tip is also examined.Translated from Problemy Prochnosti, No. 6, pp. 17–26, November–December, 1996.     

Elastic-plastic fracture mechanics assessment of test data for circumferential cracked pipes

This paper presents experimental validation of two reference stress based methods for circumferential cracked pipes. One is the R6 method where the reference stress is defined by the plastic limit load. The other is the enhanced reference stress method, recently proposed by the authors, where the reference stress is defined by the optimised reference load. Using 38 published pipe test data, the predicted maximum instability loads according to both methods are compared with the experimental ones for pipes with circumferential through-thickness cracks and with part circumferential surface cracks. It is found that the R6 method gives conservative estimates of the maximum loads for all cases. Ratios of the experimental maximum load to the predicted load range from 0.54 to 0.98. On the other hand, the proposed method gives overall closer maximum loads than R6, compared to the experimental data. However, for part through-thickness surface cracks, the estimated loads were slightly non-conservative for four cases, and possible reasons are fully discussed.     

Strength of concrete under combined compression and tension-determination of interaction curve at failure from cylinder split test

Summary Interaction curve of concrete failure in the compression— tension combination of a biaxial stress situation is determined from the stress distribution existing on the loaded diameter of a cylinder under split test. From the analysis, the strength of concrete in pure compression and tension are also determined in terms of the splitting tensile strength. The results compare favourably with those of other investigators.

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Résumé D'après la distribution des contraintes sur le diamètre de la section chargée d'un cylindre soumis à l'essai brésilien, on détermine le diagramme d'interaction de la rupture du béton sous effort combiné en compression-traction. L'analyse permet de déterminer aussi, en termes de résistance à la traction déterminée par essai brésilien, la résistance du béton en compression et traction pure. Les résultats s'accordent assez bien avec ceux d'autres expérimentateurs.

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Fatigue and fracture mechanics

The application of methods of fracture mechanics to the prediction of fatigue life presupposes the existence of a single flaw of ‘critical’ size the slow propagation of which, under repeated cyclic loading, represents the relevant damage mechanism that governs ‘fatigur’ until the flaw has grown to unstable size. The conditions under which this approach to fatigue provides a reasonable model of real behavior are, however, exactly the conditions that should be avoided by adequate fatigue design. Thus the merit of fracture mechanics with respect to fatigue is not in the realistic modeling of the fatigue process in its various aspects, but in the delimitation of the conditions under which this process approaches the model too closely for purposes of design. The paper discusses the different aspects of the fatigue process in relation to the basic concepts of fracture mechanics.     

The WST method, a fracture mechanics test method for FRC

The applicability of the wedge-splitting test method (WST), for determining fracture properties of fibre-reinforced concrete, is discussed. Experimental results, using the WST method, are compared with results from uniaxial tension tests (UTT) and three-point bending tests (3PBT) for five different FRC compositions. Furthermore, for the WST method, two different specimen sizes have been investigated. Results from this investigation demonstrate the applicability of the WST method and show that the scatter of the test results is lower than for the 3PBT. Through inverse analysis, stress–crack opening (σ–w) relationships have been determined for each mix and test method. For the two WST specimen sizes, there is no apparent difference either in the number of fibres (per cm²) crossing the fracture plane or in the fracture properties. The major factor contributing to the scatter in the test results is believed to be related to the variation in the number of fibres across the fracture plane. Furthermore, the inverse analyses indicate no systematic differences in the determined parameters between two WST specimens or between the WST and the 3PBT.

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Résumé L’article concerne l’applicabilité de la méthode d’essai d’écartement par enfoncement d’un coin (WST) pour déterminer les propriétés de fissures du béton renforcé de fibres. Les résultats expérimentaux utilisant la méthode WST sont comparés aux résultats des essais de traction uniaxiale (UTT) et des essais de flexion trois points (3PBT) pour cinq compositions différentes de béton renforcé de fibres (FRC). Par ailleurs, en ce qui concerne la méthode WST, deux différentes tailles d’éprouvettes ont été étudiées. Les résultats de cette étude confirme l’applicabilité de la méthode WST et montre que la dispersion des résultats de l’essai est inférieure à celle obtenue avec la méthode 3PBT. A partir de l’analyse inverse, la courbe de comportement du matériau contrainte–ouverture de fissure (σ–w) a été déterminée pour chaque composition et méthode d’essai. Pour les deux tailles d’éprouvette WST, aucune différence apparente n’est visible, ni dans le nombre de fibres (par cm²) traversant le plan de rupture, ni dans les propriétés de fissures. On estime que le principal facteur contribuant à la dispersion des résultats d’essai se rapporte à la variation du nombres de fibres à travers le plan de rupture. De plus, les analyses inverses n’indiquent aucunes différences systématiques dans les paramètres définis entre deux éprouvettes WST ou entre la méthode WST et 3PBT.

     

Effect of adhesive layer elasticity on the fracture mechanics of a blister test specimen

An analytical model of a blister type specimen for evaluation of adhesive bond strength is developed. The model accounts for shear deformation of both the plate and the adhesive layer in order to determine the extent to which a blister specimen may be considered to be a mixed mode fracture specimen. A method for relating the results of numerical stress analysis to energy balance concepts of fracture mechanics is presented. It is shown that the shear deformations have only a small effect on energy balance; however, the magnitude of the shear stress in the adhesive layer may reach a value of 25 percent of normal stress.

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Résumé On développe un modèle analytique d'une éprouvette simulant un cloquage pour l'évaluation de la résistance d'un joint collé. Le modèle tient compte des déformations de cisaillement de la tôle et de l'adhésif afin de déterminer dans quelle mesure l'éprouvette de cloquage peut être considérée comme une éprouvette où se produit un mode de rupture mixte. Une méthode pour relier les résultats de l'analyse numérique des contraintes au concept d'équilibre d'énergie de la mécanique de rupture est présentée. On montre que les déformations de cisaillement n'ont qu'un effet secondaire sur l'équilibre d'énergie; toutefois, la tension de cisaillement dans la couche adhésive peut atteindre une valeur de l'ordre de grandeur de 25% de la tension normale.

     

低渗砂岩力学试验及复杂断块裂缝分布预测研究

通过岩石抗拉强度试验和快速直接剪断试验对江苏油田某区块多口油井岩芯进行试验,获取相应层段的岩石抗拉强度及抗剪强度,利用测井资料中的声波等数据得到井筒附近的岩石力学参数。应用非连续模型中的接触单元描述断层系统与地层间的非连续错动,考虑沉积相引起地层的非均质性,运用多目标约束优化反分析法对江苏油田某区块含复杂断裂系统的致密砂岩油藏地应力场进行有限元数值模拟,实现地应力场的精细预测,得到复杂断块致密砂岩油藏地应力场等值线图。针对区块出现拉伸破坏和剪切破坏,利用Griffith张破裂准则和Coulomb-Navier剪破裂准则对油藏区域进行拉伸裂缝及剪切裂缝预测,最终得到裂缝综合发育分布图,确定裂缝相对发育区域。研究结果表明:断层歼灭处及复合断层交叉处地应力变化较大,对地应力场分布影响较大;断层内部裂缝较为发育,受断层系统影响复杂的区域,裂缝发育不均衡,且裂缝发育程度受沉积相的影响较大。     

Analysis of modified split Hopkinson pressure bar dynamic fracture test using an inertia model

A split Hopkinson pressure bar (two-bar set-up) has been modified to perform dynamic three-point bend tests to measure dynamic fracture toughness, and to understand the influences of various experimental parameters, as well as inertial effects, on the dynamic material response. Modeling and analysis of the dynamic three-point bend test, as loaded by a modified split Hopkinson pressure bar, is conducted. The effects of support motion, crack propagation and plastic contact stiffness on total sample deflection are investigated. The effects of crack propagation and plastic contact stiffness on the contribution of support motion to the total sample deflection are also investigated theoretically and experimentally in this paper. Further, the effects of crack propagation and plastic contact stiffness on impactor and sample load are also addressed.     

Parameters identification of polymer concrete using a fracture mechanics test method and full-field measurements

In the present work it is proposed an easy-to-implement alternative procedure to identify not only the fracture properties but also the mean elasticity modulus E and Poisson’s ratio ν of arbitrary polymer concretes. Only one test using a standard single-edge-cracked three-point bend specimen is necessary. Digital image correlation (DIC) method is used to obtain the full-field displacement close to the crack tip. The mean properties (E, ν) are determined by fitting analytical expressions for the displacement field to the experimental data. The adequacy of the proposed methodology is checked by comparing the material parameters obtained using the proposed procedure with the ones obtained through standard procedures.     

Conformal mapping and fracture mechanics

A mathematical model is applied to the problem of using a drop in electric potential to measure the size of a crack in a metal specimen. The approach is through conformal mapping and the Schwarz-Christoffel formula. The integrals and computations of maps involved are done with a computer package called SCPACK.