Showing papers on "Strain energy release rate published in 1996"

Mechanical properties of ceramics

Abstract: Preface. Acknowledgments. 1 Stress and Strain. 1.1 Introduction. 1.2 Tensor Notation for Stress. 1.3 Stress in Rotated Coordinate System. 1.4 Principal Stress. 1.4.1 Principal Stresses in Three Dimensions. 1.5 Stress Invariants. 1.6 Stress Deviator. 1.7 Strain. 1.8 True Stress and True Strain. 1.8.1 True Strain. 1.8.2 True Stress. Problems. 2 Types of Mechanical Behavior. 2.1 Introduction. 2.2 Elasticity and Brittle Fracture. 2.3 Permanent Deformation. 3 Elasticity. 3.1 Introduction. 3.2 Elasticity of Isotropic Bodies. 3.3 Reduced Notation for Stresses, Strains, and Elastic Constants. 3.4 Effect of Symmetry on Elastic Constants. 3.5 Orientation Dependence of Elastic Moduli in Single Crystals and Composites. 3.6 Values of Polycrystalline Moduli in Terms of Single-Crystal Constants. 3.7 Variation of Elastic Constants with Lattice Parameter. 3.8 Variation of Elastic Constants with Temperature. 3.9 Elastic Properties of Porous Ceramics. 3.10 Stored Elastic Energy. Problems. 4 Strength of Defect-Free Solids. 4.1 Introduction. 4.2 Theoretical Strength in Tension. 4.3 Theoretical Strength in Shear. Problems. 5 Linear Elastic Fracture Mechanics. 5.1 Introduction. 5.2 Stress Concentrations. 5.3 Griffith Theory of Fracture of a Brittle Solid. 5.4 Stress at Crack Tip: An Estimate. 5.5 Crack Shape in Brittle Solids. 5.6 Irwin Formulation of Fracture Mechanics: Stress Intensity Factor. 5.7 Irwin Formulation of Fracture Mechanics: Energy Release Rate. 5.8 Some Useful Stress Intensity Factors. 5.9 The J Integral. 5.10 Cracks with Internal Loading. 5.11 Failure under Multiaxial Stress. Problems. 6 Measurements of Elasticity, Strength, and Fracture Toughness. 6.1 Introduction. 6.2 Tensile Tests. 6.3 Flexure Tests. 6.4 Double-Cantilever-Beam Test. 6.5 Double-Torsion Test. 6.6 Indentation Test. 6.7 Biaxial Flexure Testing. 6.8 Elastic Constant Determination Using Vibrational and Ultrasonic Methods. Problems. 7 Statistical Treatment of Strength. 7.1 Introduction. 7.2 Statistical Distributions. 7.3 Strength Distribution Functions. 7.4 Weakest Link Theory. 7.5 Determining Weibull Parameters. 7.6 Effect of Specimen Size. 7.7 Adaptation to Bend Testing. 7.8 Safety Factors. 7.9 Example of Safe Stress Calculation. 7.10 Proof Testing. 7.11 Use of Pooled Fracture Data in Linear Regression Determination of Weibull Parameters. 7.12 Method of Maximum Likelihood in Weibull Parameter Estimation. 7.13 Statistics of Failure under Multiaxial Stress. 7.14 Effects of Slow Crack Propagation and R-Curve Behavior on Statistical Distributions of Strength. 7.15 Surface Flaw Distributions and Multiple Flaw Distributions. Problems. 8 Subcritical Crack Propagation. 8.1 Introduction. 8.2 Observed Subcritical Crack Propagation. 8.3 Crack Velocity Theory and Molecular Mechanism. 8.4 Time to Failure under Constant Stress. 8.5 Failure under Constant Stress Rate. 8.6 Comparison of Times to Failure under Constant Stress and Constant Stress Rate. 8.7 Relation of Weibull Statistical Parameters with and without Subcritical Crack Growth. 8.8 Construction of Strength-Probability-Time Diagrams. 8.9 Proof Testing to Guarantee Minimum Life. 8.10 Subcritical Crack Growth and Failure from Flaws Originating from Residual Stress Concentrations. 8.11 Slow Crack Propagation at High Temperature. Problems. 9 Stable Crack Propagation and R -Curve Behavior. 9.1 Introduction. 9.2 R-Curve (T-Curve) Concept. 9.3 R-Curve Effects of Strength Distributions. 9.4 Effect of R Curve on Subcritical Crack Growth. Problems. 10 Overview of Toughening Mechanisms in Ceramics. 10.1 Introduction. 10.2 Toughening by Crack Deflection. 10.3 Toughening by Crack Bowing. 10.4 General Remarks on Crack Tip Shielding. 11 Effect of Microstructure on Toughness and Strength. 11.1 Introduction. 11.2 Fracture Modes in Polycrystalline Ceramics. 11.3 Crystalline Anisotropy in Polycrystalline Ceramics. 11.4 Effect of Grain Size on Toughness. 11.5 Natural Flaws in Polycrystalline Ceramics. 11.6 Effect of Grain Size on Fracture Strength. 11.7 Effect of Second-Phase Particles on Fracture Strength. 11.8 Relationship between Strength and Toughness. 11.9 Effect of Porosity on Toughness and Strength. 11.10 Fracture of Traditional Ceramics. Problems. 12 Toughening by Transformation. 12.1 Introduction. 12.2 Basic Facts of Transformation Toughening. 12.3 Theory of Transformation Toughening. 12.4 Shear-Dilatant Transformation Theory. 12.5 Grain-Size-Dependent Transformation Behavior. 12.6 Application of Theory to Ca-Stabilized Zirconia. Problems. 13 Mechanical Properties of Continuous-Fiber-Reinforced Ceramic Matrix Composites. 13.1 Introduction. 13.2 Elastic Behavior of Composites. 13.3 Fracture Behavior of Composites with Continuous, Aligned Fibers. 13.4 Complete Matrix Cracking of Composites with Continuous, Aligned Fibers. 13.5 Propagation of Short, Fully Bridged Cracks. 13.6 Propagation of Partially Bridged Cracks. 13.7 Additional Treatment of Crack-Bridging Effects. 13.8 Additional Statistical Treatments. 13.9 Summary of Fiber-Toughening Mechanisms. 13.10 Other Failure Mechanisms in Continuous, Aligned-Fiber Composites. 13.11 Tensile Stress-Strain Curve of Continuous, Aligned-Fiber Composites. 13.12 Laminated Composites. Problems. 14 Mechanical Properties of Whisker-, Ligament-, and Platelet-Reinforced Ceramic Matrix Composites. 14.1 Introduction. 14.2 Model for Whisker Toughening. 14.3 Combined Toughening Mechanisms in Whisker-Reinforced Composites. 14.4 Ligament-Reinforced Ceramic Matrix Composites. 14.5 Platelet-Reinforced Ceramic Matrix Composites. Problems. 15 Cyclic Fatigue of Ceramics. 15.1 Introduction. 15.2 Cyclic Fatigue of Metals. 15.3 Cyclic Fatigue of Ceramics. 15.4 Mechanisms of Cyclic Fatigue of Ceramics. 15.5 Cyclic Fatigue by Degradation of Crack Bridges. 15.6 Short-Crack Fatigue of Ceramics. 15.7 Implications of Cyclic Fatigue in Design of Ceramics. Problems. 16 Thermal Stress and Thermal Shock in Ceramics. 16.1 Introduction. 16.2 Magnitude of Thermal Stresses. 16.3 Figure of Merit for Various Thermal Stress Conditions. 16.4 Crack Propagation under Thermal Stress. Problems. 17 Fractography. 17.1 Introduction. 17.2 Qualitative Features of Fracture Surfaces. 17.3 Quantitative Fractography. 17.4 Fractal Concepts in Fractography. 17.5 Fractography of Single Crystals and Polycrystals. Problems. 18 Dislocations and Plastic Deformation in Ductile Crystals. 18.1 Introduction. 18.2 Definition of Dislocations. 18.3 Glide and Climb of Dislocations. 18.4 Force on a Dislocation. 18.5 Stress Field and Energy of a Dislocation. 18.6 Force Required to Move a Dislocation. 18.7 Line Tension of a Dislocation. 18.8 Dislocation Multiplication. 18.9 Forces between Dislocations. 18.10 Dislocation Pileups. 18.11 Orowan's Equation for Strain Rate. 18.12 Dislocation Velocity. 18.13 Hardening by Solid Solution and Precipitation. 18.14 Slip Systems. 18.15 Partial Dislocations. 18.16 Deformation Twinning. Problems. 19 Dislocations and Plastic Deformation in Ceramics. 19.1 Introduction. 19.2 Slip Systems in Ceramics. 19.3 Independent Slip Systems. 19.4 Plastic Deformation in Single-Crystal Alumina. 19.5 Twinning in Aluminum Oxide. 19.6 Plastic Deformation of Single-Crystal Magnesium Oxide. 19.7 Plastic Deformation of Single-Crystal Cubic Zirconia. Problems. 20 Creep in Ceramics. 20.1 Introduction. 20.2 Nabarro-Herring Creep. 20.3 Combined Diffusional Creep Mechanisms. 20.4 Power Law Creep. 20.5 Combined Diffusional and Power Law Creep. 20.6 Role of Grain Boundaries in High-Temperature Deformation and Failure. 20.7 Damage-Enhanced Creep. 20.8 Superplasticity. 20.9 Deformation Mechanism Maps. Problems. 21 Creep Rupture at High Temperatures and Safe Life Design. 21.1 Introduction. 21.2 General Process of Creep Damage and Failure in Ceramics. 21.3 Monkman-Grant Technique of Life Prediction. 21.4 Two-Stage Strain Projection Technique. 21.5 Fracture Mechanism Maps. Problems. 22 Hardness and Wear. 22.1 Introduction. 22.2 Spherical Indenters versus Sharp Indenters. 22.3 Methods of Hardness Measurement. 22.4 Deformation around Indentation. 22.5 Cracking around Indentation. 22.6 Indentation Size Effect. 22.7 Wear Resistance. Problems. 23 Mechanical Properties of Glass and Glass Ceramics. 23.1 Introduction. 23.2 Typical Inorganic Glasses. 23.3 Viscosity of Glass. 23.4 Elasticity of Inorganic Glasses. 23.5 Strength and Fracture Surface Energy of Inorganic Glasses. 23.6 Achieving High Strength in Bulk Glasses. 23.7 Glass Ceramics. Problems. 24 Mechanical Properties of Polycrystalline Ceramics in General and Design Considerations. 24.1 Introduction. 24.2 Mechanical Properties of Polycrystalline Ceramics in General. 24.3 Design Involving Mechanical Properties. References. Index.

Microbranching instability and the dynamic fracture of brittle materials.

Abstract: We describe experiments on the dynamic fracture of the brittle plastic, PMMA. The results suggest a view of the fracture process that is based on the existence and subsequent evolution of an instability, which causes a single crack to become unstable to frustrated microscopic branching events. We demonstrate that a number of long-standing questions in the dynamic fracture of amorphous, brittle materials may be understood in this picture. Among these are the transition to crack branching, ``roughness'' and the origin of nontrivial fracture surface, oscillations in the velocity of a moving crack, the origin of the large increase in the energy dissipation of a crack with its velocity, and the large discrepancy between the theoretically predicted asymptotic velocity of a crack and its observed maximal value. Also presented are data describing both microbranch distribution and evidence of a new three-dimensional to two-dimensional transition as the ``correlation width'' of a microbranch diverges at high propagation velocities. \textcopyright{} 1996 The American Physical Society.

Energy dissipation in dynamic fracture.

Abstract: Measurements in PMMA of both the energy flux into the tip of a moving crack and the total surface area created via the microbranching instability indicate that the instability is the main mechanism for energy dissipation by a moving crack in brittle, amorphous material. Beyond the instability onset, the rate of fracture surface creation is proportional to the energy flux into the crack. At high velocities microbranches create nearly an order of magnitude larger fracture surface than smooth cracks. This mechanism provides an explanation for why the theoretical limiting velocity of a crack is never realized. PACS numbers: 68.35.Gy, 62.20.Mk, 83.50.Tq Although the subject of much research over the past decades, the fracture of brittle amorphous materials remains in many ways not understood. Of particular interest is the mechanism by which energy in the system is dissipated. Experimental measurements of the flow of energy into the tip of a running crack [1] have indicated that the fracture energy (i.e., the energy needed to create a unit extension of a crack) is a strong function of the crack’s velocity and that the majority of the energy stored in the system prior to the onset of fracture ends up as heat [2]. In this Letter we present quantitative measurements indicating that this increased dissipation is due entirely to the onset of a microbranching instability [3,4] which occurs at a critical value yc of the velocity y .A s yincreases beyond yc we find that the energy needed to create microbranches is precisely enough to account for the velocity dependence of the fracture energy. The long-standing problem of the limiting velocity of a crack is also explained by this mechanism. While linear elastic theory predicts that a crack should continuously accelerate up to the Rayleigh wave speed VR, experiments in a number of brittle materials [5] show that a crack will seldom reach even half of this value. As we will show, the total amount of fracture surface created by both the main crack and the microbranches increases rapidly with y. Thus, rather than acceleration, increased driving results in increased ramification of structure below the fracture surface. There have been a number of suggestions for the velocity dependence of fracture energy. One view is that the energy flow into the tip of a single moving crack is dissipated by plastic deformation around the crack tip. Depending on the model used to describe the area of deformation around the tip, either a nonmonotonic or monotonically increasing function [6] of the velocity of the crack can result. An alternative view of the dissipation process was suggested by Ravi-Chandar and Knauss [7]. They viewed the fracture process as the coalescence of preexisting microvoids or defects situated in the path of the crack and activated by the intense stress field at the crack tip. An increase in the energy flux to the tip, in this picture, causes an increase in the number of microcracks formed and thereby enhanced dissipation. This picture suggests that crack propagation via interacting microvoids occurs as a randomly activated process.

The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate

Abstract: The debonding problem for a composite layer that consists of a homogeneous substrate and a non-homogeneous coating is considered. It is assumed that the problem is one of plane strain or generalized plane stress and the elastic medium contains a crack along the interface. It is further assumed that the thermomechanical properties of the medium are continuous functions of the thickness coordinate with discontinuous derivatives and the kink line of the property distributions corresponds to the “interface”. The mixed-mode crack problem is formulated for arbitrary crack surface tractions and sample results are given for uniform normal and shear tractions. The main variables in the problem are two dimensionless length parameters and a nonhmogeneity constant. Calculated results consist of primarily the stress intensity factors and the strain energy release rate and are partly intended to provide benchmark solutions for further numerical studies.

The single-fibre pull-out test. 1: Review and interpretation

Abstract: For the first time, the load versus extension trace generated by the single-fibre pull-out test is thoroughly interpreted and mathematically modelled. The single-fibre pull-out test is employed experimentally to model the failure of fibre-reinforced composite materials. The interpretation of this model, however, varies between laboratories. In this paper, the test methodologies and the experimental and mathematical interpretations of various scientists are presented and discussed, as is some preliminary work employing optical fibres embedded in various neat resins. Also, a more complete description of the experimental events is presented and described mathematically via the critical strain energy release rate for crack initiation and propagation, the interfacial shear stress of the bond and the coefficient of friction.

Analysis of cracked aluminum plates repaired with bonded composite patches

Abstract: Bonded composite repair has been recognized as an efficient and economical method to extend the service life of cracked aluminum components. An accurate tool for investigating the stress intensity factor in the cracked aluminum structure after repair is needed. The use of three-dimensional finite elements is computationally expensive. A simple analysis method using Mindlin plate theory is presented. Specifically, the aluminum plate and composite patch are modeled separately by the Mindlin plate finite element, whereas the adhesive layer is modeled with effective springs connecting the patch and aluminum plate. Constraint equations are used to enforce compatibility of the patch-adhesive and adhesive-aluminum plate interfaces. Comparison of the present stress intensity factors for the aluminum crack with a plane boundary element analysis and a three-dimensional finite element analysis are made. A procedure for calculating the strain energy release rate along the debond front at the aluminum-adhesive interface is proposed.

Dynamic crack propagation in piezoelectric materials—Part I. Electrode solution

Abstract: An analysis is performed for the transient response of a semi-infinite, anti-plane crack propagating in a hexagonal piezoelectric medium. The mixed boundary value problem is solved by transform methods together with the Wiener-Hopf and Cagniard-de Hoop techniques. As a special case, a closed form solution is obtained for constant speed crack propagation under external anti-plane shear loading with the conducting electrode type of electric boundary condition imposed on the crack surface (a second type of boundary condition is considered in Part II of this work). In purely elastic, transversely isotropic elastic solids, there is no antiplane mode surface wave. However, for certain orientations of piezoelectric materials, a surface wave will occur—the BleusteindashGulyaev wave. Since surface wave speeds strongly influence crack propagation, the nature of antiplane dynamic fracture in piezoelectric materials is fundamentally different from that in purely elastic solids, exhibiting many features only associated with the indashplane modes in the elastic case. For a general distribution of crack face tractions, the dynamic stress intensity factor and the dynamic electric displacement intensity factor are derived and discussed in detail for the electrode case. As for inplane elastodynamic fracture, the stress intensity factor and energy release rate go to zero as the crack propagation velocity approaches the surface wave speed. However, the electric displacement intensity does not vanish.

Fracture Toughness of 2‐D Woven SiC/SiC CVI‐Composites with Multilayered Interphases

Abstract: Relations between fracture toughness and fiber/matrix interphases were examined on various SiC/SiC composites made by chemical vapor infiltration (CVI) and reinforced with woven fiber bundles. Strong and weak fiber/matrix bondings were obtained using multilayered interphases consisting of various combinations of carbon and SiC layers of different thickness and using fibers which had been previously treated. Fracture toughness was estimated using the J-integral and using strain energy release rate computed with a model taking into account the presence of a process zone of matrix microcracks. Both approaches evidenced similar trends. It appeared that higher toughness was obtained with those composites possessing strong interphases and subject to dense matrix microcracking.

A single leg bending test for interfacial fracture toughness determination

Abstract: A single leg bending test is described and its suitability for interfacial fracture toughness testing is evaluated. The test specimen consists of a beam-type geometry comprised of two materials, one ‘top’ and one ‘bottom’, with a split at one end along the bimaterial interface. A portion of the bottom material in the cracked section of the beam is removed and the geometry is loaded in three-point bending. Thus, the reaction force of the support at the cracked end is transmitted only into the material comprising the top portion of the beam. The test is analyzed by a crack tip element analysis and the resulting expressions for energy release rate and mode mixity are verified by comparison with finite element results. It is shown that, by varying the thicknesses of the two materials, the single leg bending test can be used to determine the fracture toughness of most bimaterial interfaces over a reasonably wide range of mode mixities.

Analysis of work-of-fracture method for measuring fracture energy of concrete

Abstract: The role that plastic-frictional energy dissipation in the fracture process zone plays in the work­ of-fracture method for measuring the fracture energy of concrete or other quasibrittle materials is analyzed, and a possible improvement of this method is proposed. It is shown that by measuring the unloading compliance at a sufficient number of states on the post-peak descending load-deflection curve, it is possible to calculate the pure fracture energy, representing the energy dissipated by the fracture process alone. However, this value of fracture energy is pertinent only if the material model (constitutive law and fracture law) used in structural analysis takes into account separately the fracture-damage deformations and the plastic-frictional deformations. Otherwise, one must use the conventional fracture energy, which includes plastic-frictional energy dissipation. Either type of fracture energy should properly be determined by extrapolation to infinite specimen size. Further, it is shown that the unloading compliancies to be used in the calculation of the pure fracture energy can be corrected to approximately eliminate the time-dependent effects (material viscoelasticity) and reverse plasticity. Finally, it is proposed to improve the work-of-fracture method by averaging the work done by fracture over only a central portion of the ligament. However, experiments are needed to check whether the specimen size required for this improved method would not be impracticably large.

Dynamic crack propagation in piezoelectric materials—Part II. Vacuum solution

Abstract: In Part I of this work, antiplane dynamic crack propagation in piezoelectric materials was studied under the condition that crack surfaces behaved as though covered with a conducting electrode. Piezoelectric surface wave phenomena were clearly seen to be critical to the behavior of the moving crack. Closed form results were obtained for stress and electric displacement intensities at the crack tip in the subsonic crack speed range; the major result is that the energy release rate vanishes as the crack speed approaches the surface (Bleustein-Gulyaev) wave speed. In this paper, an alternative assumption is made that between the growing crack surfaces there is a permeable vacuum free space, in which the electrostatic potential is nonzero. By coupling the piezoelectric equations of the solid phase with the electric charge equation in the vacuum region, a closed form solution is again obtained. In contrast to the electrode case of Part I, this case allows both applied charge and applied traction loading. In addition, the work of Part I is extended to examine piezoelectric crack propagation over the full velocity range of subsonic, transonic and supersonic speeds. Several aspects of the results are explored. The energy release rate in this case does not go to zero when the crack propagating velocity approaches the surface wave speed, even if there is only applied traction loading. When the crack exceeds the Bleustein-Gulyaev wave speed, the character of the crack-tip singularities of the physical fields depends on both speed regime and type of loading. At the other extreme, the quasi-static limit of the dynamic solution furnishes a set of new static solutions. The general permeability assumptions made here allow for fully coupled conditions that are ruled out by the a priori interfacial assumptions made in previously published solutions.

BIE fracture mechanics analysis: 25 years of developments

Abstract: The purpose of this paper is to provide a selective review of the unique properties of the boundary integral equation (BIE) method for problems in fracture mechanics. The paper draws on the extensive literature that has been developed over the past twenty-five years. Fracture mechanics problems have provided one of the most important applications of BIE formulations in solid mechanics and is one of the principal areas of application of the methods. In particular, the paper will focus on the role played by the Somigliana stress identity in providing unique algorithms and numerical results for fracture mechanics analysis not available to the finite element method.

Separation of crack extension modes in orthotropic delamination models

Abstract: In modeling a crack along a distinct interface between dissimilar elastic materials, the ratio of mode I to mode II stress intensity factors or energy release rates is typically not unique, due to oscillatory behavior of near-tip stresses and displacements. Although methods have been developed for comparing mode mixes for isotropic interfacial fracture problems, this behavior currently limits the applicability of interfacial fracture mechanics in predicting delamination in layered materials without isotropic symmetry. The virtual crack closure technique (VCCT) is a method used to extract mode I and mode II energy release rate components from numerical fracture solutions. Energy release rate components extracted from an oscillatory solution using the VCCT are not unique due to their dependence on the virtual crack extension length, Δ. In this work, a method is presented for using the VCCT to extract Δ-independent energy release rate quantities for the case of an interface crack between two in-plane orthotropic materials. The method does not involve altering the analysis to eliminate its oscillatory behavior and it is similar to existing methods for extracting a mode mix from isotropic interfacial fracture models. Knowledge of near-tip fields is used to determine the explicit Δ dependence of energy release rate parameters. Energy release rates are then defined that are separated from the oscillatory dependence on Δ. A modified VCCT using these energy release rate definitions is applied to results from finite element analyses, showing that Δ-independent energy release rate quantities result. The modified technique has potential as a consistent method for extracting a mode mix from numerical solutions. The Δ-independent energy release rate quantities extracted using this technique can also aid numerical modelers, serving as guides for testing the convergence of finite element models. Direct applications of this work include the analysis of planar composite delamination problems, where plies or debonded laminates are modeled as in-plane orthotropic materials.

The effect of exposure to elevated temperatures on the fracture energy of plain concrete

Abstract: Results are given for the fracture energy of concrete preheated to temperatures in the range of 20°C–600°C, then tested at room temperature in three-point bending, for both slowly-cooled specimens and those delivered a thermal shock. Fracture energy measurement techniques are discussed, and comparisons made across previous studies. The results suggest a significant difference between residual and ‘hot’ capacities, along with a temperature dependence on the ductile-brittle transition.