Dynamic Variational Principles for Discontinuous Elastic Fields

1979 ◽  
Vol 23 (02) ◽  
pp. 115-122 ◽  
Author(s):  
M. Cengiz Dökmeci
Keyword(s):  
Variational Principles ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Governing Equations ◽  
Natural Boundary ◽  
Dimensional Theory ◽  
Elastic Fields ◽  
Complete Set ◽  
Natural Boundary Conditions

Various forms of variational principles are derived for the three-dimensional theory of elastodynamics. The continuity requirements on the fields of stresses or strains and/or displacements are relaxed through Friedrichs's transformation. Thus, the generalized forms of certain types of earlier variational principles' are systematically constructed using a basic principle of physics. The variational principles derived herein are shown to generate, as the appropriate Euler equations, the complete set of the governing equations of linear elastodynamics, that is, the stress equations of motion, the strain displacement relations, the mixed natural boundary conditions, the constitutive equations, the natural initial conditions, and the jump conditions. Similarly, generalized variational principles are established for the nonlinear theory of elastodynamics, for the incremental motions in linear elasticity, and for an elastic Cosserat continuum, as well.

On the equations governing the motion of an anisotropic poroelastic material

2006 ◽  
Vol 462 (2072) ◽  
pp. 2373-2396 ◽  
Author(s):  
Gülay Altay ◽  
M. Cengİz Dökmecİ
Keyword(s):  
Variational Principle ◽  
Heterogeneous Media ◽  
Variational Principles ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Free Vibrations ◽  
Anisotropic Fluid ◽  
Governing Equations ◽  
Lagrange Equations ◽  
Poroelastic Material

We address Biot's equations governing the motion of an anisotropic fluid-saturated poroelastic material with certain properties. First, we investigate the uniqueness in solutions of the three-dimensional governing equations for the regular region of the poroelastic material and enumerate the conditions sufficient for the uniqueness. Next, by applying Hamilton's principle to the motion of the region, we obtain a variational principle that generates only the Biot–Newton equations and the associated natural boundary conditions. Then, by extending the variational principle for the region with an internal fixed surface of discontinuity through Legendre's transformation, we derive a six-field variational principle that operates on all the poroelastic field variables. The variational principle leads, as its Euler–Lagrange equations, to all the governing equations, including the jump conditions but the initial conditions, as a generalized version of the Hellinger–Reissner variational principle. Moreover, we consider the free vibrations of the region, and we discuss some basic properties of eigenvalues and present a variational formulation by Rayleigh's quotient. This work provides a standard tool with the features of variational principles when numerically solving the governing equations in heterogeneous media with finite element methods, treating the free vibrations and consistently deriving some one-dimensional/two-dimensional equations of the poroelastic region.

On the Vibrations of a Laminated Body

1968 ◽  
Vol 35 (4) ◽  
pp. 689-696 ◽  
Author(s):  
J. D. Achenbach ◽  
C. T. Sun ◽  
G. Herrmann
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Low Frequency ◽  
Continuum Theory ◽  
Acoustic Mode ◽  
Governing Equations ◽  
Natural Boundary ◽  
Natural Boundary Conditions ◽  
Laminated Layer

A continuum theory for a laminated medium is further developed in this paper. Constitutive equations, stress equations of motion, and natural boundary conditions are presented, and sufficient conditions for a unique solution are discussed. The governing equations and boundary conditions are employed to study the thickness-twist motion of a laminated layer. For every nodal number there is a low frequency acoustic mode and a high frequency optical mode. The frequencies of the acoustic modes are compared with the corresponding frequencies predicted by the effective modulus theory, and the relative magnitudes of the material parameters for which these frequencies are substantially at variance are indicated.

Theory of Laminated Plates

1971 ◽  
Vol 38 (1) ◽  
pp. 231-238 ◽  
Author(s):  
C. T. Sun
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Continuum Theory ◽  
Laminated Plates ◽  
Harmonic Waves ◽  
Two Dimensional ◽  
Laminated Plate ◽  
Governing Equations ◽  
Dimensional Theory ◽  
Rotatory Inertia

A two-dimensional theory for laminated plates is deduced from the three-dimensional continuum theory for a laminated medium. Plate-stress equations of motion, plate-stress-strain relations, boundary conditions, and plate-displacement equations of motion are presented. The governing equations are employed to study the propagation of harmonic waves in a laminated plate. Dispersion curves are presented and compared with those obtained according to the three-dimensional continuum theory and the exact analysis. An approximate solution for flexural motions obtained by neglecting the gross and local rotatory inertia terms is also discussed.

Meshless Integral Method for Analysis of Elastoplastic Geotechnical Materials

2010 ◽  
Author(s):  
Jianfeng Ma ◽  
Joshua David Summers ◽  
Paul F. Joseph
Keyword(s):  
Boundary Conditions ◽  
Function Approximation ◽  
Integral Method ◽  
Weak Form ◽  
Constitutive Law ◽  
Governing Equations ◽  
Natural Boundary ◽  
Pressure Sensitive ◽  
Support Domain ◽  
Natural Boundary Conditions

The meshless integral method based on regularized boundary equation [1][2] is extended to analyze elastoplastic geotechnical materials. In this formulation, the problem domain is clouded with a node set using automatic node generation. The sub-domain and the support domain related to each node are also generated automatically using algorithms developed for this purpose. The governing integral equation is obtained from the weak form of elastoplasticity over a local sub-domain and the moving least-squares approximation is employed for meshless function approximation. The geotechnical materials are described by pressure-sensitive multi-surface Drucker-Prager/Cap plasticity constitutive law with hardening. A generalized collocation method is used to impose the essential boundary conditions and natural boundary conditions are incorporated in the system governing equations. A comparison of the meshless results with the FEM results shows that the meshless integral method is accurate and robust enough to solve geotechnical materials.

scholarly journals New Analytical Model Used in Finite Element Analysis of Solids Mechanics

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1401 ◽  
Author(s):  
Sorin Vlase ◽  
Adrian Eracle Nicolescu ◽  
Marin Marin
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Elastic Solid ◽  
The Finite Element Method ◽  
Governing Equations ◽  
Element Analysis ◽  
Elastic Multibody System

In classical mechanics, determining the governing equations of motion using finite element analysis (FEA) of an elastic multibody system (MBS) leads to a system of second order differential equations. To integrate this, it must be transformed into a system of first-order equations. However, this can also be achieved directly and naturally if Hamilton’s equations are used. The paper presents this useful alternative formalism used in conjunction with the finite element method for MBSs. The motion equations in the very general case of a three-dimensional motion of an elastic solid are obtained. To illustrate the method, two examples are presented. A comparison between the integration times in the two cases presents another possible advantage of applying this method.

An Analytical and Experimental Evaluation of the Damping Capacity of Sandwich Beams With Viscoelastic Cores

1967 ◽  
Vol 89 (3) ◽  
pp. 438-443 ◽  
Author(s):  
I. W. Jones ◽  
V. L. Salerno ◽  
A. Savacchio
Keyword(s):  
Previous Investigation ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Vibration Damping ◽  
Damping Capacity ◽  
Sandwich Beams ◽  
Natural Boundary ◽  
Test Program ◽  
Test Procedures ◽  
Natural Boundary Conditions

A study is made of the free vibrations of sandwich beams with viscoelastic cores. The study, which is a generalization of a previous investigation by the authors [1] includes the equations of motion and natural boundary conditions, derivation of expressions for the modal distribution of damping based upon “small damping” assumptions, numerical examples, and a supporting test program. The generally high values calculated for beams of various materials indicate that this type of construction is efficient for vibration damping applications. It was found, however, that the calculated and test values were not in accord. This lack of agreement signifies the necessity for greater refinement in both analytical methods and test procedures.

scholarly journals An unrecognized inertial force induced by flow curvature in microfluidics

2021 ◽  
Vol 118 (29) ◽  
pp. e2103822118
Author(s):  
Siddhansh Agarwal ◽  
Fan Kiat Chan ◽  
Bhargav Rallabandi ◽  
Mattia Gazzola ◽  
Sascha Hilgenfeldt
Keyword(s):  
Entire Range ◽  
State Of The Art ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Inertial Force ◽  
Inertial Forces ◽  
Inertial Effects ◽  
Inviscid Flows ◽  
Flow Curvature ◽  
Complete Set

Modern inertial microfluidics routinely employs oscillatory flows around localized solid features or microbubbles for controlled, specific manipulation of particles, droplets, and cells. It is shown that theories of inertial effects that have been state of the art for decades miss major contributions and strongly underestimate forces on small suspended objects in a range of practically relevant conditions. An analytical approach is presented that derives a complete set of inertial forces and quantifies them in closed form as easy-to-use equations of motion, spanning the entire range from viscous to inviscid flows. The theory predicts additional attractive contributions toward oscillating boundaries, even for density-matched particles, a previously unexplained experimental observation. The accuracy of the theory is demonstrated against full-scale, three-dimensional direct numerical simulations throughout its range.

scholarly journals Stress field formulation of linear electro-magneto-elastic materials

2019 ◽  
Vol 24 (12) ◽  
pp. 3806-3822
Author(s):  
A Amiri-Hezaveh ◽  
P Karimi ◽  
M Ostoja-Starzewski
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Governing Equations ◽  
Field Equations ◽  
Elastic Solids ◽  
Elastic Materials ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Sufficient Set

A stress-based approach to the analysis of linear electro-magneto-elastic materials is proposed. Firstly, field equations for linear electro-magneto-elastic solids are given in detail. Next, as a counterpart of coupled governing equations in terms of the displacement field, generalized stress equations of motion for the analysis of three-dimensional (3D) problems Are obtained – they supply a more convenient basis when mechanical boundary conditions are entirely tractions. Then, a sufficient set of conditions for the corresponding solution of generalized stress equations of motion to be unique are detailed in a uniqueness theorem. A numerical passage to obtain the solution of such equations is then given by generalizing a reciprocity theorem in terms of stress for such materials. Finally, as particular cases of the general 3D form, the stress equations of motion for planar problems (plane strain and Generalized plane stress) for transversely isotropic media are formulated.

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