A new approach to curvature measures in linear shell theories

2020 ◽  
pp. 108128652097275
Author(s):  
Miroslav Šilhavý
Keyword(s):  
Strain Tensor ◽  
Middle Surface ◽  
Surface Strain ◽  
Affine Function ◽  
Shear Deformations ◽  
Strain Measure ◽  
Bending Strain ◽  
Curvature Measures ◽  
Tensor Formula ◽  
Strain Measures

The paper presents a coordinate-free analysis of deformation measures for shells modeled as 2D surfaces. These measures are represented by second-order tensors. As is well-known, two types are needed in general: the surface strain measure (deformations in tangential directions), and the bending strain measure (warping). Our approach first determines the 3D strain tensor E of a shear deformation of a 3D shell-like body and then linearizes E in two smallness parameters: the displacement and the distance of a point from the middle surface. The linearized expression is an affine function of the signed distance from the middle surface: the absolute term is the surface strain measure and the coefficient of the linear term is the bending strain measure. The main result of the paper determines these two tensors explicitly for general shear deformations and for the subcase of Kirchhoff-Love deformations. The derived surface strain measures are the classical ones: Naghdi’s surface strain measure generally and its well-known particular case for the Kirchhoff-Love deformations. With the bending strain measures comes a surprise: they are different from the traditional ones. For shear deformations our analysis provides a new tensor [Formula: see text], which is different from the widely used Naghdi’s bending strain tensor [Formula: see text]. In the particular case of Kirchhoff–Love deformations, the tensor [Formula: see text] reduces to a tensor [Formula: see text] introduced earlier by Anicic and Léger (Formulation bidimensionnelle exacte du modéle de coque 3D de Kirchhoff–Love. C R Acad Sci Paris I 1999; 329: 741–746). Again, [Formula: see text] is different from Koiter’s bending strain tensor [Formula: see text] (frequently used in this context). AMS 2010 classification: 74B99

DIFFERENTIAL EQUATIONS OF EQUILIBRIUM OF IDEALLY ELASTOPLASTIC CONTINUOUS MEDIUM FOR PLANE ONE-DIMENSIONAL DEFORMATION IN APPROXIMATION OF CLOSING EQUATIONS BY BIQUADRATIC FUNCTIONS

2021 ◽  
Vol 295 (2) ◽  
pp. 37-45
Author(s):  
S.V. Bakushev ◽  
Keyword(s):  
Differential Equations ◽  
Shear Deformation ◽  
Continuous Medium ◽  
Strain Tensor ◽  
Shear Deformations ◽  
One Dimensional ◽  
Volumetric Expansion ◽  
Fractional Rational Function ◽  
Secant Shear Modulus ◽  
Deformation Diagrams

The present article considers the construction of differential equations of equilibrium of geometrically and physically nonlinear ideally elastoplastic in relation to shear deformations of continuous medium under conditions of one-dimensional plane deformation, when the diagrams of volumetric and shear deformation are approximated by biquadratic functions. The construction of physical dependencies is based on calculating the secant moduli of volumetric and shear deformation. When approximating the graphs of the volumetric and shear deformation diagrams using two segments of parabolas, the secant shear modulus in the first segment is a linear function of the intensity of shear deformations; the secant modulus of volumetric expansion-contraction is a linear function of the first invariant of the strain tensor. In the second section of the diagrams of both volumetric and shear deformation, the secant shear modulus is a fractional (rational) function of the shear strain intensity; the secant modulus of volumetric expansion-compression is a fractional (rational) function of the first invariant of the strain tensor. Based on the assumption of independence, generally speaking, from each other of the volumetric and shear deformation diagrams, five main cases of physical dependences are considered, depending on the relative position of the break points of the graphs of the diagrams volumetric and shear deformation. On the basis of received physical equations, differential equations of equilibrium in displacements for continuous medium are derived under conditions of plane one-dimensional deformation. Differential equations of equilibrium in displacements constructed in the present article can be applied in determining stress and strain state of geometrically and physically nonlinear ideally elastoplastic in relation to shear deformations of continuous medium under conditions of plane one-dimensional deformation, closing equations of physical relations for which, based on experimental data, are approximated by biquadratic functions.

Compact Basis Free Relations for Stress Tensors Conjugate to Hill’s Strain Measures

2006 ◽  
Author(s):  
Reza Naghdabadi ◽  
Mohsen Asghari ◽  
Kamyar Ghavam
Keyword(s):  
Strain Tensor ◽  
Three Dimensional ◽  
Inner Product ◽  
Stress And Strain ◽  
Conjugate Pair ◽  
Stress Tensors ◽  
Plastic Materials ◽  
The Right ◽  
Elastic Plastic Materials ◽  
Strain Measures

If the double contraction of a stress tensor such as T and rate of a Lagrangean strain tensor such as E, i.e. T : E˙, produces the stress power then these stress and strain tensors are called a conjugate pair. The applications of the conjugate stress and strain measures are in the development of the basic relations in nonlinear continuum mechanics analysis such as modeling of constitutive equations of elastic-plastic materials. In this paper relations for stress tensors conjugate to an arbitrary Lagrangean strain measure of Hill’s class are obtained. The results of this paper are more compact and simpler in compare with those available in the literature. The results are valid for the three dimensional Euclidean inner product space and the case of distinct eigenvalues of the right stretch tensor U.

Bending strain in 3D topological semi-metals

2021 ◽  
Author(s):  
Jonas Diaz ◽  
Carsten Putzke ◽  
Xiangwei Huang ◽  
Amelia Estry ◽  
James Analytis ◽  
...  
Keyword(s):  
Focused Ion Beam ◽  
Ion Beam ◽  
Surface Strain ◽  
Gauge Fields ◽  
Cryogenic Temperatures ◽  
Test Bed ◽  
Strain Gradients ◽  
Topological Materials ◽  
Bending Strain ◽  
Set Up

Abstract We present an experimental set-up for the controlled application of strain gradients by mechanical piezoactuation on 3D crystalline microcantilevers that were fabricated by focused ion beam machining. A simple sample design tailored for transport characterization under strain at cryogenic temperatures is proposed. The topological semi-metal Cd3As2 serves as a test bed for the method, and we report extreme strain gradients of up to 1.3% µm-1 at a surface strain value of ≈ 0.65% at 4K. Interestingly, the unchanged quantum transport of the cantilever suggests that the bending cycle does not induce defects via plastic deformation. This approach is a first step towards realizing transport phenomena based on structural gradients, such as artificial gauge fields in topological materials.

Axisymmetric Thin Shell Analyses With Reduced Degrees-of-Freedom

1989 ◽  
Vol 42 (11S) ◽  
pp. S3-S12 ◽  
Author(s):  
C. A. Almeida ◽  
F. M. N. de Souza
Keyword(s):  
Displacement Field ◽  
Thin Shell ◽  
Degrees Of Freedom ◽  
Element Formulation ◽  
Shear Deformations ◽  
Bending Strain ◽  
Strain Components ◽  
Independent Degree ◽  
Displacement Based ◽  
Interpolation Functions

The formulation of a simple but effective unidimensional isoparametric displacement-based thin axisymmetric element is presented. The geometry is approximated by using cubic interpolation functions along the shell midsurface generatrix line and the element displacement field is represented by two spatial translation degrees-of-freedom only. The element kinematics incorporates membrane and bending strain components with the assumption of zero transverse shear deformations in the longitudinal and circumferential directions of the shell. This condition allows formulation of the element without using rotation as an independent degree-of-freedom, but continuity conditions between elements should be properly accounted for. The interaction effects between two adjoining elements or between an element and a rigid flange are modeled by using a penalty procedure to enforce continuity on the derivatives in the element midsurface radial displacements. The element formulation has been implemented and the results of various sample analyses are given to illustrate its effectiveness.

On Finite Symmetrical Deflections of Thin Shells of Revolution

1969 ◽  
Vol 36 (2) ◽  
pp. 267-270 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
Differential Equations ◽  
Nonlinear Theory ◽  
Transverse Shear ◽  
Shell Theory ◽  
Middle Surface ◽  
Thin Shells ◽  
Shells Of Revolution ◽  
Shear Deformations ◽  
Linear Shell Theory ◽  
Classical Subject

Recent simplifications of linear shell theory through consideration of transverse shear deformations and stress moments with axes normal to the shell middle surface suggest analogous approaches to the corresponding problem of nonlinear theory. As a first step in this direction consideration is given here to the classical subject of finite symmetrical deformations of shells of revolution. The principal new results of the present analysis concern the form of strain-displacement and compatibility differential equations.

scholarly journals Full-field hygro-expansion characterization of single softwood and hardwood pulp fibers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
N. H. Vonk ◽  
M. G. D. Geers ◽  
J. P. M. Hoefnagels
Keyword(s):  
Strain Tensor ◽  
Surface Strain ◽  
Pulp Fibers ◽  
Full Field ◽  
Long Duration ◽  
Large Spread ◽  
Surface Topographies ◽  
Hardwood Pulp ◽  
Fibril Angle

AbstractThe dimensional stability of paper products is a well-known problem, affecting multiple engineering applications. The macroscopic response of paper to moisture variations is governed by complex mechanisms originating in the material at all length-scales down to the fiber-level. Therefore, a recently-developed method, based on Global Digital Height Correlation of surface topographies is here exploited to measure the full-field hygro-expansion of single fibers, i. e. a surface strain tensor map over the full field of view is obtained as function of time. From the strain field, the longitudinal and transverse hygro-expansion and principle strains can be calculated. Long- and intermediate-duration dynamic tests are conducted on softwood and hardwood fibers. A large spread in the softwood fiber’s transverse and longitudinal hygro-expansion coefficient ratio was found, while hardwood fibers behave more consistently. Computing the principle strain ratios reduces this spread, as it takes into account the variations of the deformation direction, which is directly affected by the micro-fibril angle (MFA). Furthermore, long-duration tests allow identification of the half-times at which the fibers equilibrate. Finally, the determined major strain angles for all fibers are consistent with the MFA ranges reported in the literature.

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.

scholarly journals The intrinsic theory of linearly elastic plates

2018 ◽  
Vol 24 (4) ◽  
pp. 1182-1203 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare
Keyword(s):  
Boundary Condition ◽  
Elastic Plate ◽  
Tensor Field ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Middle Surface ◽  
Curvilinear Coordinates ◽  
Constant Thickness ◽  
Classical Approach ◽  
Tensor Fields

In an intrinsic approach to a problem in elasticity, the only unknown is a tensor field representing an appropriate ‘measure of strain’, instead of the displacement vector field in the classical approach. The objective of this paper is to study the displacement traction problem in the special case where the elastic body is a linearly elastic plate of constant thickness, clamped over a portion of its lateral face. In this respect, we first explicitly compute the intrinsic three-dimensional boundary condition of place in terms of the Cartesian components of the linearized strain tensor field, thus avoiding the recourse to covariant components in curvilinear coordinates and providing an interesting example of actual computation of an intrinsic boundary condition of place in three-dimensional elasticity. Second, we perform a rigorous asymptotic analysis of the three-dimensional equations as the thickness of the plate, considered as a parameter, approaches zero. As a result, we identify the intrinsic two-dimensional equations of a linearly elastic plate modelled by the Kirchhoff–Love theory, with the linearized change of metric and change of curvature tensor fields of the middle surface of the plate as the new unknowns, instead of the displacement field of the middle surface in the classical approach.

On stress rates in solid mechanics with application to elasticity theory

1974 ◽  
Vol 75 (2) ◽  
pp. 303-319 ◽  
Author(s):  
R. W. Ogden
Keyword(s):  
Elasticity Theory ◽  
Solid Mechanics ◽  
Constitutive Relations ◽  
Strain Analysis ◽  
Stress And Strain ◽  
Solid Materials ◽  
Strain Measure ◽  
Careful Choice ◽  
Further Development ◽  
Strain Measures

The efficacy of making a careful choice of strain measure in problems of solid mechanics was recognized and fully exploited by Hill(1–4). He brought coherence to fundamental stress and strain analysis by the introduction of the idea of conjugate pairs of stress and strain measures, and provided a platform for the further development of constitutive relations for solid materials.

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