Linear piezoelectricity

2020 ◽  
pp. 327-354
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Energy Balance ◽  
Energy Transport ◽  
Constitutive Relations ◽  
Electric Displacement ◽  
Balance Laws ◽  
Third Order ◽  
Static Limit ◽  
Electrical Boundary Conditions ◽  
Linear Piezoelectricity ◽  
Electromechanical Energy

This chapter presents the elements of linear piezoelectricity including mechanical and electrostatic balance laws and coupled mechanical electrical constitutive relations. The thermodynamically consistent constitutive relations are determined from a coupled electromechanical energy balance argument and expressions are given alternately considering the electric field and the electric displacement as independent fields. Appropriate electrical boundary conditions are also discussed. The theory is also specialized to poled piezoceramics. A chapter appendix provides a brief discussion of Maxwell’s equations for electromagnetics and energy transport in the quasi-static limit. A second chapter appendix discusses the properties of third order tensors.

A SPECTRAL/hp NONLINEAR FINITE ELEMENT ANALYSIS OF HIGHER-ORDER BEAM THEORY WITH VISCOELASTICITY

2012 ◽  
Vol 04 (01) ◽  
pp. 1250010 ◽  
Author(s):  
V. P. VALLALA ◽  
G. S. PAYETTE ◽  
J. N. REDDY
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Virtual Work ◽  
Constitutive Relations ◽  
Beam Theory ◽  
Element Model ◽  
Higher Order ◽  
Time Step ◽  
Third Order ◽  
The Third

In this paper, a finite element model for efficient nonlinear analysis of the mechanical response of viscoelastic beams is presented. The principle of virtual work is utilized in conjunction with the third-order beam theory to develop displacement-based, weak-form Galerkin finite element model for both quasi-static and fully-transient analysis. The displacement field is assumed such that the third-order beam theory admits C0 Lagrange interpolation of all dependent variables and the constitutive equation can be that of an isotropic material. Also, higher-order interpolation functions of spectral/hp type are employed to efficiently eliminate numerical locking. The mechanical properties are considered to be linear viscoelastic while the beam may undergo von Kármán nonlinear geometric deformations. The constitutive equations are modeled using Prony exponential series with general n-parameter Kelvin chain as its mechanical analogy for quasi-static cases and a simple two-element Maxwell model for dynamic cases. The fully discretized finite element equations are obtained by approximating the convolution integrals from the viscous part of the constitutive relations using a trapezoidal rule. A two-point recurrence scheme is developed that uses the approximation of relaxation moduli with Prony series. This necessitates the data storage for only the last time step and not for the entire deformation history.

scholarly journals A unilateral L2L^{2}-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement

2019 ◽  
Vol 12 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Matteo Negri
Keyword(s):  
Energy Balance ◽  
Phase Field ◽  
Gradient Flow ◽  
Field Variable ◽  
Discrete Solution ◽  
Vanishing Viscosity ◽  
Vanishing Viscosity Limit ◽  
Static Limit ◽  
Viscosity Limit ◽  
Minimization Scheme

AbstractWe consider an evolution in phase-field fracture which combines, in a system of PDEs, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the displacement field. We introduce a discretization in time and define a discrete solution by means of a 1-step alternate minimization scheme, with a quadratic {L^{2}}-penalty in the phase-field variable (i.e. an alternate minimizing movement). First, we prove that discrete solutions converge to a solution of our system of PDEs. Then we show that the vanishing viscosity limit is a quasi-static (parametrized) BV-evolution. All these solutions are described both in terms of energy balance and, equivalently, by PDEs within the natural framework of {W^{1,2}(0,T;L^{2})}.

A unified procedure for construction of theories of deformable media. I. Classical continuum physics

1995 ◽  
Vol 448 (1934) ◽  
pp. 335-356 ◽  
Keyword(s):  
Energy Balance ◽  
Companion Paper ◽  
Balance Laws ◽  
Kinematic Variable ◽  
Material Behaviour ◽  
Classical Formulation ◽  
Deformable Media ◽  
Basic Equations ◽  
Basic Features ◽  
Continuum Theories

A new unified procedure for constructing continuum theories of deformable media is presented and used in this and a companion paper. The procedure starts with a balance of energy and derives from it all the relevant balance laws that may also include those that are associated with thermal, electrical and magnetic effects; the basic energetic ingredients that are included in the balance of energy depend, of course, on the nature of the particular theory of material behaviour desired. The advantage of the new procedure becomes especially apparent when one considers formulation of a new theory of material behaviour for which additional balance laws (involving new kinetic quantities) are required to accompany any additional basic kinematic and thermal variables additional to those in the classical formulation. Indeed, in the formulation of such new theories, usually little or no previous information is available concerning properties of the new kinetic quantities in the additional balance laws; and, in this connection, the unified procedure of this paper provides a simple attractive setting for deriving the basic equations that are automatically consistent with the energy balance. In this paper, first the basic features of the new procedure are illustrated in the context of classical thermomechanics. Generalizations of this thermomechanical theory are then discussed in two cases: (1) in the presence of an additional kinematic variable and (2) in the presence of full electromagnetic effects. Both of these generalizations bring out some interesting novel features when new theories are being constructed.

The Influence of Rain Sensible Heat and Subsurface Energy Transport on the Energy Balance at the Land Surface

2009 ◽  
Vol 8 (4) ◽  
pp. 846-857 ◽  
Author(s):  
Stefan J. Kollet ◽  
Ivana Cvijanovic ◽  
Dirk Schüttemeyer ◽  
Reed M. Maxwell ◽  
Arnold F. Moene ◽  
...  
Keyword(s):  
Energy Balance ◽  
Land Surface ◽  
Energy Transport ◽  
Sensible Heat

scholarly journals Kinetic energy transport in Rayleigh–Bénard convection

2015 ◽  
Vol 773 ◽  
pp. 395-417 ◽  
Author(s):  
K. Petschel ◽  
S. Stellmach ◽  
M. Wilczek ◽  
J. Lülff ◽  
U. Hansen
Keyword(s):  
Energy Balance ◽  
Kinetic Energy ◽  
Prandtl Number ◽  
Viscous Dissipation ◽  
Energy Transport ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Viscous Stresses

The kinetic energy balance in Rayleigh–Bénard convection is investigated by means of direct numerical simulations for the Prandtl number range $0.01\leqslant \mathit{Pr}\leqslant 150$ and for fixed Rayleigh number $\mathit{Ra}=5\times 10^{6}$. The kinetic energy balance is divided into a dissipation, a production and a flux term. We discuss the profiles of all the terms and find that the different contributions to the energy balance can be spatially separated into regions where kinetic energy is produced and where kinetic energy is dissipated. By analysing the Prandtl number dependence of the kinetic energy balance, we show that the height dependence of the mean viscous dissipation is closely related to the flux of kinetic energy. We show that the flux of kinetic energy can be divided into four additive contributions, each representing a different elementary physical process (advection, buoyancy, normal viscous stresses and viscous shear stresses). The behaviour of these individual flux contributions is found to be surprisingly rich and exhibits a pronounced Prandtl number dependence. Different flux contributions dominate the kinetic energy transport at different depths, such that a comprehensive discussion requires a decomposition of the domain into a considerable number of sublayers. On a less detailed level, our results reveal that advective kinetic energy fluxes play a key role in balancing the near-wall dissipation at low Prandtl number, whereas normal viscous stresses are particularly important at high Prandtl number. Finally, our work reveals that classical velocity boundary layers are deeply connected to the kinetic energy transport, but fail to correctly represent regions of enhanced viscous dissipation.

The Thickness of Steady Plane Shear Flows of Circular Disks Driven by Identical Boundaries

1988 ◽  
Vol 55 (4) ◽  
pp. 969-974 ◽  
Author(s):  
D. M. Hanes ◽  
J. T. Jenkins ◽  
M. W. Richman
Keyword(s):  
Relative Motion ◽  
Constitutive Relations ◽  
Balance Laws ◽  
Plane Shear ◽  
Normal Traction ◽  
Steady Solutions ◽  
Steady Shearing ◽  
Steady Plane ◽  
Steady Motions ◽  
Circular Disks

We employ balance laws and constitutive relations for rapid, dense, plane flows of identical circular disks together with boundary conditions at a bumpy wall to analyse steady shearing flows maintained by the relative motion of two identical, parallel walls. The disks and the walls are assumed to be frictionless and nearly elastic. Given the properties of the flowing disks, those of the boundary, and the ratio of the tangential and normal tractions applied to the boundary, we determine what the distance between the walls must be for a steady solution to be possible. For these steady solutions we relate the velocity of the walls to the normal and tangential tractions applied to them. We find that in certain circumstances steady motions may be maintained even when the ratio of tangential to normal traction is much less than its value in a homogeneous simple shear. In the Appendix, the corresponding results for spheres are outlined.

scholarly journals Evaluation of energy balance closure correction methods for multiple eddy-covariance sites in different biomes

2020 ◽  
Author(s):  
Matthias Mauder ◽  
Keyword(s):  
Energy Balance ◽  
Surface Energy ◽  
Eddy Covariance ◽  
Large Scale ◽  
Surface Energy Balance ◽  
Energy Transport ◽  
Energy Balance Closure ◽  
Apparent Lack ◽  
Secondary Circulations ◽  
Measurement Height

<p>The apparent lack of surface energy balance closure is one of the most crucial challenges in the measurement of biosphere-atmosphere exchange. In principle, this issue can have a variety of potential reasons, including instrumental errors and errors introduced in the data processing chain. In addition, secondary circulations have been identified as one of the main reasons for a non-closure of the surface energy balance, since the related energy transport cannot be captured by common eddy-covariance tower flux measurements. When present, neglecting this process will result in an underestimation of turbulent fluxes. Secondary circulations can, however, be represented by means of large-eddy simulations, which have been employed to develop a novel semi-empirical model to correct for the missing large-scale flux (De Roo et al. 2018, DOI 10.1371/journal.pone.0209022). In this study, we compare the results of this process-based method with two other previously published bulk-correction methods (Mauder et al. 2013, DOI 10.1016/j.agrformet.2012.09.006; Charuchittipan et al. 2014, DOI 10.1007/s10546-014-9922-6). These three correction methods are applied for multiple sites in different biomes around the world. Independent data of energy fluxes from these sites are used to assess which of these methods leads to the most reliable results, and we discuss the limitations of these corrections methods with respect to meteorological conditions and site characteristics, such as measurement height, the landscape-scale heterogeneity and terrain complexity.</p>

scholarly journals Influences of tidal energy advection on the surface energy balance in a mangrove forest

2012 ◽  
Vol 9 (8) ◽  
pp. 11739-11765 ◽  
Author(s):  
J. G. Barr ◽  
J. D. Fuentes ◽  
M. S. DeLonge ◽  
T. L. O'Halloran ◽  
D. Barr ◽  
...  
Keyword(s):  
Energy Balance ◽  
Surface Energy ◽  
Mangrove Forest ◽  
Surface Energy Balance ◽  
Energy Transport ◽  
Tidal Energy ◽  
Water Levels ◽  
Energy Output ◽  
Tidal Flows ◽  
The Impact

Abstract. Mangrove forests are ecosystems susceptible to changing water levels and temperatures due to climate change as well as perturbations resulting from tropical storms. Numerical models can be used to project mangrove forest responses to regional and global environmental changes, and the reliability of these models depends on surface energy balance closure. However, for tidal ecosystems, the surface energy balance is complex because the energy transport associated with tidal activity remains poorly understood. This study aimed to quantify impacts of tidal flows on energy dynamics within a mangrove ecosystem. To address the research objective, an intensive study was conducted in a mangrove forest located along the Shark River in the Everglades National Park, FL. Forest-atmosphere energy exchanges were quantified with an eddy covariance system deployed on a flux tower. The lateral energy transport associated with tidal activity was calculated based on a coupled mass and energy balance approach. The mass balance included tidal flows and accumulation of water on the forest floor. The energy balance included temporal changes in enthalpy, resulting from tidal flows and temperature changes in the water column. By serving as a net sink or a source of available energy, tidal flows reduced the impact of high radiational loads on the mangrove forest. Including tidal energy advection in the surface energy balance improved the 30-min daytime energy closure from 73% to 82% over the study period. Also, the cumulative sum of energy output improved from 79% to 91% of energy input during the study period. Results indicated that tidal inundation provides an important mechanism for heat removal and that tidal exchange should be considered in surface energy budgets of coastal ecosystems. Results also demonstrated the importance of including tidal energy advection in mangrove biophysical models that are used for predicting ecosystem response to changing climate and regional freshwater management practices.

scholarly journals Symmetry types of the piezoelectric tensor and their identification

2013 ◽  
Vol 469 (2155) ◽  
pp. 20120755 ◽  
Author(s):  
W.-N. Zou ◽  
C.-X. Tang ◽  
E. Pan
Keyword(s):  
Coordinate System ◽  
Mirror Symmetry ◽  
Symmetry Type ◽  
Third Order ◽  
Order Linear ◽  
The Third ◽  
Natural Coordinate System ◽  
Linear Piezoelectricity ◽  
Rotation Transformation ◽  
Natural Coordinate

The third-order linear piezoelectricity tensor seems to be simpler than the fourth-order linear elasticity one, yet its total number of symmetry types is larger than the latter and the exact number is still inconclusive. In this paper, by means of the irreducible decomposition of the linear piezoelectricity tensor and the multipole representation of the corresponding four deviators, we conclude that there are 15 irreducible piezoelectric symmetry types, and thus further establish their characteristic web tree. By virtue of the notion of mirror symmetry and antisymmetry, we define three indicators with respect to two Euler angles and plot them on a unit disk in order to identify the symmetry type of a linear piezoelectricity tensor measured in an arbitrarily oriented coordinate system. Furthermore, an analytic procedure based on the solved axis-direction sets is also proposed to precisely determine the symmetry type of a linear piezoelectricity tensor and to trace the rotation transformation back to its natural coordinate system.

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