Influence of Boundary Conditions on Decay Rates in a Prestrained Plate1

2002 ◽  
Vol 69 (4) ◽  
pp. 515-520 ◽  
Author(s):  
B. Karp ◽  
D. Durban
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Boundary Data ◽  
Separation Of Variables ◽  
Finite Elasticity ◽  
Decay Rates ◽  
Constitutive Behavior ◽  
Hyperelastic Materials ◽  
Various Boundary Conditions ◽  
Decay Exponent

Decay of end perturbations imposed on a prestrained semi-infinite rectangular plate is investigated in the context of plane-strain incremental finite elasticity. A separation of variables eigenfunction formulation is used for the perturbed field within the plate. Numerical results for the leading decay exponent are given for three hyperelastic materials with various boundary conditions at the long faces of the plate. The study exposes a considerable sensitivity of axial decay rates to boundary data, to initial strain and to constitutive behavior. It is suggested that the results are relevant to the applicability of Saint-Venant’s principle even though the eigenfunctions are not always self-equilibrating.

End Effects in Prestrained Plates Under Compression

2004 ◽  
Vol 71 (6) ◽  
pp. 816-824 ◽  
Author(s):  
B. Karp
Keyword(s):  
Decay Rate ◽  
Separation Of Variables ◽  
Finite Elasticity ◽  
End Effects ◽  
Euler Formula ◽  
Abrupt Decrease ◽  
Incompressible Solids ◽  
Stress Free State ◽  
Low Sensitivity ◽  
Decay Exponent

The decay of end perturbations imposed on a rectangular plate subjected to compression is investigated in the context of plane-strain incremental finite elasticity. A separation of variables in the eigenfunction formulation is used for the perturbed field within the plate. Numerical results for the leading decay exponent are given for four rubbers: three compressible and one incompressible. It was found that the lowest decay rate is governed by a symmetric field that exhibits different patterns of dependence on the prestrain for compressible and for nearly incompressible solids. Compressible solids are characterized by low sensitivity of the decay rate to prestrain level up to moderate compression, beyond which an abrupt decrease of decay rate brings it to zero. Nearly incompressible solids, on the other hand, expose a different pattern involving interchange of modes with no decrease of decay rate to zero. Both patterns show that the decay rate obtained from linear elastic analysis can be considered as a good approximation for a prebuckled, slightly compressed plate, which is long enough in comparison to its width. Along with decaying modes, the eigenfunction expansion generates a nondecaying antisymmetric mode corresponding to buckling of the plate. Asymptotic expansion of that nondecaying mode near the stress free state predicts buckling according to the classical Euler formula. A consistent interpretation of end effects in the presence of a nondecaying mode is given.

On the Elastic Stability of a Rectangular Plate, when Subjected to a Variable Edge Thrust

1935 ◽  
Vol 31 (3) ◽  
pp. 368-381 ◽  
Author(s):  
D. M. A. Leggett
Keyword(s):  
Boundary Conditions ◽  
Approximate Method ◽  
Rectangular Plate ◽  
General Problem ◽  
Elastic Stability ◽  
Detailed Consideration ◽  
Various Boundary Conditions ◽  
The Stability

The stability of a rectangular plate, subjected to constant thrust over opposite pairs of edges, has been treated with some degree of completeness for various boundary conditions. The more general problem, in which the thrusts are no longer constant, has not yet received any treatment apart from the approximate method developed by E. Schwerin†, which would appear to be capable of only limited extension. The object of this paper is accordingly the detailed consideration of a simple case when the thrust is no longer constant.

The Symplectic System for Thermo-Viscoelastic Cylinders

2011 ◽  
Vol 250-253 ◽  
pp. 3662-3665
Author(s):  
Wei Xiang Zhang ◽  
Qi Xia Liu
Keyword(s):  
Boundary Conditions ◽  
Separation Of Variables ◽  
Solution Space ◽  
Governing Equations ◽  
Laplace Domain ◽  
General Solutions ◽  
Various Boundary Conditions ◽  
Temperature Boundary ◽  
The Time Domain ◽  
Symplectic System

An exact symplectic approach is presented for the isotropic viscoelastic solids subjected to external force and temperature boundary conditions. With the use of the method of separation of variables, all the general solutions of the governing equations are derived in the Laplace domain. These general solutions are expressed in concise analytical forms, and are easily to be transformed into the time domain. Accordingly, various boundary conditions can be conveniently described by the combination of the general solutions due to the completeness of the solution space. In the numerical example, the whole character of total creep of the viscoelastic solid is clearly exhibited.

scholarly journals Free Vibration Analysis of a Rectangular Plate with Kelvin Type Boundary Conditions

2007 ◽  
Vol 14 (6) ◽  
pp. 447-457 ◽  
Author(s):  
R. Kırışık ◽  
Ş. Yüksel
Keyword(s):  
Boundary Conditions ◽  
Modal Analysis ◽  
Rectangular Plate ◽  
Separation Of Variables ◽  
Free Vibration Analysis ◽  
Transverse Vibrations ◽  
Four Corners ◽  
Free Case ◽  
Type Boundary ◽  
External Forces

The transverse vibrations of a rectangular plate with the Kelvin type boundary conditions at four corners are investigated. The plate is modeled as being attached to four lumped spring-damper systems at the corners. An analytical procedure is proposed based on the modal analysis. The completely free case of the plate is first studied. The expressions for the eigenfrequencies and eigenfunctions of the plate are obtained by utilizing the separation of variables. Then, the case in which the stiffness and the viscous damping as external forces acting at the corners of the plate is studied. Following the modal analysis procedure, the general solution for the equation of motion of the rectangular plate is derived. Some numerical results are presented.

scholarly journals The Hamiltonian System Method for the Stress Analysis in Axisymmetric Problems of Viscoelastic Solids

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
W. X. Zhang ◽  
Y. Bai ◽  
F. Yuan
Keyword(s):  
Hamiltonian System ◽  
Boundary Conditions ◽  
Integral Transformation ◽  
Separation Of Variables ◽  
Expansion Method ◽  
Stress Concentrations ◽  
Viscoelastic Solids ◽  
Quasistatic Problem ◽  
General Solutions ◽  
Various Boundary Conditions

With the use of the Laplace integral transformation and state space formalism, the classical axial symmetric quasistatic problem of viscoelastic solids is discussed. By employing the method of separation of variables, the governing equations under Hamiltonian system are established, and hence, general solutions including the zero eigensolutions and nonzero eigensolutions are obtained analytically. Due to the completeness property of the general solutions, their linear combinations can describe various boundary conditions. Simply by applying the adjoint relationships of the symplectic orthogonality, the eigensolution expansion method for boundary condition problems is given. In the numerical examples, stress distributions of a circular cylinder under the end and lateral boundary conditions are obtained. The results exhibit that stress concentrations appear due to the displacement constraints, and that the effects are seriously confined near the constraints, decreasing rapidly with the distance from the boundary.

Upon Impact Numerical Modeling of Foam Materials

2017 ◽  
Vol 54 (2) ◽  
pp. 195-202
Author(s):  
Vasile Nastasescu ◽  
Silvia Marzavan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Modeling ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
General Character ◽  
Foam Material ◽  
Various Boundary Conditions ◽  
Specific Behaviour

The paper presents some theoretical and practical issues, particularly useful to users of numerical methods, especially finite element method for the behaviour modelling of the foam materials. Given the characteristics of specific behaviour of the foam materials, the requirement which has to be taken into consideration is the compression, inclusive impact with bodies more rigid then a foam material, when this is used alone or in combination with other materials in the form of composite laminated with various boundary conditions. The results and conclusions presented in this paper are the results of our investigations in the field and relates to the use of LS-Dyna program, but many observations, findings and conclusions, have a general character, valid for use of any numerical analysis by FEM programs.

The Quasi-Static Analysis for Two-Dimensional Thin Plates

2011 ◽  
Vol 255-260 ◽  
pp. 166-169
Author(s):  
Li Chen ◽  
Yang Bai
Keyword(s):  
Boundary Conditions ◽  
Eigenfunction Expansion ◽  
Solution Space ◽  
Expansion Method ◽  
Fundamental Solutions ◽  
Thin Plates ◽  
Elastic Plates ◽  
Various Boundary Conditions ◽  
Eigenfunction Expansion Method ◽  
Concentration Phenomena

The eigenfunction expansion method is introduced into the numerical calculations of elastic plates. Based on the variational method, all the fundamental solutions of the governing equations are obtained directly. Using eigenfunction expansion method, various boundary conditions can be conveniently described by the combination of the eigenfunctions due to the completeness of the solution space. The coefficients of the combination are determined by the boundary conditions. In the numerical example, the stress concentration phenomena produced by the restriction of displacement conditions is discussed in detail.

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