scholarly journals Stationary rarefaction waves in discrete materials with strain-softening behavior

2017 ◽  
Vol 31 (10) ◽  
pp. 1742005
Author(s):  
Eric B. Herbold
Keyword(s):  
Strain Softening ◽  
Equations Of Motion ◽  
Discrete Systems ◽  
Rarefaction Waves ◽  
Weakly Nonlinear ◽  
Discrete Equations ◽  
Softening Behavior ◽  
Wave Guides ◽  
Traveling Pulse ◽  
Nonlinearity Dispersion

This article details some of the techniques used to derive exact solutions from the discrete equations of motion of strongly and weakly nonlinear discrete systems. The distinction between strongly and weakly nonlinear systems is related to the amplitude of a traveling pulse and the external confining load. Materials with an anomalous strain-softening behavior will be emphasized (i.e., [Formula: see text], [Formula: see text]), though this choice does not preclude applications for strain-hardening systems like those with Hertzian potentials. Discrete materials with tunable acoustic transmission properties and novel impact mitigation capacity have gained interest in recent years due to their practical application across many scientific fields. Wave-guides comprised of discrete materials with a nonlinear interaction potential have been used to investigate the interplay between nonlinearity, dispersion and dissipation.

Semilinear Random Vibrations in Discrete Systems

1968 ◽  
Vol 35 (3) ◽  
pp. 560-564 ◽  
Author(s):  
E. T. Foster
Keyword(s):  
Equations Of Motion ◽  
Discrete Systems ◽  
Random Excitation ◽  
Solution Technique ◽  
Response Analysis ◽  
Weakly Nonlinear ◽  
Linear Set ◽  
System Output ◽  
Vibration Problems ◽  
Nonlinear Equations Of Motion

Random vibration problems are discussed for weakly nonlinear, multidegree-of-freedom discrete systems subjected to zero-mean, stationary random excitation. The semilinear solution technique developed involves substituting an optimum linear set of equations of motion for the actual nonlinear equations of motion. Parameters of this optimum linear system are selected on the basis of the system output so that a cyclic solution occurs. The cycles require parameter selection and response analysis until a convergence occurs in the sense that the answers from cycle to cycle are similar.

Rarefaction Wave Propagation in Origami-Based Mechanical Metamaterials

2015 ◽  
Author(s):  
Hiromi Yasuda ◽  
Jinkyu Yang
Keyword(s):  
Nonlinear Waves ◽  
Strain Softening ◽  
Initial Conditions ◽  
Rarefaction Waves ◽  
Efficient Manner ◽  
Softening Behavior ◽  
Mechanical Metamaterials ◽  
Tensile Wave ◽  
Potential Applications ◽  
Linkage Model

We design origami-based mechanical metamaterials composed of Tachi-Miura Polyhedron (TMP) cells, and we numerically study the propagation of nonlinear waves in them. In order to investigate the dynamics of origami structures, we model these TMP-based metamaterials into a simple multi-bar linkage model. By using this model, we find that these TMP cells exhibit strain softening behavior under compression, which can be tuned by modifying their geometrical configurations or initial conditions. By leveraging such tunable strain softening mechanisms, we verify that the origami-based metamaterials can support the propagation of rarefaction waves. These waves feature tensile wave-fronts despite the application of compressive impact to the system. Such unusual characteristics can be exploited to disintegrate shock waves in a controllable and efficient manner, thereby leading to potential applications in impact mitigation and absorption.

NONSTANDARD DISCRETIZATION SCHEMES APPLIED TO THE CONSERVATIVE HÉNON–HEILES SYSTEM

2007 ◽  
Vol 17 (03) ◽  
pp. 891-902 ◽  
Author(s):  
CHRISTOPHE LETELLIER ◽  
EDUARDO M. A. M. MENDES ◽  
RONALD E. MICKENS
Keyword(s):  
Time Reversal ◽  
Jacobian Matrix ◽  
Equations Of Motion ◽  
Hamiltonian Function ◽  
Discrete Systems ◽  
Conservative System ◽  
Discretization Scheme ◽  
Time Step ◽  
Discrete Equations ◽  
Discretization Schemes

The discretization of ordinary differential equations is investigated for the case of the conservative Hénon–Heiles system. Starting from a discrete Hamiltonian function, which is invariant under time reversal, discrete equations of motion are analytically obtained using three different discretization schemes recently proposed and investigated in the literature. In the case where the discretization scheme successfully provide discrete systems in which the trace of the Jacobian matrix corresponding to the property required by a conservative system is preserved, it is shown that they are not necessarily invariant to time reversal. Such models are however quite robust when the time step is increased. For the schemes where the trace of Jacobian matrix does not match the condition required by conservative systems, it is shown that energy conservation is not achieved and the original dynamics is lost. Steps toward the solution to this problem are given.

scholarly journals Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

2010 ◽  
Vol 2010 ◽  
pp. 1-23 ◽  
Author(s):  
Josef Diblík ◽  
Denys Ya. Khusainov ◽  
Irina V. Grytsay ◽  
Zdenĕk Šmarda
Keyword(s):  
Lyapunov Functions ◽  
Critical Case ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Simple Eigenvalue ◽  
Discrete Equations ◽  
The Matrix ◽  
The Stability ◽  
Important Characterization

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalueλ=1of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

Towards a Technique for Nonlinear Modal Analysis

2012 ◽  
Author(s):  
Simon A. Neild ◽  
Andrea Cammarano ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Transformation Process ◽  
Second Order ◽  
Identity Transformation ◽  
Modal Testing ◽  
System Response ◽  
Weakly Nonlinear ◽  
Nonlinear Modal Analysis

In this paper we discuss a theoretical technique for decomposing multi-degree-of-freedom weakly nonlinear systems into a simpler form — an approach which has parallels with the well know method for linear modal analysis. The key outcome is that the system resonances, both linear and nonlinear are revealed by the transformation process. For each resonance, parameters can be obtained which characterise the backbone curves, and higher harmonic components of the response. The underlying mathematical technique is based on a near identity normal form transformation. This is an established technique for analysing weakly nonlinear vibrating systems, but in this approach we use a variation of the method for systems of equations written in second-order form. This is a much more natural approach for structural dynamics where the governing equations of motion are written in this form as standard practice. In fact the first step in the method is to carry out a linear modal transformation using linear modes as would typically done for a linear system. The near identity transform is then applied as a second step in the process and one which identifies the nonlinear resonances in the system being considered. For an example system with cubic nonlinearities, we show how the resulting transformed equations can be used to obtain a time independent representation of the system response. We will discuss how the analysis can be carried out with applied forcing, and how the approximations about response frequencies, made during the near-identity transformation, affect the accuracy of the technique. In fact we show that the second-order normal form approach can actually improve the predictions of sub- and super-harmonic responses. Finally we comment on how this theoretical technique could be used as part of a modal testing approach in future work.

Dynamic Analysis of Mechanical Systems With Time-Varying Topologies: Part 1 — Dynamic Model and Stability Analysis

1990 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Stability Analysis ◽  
Dynamic Model ◽  
Local Stability ◽  
Global Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Time Varying ◽  
Discrete Equations ◽  
Local Stability Analysis ◽  
Multiple Collisions

Abstract A model is developed for analyzing mechanical systems with a pair of bodies with topological changes in their kinematic constraints. It is built upon the concept of Poincaré map rather than following the traditional methods of differential equations. The model provides a set of well-defined and naturally-discrete equations of motion and is capable of giving physical insights of dynamic characteristics of deadbeat convergence of multiple collisions and periodic or chaotic responses. The development of dynamic model and a local stability analysis are presented in Part 1, and the global analysis and numerical simulation are discussed in Part 2.

scholarly journals Semianalytical Solution for Large Deformation of Salt Cavern with Strain-Softening Behavior

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Lina Ran ◽  
Huabin Zhang ◽  
Qingqing Zhang
Keyword(s):  
Plastic Zone ◽  
Large Deformation ◽  
Strain Softening ◽  
Structural Characteristics ◽  
Strength Criterion ◽  
Geological Strength Index ◽  
Salt Rock ◽  
Salt Cavern ◽  
Softening Behavior ◽  
The Impact

A semianalytical solution of stress and displacement in the strain-softening and plastic flow zones of a salt cavern is presented. The solution is derived by adopting the large deformation theory, considering the nonlinear Hoek–Brown (H-B) strength criterion. The Romberg method is used to carry out numerical calculation, and then, the large deformation law of displacement is analyzed. The results are compared with those obtained by former numerical methods, and the solutions are validated. The results indicate that the displacement of the plastic zone decreases with the increase in distance away from the salt cavern. Similarly, it decreases with an increase in the geological strength index or running pressure, with the running pressure having a more significant effect on the displacement. It increases with the dilation angle, and the impact degree gradually increases. Compared with the softening parameter, h, of the plastic zone, the flow parameter, f, has little impact on the displacement. The displacement of the plastic zone obviously increased when considering the strain-softening of salt rock. When considering the shear dilation and softening behaviors of salt rock, the analytical solution obtained by employing the experiential regression Hoek–Brown (H-B) criterion, which considers many factors such as the structural characteristics of the salt formation and the rock mass quality, is safer and closer to the actual situation. This study can provide reference for many applications, including but not confined to analyzing the deformation of the surrounding rock of an underground salt cavern storage facility during construction.

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