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.

Linear Anisotropic Elastic Materials

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Elastic Constants ◽  
Elastic Material ◽  
Positive Definite ◽  
Material Symmetry ◽  
Two Dimensional ◽  
Elastic Materials ◽  
Rectangular Coordinate System ◽  
Full Symmetry ◽  
Anisotropic Elastic ◽  
Anisotropic Elastic Material

The relations between stresses and strains in an anisotropic elastic material are presented in this chapter. A linear anisotropic elastic material can have as many as 21 elastic constants. This number is reduced when the material possesses a certain material symmetry. The number of elastic constants is also reduced, in most cases, when a two-dimensional deformation is considered. An important condition on elastic constants is that the strain energy must be positive. This condition implies that the 6×6 matrices of elastic constants presented herein must be positive definite. Referring to a fixed rectangular coordinate system x1, x2, x3, let σij and εks be the stress and strain, respectively, in an anisotropic elastic material. The stress-strain law can be written as . . . σij = Cijksεks . . . . . .(2.1-1). . . in which Cijks are the elastic stiffnesses which are components of a fourth rank tensor. They satisfy the full symmetry conditions . . . Cijks = Cjiks, Cijks = Cijsk, Cijks = Cksij. . . . . . .(2.1-2). . .

Hooke’s law and elastic constants

2020 ◽  
pp. 29-54
Author(s):  
Adrian P. Sutton
Keyword(s):  
Elastic Constant ◽  
Elastic Constants ◽  
Hooke’S Law ◽  
Elastic Energy Density ◽  
Cauchy Pressure ◽  
Atomic Interactions ◽  
Hooke's Law ◽  
Isotropic Elasticity ◽  
Strain Tensors ◽  
Cauchy Relations

Hooke’s law and elastic constants are introduced. The symmetry of the elastic constant tensor follows from the symmetry of stress and strain tensors and the elastic energy density. The maximum number of independent elastic constants is 21 before crystal symmetry is considered, and this leads to the introduction of matrix notation. Neumann’s principle reduces the number of independent elastic constants in different crystal systems. It is proved that in isotropic elasticity there are only two independent elastic constants. The directional dependences of the three independent elastic constants in cubic crystalsare derived. The distinction between isothermal and adiabatic elastic constants is defined thermodynamically and shown to arise from anharmonicity of atomic interactions. Problems set 3involves the derivation of elastic constants atomistically, the numbers of independent elastic constants in non-cubic crystal symmetries, Cauchy relations, Cauchy pressure, invariants of the elastic constant tensorand compatibility stresses.

Complex Form of Hooke’s Law of Anisotropic Elastic Body

2020 ◽  
Vol 55 (4) ◽  
pp. 514-535
Author(s):  
N. I. Martynov
Keyword(s):  
Elastic Body ◽  
Complex Form ◽  
Hooke’S Law ◽  
Hooke's Law ◽  
Anisotropic Elastic

An investigation of the isotropy of epoxy polymers

1992 ◽  
Vol 7 (12) ◽  
pp. 3352-3358 ◽  
Author(s):  
Edwin M. Odom ◽  
Donald F. Adams
Keyword(s):  
Elastic Constants ◽  
Polymeric Materials ◽  
Stress And Strain ◽  
Current Investigation ◽  
Hooke’S Law ◽  
Epoxy Polymers ◽  
Hooke's Law ◽  
The Relationship

Measurements of the engineering constants E, G, and v are routinely made for polymeric materials. If these materials are isotropic, these measurements should satisfy the relationship G = E/2(1 + v). However, many past measurements have indicated that this relationship is not satisfied. This raises questions about the assumptions of material isotropy and the applicability of Hooke's law. The methods used to measure these engineering constants for a number of different polymers are first described. Then, new results obtained in the current investigation are presented, indicating that the elastic constants do in fact satisfy the isotropic relationship for strains up to 0.5%. However, it is shown that at strain levels above this level, the relationship between stress and strain is nonlinear.

INVERSION OF ANISOTROPIC ELASTIC CONSTANTS AND MUD SPEED USING BOREHOLE SONIC MODES

2020 ◽  
Author(s):  
Ting Lei ◽  
◽  
Romain Prioul ◽  
Adam Donald ◽  
Edgar Ignacio Velez Arteaga ◽  
...  
Keyword(s):  
Elastic Constants ◽  
Anisotropic Elastic

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^

Sign in / Sign up

Export Citation Format

Share Document