scholarly journals Compatibility conditions of continua using Riemann–Cartan geometry

2020 ◽  
pp. 108128652096145
Author(s):  
Christian G Böhmer ◽  
Yongjo Lee
Keyword(s):  
Differential Geometry ◽  
Continuum Mechanics ◽  
Linear Elasticity ◽  
Compatibility Condition ◽  
Three Dimensional ◽  
Geometrical Approach ◽  
Compatibility Conditions ◽  
Einstein Tensor ◽  
Linear Elasticity Theory ◽  
Cartan Geometry

The compatibility conditions for generalised continua are studied in the framework of differential geometry, in particular Riemann–Cartan geometry. We show that Vallée’s compatibility condition in linear elasticity theory is equivalent to the vanishing of the three-dimensional Einstein tensor. Moreover, we show that the compatibility condition satisfied by Nye’s tensor also arises from the three-dimensional Einstein tensor, which appears to play a pivotal role in continuum mechanics not mentioned before. We discuss further compatibility conditions that can be obtained using our geometrical approach and apply it to the microcontinuum theories.

The Deformation of O-Rings Compressed by Smooth Rigid Plates

1997 ◽  
Vol 119 (2) ◽  
pp. 73-80 ◽  
Author(s):  
W. E. Warren ◽  
J. A. Weese
Keyword(s):  
Boundary Value Problem ◽  
Contact Pressure ◽  
Linear Elasticity ◽  
Radial Displacement ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Mixed Boundary Value Problem ◽  
Linear Elasticity Theory ◽  
Cross Sectional ◽  
Ring Material

Three-dimensional effects of symmetrically compressing an O-ring with rigid plates are investigated. The O-ring is assumed to be an incompressible elastic material and the mixed boundary value problem is formulated within the framework of classical the linear elasticity theory. Transversely compressing the O-ring between flat rigid plates increases the toroidal radius and induces a reduction in the O-ring cross-sectional radius. This reduction depends strongly on the conditions of contact between plate and O-ring, and the reduced contact pressure may threaten the integrity of the seal. For O-rings compressed radially inward compressed radially inward or outward, most of the radial displacement is accommodated by a change in the toroidal radius with very little actual compression of the ring material. If this effect is not accounted for in the design of the seal, the contact pressure may not be sufficient to secure the seal.

scholarly journals Mathematical models of supersonic and intersonic crack propagation in linear elastodynamics

2021 ◽  
Author(s):  
Javier Bonet ◽  
Antonio J. Gil
Keyword(s):  
Kinetic Energy ◽  
Crack Propagation ◽  
Mathematical Models ◽  
Linear Elasticity ◽  
Strain Energy ◽  
Equations Of Motion ◽  
Two Dimensions ◽  
Discontinuous Solutions ◽  
Linear Elasticity Theory ◽  
Intersonic Crack Propagation

AbstractThis paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory.

scholarly journals Asymptotic Behavior of Stable Structures Made of Beams

2021 ◽  
Author(s):  
Georges Griso ◽  
Larysa Khilkova ◽  
Julia Orlik ◽  
Olena Sivak
Keyword(s):  
Asymptotic Behavior ◽  
Cross Section ◽  
Cross Sections ◽  
Linear Elasticity ◽  
Circular Cross Section ◽  
Linear Elasticity Theory ◽  
Linear Elastic ◽  
The Cross ◽  
Small Rotation ◽  
Stable Structures

AbstractIn this paper, we study the asymptotic behavior of an $\varepsilon $ ε -periodic 3D stable structure made of beams of circular cross-section of radius $r$ r when the periodicity parameter $\varepsilon $ ε and the ratio ${r/\varepsilon }$ r / ε simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.

Linear elasticity theory of cubic quasicrystals

1993 ◽  
Vol 48 (10) ◽  
pp. 6999-7002 ◽  
Author(s):  
Wenge Yang ◽  
Renhui Wang ◽  
Di-hua Ding ◽  
Chengzheng Hu
Keyword(s):  
Elasticity Theory ◽  
Linear Elasticity ◽  
Linear Elasticity Theory

Three-Dimensional Vibration Analysis of Twisted Cylinder With Sectorial Cross Section

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
D. Zhou ◽  
S. H. Lo
Keyword(s):  
Cross Section ◽  
Chebyshev Polynomial ◽  
Numerical Solutions ◽  
Twist Angle ◽  
Ritz Method ◽  
Three Dimensional ◽  
Cylindrical Domain ◽  
Linear Elasticity Theory ◽  
Boundary Functions ◽  
Polynomial Series

The three-dimensional (3D) free vibration of twisted cylinders with sectorial cross section or a radial crack through the height of the cylinder is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. A simple coordinate transformation is applied to map the twisted cylindrical domain into a normal cylindrical domain. The product of a triplicate Chebyshev polynomial series along with properly defined boundary functions is selected as the admissible functions. An eigenvalue matrix equation can be conveniently derived through a minimization process by the Rayleigh–Ritz method. The boundary functions are devised in such a way that the geometric boundary conditions of the cylinder are automatically satisfied. The excellent property of Chebyshev polynomial series ensures robustness and rapid convergence of the numerical computations. The present study provides a full vibration spectrum for thick twisted cylinders with sectorial cross section, which could not be determined by 1D or 2D models. Highly accurate results presented for the first time are systematically produced, which can serve as a benchmark to calibrate other numerical solutions for twisted cylinders with sectorial cross section. The effects of height-to-radius ratio and twist angle on frequency parameters of cylinders with different subtended angles in the sectorial cross section are discussed in detail.

The Boundary Contour Method for Three-Dimensional Linear Elasticity

1996 ◽  
Vol 63 (2) ◽  
pp. 278-286 ◽  
Author(s):  
A. Nagarajan ◽  
S. Mukherjee ◽  
E. Lutz
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Linear Elasticity ◽  
Three Dimensional ◽  
Contour Method ◽  
Boundary Contour ◽  
Boundary Contour Method ◽  
Novel Variant ◽  
Elasticity Problems ◽  
Element Method

This paper presents a novel variant of the boundary element method, here called the boundary contour method, applied to three-dimensional problems of linear elasticity. In this work, the surface integrals on boundary elements of the usual boundary element method are transformed, through an application of Stokes’ theorem, into line integrals on the bounding contours of these elements. Thus, in this formulation, only line integrals have to be numerically evaluated for three-dimensional elasticity problems—even for curved surface elements of arbitrary shape. Numerical results are presented for some three-dimensional problems, and these are compared against analytical solutions.

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