Hydrostatic stress

About: Hydrostatic stress is a research topic. Over the lifetime, 1568 publications have been published within this topic receiving 37773 citations.

On stress-dependent elastic moduli and wave speeds

Abstract: On the basis of the general non-linear theory of a hyperelastic material with initial stress, initially without consideration of the origin of the initial stress, we determine explicit expressions for the stress-dependent tensor of incremental elastic moduli. In considering three special cases of initial stress within the general framework, namely hydrostatic stress, uniaxial stress and planar shear stress, we then elucidate in general form the dependence of various elastic moduli on the initial stress. In each case, the effect of initial stress on the wave speed of homogeneous plane waves is studied and it is shown how various special theories from the earlier literature fit within the general framework. We then consider the situation in which the initial stress is a pre-stress associated with a finite deformation and, in particular, we discuss the specialization to the second-order theory of elasticity and highlight connections between several classical approaches to the topic, again with special reference to the influence of higher-order terms on the speed of homogeneous plane waves. Some discrepancies arising in the earlier literature are noted.

A Course in Elasticity

Abstract: 1. Kinematics of Continuous Media.- 1.1. Material and Spatial Coordinates.- 1.2. Neighborhood Transformations.- 1.3 Composition of Changes of Configuration.- 1.4 Measure of the State of Local Deformation. Green's and Jaumann's Strain.- 1.5 Rigid-Body Rotations of a Neighborhood.- 1.6 The Kinematical Decomposition of the Jacobian Matrix.- 1.7 Geometric Interpretation of Infinitesimal Strains.- 1.8 The Eulerian Viewpoint in Kinematics. Almansi's Strain.- 1.9 Eulerian Measures of Rates of Deformation and Rotation.- 1.10 Temporal, Variation of the Polar Decomposition of the Jacobian Matrix.- 2. Statics and Virtual Work.- 2.1. The Concept of Stress. True Stress.- 2.2. The Piola Stresses.- 2.3. Translational Equilibrium Equations.- 2.4. Rotational Equilibrium Equations.- 2.5. Statics and Virtual Work.- 2.6. Commutativity of the Operators ? and Di.- 2.7 Virtual Work in a Continuous Medium.- 2.8. Statics and Virtual Power for True Stresses.- 2.9. Statics and Virtual Work in Infinitesimal Changes of Configuration.- 3. Conservation of Energy.- 3.1. Constitutive Equations for Piola's Stresses.- 3.2. The Kirchhoff-Trefftz Stresses.- 3.3 The Constitutive Equations of Geometrically Linear Elasticity.- 4. Cartesian Tensors.- 4.1. Bases and Change of Basis.- 4.2 Tensors.- 4.3 Some Special Tensors.- 4.4 The Vector Product.- 4.5. Structure of Symmetric Cartesian Tensors of Order Two. Principal Axes.- 4.6. Fundamental Invariants and the Deviator.- 4.7. Structure of Skew-Symmetric Cartesian Tensors of the Second Order.- 4.8. Matrix Representation of Tensor Operations.- 5. The Equations of Linear Elasticity.- 5.1. Compatibility of Strains in a Simply Connected Region.- 5.2. Compatibility of Strains in a Multiply Connected Region.- 5.3. Principal Elongations and Fundamental Invariants of Strain.- 5.4. Principal Stresses and Fundamental Invariants of the Stress State.- 5.5. Octahedral Stresses and Strains.- 5.6. Mohr's Circles.- 5.7. Statics and Virtual Work.- 5.8. Taylor's Development of the Strain Energy.- 5.9. Infinitesimal Stability.- 5.10. Hadamard's Condition for Infinitesimal Stability.- 5.11. Isotropy and Anisotropy.- 5.12. Criteria for Elastic Limits.- 5.13. Navier's Equations.- 5.14. The Beltrami-Michell Equations.- 6. Extension, Bending, and Torsion of Prismatic Beams.- 6.1. Green's and Stokes' Formulas.- 6.2. The Centroid.- 6.3. Moments of Inertia.- 6.4. The Semi-Inverse Method of Saint-Venant.- 6.5. Resultants of Stresses on a Cross Section.- 6.6. Calculation of the Transverse Displacements.- 6.7. Equations Governing the Shear Stresses.- 6.8. Calculation of the Longitudinal Displacement.- 6.9. Separation of Solutions.- 6.10. Pure Torsion.- 6.11. The Center of Torsion for a Fully Constrained Section.- 6.12. Bending without Torsion.- 6.13. The Stiffness Relation for the Twist.- 6.14. Total Energy as a Function of the Deformations of the Fibers.- 6.15. Total Energy as a Function of Generalized Forces.- 6.16. The Generalized Constitutive Equations for Bending and Torsion of Beams.- 6.17. One-Dimensional Formulation of Bending and Torsion of Beams.- 6.18. Applications.- A. Stress function for torsion of the elliptic bar.- B. Stress functions for torsion of the circular bar.- C. Stress functions with poles.- D. Torsion of a triangular bar.- E. Torsion of a rectangular bar.- F. Bending of a circular bar.- G. Bending of a circular tube.- H. Bending of a rectangular bar.- 7. Plane Stress and Plane Strain.- 7.1. Lemmas for the Integration of Partial Differential Equations in Complex Form.- 7.2. The Structure of a Biharmonic Function.- 7.3. Structure of the Solution of the Problems of Plane Strain.- 7.4.Structure of the Solution of the Problem of Plane Stress.- 7.5. Generalized Plane Stress.- 7.6. Airy's Stress Function.- 7.7. Complex Representation of Airy's Function.- 7.8. Polar Coordinates.- 7.9. Applications in Cartesian Coordinates.- A. The state of hydrostatic stress.- B. Uniform gradient of areal dilation.- C. Pure uniform shear.- D. Linear variation of a normal stress.- E. Simple extension.- F. Pure bending.- G. Shear lag.- H. Bending by shear forces.- I. Saint-Venant's bending of a rectangular beam with flanges.- J. Transverse loading of a beam with flanges.- 7.10. Applications in Polar Coordinates.- A. Circular aperture with traction-free circumference in a plate in plane stress.- B. Volterra's dislocation of the circular ring.- C. Bending of beams with constant curvature.- D. The annular ring loaded by shear tractions.- E. The thick tube under pressure.- F. Concentric cylindrical tubes and rings.- G. Force concentrated at the origin in an infinite plate.- 8. Bending of Plates.- 8.1. Basic Hypotheses.- 8.2. Application of the Canonical Variational Principle.- 8.3. The Two-Dimensional Canonical Principle.- 8.4. Further Connections Between the Two- and Three-Dimensional Theories.- 8.5. Other Types of Approximations.- 8.6. Kirchhoff's Hypothesis.- 8.7. Boundary Conditions in Kirchhoff's Theory.- 8.8. Kirchhoff's Variational Principle.- 8.9. Structure of the Solution of the Equations of Plates of Moderate Thickness.- 8.10. The Edge Effect.- 8.11. Torsion of a Plate.- 8.12. Saint-Venant's Bending of a Plate.- 8.13. Particular Solutions for Transverse Load.- 8.14. Solutions in Polar Coordinates.- 8.15. Axisymmetric Bending.

Shock-induced crystal-plastic deformation and post-shock annealing of quartz microstructural evidence from crystalline target rocks of the Charlevoix impact structure, Canada

Abstract: Two distinct types of shock-induced quartz microstructure in charnockitic target rocks and quartz veins of the Charlevoix impact structure are described. The dominant shock effects in the type 1 microstructure in charnockites at ∼2–4 km from the centre of the structure are planar deformation features (PDFs) parallel to rhombohedral planes of quartz. The abundance of different sets of these PDFs indicates a high hydrostatic component of the shock wave-associated stress (∼10–15 GPa). Evidence of crystal-plastic deformation due to high deviatoric stresses is absent. In contrast, PDFs parallel to the basal plane are predominant in the type 2 microstructure developed in rocks at ∼4–9 km from the centre of the structure, whereas rhombohedral PDFs are rare. This indicates a lower hydrostatic stress component (∼7–8 GPa), which correlates with a radial decrease in recorded peak shock pressure. The basal PDFs are revealed by transmission electron microscopy to represent mechanical Brazil twins, which record crystal-plastic deformation at high deviatoric stresses (McLaren et al. , 1967). These findings imply that the deviatoric component of the shock wave-associated stress increases relative to the hydrostatic component with increasing distance from the centre of the impact structure. In the type 2 microstructure, numerous deformation bands, strong undulose extinction and cataclastic zones at the optical scale, as well as glide-dislocations and microcracks at the TEM scale, occur in association with basal PDFs, and are therefore also interpreted to be shock-induced. This is consistent with the observation that quartz from the outer part of the impact structure is devoid of similar features. Thus, the highly heterogeneous and localised glide-controlled deformation accompanied by mechanical twinning and microcracking recorded by the type 2 microstructure is suspected to be induced by the high deviatoric stresses and high loading rates during shock. Post-shock recovery is indicated in the type 1 microstructure by the actual microstructure of rhombohedral PDFs, dislocations in climb configuration and well-ordered low angle grain boundaries, as well as in the type 2 microstructure by the occurrence of small elongate subgrains with low angle grain boundaries paralleling low-index planes. This has probably taken place during annealing shortly after the impact event at quasi-static conditions and still sufficiently high post-shock temperatures, rather than during a separate regional thermal event.

Role of hydrostatic stress in hydrogen diffusion in pearlitic steel

Abstract: The relevant role of hydrostatic stress in hydrogen diffusion in pearlitic steel is outlined from both theoretical and experimental points of view. The theoretical development offers the formulation of hydrogen diffusion equations where hydrogen flux density depends not only on the concentration gradient, but also on the hydrostatic stress distribution in the sample. The experimental programme consisted of slow strain-rate tests on axisymmetric notched samples at different strain rates under simultaneous hydrogen charging by cathodic polarization. The use of different notch geometries allows a study of the influence on hydrogen diffusion of the hydrostatic stress state in the vicinity of the notch tip. A specific microscopic mode of fracture different from classical cleavage was found, associated with hydrogen effects: the tearing topography surface. In the quasi-instantaneous tests, the value of hydrostatic stress at the sample boundary (just the notch tip) at the failure instant is relevant from the fracture point of view. In the quasi-static tests, the tearing topography surface depth equals that of the maximum hydrostatic stress point, and the maximum value of the stress triaxiality in each geometry (ratio of the hydrostatic to the equivalent stress, almost constant during the tests) seems to govern the diffusion process. These facts emphasize the relevant role of hydrostatic stress in the vicinity of the notch in hydrogen diffusion.

On the development of constitutive relations for metallic powders

Abstract: This article describes the development of elastic and plastic constitutive relations as functions of relative density for partially consolidated —100 mesh aluminum powder. First, measurements of yield stress as a function of stress state and relative density are described. Measurements of the plastic strain increments associated with yielding in unconstrained compression tests and elastic properties, both as functions of relative density, are also described. The experimental results are combined with the associated flow rule to show that the yield surface is asymmetric with respect to hydrostatic tension and compression. Second, it is shown that the yield stress results can be represented by a two-part (capped Drucker-Prager) yield surface. The consoli-dation yield surface moves along the hydrostatic stress axis during densification, while the shear yield surface approaches the Mises yield surface. For the Al powder used in the present inves-tigation, superposition of shear stress on a hydrostatic stress state aids the densification process. However, the hydrostatic stress requirement was found to be reduced by only about 20 pct for relative densities below 0. 98.