# Hybrid Finite Element Formulation for Electrostrictive Materials: Dynamic Analysis of Beam

TL;DR: In this article, a nonlinear electromechanical coupled dynamic finite element formulation for electrostrictive materials is proposed, which includes the quadratic dependence of strain with polarization, valid at a constant temperature and excludes hysteresis.

Abstract: A nonlinear electromechanical coupled dynamic finite element formulation for electrostrictive materials is proposed. This formulation includes the quadratic dependence of strain with polarization, valid at a constant temperature and excludes hysteresis. The present formulation uses linear finite element analysis along with the numerical solution of the nonlinear constitutive equation using Newton-Raphson technique only within each electrostrictive elements. Therefore this formulation is capable of handling nonlinearity only on the electrostrictive domain instead of making the whole finite element analysis as nonlinear. The Newton-Raphson technique is specially modified in order to guarantee the convergence of the solution. A simple technique for obtaining the initial guess of the solution for Newton-Raphson technique is also proposed which gives faster convergence of the solution. The assumption made in most of the finite element formulations, that polarization is approximately equal to electric displacem...

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TL;DR: Slepian used an operational "definition" of stress and concluded that the compensating mechanical forces which must be introduced operationally are not derivable from a tensor.

Abstract: for the force on a dielectric in an electrical field E and bounded by a surface S with unit normal n. He used an operational \"definition\" of stress and concluded that the compensating mechanical forces which must be introduced operationally are not derivable from a tensor. It is suggested here that Slepian's analysis is essentially correct and that the difficulty arises because of the choice of an operationally \"defined\" stress. This choice is inconsistent with the existence of an electrical surface stress-which is familiar, in the magnetic analogue, in studies of the form effect-and it is argued here that the Euler-Cauchy definition of stress is the appropriate one. The Definition of Stress.-In authoritative works on continuum mechanics stress is introduced by means of the stress hypothesis of Euler and Cauchy,2 that is, by asserting that, acting upon any imagined closed geometrical surface a within the body, there exists a field of stress vectors t which has an equivalent effect to the (interparticle) forces exerted by the material outside aupon the material within. For a dielectric material the interaprticle (i.e., intermolecular) forces are partly long-range in character and they may therefore contribute not only to t but also to f, the body force per unit volume. For the present purpose, however, the important point to note is that ais an imagined geometrical surface and not a physical surface of separation within the material. An alternative procedure is to use the operational definition of stress in which it is imagined that a physical cut is made in the material along an internal element of surface dd = nda. If means are then introduced for keeping the strains in the material on both sides of the cut the same as they were before the cut was made, then the force introduced by these means is t'do-, where t' is the operationally defined stress vector. In adopting this operational definition, Slepian commented: \"It is not assumed that the cut and the introduced means do not disturb the microstructure and micromechanics of the material. For example, in the case of a fluid the cut and means would cause molecules to be reflected which would otherwise pass through the geometric element of surface dS. It is assumed, however, that in spite of the change in the micromechanics, there is no change in the observable macromechanics.\"' It may also be noted there is a further element of idealization involved in that the cut is imagined to be of finite extent: in practice, as discussed later in this paper, it is only possible to measure the force on an element of volume when the element is completely separated from the rest of the body. For an ordinary elastic material the stress acting at a physical surface of separa-

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01 Jan 1974

TL;DR: In this article, the authors present a formal notation for one-dimensional elements in structural dynamics and vibrational properties of a structural system, including the following: 1. Isoparametric Elements.

Abstract: Notation. Introduction. One-Dimensional Elements, Computational Procedures. Basic Elements. Formulation Techniques: Variational Methods. Formulation Techniques: Galerkin and Other Weighted Residual Methods. Isoparametric Elements. Isoparametric Triangles and Tetrahedra. Coordinate Transformation and Selected Analysis Options. Error, Error Estimation, and Convergence. Modeling Considerations and Software Use. Finite Elements in Structural Dynamics and Vibrations. Heat Transfer and Selected Fluid Problems. Constaints: Penalty Forms, Locking, and Constraint Counting. Solid of Revolution. Plate Bending. Shells. Nonlinearity: An Introduction. Stress Stiffness and Buckling. Appendix A: Matrices: Selected Definition and Manipulations. Appendix B: Simultaneous Algebraic Equations. Appendix C: Eigenvalues and Eigenvectors. References. Index.

6,126 citations

### "Hybrid Finite Element Formulation f..." refers methods in this paper

...4 Time Domain Integration An explicit direct time integration scheme has been employed to obtain the dynamic response [20]....

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01 Jan 1981TL;DR: In this paper, the NavierStokes Equations are used to define linear elasticity for tensor analysis, and the invariance of material response is established. But the analysis is restricted to finite elasticity and cannot be extended to infinite elasticity.

Abstract: Preface. Acknowledgments. Tensor Algebra. Tensor Analysis. Kinematics. Mass. Momentum. Force. Constitutive Assumptions. Inviscid Fluids. Change in Observer. Invariance of Material Response. Newtonian Fluids. The NavierStokes Equations. Finite Elasticity. Linear Elasticity. Appendix. References. Hints for Selected Exercises. Index.

1,751 citations

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01 Jan 2010

TL;DR: This new edition offers expanded coverage of the subject matter both in terms of details and contents, providing greater flexibility for either a one or two-semester course in either continuum mechanics or elasticity, and contains expanded and improved problem sets providing both intellectual challenges and engineering applications.

Abstract: Continuum Mechanics is a branch of physical mechanics that describes the macroscopic mechanical behavior of solid or fluid materials considered to be continuously distributed. It is fundamental to the fields of civil, mechanical, chemical and bioengineering. This time-tested text has been used for over 35 years to introduce junior and senior-level undergraduate engineering students, as well as graduate students, to the basic principles of continuum mechanics and their applications to real engineering problems. The text begins with a detailed presentation of the coordinate invariant quantity, the tensor, introduced as a linear transformation. This is then followed by the formulation of the kinematics of deformation, large as well as very small, the description of stresses and the basic laws of continuum mechanics. As applications of these laws, the behaviors of certain material idealizations (models) including the elastic, viscous and viscoelastic materials, are presented. This new edition offers expanded coverage of the subject matter both in terms of details and contents, providing greater flexibility for either a one or two-semester course in either continuum mechanics or elasticity. Although this current edition has expanded the coverage of the subject matter, it nevertheless uses the same approach as that in the earlier editions - that one can cover advanced topics in an elementary way that go from simple to complex, using a wealth of illustrative examples and problems. It is, and will remain, one of the most accessible textbooks on this challenging engineering subject. It offers significantly expanded coverage of elasticity in Chapter 5, including solutions of some 3-D problems based on the fundamental potential functions approach. It includes a new section at the end of Chapter 4 devoted to the integral formulation of the field equations. Seven new appendices appear at the end of the relevant chapters to help make each chapter more self-contained. This book contains expanded and improved problem sets providing both intellectual challenges and engineering applications.

1,268 citations

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General Dynamics

^{1}TL;DR: In this paper, a finite element formulation which includes the piezoelectric or electroelastic effect is given, a strong analogy is exhibited between electric and elastic variables, and a stiffness finite element method is deduced.

Abstract: A finite element formulation which includes the piezoelectric or electroelastic effect is given. A strong analogy is exhibited between electric and elastic variables, and a ‘stiffness’ finite element method is deduced. The dynamical matrix equation of electroelasticity is formulated and found to be reducible in form to the well-known equation of structural dynamics, A tetrahedral finite element is presented, implementing the theorem for application to problems of three-dimensional electroelasticity.

972 citations

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765 citations

### "Hybrid Finite Element Formulation f..." refers methods in this paper

...Devonshire [3] formulated a phenomenological model of ferroelectricity that used polarization, stress and temperatures as independent variables and included electrostrictive effect....

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