Showing papers on "Stress–strain curve 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.

Degree of hydration-based description of mechanical properties of early age concrete

Abstract: For the evaluation of the risk of thermal cracking in hardening massive concrete elements, knowledge of the development of strength and deformability of early-age concrete is extremely important. Based on an extensive experimental research program on hardening concrete elements, a degree of hydration-based description for the compressive strength, Young's modulus, the uniaxial tensile strength, the splitting tensile strength, the flexural tensile strength, Poisson's ratio and the peak strain are all worked out. An extension of the formulation of Sargin for the stress-strain relation for short-term compressive loading leads to a degree of hydration-based stress-strain relation for hardening concrete. Good agreement with experimental results is reported.

Concrete as a new strain/stress sensor

Abstract: A new strain/stress sensor technology was developed, based on the concept of short electrically conducting fiber pull-out that accompanies slight and reversible crack opening. The fiber pull-out reversibly increases the composite's electrical resistance, which is the signal provided by the sensor under static or cyclic loading. The new technology was manifested in concrete and mortar containing electrically conducting short fibers (e.g. carbon fibers and steel fibers), but not in those containing no fibers or those containing non-conducting (polyethylene) fibers. Carbon fibers worked best. They served to greatly decrease the crack height, so that reversible pull-out of the crack bridging fibers occurred. Even in the elastic regime, a part of the resistance change was irreversible, such that it provided memory of the first deformation; this is due to permanent damage, probably associated with the increase in fiber/matrix contact electrical resistivity due to the interface bond weakening. The stress at which this damage began was much lower under tension than compression. The ratio of the contribution to the fractional resistance increase by the reversible part to that by the reversible part was much higher under tension than compression. The irreversible part increased with increasing irreversible strain, which increased with increasing stress amplitude. The fractional increase in resistance at fracture was much larger under compression than under tension, was much larger for mortars than concretes at similar volume fractions, and was quite independent of the loading rate.

Experimental investigation of concrete fracture under uniaxial compression

Abstract: Localization of deformations has been investigated in a series of displacement controlled uniaxial compression experiments. Of main interest are the effects of specimen slenderness and friction between loading platen and specimen. Both effects have a direct influence on the development of localized fracture zones in the specimen. The results indicate that the use of a double layer of teflon with an intermediate layer of grease yields size-independent results as far as the pre-peak stress–strain behaviour and the peak strength are concerned. However, in terms of stress and strain, a significant influence of both the specimen slenderness and the amount of boundary restraint has been observed in the post-peak regime. It is found that the post-peak curves become almost completely identical when they are plotted in terms of nominal stress and post-peak displacement. For any type of loading platen used, the post-peak relative stress-displacement curves are found to be independent of the specimen height. Furthermore, since during post-peak localization relative sliding and movements of larger parts of the specimen are observed, the definition of a unique Poisson's ratio is virtually impossible.

The cracking mechanism of silicon particles in an A357 aluminum alloy

Abstract: The cracking of Si particles in an A357 Al alloy has been investigated over a spectrum of stress and strain by varying aging strength and applying different tensile strains. The variation of the fraction of broken Si particles with stress, strain, and cleavage plane orientation has been obtained. The features of cracking reveal that cracking of Si particles is a very localized event. A dislocation pileup mechanism is the most probable one among all crack-initiation theories for explaining the behavior. Based on this mechanism, further deduction has been made to obtain the relationship between the fraction of broken particles and metallurgical factors. The present data, along with Gurlandrss and that of Lowet al., have been found to verify this relationship for the effect of stress, strain, and cleavage plane orientation.

Elastic Properties of Soils

Abstract: A triaxial apparatus to measure small strains in the range between 10 −6 and 10 −2 has been set up to investigate the reversible behavior of various types of soils (sands, gravels, clays). Reversible behavior, along compression and extension triaxial paths, was demonstrated for strain amplitudes lower than 10 −5 . Elasticity was found to be nonlinear, since the Young's modulus depended on the mean effective stress. We also studied the influence of such parameters as deviatoric stresses, void ratio, stress and strain history, and soil structure on the elastic coefficients of these soils. A general equation was proposed for determining the Young's modulus E in initially isotropic soils, which was modified by further loading. In granular materials, strain hardening increased the value of E , while in structured materials, a decrease of E was observed due to internal structure damage.

On the analysis of transverse stress gauge data from shock loading experiments

Abstract: The importance of understanding the variation of shear strength with pressure in the formulation of constitutive models has long been recognized. Previously, this had been deduced by measurements of the offset of the Hugoniot curve for a material from the calculated hydrostat. In recent papers, a direct measurement technique has been suggested in which both principal components of stress are measured using piezoresistive gauges. The reduction of the data collected from transverse stress gauges has attracted some debate and is reviewed here. In particular, the stress and strain states experienced by the gauge must be considered. The hardening of the gauge with longitudinal stress or pressure was investigated. Examples from experiments in metals and ceramics are given. The effect of gauge geometry was assessed and results show that the measured stresses from either gauge were within 0.5% of each other when subjected to identical impact conditions. An investigation was also performed on the effect of insulation thickness around the gauge and again no effect on the measured stress was found.

A new approach to measuring transverse properties of structural tubing by a ring test

Abstract: A new approach to the ring test is presented in this paper. Three-dimensional elastoplastic finite element modeling with contact has been performed to analyze the stress and strain distribution in the ring, to optimize the ring testing system, and to investigate the effect of friction between the ring specimen and the fixture. Based on the numerical results, a new design of a holding device that creates a uniaxially stressed zone is proposed in order to determine transverse behavior of tubular products, such as the modulus and the stress-strain curve. A case study is presented on nuclear cladding tubes in Zircaloy. The approach shows promise for testing various kinds of materials in structural tubing, including ductile or brittle materials, metals, composites, or polymers.

Modeling of Stress - Strain Relationships for Steel and Concrete in Concrete Filled Circular Steel Tubular Columns

Abstract: In this paper, a stress-strain relationship in concrete which takes account of the confining effect on the filled concrete is proposed to evaluate ultimate strength and deformation capacity of concrete filled circular steel tubular columns rationally. The confining effect is quantitatively derived from the characterization between Poisson's ratio and axial strain in steel and concrete filled tube under concentric loading. The applicability of the proposed stress-strain relationship in concrete as well as the influence of strain hardening in steel is discussed. Ultimate strength, curvature and deflection of the concrete filled circular columns are calculated on the basis of the proposed model in steel and concrete, comparing with the past test results. It is concluded that the predicted relations agree quite well with the test results.

Strain Rate and Preshear Effects in Cyclic Resistance of Soft Clay

Abstract: Cyclic constant-volume direct simple shear tests were performed on intact specimens of a sensitive clay reconsolidated to the in-situ stress state resulting in an overconsolidation ratio (OCR) of 2.2. The tests were conducted for different values of initial static undrained shear stress. The results confirmed that the high strain rate associated with cyclic loading has a significant effect on the cyclic resistance of soft clays. For up to 12 cycles, the strain-rate effect fully compensates the degradation of shear strength associated with cycling. The presence of an initial static shear stress decreases the cyclic resistance but increases the total undrained shear resistance, due to the partial or total disappearance of stress and strain reversal. The results indicate that for such a clay, the static undrained shear strength provides a conservative estimate in a pseudostatic analysis.

Stress-strain analysis of single-lap composite joints under tension

Abstract: Based on the laminated anisotropic plate theory, an analytical model is proposed to determine the stress and strain distributions of adhesive-bonded composite single-lap joints under tension. The laminated anisotropic plate theory is applied in the derivation of the governing equations of the two bonded laminates. The entire coupled system is then obtained through assuming the peel stress between the two laminates. With the Fourier series and appropriate boundary conditions, the solutions of the system are obtained. Based on the proposed model, the stress and strain distributions of the adherends and the adhesive can be predicted. The coupling effect between the external tension and the induced bending due to the asymmetry of composite laminates are also included. The two adherends can also have different materials and properties. An existing FEA code, ALGOR, is used as a comparison with this proposed analytical model. Results from this developed model are also compared with Goland and Reissner`s as well as Hart-Smith`s papers.

Elastic-plastic stress-strain calculation in notched bodies subjected to non-proportional loading

Abstract: An analytical method for calculating notch tip stresses and strains in elastic-plastic isotropic bodies subjected to non-proportional loading sequences is presented. The key elements of the two proposed models are generalized relationships between elastic and elastic-plastic strain energy densities, and the material constitutive relations. These two models form the lower and the upper limits of the actual energy densities at the notch tip. Each method consists of a set of seven linear algebraic relations that can easily be solved for elastic-plastic strain and stress increments, knowing the hypothetical notch tip elastic stress history and the material stress-strain curve. Results of the validation show that the proposed methods compare well with finite element data and each solution set forms the limits of a band within which actual notch tip strains fall.

Cracks in gradient elastic bodies with surface energy

Abstract: In the present paper the effect of higher order gradients on the structure of line-crack tips is determined. In particular we introduce in the constitutive equations of the linear deformation of an elastic solid a volumetric energy term, which includes the contribution of the strain gradient, and a surface energy gradient dependent term and then determine the effect of these terms on the structure of the mode-III crack tip and the associated stress and strain fields. By making use of the solution in terms of Fourier transform of the equation of elastic equilibrium we solve the half-plane boundary value problems of: (a) specified tractions, and (b) prescribed displacements, along the crack surface, respectively.

Strain softening and shear band formation of sand in multi-axial testing

Abstract: The strain softening of a granular soil under a σ1 ≠ σ2 ≠ σ3 condition and along a wide spectrum of paths was investigated experimentally. To ensure that the observed behaviour was not an aberration due to boundary imperfections, a specially designed multi-axial cell was used. Microprocessor control of both stress and strain paths was incorporated into the testing programme. In particular, strain path testing where the strain increment ratio dev/de1 was controlled to a specified value was used extensively. A photographic technique was used to detect the initiation of shear band. Strain softening was found to be path-dependent. It can take the form of shear band formation, or it can occur without the development of any non-homogeneity. Shear band formation is not necessarily a consequence of boundary imperfections, but can occur as the inevitable response of a sample to certain stress states and shear paths. The conditions for shear band formation were established and three types of strain softening identi...

Deformation fields around a fault embedded in a non-linear ductile medium

Abstract: SUMMARY The geological record of deformation is often characterized by a combination of discontinuous deformation, in which strain is concentrated in faults, and continuous deformation, in which strain is distributed through the material. Where slip occurs on a fault that terminates, the surrounding material is deformed. In the lower crust and in cases where large strains occur over long geological time-scales, it is appropriate to model the deformation using a viscous (probably non-linear viscous) rheology. We describe a method for practical finite-element solution of this problem using a dynamically self-consistent formulation for stress and displacement on a fault of arbitrary geometry; the accuracy of the method is tested by comparison with an analytical solution for the linear rheology. We describe here the instantaneous deformation fields around a mode II fault under both plane-strain and plane-stress conditions, and a range of rheological exponents n (where strain rate is proportional to deviatoric stress to the nth power). the distributions of stress and strain rate around the fault tip are controlled primarily by the rheological exponent n. A localized zone of high strain rate projects beyond the end of the fault if n is about 3 or greater, and the degree of localization of deformation increases with the value of n. the zone of high shear-strain rate can be defined in practical terms by considering (1) the region in which the creep velocity differs by more than 20 per cent from the velocity on the nearby external boundary and (2) the region in which the maximum shear-strain rate is greater than about twice the externally imposed shear-strain rate. For n= 1, the volumes so defined differ considerably, but for large values of n, the two definitions both describe the same narrow zone of deformation beyond the end of the fault. Evaluation of the Navier-Coulomb criterion for brittle failure of the medium surrounding the fault tip shows first that brittle failure is much more likely on the extensional side of the fault than the compressional side. It also shows that the volume of material subject to brittle failure decreases rapidly with increasing n because of the relatively weaker stress singularity. We analyse previously published displacement versus distance data for faults terminating in sedimentary rocks at 0.1 to 100 m length-scales under different tectonic conditions, in order to determine the rheological exponent n. These analyses result in n values between approximately 0.85 and 5 for the different faults, with error bounds on n typically pL 1. the variation in n values may result from differences in pressure, temperature and fluid conditions at the time of faulting. More importantly, the analysis demonstrates a new method for the determination of the effective rheological exponent under in situ geological conditions.

Internal stress and strain in heavily boron-doped diamond films grown by microwave plasma and hot filament chemical vapor deposition

Abstract: The internal stress and strain in boron‐doped diamond films grown by microwave plasma chemical vapor deposition (MWCVD) and hot filament CVD (HFCVD) were studied as a function of boron concentration. The total stress (thermal+intrinsic) was tensile, and the stress and strain increased with boron concentration. The stress and the strain measured in HFCVD samples were greater than those of MWCVD samples at the same boron concentration. The intrinsic tensile stress, 0.84 GPa, calculated by the grain boundary relaxation model, was in good agreement with the experimental value when the boron concentration in the films was below 0.3 at.%. At boron concentrations above 0.3 at.%, the tensile stress was mainly caused by high defect density, and induced by a node‐blocked sliding effect at the grain boundary.

Analytical formulation of the complete stress-strain curve for high strength concrete

Abstract: In this paper, some available formulations for the complete stress-strain curve for high strength concrete under uniaxial compression are examined and compared with results of deformation-controlled compression tests on high strength concrete cylinders. Based on these findings, a new analytical formulation for the complete stress-strain curve is elaborated, with emphasis on the softening branch. The new formulation is based on the curve originally introduced by Sargin and adopted in the CEB-FIP Model Code for Concrete Structures with appropriate parametric values.