General forms of elastic-plastic matching equations for mode-Ⅲ cracks near crack line

Crack line analysis is an effective way to solve elastic-plastic crack problems. Application of the method does not need the traditional small-scale yielding conditions and can obtain sufficiently accurate solutions near the crack line. To address mode- Ⅲ crack problems under the perfect elastic-plastic condition, matching procedures of the crack line analysis method axe summarized and refined to give general forms and formulation steps of plastic field, elastic-plastic boundary, and elastic-plastic matching equations near the crack line. The research unifies mode-III crack problems under different conditions into a problem of determining four integral constants with four matching equations. An example is given to verify correctness, conciseness, and generality of the procedure.     

The singularity analysis of the stress and strain fields for Mode I fracture in elastic-plastic state

The stress and strain singularities of power hardening material for Mode I fracrure are analysed according to the fundamental equations of elastic-plastic mechanics. It is found that the singularities of all stress and strain components do not change in the thick direction, and neither the six stress components nor the six strain components have the same singularity.     

A plane stress formulation for elastic-plastic deformation of unidirectional composites

A plane stress analysis of the elastic-plastic deformation of unidirectional composites is presented. A continuum model based on the solid-mixture concept is selected as the basis for the analysis. Model parameters, including process-dependent variables, are deduced from experiments performed on unidirectional composites. A computer program MET1MAT has been developed accordingly and tested for a few simple in-plane loading cases. Experimental data for uniaxial tests performed in longitudinal and transverse directions and for a few biaxial tests are presented to substantiate the analysis. And, finally, application of the results to laminated metal matrix composites is discussed.     

Finite element analysis of elastic-plastic plane stress problems based upon Tresca yield criterion

Summary A finite element technique for the elastic-plastic analysis of two dimensional structures subjected to conditions of plane stress and monotonically increasing loads is presented. The complete load deformation history as well as the propagation of the yield zones in the structure up to plastic collapse are studied. The material is assumed to be elastic-perfectly plastic and yielding is governed by Tresca yield condition. Plastic stress-strain relations for the sides and corners of Tresca yield condition are derived in terms of the components of the stresses and strains along a fixed reference coordinate system and the direction of the principal stress. The load is applied in small increments and the principal stress direction for each plastic element during a load increment is determined by an interpolation technique which leads to stresses that satisfy the yield condition. Numerical examples are given to illustrate the accuracy of the results obtained by the proposed method.

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Übersicht Es wird eine finite element-Methode vorgeschlagen zur Berechnung zweidimensionaler elastischplastischer Baukonstruktionen, die einem ebenen Spannungszustand mit monoton wachsender Last unterworfen sind. Das Belastungs-Verformungs-Verhalten und die Ausbreitung der Fließbereiche bis zum plastischen Zusammenbruch werden untersucht. Das Material soll der Fließbedingung von Tresca genügen. Spannungs-Verformungs-Berechnungen für die Ränder und Ecken des Tresca-Bereiches werden in Komponenten der Spannungen und Dehnungen längs der Achsriclitungen eines festen Bezugssystems und der Hauptspannungsrichtungen ausgedrückt. Die Belastung wird in kleinen Stufen aufgebracht, und die Hauptspannungsrichtungen werden für jedes plastische Teilchen während des Lastaufbringens durch Interpolation bestimmt. Die so erhaltenen Spannungen genügen der Fließbedingung. Durch numerische Beispiele wird die Genauigkeit des vorgeschlagenen Verfahrens demonstriert.

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On leave from Northwestern University, Evanston, Illinois     

Experimental methods of X-ray stress analysis

A technique is described for measuring residual stresses in steels and composite materials by X-ray diffraction. Specimen preparation, X-ray diffractometer alignment, diffraction-peak location, and the determination of the elastic modulus, Poisson's ratio and stress factor are covered. Application examples include measurement of heat-treating and shot-peening stresses in steels and grinding and temperature stresses in WC-Co composites.     

Combining photoelasticity and finite-element methods for stress analysis using least squares

Photoelastic data are combined with the finite-element method for stress solutions over regions partially bounded by free surfaces and axes of symmetry. Least-squares solutions are obtained without presumed values of applied forces at element nodes and without isoclinic data. Varied example problems are used to compare the results to independent photoelastic and finite-element solutions and to theoretical stress values.Paper was presented at the 1989 SEM Spring Conference on Experimental Mechanics held in Cambridge, MA on May 28–June 1.     

A pattern matching technique for measuring sediment displacement levels

This paper describes a novel technique for obtaining accurate, high (spatial) resolution measurements of sediment redeposition levels. A sequence of different random patterns are projected onto a sediment layer and captured using a high-resolution camera, producing a set of reference images. The same patterns are used to obtain a corresponding sequence of deformed images after a region of the sediment layer has been displaced and redeposited, allowing the use of a high-accuracy pattern matching algorithm to quantify the distribution of the redeposited sediment. A set of experiments using the impact of a vortex ring with a glass ballotini particle layer as the resuspension mechanism are described to test and illustrate the technique. The accuracy of the procedure is assessed using a known crater profile, manufactured to simulate the features of the craters observed in the experiments.

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Cavity nucleation, growth and stress field in elastic-plastic medium

A finite element method is used to study the cavity nucleation and growth in an elastic-plastic medium. The critical cohesion strength, _c at IPM (interface between second phase particle and matrix) is employed as the criterion of cavity formation. Three different values of _c are taken to examine their influence on the overall mechanical behaviour and process of cavity formation.     

A study of the hybrid element methods for stress analysis

A new functional which forms the basis of an improved hybrid element formulation is proposed. The variables for the functional include stresses, strains and displacements, and the displacements and stresses are further decomposed into two parts respectively. The proposed new formulation appears to be particularly suitable for improving conforming models.The relationship between the new hybrid elements and the conventional displacement elements are also explored in this paper.     

A maximum principle for the analysis of elastic-plastic systems

Summary We consider an elastic-plastic body subjected to a specific programme of static loading. In the analysis of the behaviour of the body in the course of the individual steps of the loading programme some difficulties arise if the variational principles of the theory of plasticity are applied. We then propose a maximum principle which appears suitable for the formulation of simple and direct computing procedures. For an elastic-perfectly plastic material the function to be maximized represents the differential energy dissipated in the single infinitesimal step starting from the elastic solution. In that function the variables are the plastic distortions. Since the energy dispersed by the effect of these latter must at every point be positive or zero, the maximum in question is a field maximum and therefore the property is not variational.The principle is demonstrated both for elastic-perfectly plastic materials and for elastic—work-hardening materials. Materials with regular yield surfaces are at first considered; the demonstration is then extended to the case of singular yield surfaces.First published in Rendiconti dell'Istituto Lombardo, Classe di Scienze, A 99, 125–140, 1965.     

A weighted residual method for elastic-plastic analysis near a crack tip and the calculation of the plastic stress intensity factors

In this paper,a weighted residual method for the elastic-plastic analysis near a crack tip is systematically given by taking the model of power-law hardening under plane strain condition as a sample.The elastic-plastic solutions of the crack tip field and an approach based on the superposition of the nonlinear finite element method on the complete solution in the whole crack body field,to calculate the plastic stress intensity factors,are also developed.Therefore,a complete analysis based on the calculation both for the crack tip field and for the whole crack body field is provided.     

Refractive index matching methods for liquid flow investigations

A difficulty common to most optical diagnostic techniques that are applied to fluid dynamics studies is the refraction of light passing through model and/or test section walls. The method of choice to eliminate refraction problems in liquid flows is to match refractive index. This paper presents techniques for refractive index matching including, (i) arrangement of test section and model, (ii) choice of solid and liquid materials, and (iii) methods for tuning the match. In addition, a new application of refractive index matching to liquid-liquid droplet studies is presented.The author would like to thank Rick James and Jon Martinez for their assistance in conducting the matching experiments and Professor Mickey Gunter for his comments on refractometry. We are grateful to the United States Department of Energy for financial support of the droplet work under Grant DE-FG07-86ER13572     

CCD automated polariscope system and the stress analysis methods

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Hybrid stress analysis by digitized photoelastic data and numerical methods

A method is presented that determines photoelastic isochromatic values at the nodal points of a grid mesh which in turn is generated by a computer program that accepts digitized input. Values of σ₁ - σ₂ are computed from the digitized fringe orders. The Laplace equation is solved to separate the principal stresses at each nodal point. The method is extended to digitize isoclinics. Subsequently, σ _x - σ _y and τ _xy are calculated to be used as starting values for the solution of the pertaining partial differential equations to enhance convergence. For further accelerating the rate of convergence, superfluous boundary conditions are added from the digitized data; significant improvement is demonstrated. Estimated values of σ _x - σ _y from the digitized data are further used in conjunction with the solution of the Laplace equation to determine the state of stress without solving the boundary value problems. Paper was presented at the 1988 SEM Spring Conference on Experimental Mechanics held in Portland, OR on June 5–10.     

THE EXACT SOLUTIONS OF ELASTIC－PLASTIC CRACK LINE FIELD FOR MODE ⅡPLANE STRESS CRACK

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Displacement and strain measurement with automated grid methods

An image-processing-based automated grid method is investigated to determine the method's displacement and strain accuracy limits, and how these limits are influenced by the choice of camera-calibration models. A CCD camera and a PC-based frame grabber are used to record grid spot motion, then ordering and centroiding are used to identify each spot and calculate their individual displacements. The displacements are fitted with a moving biquadratic surface, and the strains are obtained by analytical differentiation of that surface. Camera-calibration models which are considered include various combinations of image-perspective transformation, image stretching, and elliptical-lens distortion. The strain and displacement accuracy are explored through rigid-body motion and uniaxial tension tests. In the process, sensitivity to in-plane and out-of-plane rigid-body translation, and extreme sensitivity to in-plane rigid-body rotation (for non-synchronized frame grabbers) are confirmed. It is found that under the best conditions the displacement accuracy is 015 pixels and that the strain accuracy is 120 microstrain. Finally, the automated grid method is used to investigate the strains developed in an aluminum perforated strip subjected to uniaxial tension.     

Sensitivity analysis and optimum design of elastic-plastic structural systems

The paper deals with elastic-plastic optimization of flexural structural systems, subjected to kinematic restrictions. A finite holonomic piecewise linear elastic-hardening constitutive law is adopted. Sensitivity analysis for the displacement field is also performed, and a suitable finite element formulation, allowing for the spreading of plasticity, is also given. Finally, some meaningful numerical applications, together with their physical interpretation, are presented.Paper presented at OPT189: Computer Aided Optimum Design of Structures, Southampton, UK, 20–23 June 1989.     

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.     

Refractive index matching and marking methods for highly concentrated solid–liquid flows

 Two index-matched systems of solid particles in liquid have been developed to enable the study of velocity and concentration distributions in the highly concentrated solid–liquid flows. The mixtures have excellent optical transparency up to depths exceeding 80 mm at concentrations up to 50% solids by volume. Highly visible marker particles with nearly the same mechanical properties as the index-matched particles are formed by metal plating or substitution. Good quality images of marker particles are obtained in both stationary and moving two-phase mixtures, and permit accurate tracking of the individual markers. Reeceived: 4 March 1996/Accepted: 14 June 1996     

Testing theoretical bases of caustic methods in fracture mechanics and stress analysis

Determination of stress intensity factors using shadow optical method of caustics attracts increasing attention. However, the reliability and ranges of applicability of this method are seldom discussed. This paper presents such an analysis regarding the reliability and predictive power of caustic method in fracture mechanics. This analysis is performed according to the accepted methodology of testing any analytical, numerical, or experimental procedure, namely by testing its theoretical bases. The following four basic assumptions of the shadow optical method of caustics in fracture mechanics, and their consequences, are analyzed: assumption of a generalized plane stress state near crack tip; assumption that the radiant energy used to produce shadow images propagates rectilinearly inside a stressed body; assumption that the light velocity inside a stressed body depends only on stress components in planes normal to wave normals; assumption that there is no alteration in state of polarization of radiation impinging upon a stressed plate at oblique incidence.The first two basic assumptions are tested using results obtained by means of three analytical-experimental procedures, namely isodynes, gradient index method, and classical strain gages. It is known that the inaccuracies of the fourth assumption are within the noise level only when the angles of incidence are small.It is shown that the magnitudes of the effects caused by the geometric lens effect and the gradient index lens effect are comparable, and that the stress states near crack tips and bottoms of notches, which produce the geometric lens effect, are clearly three-dimensional.It is also indicated that the gradient index lens effect is caused jointly by the stress/strain-induced alteration of the optical path and by the bending of the light path caused by strain gradient.Obtained empirical evidence shows the existence of a particular relationship between the observed gradient index lens effect and the order of singularity in a particular singular solution of linear fracture mechanics, with the exception of the vicinity of the crack tip where singularities are inadmissible.