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.

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.

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.

scholarly journals Volume free strain energy density method for applications to blunt V-notches

2020 ◽  
Vol 28 ◽  
pp. 734-742
Author(s):  
Pietro Foti ◽  
Seyed Mohammad Javad Razavi ◽  
Liviu Marsavina ◽  
Filippo Berto
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Density Method ◽  
Free Strain

scholarly journals On the application of the volume free strain energy density method to blunt V-notches under mixed mode condition

2021 ◽  
Vol 230 ◽  
pp. 111716
Author(s):  
Pietro Foti ◽  
Seyed Mohammad Javad Razavi ◽  
Majid Reza Ayatollahi ◽  
Liviu Marsavina ◽  
Filippo Berto
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Mixed Mode ◽  
Strain Energy ◽  
Density Method ◽  
Free Strain

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.

Storing Energy in the Mechanical Domain

2014 ◽  
Vol 1679 ◽  
Author(s):  
O.G. Súchil ◽  
G. Abadal ◽  
F. Torres
Keyword(s):  
Energy Source ◽  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Critical Strain ◽  
Substrate Surface ◽  
Maximum Load ◽  
Linear Buckling ◽  
Initial Ph ◽  
Self Powered

ABSTRACTSelf-powered microsystems as an alternative to standard systems powered by electrochemical batteries are taking a growing interest. In this work, we propose a different method to store the energy harvested from the ambient which is performed in the mechanical domain. Our mechanical storage concept is based on a spring which is loaded by the force associated to the energy source to be harvested [1]. The approach is based on pressing an array of fine wires (fws) grown vertically on a substrate surface. For the fine wires based battery, we have chosen ZnO fine wires due the fact that they could be grown using a simple and cheap process named hydrothermal method [2]. We have reported previous experiments changing temperature and initial pH of the solution in order to determine the best growth [3]. From new experiments done varying the compounds concentration the best results of fine wires were obtained. To characterize these fine wires we have considered that the maximum load we can apply to the system is limited by the linear buckling of the fine wires. From the best results we obtained a critical strain of εc = 3.72 % and a strain energy density of U = 11.26 MJ/m3, for a pinned-fixed configuration [4].

On the Crack Initiated from the Bi-Material Notch Tip

2010 ◽  
Vol 452-453 ◽  
pp. 441-444 ◽  
Author(s):  
Tomáš Profant ◽  
Jan Klusák ◽  
Michal Kotoul
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Local Minimum ◽  
Plane Elasticity ◽  
Mean Value ◽  
Tangential Stresses ◽  
Density Factor ◽  
The Mean ◽  
Energy Density Factor

The bi-material notch composed of two orthotropic parts is considered. The radial and tangential stresses and strain energy density is expressed using the Stroh-Eshelby-Lekhnitskii formalism for the plane elasticity. The potential direction of the crack initiation is determined from the maximum mean value of the tangential stresses and local minimum of the mean value of the generalized strain energy density factor in both materials. Matched asymptotic procedure is used to derive the change of potential energy for the debonding crack and the crack initiated in the determined direction.

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