Finite Elastic Deformation Governed by Linear Equations

1986 ◽  
Vol 53 (4) ◽  
pp. 819-820 ◽  
Author(s):  
Joseph B. Keller
Keyword(s):  
Stress Tensor ◽  
Elastic Deformation ◽  
Finite Deformation ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Deformation Gradient ◽  
Kirchhoff Stress ◽  
Kirchhoff Stress Tensor ◽  
Three Components

It is observed that the equations of motion governing finite elastic deformation are linear if and only if the Piola-Kirchhoff stress tensor is linear in the deformation gradient. Then the three components of displacement satisfy uncoupled linear equations. These equations are used to solve some problems of finite deformation.

Fractional Derivative Constitutive Models for Finite Deformation of Viscoelastic Materials

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Fractional Derivative ◽  
Stress Tensor ◽  
Finite Deformation ◽  
Dynamical Behavior ◽  
Strain Tensor ◽  
Constitutive Models ◽  
Cauchy Stress ◽  
Viscoelastic Materials ◽  
Kirchhoff Stress Tensor ◽  
Biot Stress

A methodology to derive fractional derivative constitutive models for finite deformation of viscoelastic materials is proposed in a continuum mechanics treatment. Fractional derivative models are generalizations of the models given by the objective rates. The method of generalization is applied to the case in which the objective rate of the Cauchy stress is given by the Truesdell rate. Then, a fractional derivative model is obtained in terms of the second Piola–Kirchhoff stress tensor and the right Cauchy-Green strain tensor. Under the assumption that the dynamical behavior of the viscoelastic materials comes from a complex combination of elastic and viscous elements, it is shown that the strain energy of the elastic elements plays a fundamental role in determining the fractional derivative constitutive equation. As another example of the methodology, a fractional constitutive model is derived in terms of the Biot stress tensor. The constitutive models derived in this paper are compared and discussed with already existing models. From the above studies, it has been proved that the methodology proposed in this paper is fully applicable and effective.

scholarly journals Correction to: An efficient and stabilised SPH method for large strain metal plastic deformations

2019 ◽  
Vol 7 (3) ◽  
pp. 541-541
Author(s):  
Giorgio Greto ◽  
Sivakumar Kulasegaram
Keyword(s):  
Stress Tensor ◽  
Large Strain ◽  
Sph Method ◽  
Plastic Deformations ◽  
Kirchhoff Stress ◽  
Time Stepping ◽  
Kirchhoff Stress Tensor

The symbol was introduced incorrectly inside the “Time-stepping the solution” box, directly under the “Compute first Piola–Kirchhoff stress tensor Pi” as in “Appendix A” listing.

Recursive Linearization of Multibody Dynamics and Application to Control Design

1990 ◽  
Author(s):  
Tsung-Chieh Lin ◽  
K. Harold Yae
Keyword(s):  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Control Design ◽  
Numerical Differentiation ◽  
Difficult Problem ◽  
Computational Method ◽  
Computational Error ◽  
Dynamics Modeling ◽  
Analytical Approximations

Abstract The non-linear equations of motion in multi-body dynamics pose a difficult problem in linear control design. It is therefore desirable to have linearization capability in conjunction with a general-purpose multibody dynamics modeling technique. A new computational method for linearization is obtained by applying a series of first-order analytical approximations to the recursive kinematic relationships. The method has proved to be computationally more efficient. It has also turned out to be more accurate because the analytical perturbation requires matrix and vector operations by circumventing numerical differentiation and other associated numerical operations that may accumulate computational error.

scholarly journals INVESTIGATION OF THE DYNAMICS OF AN OVERHEAD CRANE LIFTING PROCESS IN A VERTICAL PLANE

Transport ◽  
2005 ◽  
Vol 20 (5) ◽  
pp. 176-180 ◽  
Author(s):  
Marijonas Bogdevičius ◽  
Aleksandr Vika
Keyword(s):  
Dynamic Behaviour ◽  
Equations Of Motion ◽  
Asynchronous Motor ◽  
Linear Equations ◽  
Vertical Plane ◽  
Overhead Crane ◽  
Dynamic Forces ◽  
Planet Gear ◽  
Linear Equations Of Motion ◽  
Non Linear Equations

The paper analyses the dynamic behaviour of supporting structure of an overhead crane during the operation of a hoisting mechanism. The crane is expected to operate with a hook and to carry 50 kN of weight. The electric hoist consists of an asynchronous motor with a magnetic brake, a two‐level planet gear, a load drum and an upper block. Non‐linear equations of motion of a crane hoisting mechanism are derived. Real dynamic forces and their influence on the hoisting crane behaviour are obtained. Numerical results of the crane are derived considering two hoisting regimes during the operation of the hoisting.

scholarly journals NON-RIGID KINEMATIC EXCITATION FOR MULTIPLY-SUPPORTED SYSTEM WITH HOMOGENEOUS DAMPING

2018 ◽  
Vol 14 (4) ◽  
pp. 152-157
Author(s):  
Alexander G. Tyapin
Keyword(s):  
Mass Matrix ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Internal Damping ◽  
Static Response ◽  
Kinematic Excitation ◽  
Hand Part ◽  
Internal Forces ◽  
The Right ◽  
Damping Model

This paper continues the discussion of linear equations of motion. The author considers non-rigid kinematic excitation for multiply-supported system leading to the deformations in quasi-static response. It turns out that in the equation of motion written down for relative displacements (relative displacements are defined as absolute displacements minus quasi-static response) the contribution of the internal damping to the load in some cases may be zero (like it was for rigid kinematical excitation). For this effect the system under consideration must have homogeneous damping. It is the often case, though not always. Zero contribution of the internal damping to the load is different in origin for rigid and non-rigid kinematic excitation: in the former case nodal loads in the quasi-static response are zero for each element; in the latter case nodal loads in elements are non-zero, but in each node they are balanced giving zero resulting nodal loads. Thus, damping in the quasi-static response does not impact relative motion, but impacts the resulting internal forces. The implementation of the Rayleigh damping model for the right-hand part of the equation leads to the error (like for rigid kinematic excitation), as damping in the Rayleigh model is not really “internal”: due to the participation of mass matrix it works on rigid displacements, which is impossible for internal damping

Nonuniform Stress Field Determination Based on Deformation Measurement

2021 ◽  
pp. 1-26
Author(s):  
Cheng Liu
Keyword(s):  
Principal Stress ◽  
Deformation Gradient ◽  
Deformation Measurement ◽  
Cauchy Stress ◽  
Kirchhoff Stress ◽  
Nonuniform Deformation ◽  
Principal Stretch ◽  
Stress Orientation ◽  
Lagrangian Strain ◽  
Stretch Orientation

Abstract We demonstrate a technique that, under certain circumstances, will determine stresses associated with a nonuniform deformation field without knowing the detailed constitutive behavior of the deforming material. This technique is based on (1) a detailed deformation measurement of a domain and (2) the observation that for isotropic materials, the strain and the stress, which form the so-called work-conjugate pair, are co-axial, or their eigenvectors share the same direction. The particular measures for strain and stress considered are the Lagrangian strain and the second Piola-Kirchhoff stress. The deformation measurement provides the field of the principal stretch orientation θλ and since the Lagrangian strain and the second Piola-Kirchhoff stress are co-axial, the principal stress orientation θs of the second Piola-Kirchhoff stress is determined. The Cauchy stress is related to the second Piola-Kirchhoff stress through the deformation gradient tensor, which can be measured experimentally. We then show that the principal stress orientation θσ of the Cauchy stress is the sum of the principal stretch orientation θλ and the local rigid-body rotation θq, which is determinable by the deformation gradient through polar decomposition. With the principal stress orientation θσ known, the equation of equilibrium, now in terms of the two principal stresses σ1 and σ2, and θσ, can be solved numerically with appropriate traction boundary conditions. The technique is then applied to the experimental case of nonuniform deformation of a PVC sheet with a circular hole and subject to tension. Limitations and restrictions of the technique and possible extensions will be discussed.

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