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.

Extending Koiter’s Simplified Equations to Shells of Arbitrary Closed Cross Section

2005 ◽  
Vol 73 (4) ◽  
pp. 709-711
Author(s):  
James G. Simmonds
Keyword(s):  
Cylindrical Shell ◽  
Energy Density ◽  
Error Estimate ◽  
Cross Section ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Circular Cylindrical Shell ◽  
Cylindrical Shells ◽  
Equilibrium Equations ◽  
Pointwise Error

The techniques used by Koiter in 1968 to derive a simplified set of linear equilibrium equations for an elastically isotropic circular cylindrical shell in terms of displacements and the associated pointwise error estimate engendered in Love’s uncoupled strain-energy density are here extended to derive analogous simplified equilibrium equations and an error estimate for elastically isotropic cylindrical shells of arbitrary closed cross section.

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.

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.

True bounds on Poisson's ratios for transversely isotropic solids

1992 ◽  
Vol 27 (1) ◽  
pp. 43-44 ◽  
Author(s):  
P S Theocaris ◽  
T P Philippidis
Keyword(s):  
Energy Density ◽  
Basic Principle ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Elastic Solid ◽  
Transversely Isotropic ◽  
Linear Elastic ◽  
Positive Strain ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

The basic principle of positive strain energy density of an anisotropic linear or non-linear elastic solid imposes bounds on the values of the stiffness and compliance tensor components. Although rational mathematical structuring of valid intervals for these components is possible and relatively simple, there are mathematical procedures less strictly followed by previous authors, which led to an overestimation of the bounds and misinterpretation of experimental results.

Large Deformation Analysis of the Arterial Cross Section

1971 ◽  
Vol 93 (2) ◽  
pp. 138-145 ◽  
Author(s):  
B. R. Simon ◽  
A. S. Kobayashi ◽  
D. E. Strandness ◽  
C. A. Wiederhielm
Keyword(s):  
Finite Element ◽  
Energy Density ◽  
Numerical Integration ◽  
Density Function ◽  
Cross Section ◽  
Finite Deformation ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Exponential Form ◽  
Strain Energy Density Function

Possible relations between arterial wall stresses and deformations and mechanisms contributing to atherosclerosis are discussed. Necessary material properties are determined experimentally and from available data in the literature by assuming the arterial response to be a static finite deformation of a thick-walled cylinder constrained in a state of plane strain and composed of an incompressible, nonlinear elastic, transversely isotropic material. Experimental justification from the literature and supporting theoretical considerations are presented for each assumption. The partial derivative of the strain energy density function δW1/δI , necessary for in-plane stress calculation, is determined to be of exponential form using in situ biaxial test results from the canine abdominal aorta. An axisymmetric numerical integration solution is developed and used as a check for finite element results. The large deformation finite element theory of Oden is modified to include aortic material nonlinearity and directional properties and is used for a structural analysis of the aortic cross section. Results of this investigation are: (a) Fung’s exponential form for the strain energy density function of soft tissues is found to be valid for the aorta in the biaxial states considered; (b) finite deformation analyses by the finite element method and numerical integration solution reveal that significant tangential stress gradients are present in arteries commonly assumed to be “thin-walled” tubes using linear theory.

Mechanical Behavior Modeling of Hyperelastic Transversely Isotropic Materials Based on a New Polyconvex Strain Energy Function

2018 ◽  
Vol 10 (09) ◽  
pp. 1850104 ◽  
Author(s):  
D. M. Taghizadeh ◽  
H. Darijani
Keyword(s):  
Energy Density ◽  
Mechanical Behavior ◽  
Test Data ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Transversely Isotropic ◽  
Material Behavior ◽  
Isotropic Materials ◽  
Proposed Model ◽  
Transversely Isotropic Materials

In this paper, the mechanical behavior of incompressible transversely isotropic materials is modeled using a strain energy density in the framework of Ball’s theory. Based on this profound theory and with respect to physical and mathematical aspects of deformation invariants, a new polyconvex constitutive model is proposed for the mechanical behavior of these materials. From the physical viewpoint, it is assumed that the proposed model is additively decomposed into three parts nominally representing the energy contributions from the matrix, fiber and fiber–matrix interaction where each of the parts should be presented in terms of the invariants consistent with the physics of the deformation. From the mathematical viewpoint, the proposed model satisfies the fundamental postulates on the form of strain energy density, specially polyconvexity and coercivity constraints. Indeed, polyconvexity ensures ellipticity condition, which in turn provides material stability and in combination with coercivity condition, guarantees the existence of the global minimizer of the total energy. In order to evaluate the performance of the proposed strain energy density function, some test data of incompressible transverse materials with pure homogeneous deformations are used. It is shown that there is a good agreement between the test data and the obtained results from the proposed model. At the end, the performance of the proposed model in the prediction of the material behavior is evaluated rather than other models for two representative problems.

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