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.

The Strain-Energy Density of Rubber-Like Shells of Revolution Undergoing Torsionless, Axisymmetric Deformation (Axishells)

1986 ◽  
Vol 53 (3) ◽  
pp. 593-596 ◽  
Author(s):  
J. G. Simmonds
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Shell Theory ◽  
Three Dimensional ◽  
Axisymmetric Deformation ◽  
Shells Of Revolution ◽  
Characteristic Wavelength ◽  
Transverse Shearing ◽  
Shearing Strain

We consider a shell of revolution made of an incompressible elastically isotropic material. Assuming a torsionless, axisymmetric three-dimensional displacement field that permits large normal strains (i.e., large thickness changes) but small transverse shearing strains, we construct a two-dimensional strain-energy density for a first-approximation shell theory in which the extensional strains may be O(1). The bending strains, however, are small, as in Reissner’s nonlinear theory. An error estimate is given that depends on the undeformed thickness and curvatures, the bending strains, the transverse shearing strain, and the characteristic wavelength of the shell theory solutions.

Fatigue Strength of Three Dimensional Welded Joints: A Synthesis Based on the Strain Energy Density over a Control Volume

2007 ◽  
Vol 348-349 ◽  
pp. 413-416
Author(s):  
M. Zappalorto ◽  
Filippo Berto ◽  
Paolo Lazzarin
Keyword(s):  
Experimental Data ◽  
Energy Density ◽  
Welded Joints ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Complex Geometry ◽  
Control Volume ◽  
Three Dimensional ◽  
Weld Toe ◽  
Weld Root

A recent approach based on the local strain energy density (SED) averaged over a given control volume is applied to well documented experimental data taken from the literature, all related to steel welded joints of complex geometry. This small size volume embraces the weld root or the weld toe, both regions modelled as sharp (zero notch radius) V-notches with different opening angles. The SED is evaluated from three-dimensional finite element models by using a circular sector with a radius equal to 0.28 mm. The data expressed in terms of the local energy fall in a scatter band recently reported in the literature, based on about 650 experimental data related to fillet welded joints made of structural steel with failures occurring at the weld toe or at the weld root.

scholarly journals On Dynamic Extension of a Local Material Symmetry Group for Micropolar Media

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1632
Author(s):  
Victor A. Eremeyev ◽  
Violetta Konopińska-Zmysłowska
Keyword(s):  
Energy Density ◽  
Kinetic Energy ◽  
Symmetry Group ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Constitutive Relations ◽  
Kinetic Energy Density ◽  
Material Symmetry ◽  
Local Material ◽  
Micropolar Media

For micropolar media we present a new definition of the local material symmetry group considering invariant properties of the both kinetic energy and strain energy density under changes of a reference placement. Unlike simple (Cauchy) materials, micropolar media can be characterized through two kinematically independent fields, that are translation vector and orthogonal microrotation tensor. In other words, in micropolar continua we have six degrees of freedom (DOF) that are three DOFs for translations and three DOFs for rotations. So the corresponding kinetic energy density nontrivially depends on linear and angular velocity. Here we define the local material symmetry group as a set of ordered triples of tensors which keep both kinetic energy density and strain energy density unchanged during the related change of a reference placement. The triples were obtained using transformation rules of strain measures and microinertia tensors under replacement of a reference placement. From the physical point of view, the local material symmetry group consists of such density-preserving transformations of a reference placement, that cannot be experimentally detected. So the constitutive relations become invariant under such transformations. Knowing a priori a material’s symmetry, one can establish a simplified form of constitutive relations. In particular, the number of independent arguments in constitutive relations could be significantly reduced.

A Geometrically Exact Rod Model Including In-Plane Cross-Sectional Deformation

2010 ◽  
Vol 78 (1) ◽  
Author(s):  
Ajeet Kumar ◽  
Subrata Mukherjee
Keyword(s):  
Energy Density ◽  
Cross Section ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Three Dimensional ◽  
Rigid Motion ◽  
Transversely Isotropic ◽  
Equilibrium Equations ◽  
Cross Sectional ◽  
Deformation Problem

We present a novel approach for nonlinear, three dimensional deformation of a rod that allows in-plane cross-sectional deformation. The approach is based on the concept of multiplicative decomposition, i.e., the deformation of a rod’s cross section is performed in two steps: pure in-plane cross-sectional deformation followed by its rigid motion. This decomposition, in turn, allows straightforward extension of the special Cosserat theory of rods (having rigid cross section) to a new theory allowing in-plane cross-sectional deformation. We then derive a complete set of static equilibrium equations along with the boundary conditions necessary for analytical/numerical solution of the aforementioned deformation problem. A variational approach to solve the relevant boundary value problem is also presented. Later we use symmetry arguments to derive invariants of the objective strain measures for transversely isotropic rods, as well as for rods with inbuilt handedness (hemitropy) such as DNA and carbon nanotubes. The invariants derived put restrictions on the form of the strain energy density leading to a simplified form of quadratic strain energy density that exhibits some interesting physically relevant coupling between the different modes of deformation.

An Innovative Yield Criterion Considering Strain Rates Based on Von Mises Stress

2019 ◽  
Vol 142 (1) ◽  
Author(s):  
Jingwei Yu ◽  
Qingguo Fei ◽  
Peiwei Zhang ◽  
Yanbin Li ◽  
Dahai Zhang ◽  
...  
Keyword(s):  
Energy Density ◽  
Yield Strength ◽  
Constitutive Model ◽  
Strain Rate ◽  
Dynamic Load ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Yield Criterion ◽  
Von Mises Stress ◽  
Von Mises

Abstract An innovative yield criterion based on von Mises stress is proposed to represent the strain rate-dependent behavior under dynamic load. Considering the strain rate in the constitutive model, the distortional strain energy density is derived and the yield criterion is established. A plot of yield strength for a range of strain rate reveals that despite the differences in material properties and test methods, the yield strength rise can be represented by a unified criterion. The overall yield behavior of the material under dynamic load can be explained by introducing the strain rate into the constitutive model and threshold distortional strain energy density. This criterion is in a simple form that may be widely applied.

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.

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