On Stress-Strain Relations and Strain-Energy Expressions in the Theory of Thin Elastic Shells

1960 ◽  
Vol 27 (1) ◽  
pp. 104-106 ◽  
Author(s):  
J. K. Knowles ◽  
Eric Reissner
Keyword(s):  
Strain Energy ◽  
Middle Surface ◽  
Stress Strain ◽  
Elastic Shells ◽  
Energy Expression ◽  
Energy Formula

The stress-strain relations of Flu¨gge and Byrne for thin elastic shells are inverted to express strain quantities, and therewith the strain energy, in terms of stress resultants and couples. In this form, and upon omission of terms which are small of order h2/R2, the stress-strain relations and the strain-energy expression are shown to be simply related to corresponding results of Trefftz. The strain-energy formula of Trefftz is generalized to arbitrary orthogonal middle surface co-ordinates.

The Strain-Energy Expression for Thin Elastic Shells

1958 ◽  
Vol 25 (4) ◽  
pp. 546-552
Author(s):  
J. H. Haywood ◽  
L. B. Wilson
Keyword(s):  
Circular Cylinder ◽  
Plane Stress ◽  
Strain Energy ◽  
Middle Surface ◽  
Thin Shells ◽  
Elastic Shells ◽  
Energy Expression ◽  
Small Deflection ◽  
Deflection Theory

Abstract A strain-energy expression is derived for thin isotropic elastic shells in terms of the displacements of the middle surface of the shell. This expression is confined to small-deflection theory, and the condition of plane stress previously used in the theory of thin shells is retained. A simplified expression is also obtained by the introduction of the Kirchhoff-Love hypothesis, and the relative merits of these two expressions are discussed. The strain-energy expression is applied to the thin circular cylinder, and the result is compared with various strain-energy expressions developed by previous authors.

A Strain-Energy Expression for Thin Elastic Shells

1949 ◽  
Vol 16 (2) ◽  
pp. 183-189
Author(s):  
H. L. Langhaar
Keyword(s):  
Strain Energy ◽  
Elastic Shell ◽  
Strain Tensor ◽  
Middle Surface ◽  
Circular Cylinders ◽  
Flat Plates ◽  
Energy Expression ◽  
Deflection Theory ◽  
Large Deflection Theory ◽  
Elliptical Cylinders

Abstract A derivation is given for the strain energy of an isotropic elastic shell whose radii of curvature are sufficiently large that strains may be assumed to vary linearly throughout the thickness. The work of Love (1) has been the only previous general investigation which expresses the strain energy in terms of the displacements of the middle surface. The effects of the tangential displacements upon the energy due to bending are found to differ appreciably from Love’s results in the first-order terms. As in the classical large-deflection theory of flat plates, quadratic terms in the derivatives of the normal deflection are retained in the strain tensor, but quadratic terms which involve the tangential displacements are neglected. Special forms of the general energy expression derived in this paper are given for shells in the shapes of flat plates, circular cylinders, elliptical cylinders, ellipsoids of revolution, and spheres. These applications, as well as certain intuitive observations, provide checks on the theory.

Mechanical characterization of a woven multi-layered hyperelastic composite laminate under uniaxial loading

2021 ◽  
pp. 002199832110115
Author(s):  
Shaikbepari Mohmmed Khajamoinuddin ◽  
Aritra Chatterjee ◽  
MR Bhat ◽  
Dineshkumar Harursampath ◽  
Namrata Gundiah
Keyword(s):  
Composite Materials ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Mechanical Tests ◽  
Stress Strain ◽  
Aerospace Applications ◽  
Envelope Material ◽  
The Individual ◽  
High Dielectric

We characterize the material properties of a woven, multi-layered, hyperelastic composite that is useful as an envelope material for high-altitude stratospheric airships and in the design of other large structures. The composite was fabricated by sandwiching a polyaramid Nomex® core, with good tensile strength, between polyimide Kapton® films with high dielectric constant, and cured with epoxy using a vacuum bagging technique. Uniaxial mechanical tests were used to stretch the individual materials and the composite to failure in the longitudinal and transverse directions respectively. The experimental data for Kapton® were fit to a five-parameter Yeoh form of nonlinear, hyperelastic and isotropic constitutive model. Image analysis of the Nomex® sheets, obtained using scanning electron microscopy, demonstrate two families of symmetrically oriented fibers at 69.3°± 7.4° and 129°± 5.3°. Stress-strain results for Nomex® were fit to a nonlinear and orthotropic Holzapfel-Gasser-Ogden (HGO) hyperelastic model with two fiber families. We used a linear decomposition of the strain energy function for the composite, based on the individual strain energy functions for Kapton® and Nomex®, obtained using experimental results. A rule of mixtures approach, using volume fractions of individual constituents present in the composite during specimen fabrication, was used to formulate the strain energy function for the composite. Model results for the composite were in good agreement with experimental stress-strain data. Constitutive properties for woven composite materials, combining nonlinear elastic properties within a composite materials framework, are required in the design of laminated pretensioned structures for civil engineering and in aerospace applications.

Analytical examination of mesh-dependency issue for uniaxial RC elements and new fracture energy-based regularization technique

2021 ◽  
pp. 105678952110392
Author(s):  
De-Cheng Feng ◽  
Xiaodan Ren
Keyword(s):  
Reinforced Concrete ◽  
Fracture Energy ◽  
Regularization Method ◽  
Strain Energy ◽  
Plain Concrete ◽  
Comprehensive Analysis ◽  
Stress Strain ◽  
Regularization Technique ◽  
Mesh Dependency ◽  
Energy Equivalence

This paper presents a comprehensive analysis of the mesh-dependency issue for both plain concrete and reinforced concrete (RC) members under uniaxial loading. The detailed mechanisms for each case are firstly derived, and the analytical and numerical strain energies for concrete in different cases are compared to explain the phenomena of mesh-dependency. It is found that the mesh-dependency will be relieved or even eliminated with the increasing of the reinforcing ratio. Meanwhile, a concept of the critical reinforcing ratio is proposed to identify the corresponding boundary of mesh-dependency of RC members. In order to verify the above findings, several illustrative examples are performed and discussed. Finally, to overcome the mesh-dependency issue for RC members with lower reinforcing ratios, we propose a unified regularization method that modifies both stress-strain relations of steel and concrete based on the strain energy equivalence. The method is also applied to the illustrative examples for validation, and the numerical results indicate that the developed method can obtain objective results for cases with different meshes and reinforcing ratios.

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