A Nonlinear Catastrophic Model of Isolated Rock Pillar Instability and its Chaotic Behaviour Based on Material Properties

2014 ◽  
Vol 908 ◽  
pp. 98-102
Author(s):  
Xiao Hu Zhang ◽  
Xiao Long Ren ◽  
De Chao Liu
Keyword(s):  
Differential Equation ◽  
Lyapunov Exponent ◽  
Material Properties ◽  
Constitutive Relations ◽  
Nonequilibrium State ◽  
Uniaxial Stress ◽  
Rock Pillar ◽  
Rock Pillars ◽  
Static Criterion ◽  
Catastrophic Model

A cusp catastrophic model of rock pillar instability was developed under uniaxial stress conditions, based on the assumptive softening constitutive relations under static mechanics. The static criterion of rock pillar instability is derived. The dynamical nonlinear differential equation under nonequilibrium state followed by the cusp catastrophic model is introduced. The characteristics of rock pillars chaotic evaluation are studied under the change of linear stiffness. The results show that the rock pillars motion is chaotic when the linear stiffness is in a certain interval, which are proved by the calculated max Lyapunov exponent.

Theoretical Meso-Model of Al2O3/ZrO2 Ceramic Response under Compression

2014 ◽  
Vol 601 ◽  
pp. 92-95
Author(s):  
Tomasz Sadowski ◽  
Liviu Marsavina
Keyword(s):  
Grain Boundary ◽  
Material Properties ◽  
Mechanical Model ◽  
Constitutive Relations ◽  
Ceramic Composites ◽  
Theoretical Modeling ◽  
Polycrystalline Structure ◽  
Two Phase

This paper presents theoretical modeling of two-phase ceramic composites subjected to compression. The meso-mechanical model allows for inclusion of all microdefects in the polycrystalline structure that exists at the grain boundary interfaces and inside the grains. The constitutive relations for the Al2O3/ZrO2composite with the gradual degradation of the material properties due to different defects development were formulated.

scholarly journals Viscoelastic Characterization of Peripapillary Sclera: Material Properties by Quadrant in Rabbit and Monkey Eyes

2003 ◽  
Vol 125 (1) ◽  
pp. 124-131 ◽  
Author(s):  
J. Crawford Downs ◽  
J-K. Francis Suh ◽  
Kevin A. Thomas ◽  
Anthony J. Bellezza ◽  
Claude F. Burgoyne ◽  
...  
Keyword(s):  
Optic Nerve ◽  
Viscoelastic Material ◽  
Optic Nerve Head ◽  
Material Properties ◽  
Uniaxial Stress ◽  
Species Group ◽  
Linear Viscoelastic ◽  
Viscoelastic Theory ◽  
Linear Viscoelastic Theory

In this report we characterize the viscoelastic material properties of peripapillary sclera from the four quadrants surrounding the optic nerve head in both rabbit and monkey eyes. Scleral tensile specimens harvested from each quadrant were subjected to uniaxial stress relaxation and tensile ramp to failure tests. Linear viscoelastic theory, coupled with a spectral reduced relaxation function, was employed to characterize the viscoelastic properties of the tissues. We detected no differences in the stress-strain curves of specimens from the four quadrants surrounding the optic nerve head (ONH) below a strain of 4 percent in either the rabbit or monkey. While the peripapillary sclera from monkey eyes is significantly stiffer (both instantaneously and in equilibrium) and relaxes more slowly than that from rabbits, we detected no differences in the viscoelastic material properties (tested at strains of 0–1 percent) of sclera from the four quadrants surrounding the ONH within either species group.

Modeling of Segmented Peltier Cooling with Discrete and Continuous Concentration Function

2005 ◽  
Vol 492-493 ◽  
pp. 507-516 ◽  
Author(s):  
S. Walczak ◽  
Winfried Seifert ◽  
Eckhard Müller
Keyword(s):  
Differential Equation ◽  
Seebeck Coefficient ◽  
Temperature Profile ◽  
Material Properties ◽  
Performance Estimation ◽  
Cooling Power ◽  
One Dimensional ◽  
Peltier Cooling ◽  
Cooling Devices ◽  
Te Material

Commercialization of Peltier coolers has progressed during last years and special efforts have been undertaken to enhance the efficiency of thermoelectric (TE) devices. Along with the continued search for advanced TE materials, the concept of FGM offers a strategy of gradual improvement of device performance. In reality a functional gradient in a TE material means a related spatial variation of all TE properties – Seebeck coefficient, electrical, and thermal conductivity – whereas the most relevant effect is linked to the gradient of the Seebeck coefficient. Due to the spatial dependence of the Seebeck coefficient, Peltier heat is absorbed or released inside the TE element under current flow (distributed Peltier effect) which can be exploited to shape the internal temperature profile in a desired manner. Starting from the first principles of thermoelectricity, a differential equation governing the coupling of thermal and electrical transport is derived within the frame of a one-dimensional model. It is shown that this approach can be also used to model multi-segment Peltier cooling devices. Temperature profiles T(x) have been calculated for a segmented TE element within the framework of a constant parameters theory. The work presents an analytical model for performance evaluation of multiply-segmented Peltier elements. The problem is treated in a one-dimensional approach for a p-type stack containing N segments of different properties. Assuming constant TE material properties in each of the segments, the differential equation of TE transports has been solved to obtain the temperature profile T(x) in each segment. With the material properties values in each segment representing volume average values this model gives an excellent approximation also for continuously graded elements. The boundary conditions of the TE problem set-up, as conservation of heat at any intermediate junction between the segments, and fixed temperature at the cold and hot end of the element, lead to a linear equation system, which can be easily solved by means of standard methods. From the solution, all desired performance parameters can be deduced. Based on realistic material data exemplary calculations are presented for stacked and continuously graded elements. To demonstrate the developed numerical algorithm, gradients of the Seebeck coefficient are mainly considered. Calculations have been performed for N = 2, 5, 10, and continuous gradients. As target parameters, the C.O.P. and the cooling power have been calculated as functions of the electric current. As well, the minimum temperature of the cold side has been determined for various shape of the Seebeck gradient. It is shown that the TE FGM effect can be almost completely utilized already by a stack of two to five homogeneous segments. The results allow for giving an estimation on the order of magnitude of performance improvement of both discontinuously and continuously graded Peltier cooling devices. The model calculation was implemented with the software tool MATHEMATICA. The code provides an easy to handle convenient instrument for performance estimation of non-homogeneous Peltier pellets. Technological studies for controlled fabrication of those pellets are underway.

On fractional lyapunov exponent for solutions of linear fractional differential equations

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Nguyen Cong ◽  
Doan Son ◽  
Hoang Tuan
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Lyapunov Exponent ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Lyapunov Spectrum ◽  
Bounded Linear ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

AbstractOur aim in this paper is to investigate the asymptotic behavior of solutions of linear fractional differential equations. First, we show that the classical Lyapunov exponent of an arbitrary nontrivial solution of a bounded linear fractional differential equation is always nonnegative. Next, using the Mittag-Leffler function, we introduce an adequate notion of fractional Lyapunov exponent for an arbitrary function. We show that for a linear fractional differential equation, the fractional Lyapunov spectrum which consists of all possible fractional Lyapunov exponents of its solutions provides a good description of asymptotic behavior of this equation. Consequently, the stability of a linear fractional differential equation can be characterized by its fractional Lyapunov spectrum. Finally, to illustrate the theoretical results we compute explicitly the fractional Lyapunov exponent of an arbitrary solution of a planar time-invariant linear fractional differential equation.

scholarly journals State Variables in Saturated-Unsaturated Soil Mechanics

2016 ◽  
pp. 3-17
Author(s):  
Delwyn G. Fredlund
Keyword(s):  
Stress State ◽  
Continuum Mechanics ◽  
Unsaturated Soil ◽  
Material Properties ◽  
Soil Mechanics ◽  
Constitutive Relations ◽  
Saturated Soil ◽  
Clear Distinction ◽  
State Variables ◽  
Unsaturated Soil Mechanics

The description of the stress state in soils is the foundational point around which an applied science should be built for engineering practice. The stress state description has proven to be pivotal for saturated soil mechanics and the same should be true for unsaturated soil mechanics. Continuum mechanics sets forth a series of principles upon which a common science base can be developed for a wide range of materials. The principles require that there be a clear distinction between state variables and constitutive relations. Constitutive relations relate state variables and incorporate material properties. State variables, on the other hand, are independent of the material properties. It has been possible to maintain a clear distinction between variables of state and constitutive relations in the development of saturated soil mechanics and the same should be true for unsaturated soil mechanics. This paper presents a description of the source and character of stress state variables for saturated and unsaturated soils. The descriptions are consistent with the principles of multiphase continuum mechanics and provide an understanding of the source and importance of stress state variables.

A Note on Tire Carcass Mechanical Properties

1978 ◽  
Vol 6 (4) ◽  
pp. 248-262
Author(s):  
J. T. Tielking ◽  
R. E. Martin ◽  
R. A. Schapery
Keyword(s):  
Mechanical Properties ◽  
Material Properties ◽  
Geometric Nonlinearity ◽  
Elastic Stress ◽  
Uniaxial Stress ◽  
Element Model ◽  
Stress Tests ◽  
Linear Elastic ◽  
Constituent Material

Abstract Uniaxial stress tests were conducted on composite specimens cut from two different locations on a bias tire carcass. These data together with cord data, the Halpin-Tsai “micromechanics” equations, and the linear laminate constitutive equations are used to derive the in-situ rubber modulus as a function of time and to check for consistency among the specimens tested. The main purpose of the first part of the study was to obtain constituent material properties for use in a finite element model of a tire. This model is then employed in the investigation of the influence of uniform rubber modulus on the shape of an inflated tire carcass, and it is concluded that the strain and time dependence of the rubber modulus will introduce some error in a tire structural analysis that uses linear elastic stress-strain equations and permits geometric nonlinearity. It appears that the error will be minimal in a low strain region such as in the sidewall.

scholarly journals Viscoelastic Material Properties of the Myocardium and Cardiac Jelly in the Looping Chick Heart

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
Jiang Yao ◽  
Victor D. Varner ◽  
Lauren L. Brilli ◽  
Jonathan M. Young ◽  
Larry A. Taber ◽  
...  
Keyword(s):  
Viscoelastic Material ◽  
Material Properties ◽  
Constitutive Relations ◽  
Computational Method ◽  
Heart Tube ◽  
Embryonic Tissues ◽  
Strain Energy Density Function ◽  
Modeling Techniques ◽  
Chick Heart ◽  
Intact Heart

Accurate material properties of developing embryonic tissues are a crucial factor in studies of the mechanics of morphogenesis. In the present work, we characterize the viscoelastic material properties of the looping heart tube in the chick embryo through nonlinear finite element modeling and microindentation experiments. Both hysteresis and ramp-hold experiments were performed on the intact heart and isolated cardiac jelly (extracellular matrix). An inverse computational method was used to determine the constitutive relations for the myocardium and cardiac jelly. With both layers assumed to be quasilinear viscoelastic, material coefficients for an Ogden type strain-energy density function combined with Prony series of two terms or less were determined by fitting numerical results from a simplified model of a heart segment to experimental data. The experimental and modeling techniques can be applied generally for determining viscoelastic material properties of embryonic tissues.

Innovative design of mechanical structures from first principles

1987 ◽  
Vol 1 (3) ◽  
pp. 169-189 ◽  
Author(s):  
Jonathan Cagan ◽  
Alice M. Agogino
Keyword(s):  
Material Properties ◽  
First Principles ◽  
Constitutive Relations ◽  
Solution Space ◽  
Optimization Techniques ◽  
Design System ◽  
First Principle ◽  
Innovative Design ◽  
The Creation ◽  
Monotonicity Analysis

In this paper a unique design methodology known as 1stPRINCE (FIRST PRINciple Computational Evalualor) is developed to perform innovative design of mechanical structures from first principle knowledge. The method is based on the assumption that the creation of innovative designs of physical significance, concerning geometric and material properties, requires reasoning from first principles. The innovative designs discovered by 1stPRINCE differ from routine designs in that new primitives are created. Monotonicity analysis and computer algebra are utilized to direct design variables in a globally optimal direction relative to the goals specified. In contrast to strict constraint propagation approaches, formal qualitative optimization techniques efficiently search the solution space in an optimizing direction, eliminate infeasible and suboptimal designs, and reason with both equality and inequality constraints. Modification of the design configuration space and the creation of new primitives, in order to meet the constraints or improve the design, are achieved by manipulating mathematical quantities such as the integral. The result is a design system which requires a knowledge base only of fundamental equations of deformation with physical constraints on variables, constitutive relations, and fundamental engineering assumptions; no pre-compiled knowledge of mechanical behavior is needed. Application of this theory to the design of a beam under torsion leads to designs of a hollow tube and a composite rod exhibiting globally optimal behavior. Further, these optimally-behaved designs are described symbolically as a function of the material properties and system parameters. This method is implemented in a LISP environment as a module in a larger intelligent CAD system that integrates qualitative, functional and numerical computation for engineering applications.

Plastic behaviour of two-dimensional regular hexagonal structures with bilinear and uniaxial strength asymmetry in cellular materials

2008 ◽  
Vol 222 (12) ◽  
pp. 2365-2372 ◽  
Author(s):  
Qing Hang Zhang ◽  
Soon Huat Tan ◽  
Siaw Meng Chou
Keyword(s):  
Volume Fraction ◽  
Constitutive Relations ◽  
Micromechanical Model ◽  
Uniaxial Stress ◽  
Hexagonal Structure ◽  
Uniaxial Strain ◽  
Two Dimensional ◽  
Loading Conditions ◽  
Strength Asymmetry ◽  
Uniaxial Strength

An elasto-plastic micromechanical model of the two-dimensional regular hexagonal structure was developed. General analytical expressions for the incremental constitutive relations were derived in terms of parameters defining the architecture and material of an internal beam. Non-linearity of the structure was introduced by considering the elastic—linear strain hardening behaviour of each internal beam, in which uniaxial strength asymmetry of the cellular material was accounted for. The plastic stress—strain relationship of the structure under any loading conditions can therefore be analysed by localized beam deformation. The results show that the bending deformation of the internal beam dominates under uniaxial stress loading conditions, however, the axial displacement dominates under the uniaxial strain conditions. The structure will present different behaviours under different loading conditions. The corresponding stresses under the uniaxial strain condition are greater than those under the uniaxial stress condition. The analyses also show that the volume fraction is highly correlated with the elastic constants and yield stresses of the structure. The denser the structure, the higher the moduli and yield stresses.

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