A constrained theory of a Cosserat point for the numerical solution of dynamic problems of non-linear elastic rods with rigid cross-sections

2007 ◽  
Vol 42 (2) ◽  
pp. 216-232 ◽  
Author(s):  
M. Brand ◽  
M.B. Rubin
Keyword(s):  
Numerical Solution ◽  
Cross Sections ◽  
Elastic Rods ◽  
Dynamic Problems ◽  
Linear Elastic ◽  
Non Linear ◽  
Cosserat Point

On the Numerical Solution of One-Dimensional Continuum Problems Using the Theory of a Cosserat Point

1985 ◽  
Vol 52 (2) ◽  
pp. 373-378 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Numerical Solution ◽  
Body Force ◽  
Closed Form Solution ◽  
Form Solution ◽  
Element Approximation ◽  
One Dimensional ◽  
Linear Elastic ◽  
Elastic Bar ◽  
Cosserat Point ◽  
Nonlinear Elastic

The theory of a Cosserat point is specialized to describe the motion of a one-dimensional continuum. Attention is focused on two problems of an elastic bar. Vibration of a linear-elastic bar is considered in the first problem and static deformation of a nonlinear-elastic bar subjected to a uniform body force is considered in the second problem. A closed-form solution is derived for each problem by dividing the bar into two elements, each of which is modeled as a Cosserat point. The predictions of the two-element approximation are shown to be very accurate.

scholarly journals A NON-LINEAR PROBLEM ARISING FROM THE DESCRIPTION OF THE WAVE PROPAGATION IN LINEAR ELASTIC RODS

2019 ◽  
Vol 49 (1) ◽  
pp. 61-63
Author(s):  
R.M. S. GAMA
Keyword(s):  
Wave Propagation ◽  
Shock Waves ◽  
Continuum Mechanics ◽  
Linear Problem ◽  
Elastic Rods ◽  
Elastic Rod ◽  
Kinematic Constraint ◽  
Linear Elastic ◽  
Non Linear

In this work it is presented the modeling and the simulation of the dynamics of an elastic rod, taking into account the kinematic constraint arising from the Classical Continuum Mechanics. The simulation involves shock waves that consists of contact shocks when the kinematic constraint does not need to be imposed.

On the Theory of a Cosserat Point and Its Application to the Numerical Solution of Continuum Problems

1985 ◽  
Vol 52 (2) ◽  
pp. 368-372 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Numerical Solution ◽  
Constitutive Equations ◽  
Isotropic Material ◽  
Small Volume ◽  
Material Point ◽  
Balance Laws ◽  
Linear Elastic ◽  
Linearized Theory ◽  
Cosserat Point ◽  
Constitutive Properties

The theory of a Cosserat point is developed to describe motion of a body that is essentially a material point surrounded by some small volume. The development of this theory is motivated mainly by its applicability to the numerical solution of continuum problems. Attention is confined to the purely mechanical theory and nonlinear balance laws are proposed for Cosserat points with arbitrary constitutive properties. The linearized theory is developed and constitutive equations for an elastic material are discussed within the context of both the nonlinear and linear theories. Explicit constitutive equations for a linear-elastic isotropic Cosserat point are developed to model a parallelepiped composed of a linear-elastic homogeneous isotropic material.

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.

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