Constitutive Equations, Governing Equations, Elastodynamic Energy Conservation

2012 ◽  
pp. 131-160
Keyword(s):  
Energy Conservation ◽  
Constitutive Equations ◽  
Governing Equations

scholarly journals The Effects of Piezoelectricity on the Interaction of Waves in Fluid-Loaded Poroelastic Half-Space

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Vishakha Gupta ◽  
Anil K. Vashishth
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Constitutive Equations ◽  
Vertical Component ◽  
Amplitude Ratios ◽  
Governing Equations ◽  
Solid Interface ◽  
Interaction Of Waves ◽  
Electric Potentials

The effects of piezoelectricity on the interaction of waves at fluid-poroelastic interface are studied. The constitutive equations and governing equations are formulated and their solution is obtained. The boundary conditions are described at fluid-solid interface. The effects of various parameters on the angle of refraction, amplitude ratios, displacements, electric potentials, and vertical component of slowness are studied numerically for a particular model. The results obtained are in agreement with the general laws of physics.

scholarly journals 50 Years of the K-BKZ Constitutive Relation for Polymers

2013 ◽  
Vol 2013 ◽  
pp. 1-22 ◽  
Author(s):  
Evan Mitsoulis
Keyword(s):  
Mathematical Modeling ◽  
Constitutive Model ◽  
Particle Tracking ◽  
Constitutive Equations ◽  
Historical Perspective ◽  
Governing Equations ◽  
Dimensionless Numbers ◽  
Flow Problems ◽  
Polymer Flows ◽  
Method Of Solution

The K-BKZ constitutive model is now 50 years old. The paper reviews the connections of the model and its variants with continuum mechanics and experiment, presenting an up-to-date recap of research and major findings in the open literature. In the Introduction a historical perspective is given on developments in the last 50 years of the K-BKZ model. Then a section follows on mathematical modeling of polymer flows, including governing equations of flow, rheological constitutive equations (with emphasis on viscoelastic integral constitutive equations of the K-BKZ type), dimensionless numbers, and boundary conditions. The Method of Solution section reviews the major developments of techniques necessary for particle tracking and calculation of the integrals for the viscoelastic stresses in flow problems. Finally, selected examples are given of successful application of the K-BKZ model in polymer flows relevant to rheology.

Rigid-Elastic Coupled Dynamics and its Application

2012 ◽  
Vol 629 ◽  
pp. 587-592
Author(s):  
Hai Yan Song ◽  
Li Fu Liang ◽  
Hai Bo Li
Keyword(s):  
Variational Principle ◽  
Energy Conservation ◽  
Conservation Law ◽  
Vibration Mode ◽  
Coupled Vibration ◽  
Governing Equations ◽  
Coupled Dynamics ◽  
Energy Conservation Law

According to energy conservation law, the functional of quasi-variational principle of rigid-elastic coupled dynamics is established. Quasi-stationary value conditions of quasi-variational principle of rigid-elastic coupled dynamics are derived. The governing equations of rigid-elastic coupled dynamics are obtained. As an application of rigid-elastic coupled dynamics, an approach of coupled vibration mode determined of unrestrained beam is established. The coupled vibration mode and coupled frequency are studied by the approach.

Study on General Governing Equations of Computational Heat Transfer and Fluid Flow

2012 ◽  
Vol 12 (5) ◽  
pp. 1482-1494 ◽  
Author(s):  
Wang Li ◽  
Bo Yu ◽  
Yi Wang ◽  
Xin-Ran Wang ◽  
Qing-Yuan Wang ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Heat Capacity ◽  
Fluid Flow ◽  
Energy Conservation ◽  
Specific Heat ◽  
Specific Heat Capacity ◽  
Great Success ◽  
Governing Equations ◽  
Science And Engineering

AbstractThe governing equations for heat transfer and fluid flow are often formulated in a general form for the simplification of discretization and programming, which has achieved great success in thermal science and engineering. Based on the analysis of the popular general form of governing equations, we found that energy conservation cannot be guaranteed when specific heat capacity is not constant, which may lead to unreliable results. A new concept of generalized density is put forward, based on which a new general form of governing equations is proposed to guarantee energy conservation. A number of calculation examples have been employed to verify validation and feasibility.

On governing equations for a nanoplate derived from the 3D gradient theory of elasticity

2021 ◽  
pp. 108128652110575
Author(s):  
Gennadi Mikhasev
Keyword(s):  
Constitutive Equations ◽  
Three Dimensional ◽  
Low Frequency ◽  
Gradient Elasticity ◽  
Theory Of Elasticity ◽  
Gradient Theory ◽  
Governing Equations ◽  
Nonlocal Effects ◽  
Internal Length ◽  
Variable Surface

The paper is concerned with the asymptotically consistent theory of nanoscale plates capturing the spatial nonlocal effects. The three-dimensional (3D) elasticity equations for a thin plate are used as the governing equations. In the general case, the plate is acted upon by dynamic body forces varying in the thickness direction, and by variable surface forces. The thickness of the plate is assumed to be greater than the characteristic micro/nanoscale measure and much smaller than the in-plane characteristic dimension (e.g., the wave or deformation length). The 3D constitutive equations of gradient elasticity are used to link the fields of nonlocal stresses and strains. Using the asymptotic approach, a sequence of relations for stresses and displacements in the form of polynomials in the transverse coordinate with coefficients depending on time and in-plane coordinates was obtained. The asymptotically consistent 2D differential equation governing vibration (or static deformation) of a plate accounting for both transverse shears and the spatial nonlocal contribution of the stress and strain fields was derived. It was revealed that capturing nonlocal effects in all directions leads to an increase in the correction factor compared with the well-known 2D theories based on kinematic hypotheses and the Eringen-type gradient constitutive equations. The effect of the internal length scales parameters on free low-frequency vibrations and displacements of a plate is discoursed.

Dynamic Finite Element Analysis of a Piezoelectric Bimorph Beam Deflector With Consideration of Hysteresis Effect

2003 ◽  
Author(s):  
Paul C. P. Chao ◽  
C. W. Chiu ◽  
J. S. Huang ◽  
H. C. Tseng
Keyword(s):  
Finite Element ◽  
Constitutive Equations ◽  
Hysteresis Effect ◽  
Piezoelectric Bimorph ◽  
Governing Equations ◽  
Element Analysis ◽  
Dynamic Finite Element Analysis ◽  
Polarization Term ◽  
Bimorph Beam ◽  
Element Technique

This study is devoted to propose a method of finite element technique to account for the hysteresis effect of a piezoelectric bimorph beam deflector. To this end, the constitutive equations of a general piezoelectric material are first modified to include the hysteresis effect by adding a polarization term in one of constitutive equations. Based on these modified constitutive equations and employment of Preisach model for hysteresis, the governing equations of the bimorph beam are derived through the utilization of Hamilton’s principle and calculus of variation. In addition, according to the common physical rules, boundary, transition and continuous conditions are next formulated to complement the governing equations. Simulations are finally conducted to show the effectiveness of the proposed modeling technique and decipher the dynamic behavior of the piezoelectric beam with consideration of hysteresis effect.

Mechanics and thermodynamics of saturated/unsaturated porous materials and quantitative solutions*

2002 ◽  
Vol 55 (4) ◽  
pp. 351-388 ◽  
Author(s):  
BA Schrefler
Keyword(s):  
Porous Media ◽  
Mechanical Behavior ◽  
Porous Materials ◽  
Constitutive Equations ◽  
Review Article ◽  
Unsaturated Porous Media ◽  
Governing Equations ◽  
Balance Equations ◽  
Thermodynamic Equations

Models for thermo-hydro-mechanical behavior of saturated/unsaturated porous media are reviewed. The necessary balance equations are derived using averaging theories. Constitutive equations are obtained using the Coleman-Noll procedure and thermodynamic equations for the model closure are introduced. A particular form of the governing equations is then solved numerically and the numerical properties are discussed. Application examples conclude the paper. There are 165 references in this review article.

Core Variation in an Externally Pressurized Converging Thrust Bearing with Bingham Lubricant

2016 ◽  
Vol 852 ◽  
pp. 428-434 ◽  
Author(s):  
I. Jayakaran Amalraj ◽  
G. Alexander Raymand
Keyword(s):  
Film Thickness ◽  
Boundary Conditions ◽  
Yield Surface ◽  
Constitutive Equations ◽  
Thrust Bearing ◽  
Numerical Solutions ◽  
Flow Region ◽  
Core Region ◽  
Governing Equations ◽  
The Core

The effects of angle of convergence on the shape and thickness of the core are analyzed theoretically by considering variable film thickness in an externally pressurized circular thrust bearing. Using the assumptions of the lubrication theory, modified Reynold’s equation and the governing equations are obtained. Using the boundary conditions of the problem in the constitutive equations we get the velocity of the core region as well as flow region. By considering the equilibrium of an element in the yield surface, an algebraic equation to determine the thickness of the yield surface is derived. Numerical solutions are obtained for the thickness of yield surface and velocities for various values of Bingham Numbers and the angle of convergence.

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