Intrinsic formulation of the displacement-traction problem in linearized elasticity

2014 ◽  
Vol 24 (06) ◽  
pp. 1197-1216 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Elastic Body ◽  
Displacement Field ◽  
Tensor Field ◽  
Strain Tensor ◽  
Linearized Strain ◽  
Linearized Elasticity

The displacement-traction problem of linearized elasticity is a system of partial differential equations and boundary conditions whose unknown is the displacement field inside a linearly elastic body. We explicitly identify here the corresponding boundary conditions satisfied by the linearized strain tensor field associated with such a displacement field. Using this identification, we are then able to provide an intrinsic formulation of the displacement-traction problem of linearized elasticity, by showing how it can be recast into a boundary value problem whose unknown is the linearized strain tensor field.

A Mixed Method in Elasticity

1977 ◽  
Vol 44 (1) ◽  
pp. 51-56 ◽  
Author(s):  
N. S. V. Kameswara Rao ◽  
Y. C. Das
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Elastic Body ◽  
Mixed Method ◽  
Forced Vibration ◽  
Three Dimensional ◽  
Dynamic Problems ◽  
Partial Differential ◽  
Theory Of Plates

A mixed method for three-dimensional elasto-dynamic problems has been formulated which gives a complete choice in prescribing the boundary conditions in terms of either stresses, or displacements, or partly stresses and partly displacements. The general expressions for the responses of the elastic body have been derived in the form of transcendental partial differential equations of a set of initial functions, which can be evaluated in terms of the prescribed boundary conditions. The method so-formulated has been illustrated by applying it to the theory of plates. Only plates subjected to antisymmetric loads have been considered for illustration. Some examples of free and forced vibration of plates have been presented. Results are compared with solutions from existing theories.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

scholarly journals On the theory and practice of thin-walled structures

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tamaz Vashakmadze
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Complex Analysis ◽  
Theory And Practice ◽  
Linear Case ◽  
Wide Sense ◽  
Partial Differential ◽  
Thin Walled Structures ◽  
Thin Walled

Abstract The basic problem of satisfaction of boundary conditions is considered when the generalized stress vector is given on the surfaces of elastic plates and shells. This problem has so far remained open for both refined theories in a wide sense and hierarchic type models. In the linear case, it was formulated by I. N. Vekua for hierarchic models. In the nonlinear case, bending and compression-expansion processes do not split and in this context the exact structure is presented for the system of differential equations of von Kármán–Mindlin–Reisner (KMR) type, constructed without using a variety of ad hoc assumptions since one of the two relations of this system in the classical form is the compatibility condition, but not the equilibrium equation. In this paper, a unity mathematical theory is elaborated in both linear and nonlinear cases for anisotropic inhomogeneous elastic thin-walled structures. The theory approximately satisfies the corresponding system of partial differential equations and the boundary conditions on the surfaces of such structures. The problem is investigated and solved for hierarchic models too. The obtained results broaden the sphere of applications of complex analysis methods. The classical theory of finding a general solution of partial differential equations of complex analysis, which in the linear case was thoroughly developed in the works of Goursat, Weyl, Walsh, Bergman, Kolosov, Muskhelishvili, Bers, Vekua and others, is extended to the solution of basic nonlinear differential equations containing the nonlinear summand, which is a composition of Laplace and Monge–Ampére operators.

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

scholarly journals Repeated alternating loading of a elastoplastic three-layer plate in a temperature field

2021 ◽  
Vol 21 (1) ◽  
pp. 60-75
Author(s):  
Eduard I. Starovoitov ◽  
◽  
Denis V. Leonenko ◽  
Keyword(s):  
Temperature Field ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Plastic Region ◽  
Local Load ◽  
Natural State ◽  
Elastic Plastic

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

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