scholarly journals Solution of Extended Kelvin-Voigt Model

2012 ◽  
Vol 10 (1) ◽  
pp. 119-130
Author(s):  
Sławomir Karaś
Keyword(s):  
Constitutive Equations ◽  
Civil Engineering ◽  
Initial Conditions ◽  
Complete Solution ◽  
Highway Engineering ◽  
Integral Forms ◽  
Higher Rank ◽  
Voigt Model

The great usefulness of uniaxial visco-elastic models, especially in highway engineering pavement theory, composites and other civil engineering disciplines were the reason for undertaking the trial to find a complete solution for the generalization of Kelvin-Voigt body. Here the elements of higher rank than velocities of strain and stress are considered. Carson’s transformation simultaneously with residuum theorem are used for solutions derivation. The introduced procedure can be also used for more complicated differential or integral forms of constitutive equations, as well as for non homogenous initial conditions. The Burgers’ body is examined. Finally, as an example the vibration of simple beam is shown.

Nonlinear Instabilities, Negative Energy Modes and Generalized Cherry Oscillators

1990 ◽  
Vol 45 (7) ◽  
pp. 839-846 ◽  
Author(s):  
D. Pfirsch
Keyword(s):  
Initial Conditions ◽  
Complete Solution ◽  
Continuum Theory ◽  
Wave Interaction ◽  
Negative Energy ◽  
Solution Set ◽  
Maxwell Theory ◽  
Resonant Oscillators ◽  
Simple Physical Interpretation ◽  
Two Parameter

AbstractIn 1925 Cherry [1] discussed two oscillators of positive and negative energy that are nonlinearly coupled in a special way, and presented a class of exact solutions of the nonlinear equations showing explosive instability independent of the strength of the nonlinearity and the initial amplitudes. In this paper Cherry's Hamiltonian is transformed into a form which allows a simple physical interpretation. The new Hamiltonian is generalized to three nonlinearly coupled oscillators; it corresponds to three-wave interaction in a continuum theory, like the Vlasov-Maxwell theory, if there exist linear negative energy waves [2-4, 5, 6], Cherry was able to present a two-parameter solution set for his example which would, however, allow a four-parameter solution set, and, as a first result, an analogous three-parameter solution set for the resonant three-oscillator case is obtained here which, however, would allow a six-parameter solution set. Nonlinear instability is therefore proven so far only for a very small part of the phase space of the oscillators. This paper gives in addition the complete solution for the three-oscillator case and shows that, except for a singular case, all initial conditions, especially those with arbitrarily small amplitudes, lead to explosive behaviour. This is true of the resonant case. The non-resonant oscillators can sometimes also become explosively unstable, but the initial amplitudes must not be infinitesimally small. A few examples are presented for illustration.

Dynamics Simulation of a Circular Membrane With an Eccentric Circular Areal Constraint

2011 ◽  
Author(s):  
Assaad AlSahlani ◽  
Ranjan Mukherjee
Keyword(s):  
Initial Conditions ◽  
Complete Solution ◽  
Outer Boundary ◽  
Dynamics Simulation ◽  
Circular Membrane ◽  
Circular Area ◽  
Orthogonality Conditions ◽  
Vibratory Motion ◽  
Modes Of Vibration ◽  
Symmetric And Antisymmetric Modes

We investigate the dynamics of a circular membrane with an eccentric circular areal constraint under arbitrary initial conditions. The membrane is assumed to be fixed at its outer boundary and the constraint is assumed to impose zero displacement over a circular area of the membrane. The symmetric and antisymmetric modes of vibration of the membrane are derived and their orthogonality properties are established. Using the orthogonality conditions established in this paper, the complete solution to the constrained vibratory motion of the membrane is determined for arbitrary initial conditions. Two sets of numerical simulation results are presented.

Study on Mechanics Behaviors of Composite and Isotropic Shells

2011 ◽  
Vol 368-373 ◽  
pp. 215-218 ◽  
Author(s):  
Yao Peng Wu
Keyword(s):  
Shell Structure ◽  
Constitutive Equations ◽  
Thin Shell ◽  
Civil Engineering ◽  
Initial Configuration ◽  
Isotropic Shell ◽  
Broad Application ◽  
Isotropic Materials ◽  
Deployable Structure ◽  
Laminate Theory

Thin shell structure can show interesting bi-stable behavior. As a novel deployable structure, it shows a broad application prospect in the field of aeronautics and civil engineering, etc. The thesis deduces the general constitutive equations of thin shell structure on the basis of classical laminate theory. If the layup of the composite is anti-symmetric, the results show that there exist tension-bend coupling in the deformation of the shell structure; if the layup is symmetric, there exist bend-twist coupling. For isotropic shell, it has no tension-bend and bend-twist coupling, but if made unstressed from isotropic materials it is only stable in the initial configuration, which coincides with the known conclusion.

Modal Representation of Second Order Flexible Structures With Damped Boundaries

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Lea Sirota ◽  
Yoram Halevi
Keyword(s):  
Initial Conditions ◽  
Separation Of Variables ◽  
Complete Solution ◽  
Flexible Structures ◽  
Second Order ◽  
Free Response ◽  
Laplace Domain ◽  
Boundary Case ◽  
Sum Of Products ◽  
Infinite Sum

The problem of obtaining a modal (i.e., infinite series) solution of second order flexible structures with viscous damping boundary conditions is considered. In conservative boundary systems, separation of variables is well established and there exist closed form modal solutions. However, no counterpart results exist for the damped boundary case and previous publications fall short of providing a complete solution for the series, in particular, its coefficients. The paper presents the free response of damped boundary structures to general initial conditions in the form of an infinite sum of products of spatial and time functions. The problem is attended via Laplace domain approach, and explicit expressions for the series components and coefficients are derived. The modal approach is useful in finite dimension modeling, since it provides a convenient framework for truncation. It is shown via examples that often few modes suffice for approximation with good accuracy.

scholarly journals Mechanical Models of Pattern and Form in Biological Tissues: The Role of Stress–Strain Constitutive Equations

2021 ◽  
Vol 83 (7) ◽  
Author(s):  
Chiara Villa ◽  
Mark A. J. Chaplain ◽  
Alf Gerisch ◽  
Tommaso Lorenzi
Keyword(s):  
Pattern Formation ◽  
Balance Equation ◽  
Constitutive Equations ◽  
Constitutive Models ◽  
Maxwell Model ◽  
Biological Tissues ◽  
Force Balance ◽  
Stress Strain ◽  
Stress Strain Relation ◽  
Voigt Model

AbstractMechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. Under the assumption that the cell-ECM system can be regarded as an isotropic linear viscoelastic material, the force-balance equation is often defined using the Kelvin–Voigt model of linear viscoelasticity to represent the stress–strain relation of the ECM. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. In this paper, we systematically assess the pattern formation potential of different stress–strain constitutive equations for the ECM within a mechanical model of pattern formation in biological tissues. The results obtained through linear stability analysis and the dispersion relations derived therefrom support the idea that fluid-like constitutive models, such as the Maxwell model and the Jeffrey model, have a pattern formation potential much higher than solid-like models, such as the Kelvin–Voigt model and the standard linear solid model. This is confirmed by the results of numerical simulations, which demonstrate that, all else being equal, spatial patterns emerge in the case where the Maxwell model is used to represent the stress–strain relation of the ECM, while no patterns are observed when the Kelvin–Voigt model is employed. Our findings suggest that further empirical work is required to acquire detailed quantitative information on the mechanical properties of components of the ECM in different biological tissues in order to furnish mechanical and mechanochemical models of pattern formation with stress–strain constitutive equations for the ECM that provide a more faithful representation of the underlying tissue rheology.

The numerical solution of systems of stiff ordinary differential equations

1974 ◽  
Vol 76 (2) ◽  
pp. 443-456
Author(s):  
J. R. Cash
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Complete Solution ◽  
Step Length ◽  
Rate Of Change ◽  
First Order ◽  
Stiff Ordinary Differential Equations ◽  
Iterative Procedures

AbstractAlgorithms are developed for the numerical solution of systems of first-order ordinary differential equations, the solutions of which have widely different rates of variation. The iterative procedures described use a step length of integration proportional to the rate of change of the required slowly varying solution in a region of integration, where either the transient components of the complete solution have become negligible compared with the chosen working accuracy or in a region where rapidly increasing components of the solution are theoretically possible but are made absent by the initial conditions. Several numerical examples are given to demonstrate the algorithms.

scholarly journals A mixture theory for geophysical fluids

2004 ◽  
Vol 11 (1) ◽  
pp. 75-82 ◽  
Author(s):  
A. C. Eringen
Keyword(s):  
Constitutive Equations ◽  
Degrees Of Freedom ◽  
Oil Spills ◽  
Initial Conditions ◽  
Mixture Theory ◽  
Continuum Theory ◽  
Second Law Of Thermodynamics ◽  
Heat Conducting ◽  
Field Equations ◽  
The Individual

Abstract. A continuum theory is developed for a geophysical fluid consisting of two species. Balance laws are given for the individual components of the mixture, modeled as micropolar viscous fluids. The continua allow independent rotational degrees of freedom, so that the fluids can exhibit couple stresses and a non-symmetric stress tensor. The second law of thermodynamics is used to develop constitutive equations. Linear constitutive equations are constituted for a heat conducting mixture, each species possessing separate viscosities. Field equations are obtained and boundary and initial conditions are stated. This theory is relevant to an atmospheric mixture consisting of any two species from rain, snow and/or sand. Also, this is a continuum theory for oceanic mixtures, such as water and silt, or water and oil spills, etc.

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