On the Form of Variationally Derived Shell Equations

1964 ◽  
Vol 31 (2) ◽  
pp. 233-238 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
Variational Problem ◽  
Shell Theory ◽  
Three Dimensional ◽  
Torsion Problem ◽  
Considerable Influence ◽  
Equilibrium Equations ◽  
Circular Cylindrical Shells ◽  
Shell Equations ◽  
Basic Equations ◽  
Three Dimensional Elasticity

In the first part of the paper the basic equations of three-dimensional elasticity are formulated as a system of four equilibrium equations and seven stress displacement relations, together with a variational problem which has these eleven equations as Euler equations. In the second part of the paper, the new variational problem is used for a derivation of shell theory which accounts particularly simply for the differences between the resultants N12 and N21 and the couples M12 and M21. In the third part of the paper a solution is given of the torsion problem for circumferentially nonhomogeneous circular cylindrical shells, as an explicit demonstration of the fact that certain terms in the shell equations which are often of negligible influence sometimes are of considerable influence.

scholarly journals Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Mircea Bîrsan
Keyword(s):  
Energy Density ◽  
Constitutive Model ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Shell Theory ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
General Method

AbstractIn this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform the integration over the thickness and obtain a consistent shell model of order $$ O(h^5) $$ O ( h 5 ) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

scholarly journals Basic Solutions of Three Dimensional Elasticity

2021 ◽  
Vol 236 ◽  
pp. 05039
Author(s):  
Wx Zhang
Keyword(s):  
Boundary Conditions ◽  
Three Dimensional ◽  
Practical Calculation ◽  
Basic Solution ◽  
Final Solution ◽  
Theoretical Derivation ◽  
Basic Solutions ◽  
Special Solutions ◽  
Basic Equations ◽  
Three Dimensional Elasticity

Elastic calculation method is an important research content of computational mechanics. The problems of elasticity include basic equations and boundary conditions. Therefore, the final solution consists of the general solutions of the basic equations and the special solutions satisfying the boundary conditions. Numerical method is often used in practical calculation, but the analytical solution is also an important subject for researchers. In this paper, the basic solution of three-dimensional elastic materials is given by theoretical derivation.

Conservation laws of linear elasticity in stress formulations

2005 ◽  
Vol 461 (2053) ◽  
pp. 99-116 ◽  
Author(s):  
Shaofan Li ◽  
Anurag Gupta ◽  
Xanthippi Markenscoff
Keyword(s):  
Conservation Laws ◽  
Linear Elasticity ◽  
Three Dimensional ◽  
Cauchy Stress ◽  
General Boundary Conditions ◽  
Cauchy Stress Tensor ◽  
Equilibrium Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Three Dimensional Elasticity

In this paper, we present new conservation laws of linear elasticity which have been discovered. These newly discovered conservation laws are expressed solely in terms of the Cauchy stress tensor, and they are genuine, non–trivial conservation laws that are intrinsically different from the displacement conservation laws previously known. They represent the variational symmetry conditions of combined Beltrami–Michell compatibility equations and the equilibrium equations. To derive these conservation laws, Noether's theorem is extended to partial differential equations of a tensorial field with general boundary conditions. By applying the tensorial version of Noether's theorem to Pobedrja's stress formulation of three–dimensional elasticity, a class of new conservation laws in terms of stresses has been obtained.

scholarly journals Derivation of a refined six-parameter shell model: descent from the three-dimensional Cosserat elasticity using a method of classical shell theory

2020 ◽  
Vol 25 (6) ◽  
pp. 1318-1339 ◽  
Author(s):  
Mircea Bîrsan
Keyword(s):  
Shell Model ◽  
Shell Theory ◽  
Local Equilibrium ◽  
Three Dimensional ◽  
Simple Expression ◽  
Equilibrium Equations ◽  
Cosserat Elasticity ◽  
Shell Models ◽  
Classical Shell Theory ◽  
Derivation Method

Starting from the three-dimensional Cosserat elasticity, we derive a two-dimensional model for isotropic elastic shells. For the dimensional reduction, we employ a derivation method similar to that used in classical shell theory, as presented systematically by Steigmann (Koiter’s shell theory from the perspective of three-dimensional nonlinear elasticity. J Elast 2013; 111: 91–107). As a result, we obtain a geometrically nonlinear Cosserat shell model with a specific form of the strain energy density, which has a simple expression, with coefficients depending on the initial curvature tensor and on three-dimensional material constants. The explicit forms of the stress–strain relations and the local equilibrium equations are also recorded. Finally, we compare our results with other six-parameter shell models and discuss the relation to the classical Koiter shell model.

On a Variational Theorem in Elasticity and Its Application to Shell Theory

1964 ◽  
Vol 31 (4) ◽  
pp. 647-653 ◽  
Author(s):  
P. M. Naghdi
Keyword(s):  
Boundary Conditions ◽  
Shell Theory ◽  
Three Dimensional ◽  
Transverse Shear Deformation ◽  
Special Significance ◽  
Field Equations ◽  
Variational Theorem ◽  
Three Dimensional Elasticity ◽  
Strain Measures

After stating a variational theorem which is a further generalization of known variational theorems and which has as its Euler equations all of the field equations and the boundary conditions of classical linear three-dimensional elasticity, the remainder of the paper deals with its application to shell theory. A new characterization of the basic system of field equations and the boundary conditions of the linear theory of elastic shells is derived which includes the effect of transverse shear deformation and involves only symmetric resultants and symmetric shell-strain measures. These results are of special significance in relation to those of a number of recent investigations in shell theory under the Kirchhoff-Love hypothesis in which the boundary-value problem of shell theory is recast in terms of symmetric (but not necessarily the same) variables.

ELASTIC THIN SHELLS: ASYMPTOTIC THEORY IN THE ANISOTROPIC AND HETEROGENEOUS CASES

1995 ◽  
Vol 05 (04) ◽  
pp. 473-496 ◽  
Author(s):  
D. CAILLERIE ◽  
E. SANCHEZ-PALENCIA
Keyword(s):  
Shell Theory ◽  
Asymptotic Theory ◽  
Order Term ◽  
Three Dimensional ◽  
Leading Order ◽  
Three Dimensional Elasticity ◽  
Thin Shell Theory ◽  
Variational Formulations ◽  
Natural Way

Asymptotic (two-scale) methods are used to derive thin shell theory from three-dimensional elasticity. The asymptotic process is done directly for the variational formulations, and existence and uniqueness theorems are given for the shell problem. The asymptotic behavior is the same as that recently derived by the authors using classical hypotheses of shell theory. The role of the subspace G of pure bendings (inextensional motions) appears in a natural way. The asymptotic is basically described by a leading order term contained in G and a lower order term contained in the orthogonal to G. As in anisotropic heterogeneous plates, which exhibit a coupling between flexion and traction, in heterogeneous shells there is coupling between the terms in G and in its orthogonal.

scholarly journals STRESS-STRAIN STATE IN BOUNDARY LAYER OF THE CIRCULAR PLATES WITH VARIOUS THICKNESSES BASED ON THE REFINED THEORY

2020 ◽  
Vol 82 (1) ◽  
pp. 32-42
Author(s):  
Val.V. Firsanov ◽  
Q.H. Doan ◽  
N.D. Tran
Keyword(s):  
Boundary Layer ◽  
Stress State ◽  
Strain State ◽  
Three Dimensional ◽  
Refined Theory ◽  
Circular Plates ◽  
Equilibrium Equations ◽  
Stress Strain ◽  
Stress Strain State ◽  
Three Dimensional Elasticity

A variant of the refined theory on calculation of the stress-strain state of circular plates with symmetrically various thicknesses according to an arbitrary law in the radial direction was presented. Equations of the plate state were established by using the three-dimensional elasticity theory. The required displacements were approximately calculated according to upright direction to the middle plane by polynomials with two degrees higher than in the classical Kirchhoff - Love theory. The differential equation at equilibrium in displacements with various coefficients was obtained by using means of the Lagrange variational principle. The direct integration of the equilibrium equations in the three-dimensional elasticity theory was used to determine the transverse normal and shear stresses. Of an isotropic circular plate with changing in thickness by using the analyzing Fourier chain, the obtained differential equilibrium equations in displacements with variable coefficients containing supplement components and taking into account of the effect of thickness on the stress-strain state of the plate. Examples of calculating the stress state of a circular plate with a thickness varying according to linear and parabolic laws under the action of a uniformly distributed load were considered. The limited difference method was employed to solve the boundary value problem. Comparison results of the refined and classical theories were investigated. It is demonstrated that the study on the stress state in the zones of its distortion (compounds, local loading zones, etc.) should use a refined theory, since the additional corresponding stresses of the “boundary layer” type are of the same order with the values of the main (internal) stress state. This is important to increase the reliability of strength calculations of such elements of aircraft-rocket structures as the power housings of aircraft, their various transition zones and connections, as well as objects in various engineering industries.

Three-Dimensional Analysis of an Elastic Plate-Cylinder Intersection by the Boundary-Point-Least-Squares Technique

1977 ◽  
Vol 99 (1) ◽  
pp. 17-25 ◽  
Author(s):  
D. Redekop
Keyword(s):  
Boundary Conditions ◽  
Least Squares ◽  
Boundary Point ◽  
Shell Theory ◽  
Three Dimensional ◽  
Middle Part ◽  
Elasticity Equations ◽  
Three Dimensional Elasticity ◽  
Tension Loading ◽  
Theoretical Results

The boundary-point-least-squares technique is applied to the axisymmetric three-dimensional elasticity problem of a hollow circular cylinder normally intersecting with a perforated flat plate. The geometry of the intersection is partitioned into three parts. Boundary conditions on the middle part and continuity conditions between adjacent parts are satisfied using the numerical boundary-point-least-squares technique while the governing elasticity equations and all other boundary conditions are satisfied exactly. Sample theoretical results are presented for the case of axisymmetric radial tension loading on the plate. The results compare favorably with previously published experimental data and provide supplementary data to theoretical results obtained using existing shell theory solutions.

Exact Elasticity Solution for Laminated Anisotropic Cylindrical Shells

1993 ◽  
Vol 60 (1) ◽  
pp. 41-47 ◽  
Author(s):  
K. Bhaskar ◽  
T. K. Varadan
Keyword(s):  
Shell Theory ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Elasticity Solution ◽  
Transverse Loading ◽  
Cylindrical Bending ◽  
Simply Supported ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
Anisotropic Cylindrical Shell

An exact three-dimensional elasticity solution is obtained for cylindrical bending of simply-supported laminated anisotropic cylindrical shell strips subjected to transverse loading. Displacements and stresses are presented for different angle-ply layups and radius-to-thickness ratios, so as to serve as useful benchmark results for the assessment of various two-dimensional shell theories. Finally, in the light of these results, the accuracy of the Love-type classical shell theory is examined.

Reissner’s New Mixed Variational Principle Applied to Laminated Cylindrical Shells

1992 ◽  
Vol 114 (1) ◽  
pp. 115-119 ◽  
Author(s):  
K. Bhaskar ◽  
T. K. Varadan
Keyword(s):  
Cylindrical Shells ◽  
Three Dimensional ◽  
Order Theory ◽  
Equilibrium Equations ◽  
Mixed Variational Principle ◽  
Elasticity Solutions ◽  
Transverse Stresses ◽  
Higher Order Theory ◽  
Displacement Approach ◽  
Three Dimensional Elasticity

Reissner’s new mixed variational theorem, which allows independent interpolation, through the thickness, of the three transverse stresses besides that of the three displacements, is applied here to derive a higher-order theory of laminated orthotropic cylindrical shells. The accuracy of the theory is verified by comparison with three-dimensional elasticity solutions. It is shown that Reissner’s principle does not directly lead to accurate transverse shear stress predictions, but requires the use of the equilibrium equations of three-dimensional elasticity as is common in the conventional displacement approach.

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