A Theorem of Maximum Strain Energy

1959 ◽  
Vol 26 (1) ◽  
pp. 73-76
Author(s):  
E. H. Brown
Keyword(s):  
Strain Energy ◽  
Elastic Structure ◽  
Maximum Strain ◽  
Elastic Structures ◽  
Hooke’S Law ◽  
Rigid Structure ◽  
Maximum Value ◽  
Hooke's Law ◽  
External Loads

Abstract A new theorem for elastic structures obeying Hooke’s law is proved and enunciated as follows: If the strain energy can be expressed in terms of given external loads and the positions of points constrained to have no displacement, then the expression will have a maximum value when the positions are such as make the constraining forces zero. The theorem is generalized to allow a wider class of constraint, such as the attachment of a second, rigid structure to the elastic structure under analysis, in such a way that the two have certain displacement components in common. The principle has been applied by E. H. Mansfield to certain problems in the bending of plates.

Rock Failure Under Dynamic Loading Conditions

1963 ◽  
Vol 3 (01) ◽  
pp. 1-8 ◽  
Author(s):  
N.T. Burdine
Keyword(s):  
The State ◽  
Cumulative Damage ◽  
Material Behavior ◽  
Particle Orientation ◽  
Loading Conditions ◽  
Hooke’S Law ◽  
Rock Samples ◽  
Cyclic Stresses ◽  
Hooke's Law ◽  
Failure Theories

BURDINE, N.T., SOCONY MOBIL OIL CO., INC., DALLAS, TEX Abstract The present investigation is concerned with the cumulative damage to rock samples when exposed to cyclic stresses under various loading conditions. Information on the response of rocks to repetitive deformational forces is an essential prerequisite to an understanding of the fundamentals of drilling. Using a laboratory designed and constructed dynamic-stress apparatus, preliminary data were obtained on cylindrical rock samples. The experiments consist of measuring the number of cycles to failure for a given axial load ( static plus dynamic). Data were obtained for various confining and pore pressures, pore fluids (air and water), frequencies of stress application and loading procedures. The results are related to failure theories and dynamic fatigue properties of other materials. Introduction In most conventional and new drilling processes, repetitive forces are applied to the bottom of the borehole. Furthermore, in hard-rock drilling the number of applications of the forces to a particular section of rock may become excessively large. The present investigation is concerned with the cumulative damage to rocks when exposed to cyclic stresses under various loading conditions. It is believed that the experiments will lead to a better understanding of the mechanical response of rocks to particular deformational forces and to a more efficient drillingprocedure.Thepresent investigation is the initial part of a general study of the behavior of inelastic materials under static and dynamic conditions, including both theoretical and experimental studies. SURVEY OF FAILURE THEORIES OF MATERIALS Few, even phenomenological, theories on rock deformation have been established because the state of knowledge of flow, fracture and strength of rocks is largely empirical. Most of the theories that do exist were originally formulated for other materials. HOOKE'S LAW The state of stress in continuous media is completely determined by the stress tensor and the state of deformation by the strain tensor . In the linear theory of elasticity the generalized Hooke's law is ..........................(1) where the coefficients are the components of the elasticity tensor. For homogeneous and isotropic conditions the number of independent coefficients reduce to two, and Eq. 1 becomes ..................(2) in which and are Lame's constants; is the kronecker delta; and is the dilation. This simplified version of Hooke's law has been used quite extensively in geophysical research where most of the information about the mechanical properties of the earth have been obtained. However, it has only limited application in rock fatigue studies. MATERIAL BEHAVIOR Many solids obey Hooke's law at small stresses, but for higher stresses a hysteretic effect occurs due to temporary or permanent residual deformation of the solid (inelastic deformation). Such deviations in mechanical behavior exist in varying degrees in different classes of materials. Most elastic materials have a microscopic heterogeneity due either to random distribution of anisotropic particles, or due to some preferred particle orientation, or both. Other materials are quite grossly heterogeneous. And the method of formation, particularly in rocks, oftentimes creates residual stress concentrations which have complicated states of imperfect equilibrium. Also, the thermal effects resulting from structural behavior give rise to nonuniform temperature distributions and the degradation of mechanical energy. When such bodies are exposed to certain large loading conditions, the inelastic behavior is intensified so strongly that the deformation, normally brittle, becomes ductile. SPEJ P. 1^

The Theoretical Treatment of Hooke's Law

1939 ◽  
Vol 7 (2) ◽  
pp. 134-134
Author(s):  
Zigmond Wilchinsky
Keyword(s):  
Theoretical Treatment ◽  
Hooke’S Law ◽  
Hooke's Law

scholarly journals Quantum Hooke's Law to Classify Pulse Laser Induced Ultrafast Melting

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
Hao Hu ◽  
Hepeng Ding ◽  
Feng Liu
Keyword(s):  
Pulse Laser ◽  
Hooke’S Law ◽  
Hooke's Law

Representation of the Microstructural Dependence of the Anisotropic Elastic Constants of Textured Materials

1990 ◽  
Vol 207 ◽  
Author(s):  
Stephen C. Cowin
Keyword(s):  
Elastic Constants ◽  
Material Symmetry ◽  
Hooke’S Law ◽  
Mechanical Models ◽  
Hooke's Law ◽  
Micromechanical Models ◽  
Anisotropic Elastic ◽  
Anisotropic Form ◽  
Textured Materials ◽  
Elastic Coefficients

AbstractThis paper addresses the question of representing the dependence of the elastic coefficients in the anisotropic form of Hooke's law upon the microstructure of a material. The concern is with textured material symmetries, that is to say materials such as natural and man-made composites whose material symmetry is determined by microstructural organization. The approach is to relate the anisotropic elastic coefficients to local geometric or stereological measures of the microstructure. The predictions of micromechanical models and continuum mechanical models are compared and are found to be consistent with each other.

scholarly journals Hooke’s law in statistical manifolds and divergences

2000 ◽  
Vol 159 ◽  
pp. 1-24 ◽  
Author(s):  
Masayuki Henmi ◽  
Ryoichi Kobayashi
Keyword(s):  
Classical Mechanics ◽  
Riemannian Metric ◽  
Legendre Transform ◽  
Point Function ◽  
Hooke’S Law ◽  
Affine Connections ◽  
Statistical Manifolds ◽  
Dual Affine ◽  
Hooke's Law

The concept of the canonical divergence is defined for dually flat statistical manifolds in terms of the Legendre transform between dual affine coordinates. In this article, we introduce a new two point function defined for any triple (g,∇, ∇*) of a Riemannian metric g and two affine connections ∇ and ∇*. We show that this interprets the canonical divergence without refering to the existence of special coordinates (dual affine coordinates) but in terms of only classical mechanics concerning ∇- and ∇*-geodesics. We also discuss the properties of the two point function and show that this shares some important properties with the canonical divergence defined on dually flat statistical manifolds.

scholarly journals Large elastic deformations of isotropic materials. I. Fundamental concepts

1948 ◽  
Vol 240 (822) ◽  
pp. 459-490 ◽  
Keyword(s):  
The Body ◽  
Elastic Behaviour ◽  
Stress And Strain ◽  
Elastic Deformations ◽  
Hooke’S Law ◽  
Widespread Agreement ◽  
Wide Range ◽  
Hooke's Law ◽  
High Degree

The mathematical theory of small elastic deformations has been developed to a high degree of sophistication on certain fundamental assumptions regarding the stress-strain relationships which are obeyed by the materials considered. The relationships taken are, in effect, a generalization of Hooke’s law— ut tensio, sic vis . The justification for these assumptions lies in the widespread agreement of experiment with the predictions of the theory and in the interpretation of the elastic behaviour of the materials in terms of their known structure. The same factors have contributed to our appreciation of the limitations of these assumptions. The principal problems, which the theory seeks to solve, are the determination of the deformation which a body undergoes and the distribution of stresses in it, when certain forces are applied to it, and when certain points of the body are subjected to specified displacements. These problems are always dealt with on the assumption that the generalization of Hooke’s law is obeyed by the material of the body and that the deformation is small, i.e. the change of length, in any linear element in the material, is small compared with the length of the element in the undeformed state. Apart from the fact that the generalization of Hooke’s law is obeyed accurately by a very wide range of materials, under a considerable variety of stress and strain conditions, it has the further advantage that it leads to a mathematically tractable theory.

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