A Multiscale Approximation Method to Describe Diatomic Crystalline Systems: Constitutive Equations

2018 ◽  
Vol 09 (03) ◽  
pp. 1840001
Author(s):  
Pasquale Giovine
Keyword(s):  
Approximation Method ◽  
Constitutive Equations ◽  
Mixture Theory ◽  
Theory Of Elasticity ◽  
Representation Theorems ◽  
Dynamical Equations ◽  
Elastic Bodies ◽  
Multiscale Approximation ◽  
Theory Use ◽  
Final Expression

We model the mechanical behavior of diatomic crystals in the light of mixture theory. Use is made of an approximation method similar to one proposed by Signorini within the theory of elasticity, by supposing that the relative motion between phases is infinitesimal. The constitutive equations for a mixture of elastic bodies in the absence of diffusion are adapted to the partially linearized case considered here, and the representation theorems for constitutive fields are applied to obtain the final expression of dynamical equations in the form which appears in theories of continua with vectorial microstructure. Comparisons are made with results of lattice theories.

Fundamental Concepts of Strength of Materials

2015 ◽  
pp. 10-30
Keyword(s):  
Constitutive Model ◽  
Constitutive Equations ◽  
Virtual Work ◽  
Elementary Theory ◽  
Elastic Beam ◽  
Theory Of Elasticity ◽  
Principle Of Virtual Work ◽  
Basic Tool ◽  
Strength Of Materials

This chapter presents the concepts of strength of materials that are relevant to the analysis of frames. These are the modified Timoshenko theory of elastic beams (Sections 2.1-2.3) and the Euler-Bernoulli one (Section 2.4). These concepts are not presented as in the conventional textbooks of strength of materials. Instead, the formulations are described using the scheme that is customary in the theory of elasticity and that was described in Chapter 1 (Section 1.1.1) (i.e. in terms of kinematics, statics, and constitutive equations). Kinematics is the branch of mechanics that studies the movement of solids and structures without considering its causes. Statics studies the equilibrium of forces; the basic tool for this analysis is the principle of virtual work. The constitutive model that describes a one-to-one relationship between stresses and deformations completes the formulation of the elastic beam problem. Finally, in Section 2.5, some concepts of the elementary theory of torsion needed for the formulation of tridimensional frames are recalled.

scholarly journals Unilateral Contact of Elastic Bodies (Moment Theory)

2001 ◽  
Vol 8 (4) ◽  
pp. 753-766
Author(s):  
R. Gachechiladze
Keyword(s):  
Contact Zone ◽  
Couple Stress Theory ◽  
Contact Problems ◽  
Theory Of Elasticity ◽  
Unilateral Contact ◽  
Stress Theory ◽  
Nonhomogeneous Media ◽  
Elastic Bodies ◽  
The Moment ◽  
Boundary Contact

Abstract Boundary contact problems of statics of the moment (couple-stress) theory of elasticity are studied in the case of a unilateral contact of two elastic anisotropic nonhomogeneous media. A problem, in which during deformation the contact zone lies within the boundaries of some domain, and a problem, in which the contact zone can extend, are given a separate treatment. Concrete problems suitable for numerical realizations are considered.

Advanced Mechanics of Solids

2021 ◽  
Author(s):  
Lester W. Schmerr Jr.
Keyword(s):  
Undergraduate Students ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Mechanics Of Solids ◽  
Stress Strain ◽  
Elastic Bodies ◽  
Mechanics Of Materials ◽  
Modern Textbook ◽  
Elementary Mechanics ◽  
Matrix Vector

Build on the foundations of elementary mechanics of materials texts with this modern textbook that covers the analysis of stresses and strains in elastic bodies. Discover how all analyses of stress and strain are based on the four pillars of equilibrium, compatibility, stress-strain relations, and boundary conditions. These four principles are discussed and provide a bridge between elementary analyses and more detailed treatments with the theory of elasticity. Using MATLAB® extensively throughout, the author considers three-dimensional stress, strain and stress-strain relations in detail with matrix-vector relations. Based on classroom-proven material, this valuable resource provides a unified approach useful for advanced undergraduate students and graduate students, practicing engineers, and researchers.

Steady-State Frictional Sliding of Two Elastic Bodies With a Wavy Contact Interface

2000 ◽  
Vol 122 (3) ◽  
pp. 490-495 ◽  
Author(s):  
M. Nosonovsky ◽  
G. G. Adams
Keyword(s):  
Steady State ◽  
Friction Coefficient ◽  
Jacobi Polynomials ◽  
Integral Transform ◽  
Theory Of Elasticity ◽  
Sliding Velocity ◽  
Zone Length ◽  
Frictional Sliding ◽  
Flat Surfaces ◽  
Elastic Bodies

Dry frictional sliding of two elastic bodies, one of which has a periodic wavy surface, is considered. Such a model represents the frictional sliding of two nominally flat surfaces, one of which has periodically spaced asperities. The dependence of the true contact area on loading is analyzed by using the plane strain theory of elasticity. Fourier series and integral transform techniques are applied to reduce the problem to an integral equation which is solved using a series of Jacobi polynomials. For steady-state dynamic frictional sliding with given values of the friction coefficient, materials constants, and sliding velocity, the dependence of the contact zone length on the remotely applied tractions is determined. The results indicate a decrease of the minimum applied traction required to close the gap between the bodies, with an increase of the friction coefficient and/or the sliding velocity. A resonance exists as the sliding velocity approaches the Rayleigh wave speed of the flat body. [S0742-4787(00)01403-X]

A Discrete Quasi-Coordinate Formulation for the Dynamics of Elastic Bodies

2006 ◽  
Vol 74 (2) ◽  
pp. 231-239 ◽  
Author(s):  
G. M. T. D’Eleuterio ◽  
T. D. Barfoot
Keyword(s):  
Equations Of Motion ◽  
Computational Cost ◽  
Dynamical Equations ◽  
Generalized Coordinates ◽  
Order Form ◽  
Elastic Bodies ◽  
Alternative Approach ◽  
Elastic Systems ◽  
Computationally Intensive ◽  
Derivatives Of

The discretized equations of motion for elastic systems are typically displayed in second-order form. That is, the elastic displacements are represented by a set of discretized (generalized) coordinates, such as those used in a finite-element method, and the elastic rates are simply taken to be the time-derivatives of these displacements. Unfortunately, this approach leads to unpleasant and computationally intensive inertial terms when rigid rotations of a body must be taken into account, as is so often the case in multibody dynamics. An alternative approach, presented here, assumes the elastic rates to be discretized independently of the elastic displacements. The resulting dynamical equations of motion are simplified in form, and the computational cost is correspondingly lessened. However, a slightly more complex kinematical relation between the rate coordinates and the displacement coordinates is required. This tack leads to what may be described as a discrete quasi-coordinate formulation.

Hypersingular Integral Equations Arising in the Boundary Value Problems of the Elasticity Theory

2020 ◽  
Vol 73 (1) ◽  
pp. 51-75
Author(s):  
S M Mkhitaryan ◽  
M S Mkrtchyan ◽  
E G Kanetsyan
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Canonical Forms ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations ◽  
Hadamard Finite Part ◽  
Elastic Bodies

Summary The exact solutions of a class of hypersingular integral equations with kernels $\left( {s-x} \right)^{-2}$, $\left( {\sin \frac{s-x}{2}} \right)^{-2}$, $\left( {\sinh \frac{s-x}{2}} \right)^{-2},\cos \frac{s-x}{2}\left( {\sin \frac{s-x}{2}} \right)^{-2}$, $\cosh \frac{s-x}{2}\left( {\sinh \frac{s-x}{2}} \right)^{-2}$ are obtained where the integrals must be interpreted as Hadamard finite-part integrals. Problems of cracks in elastic bodies of various canonical forms under antiplane and plane deformations, where the crack edges are loaded symmetrically, lead to such equations. These problems, in turn, lead to mixed boundary value problems of the mathematical theory of elasticity for a half-plane, a circle, a strip and a wedge.

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