Incompressible limit for a magnetostrictive energy functional

In general, materials with linear memory constitutive relations are characterized by a relaxation function. This leads to a situation where the free energy for most materials with memory is not unique. There is a convex set of free energy functionals with a minimum and a maximum element. An alternative procedure is proposed which characterizes a material by the kernel of the rate of dissipation functional. Using some recent results, we find that a unique free energy and relaxation function may then be deduced. An example is given for discrete spectrum materials. Also, the new results are used to show that a previously derived general representation of rate of dissipation and free energy functionals is not complete, in the sense that there are valid functionals which cannot be described by this general formula.

The Hencky medium with softening under isothermal and quasistatic deformation is considered. It is believed that volume deformation is not positive. In this case softening is characterized by the part of union curve with negative slope. For aforementioned conditions function of free energy is presented. For all stages of deformation in the space “volume deformation - intensity of shear’s deformation” level lines of the free energy are constructed. It is established that level lines are ellipses in hardening and function of free energy increases with distance from their centers, while in softening hyperbolas are level lines and function of free energy decreases with distance from their centers. Obtained results indirectly confirm the change in type of boundary value problem from elliptic to hyperbolic under material transition to the softening.

This paper aims to investigate the pull-in phenomenon of functionally graded (FG) capacitive nanocantilevers subjected to an electrostatic force and thermal moment due to an applied voltage and thermal shock considering intermolecular force within the framework of nonlocal elasticity theory to account for the small scale effect. The FG nano-beam is made of mixture of metal and ceramic which the material properties vary continuously through the thickness according to an exponential distribution law (E-FGM). The nonlocal elastic behavior is described by the differential constitutive model of Eringen, which enables the present model to become effective in the analysis and design of nanosensors. The nano-beam is modeled assuming the Euler–Bernoulli beam theory and the equations are derived using the equilibrium of an element. A Galerkin-based step by step linearization method has been used to solve the governing static deflection equation. The present solution is validated with existing results reported in previous studies. The effects of temperature change, Van der Waals (VdW) or Casimir force and small scale factor on the five types of FG nano-beams are discussed in detail. The results indicate that VdW/Casimir force and thermal moment reduce the pull-in voltage, however, on the contrary, small scale effect cause to slightly increase the amount of pull-in voltage.

Smart materials are active and multifunctional materials, which play an important part for sensor and actuator applications. These materials have the potential to transform passive structures into adaptive systems. However, a prerequisite for the design and the optimization of these materials is, that reliable models exist, which incorporate the interaction between the different combinations of thermal, electrical, magnetic, optical and mechanical effects. Polymeric electroelastic materials, so-called (EAP), own the characteristic to deform if an electric field is applied. EAP's possesses the benefit that they share the characteristic of polymers, these are lightweight, inexpensive, fracture tolerant, elastic, and the chemical and physical structure is well understood. However, the description electroactive polymer is a generic term for many kinds of different microscopic mechanisms and polymeric materials. Based on the laws of electromagnetism and elasticity, a visco-electroelastic model is developed and implemented into the finite element method (FEM). The presented three-dimensional solid element has eight nodes and trilinear interpolation functions for the displacement and the electric potential. The continuum mechanics model contains finite deformations, the time dependency and the nearly incompressible behavior of the material. To describe the possible, large time dependent deformations, a finite viscoelastic model with a split of the deformation gradient is used. Thereby the time dependent characteristic of polymeric materials is incorporated through the free energy function. The electromechanical interactions are considered by the electrostatic forces and inside the energy function.

The interactions of amino acid side-chains confer diverse energetic contributions and physical properties to a protein's stability and function. Various computational tools estimate the effect of changing a given amino acid on the protein's stability based on parametrized (free) energy functions. When parametrized for the prediction of protein stability in water, such energy functions can lead to suboptimal results for other solvents, such as ionic liquids (IL), aqueous ionic liquids (aIL), or salt solutions. However, to our knowledge, no comprehensive data are available describing the energetic effects of aIL on intramolecular protein interactions. Here, we present the most comprehensive set of potential of mean force (PMF) profiles of pairwise protein-residue interactions to date, covering 50 relevant interactions in water, the two biotechnologically relevant aIL [BMIM/Cl] and [BMIM/TfO], and [Na/Cl]. These results are based on a cumulated simulation time of >1 ms. aIL and salt ions can weaken, but also strengthen, specific residue interactions by more than 3 kcal mol-1, depending on the residue pair, residue-residue configuration, participating ions, and concentration, necessitating considering such interactions specifically. These changes originate from a complex interplay of competitive or cooperative noncovalent ion-residue interactions, changes in solvent structural dynamics, or unspecific charge screening effects and occur at the contact distance but also at larger, solvent-separated distances. This data provide explanations at the atomistic and energetic levels for complex IL effects on protein stability and should help improve the prediction accuracies of computational tools that estimate protein stability based on (free) energy functions.

If the hydrostatic tensile load is applied to a hyper-elastic material, the void initiates when the load exceeds the critical value. On the other hand, it is important to study multiple void growth phenomena, in order to consider the fracture by coalescence of voids. In this paper, we study the growth of multiple voids in the hyper-elastic material. The material is characterized by the energy function as the compressive material. Some experiments for the rubber, as a hyper-elastic material, are proceeded, in order to obtain these mechanical properties in the energy function. A square cell with some small voids is constructed and applied with tensile deformation by moving outer surface. The large deformation and the non-linear simulations are proceeded by using FEM. If there is only one seed, one void grows from the seed. However, when there are some seeds, we observed the void growing and the void vanishing by the influence from the other voids. The influence of the initial voids scale to the void growth is studied.

In many aspects, elastomers and soft biological tissues exhibit similar mechanical properties such as a pronounced nonlinear stress–strain relation and a viscoelastic response to external loads. Consequently, many models use the same rheological framework and material functions to capture their behavior. The viscosity function is thereby often assumed to be constant and the corresponding free energy function follows that one of the long-term equilibrium response. This work questions this assumption and presents a detailed study on non-Newtonian viscosity functions for elastomers and brain tissues. The viscosity functions are paired with several commonly used free energy functions and fitted to two different types of elastomers and brain tissues in cyclic and relaxation experiments, respectively. Having identified suitable viscosity and free energy functions for the different materials, numerical aspects of viscoelasticity are addressed. From the multiplicative decomposition of the deformation gradient and ensuring a non-negative dissipation rate, four equivalent viscoelasticity formulations are derived that employ different internal variables. Using an implicit exponential map as time integration scheme, the numerical behavior of these four formulations are compared among each other and numerically robust candidates are identified. The fitting results demonstrate that non-Newtonian viscosity functions significantly enhance the fitting quality. It is shown that the choice of a viscosity function is even more important than the choice of a free energy function and the classical neo-Hooke approach is often a sufficient choice. Furthermore, the numerical investigations suggest the superiority of two of the four viscoelasticity formulations, especially when complex finite element simulations are to be conducted.

The paper presents a continuum mechanics approach in a mathematical formulation of a class of constitutive relations which tend to descibe mechanical as well as thermo-rheological behavior of a material with two limiting cases of consitutive behavior: thermo-hyperelastic representing the rate independent behavior and thermo-viscoelastic in a general rheological sense. In the rate-independent limit the mechanical part can be represented by appropriate strain energy functions. We consider potential functions of Murnagan’s type for compressible and Signorini’s type for incompressible materials. In addition to the standard kinematical variables, the time dependent part of the assumed general form of constitutive relation is postulated to depend on an internal variable similar to the reduced time scaled by a shift function as in classical thermo-viscoelasticity, but with much broader functional relationship. The corresponding expressions for stress, entropy, and dissipation inequality are derived from strict thermodynamic considerations and are given in an explicit form. Development is based on the principle of fading memory established by (Coleman 1961), (Coleman & Noll 1964) which was subsequently specialized for the class of viscoelastic materials by (Lianis 1965, 1970). A fundamental assumption of the proposed procedure is that in the expansion of the rheological time-dependent part of a free energy function small differences in histories of strain and temperature fields enter the thermodynamic potential or the free energy function only up to quadratic terms. Consequently, the derived expressions are quite amenable for further development. We discuss a thermo-rheologically simple constitutive model of a material which is described by a constitutive relation of a single-integral equation type.

Two families of explicit models constructed from a homogenization solution for the magnetoelastic response of MREs containing iron and ferrofluid particles

In this paper, we investigate the influence of the extended uncertainty principle (EUP) on the thermodynamics of a charged rotating Bañados, Teitelboim and Zanelli (BTZ) black hole in ([Formula: see text])-dimensional anti-de Sitter (AdS) space–time. To this end, we derive EUP-corrected Hawking temperature, pressure, entropy, Gibbs free energy and heat capacity functions. We observe that EUP-based quantum corrections decrease pressure and entropy functions in AdS space–time. Such decreases show up itself only in physically meaningful region for Gibbs free energy and specific heat functions.

In this chapter we provide an overview of the basic equations governing the mechanical response of rubberlike materials capable of finite deformations. We use the term rubberlike materials to refer to highly deformable, nonlinear elastic continua, which respond differently in their reference state and their deformed configuration. An elastic material, by definition, will always return to its original state after the application and removal of forces. Nonlinear elastic materials, in particular, exhibit a change of material behavior during deformation, but assume the original shape after removal of all applied forces. Therefore, we do not include the effects of time and rate dependence, nor do we assume that the applied forces are of such magnitude to induce damage to the material.We first introduce the mathematics of deformation and different stress tensors for the analysis of rubberlike materials. Subsequently, we summarize the general constitutive theory and specialize its use to isotropic compressible and incompressible materials. A large number of constitutive functions are available in the literature to describe the nonlinear elastic response. We provide an overview of the most frequently used formulations.The last section of this chapter covers the solution of some boundary value problems. In particular, we consider incompressible, isotropic and elastic materials to illustrate the mechanical response when different strain energy functions are used. We apply the theories to circular, cylindrical thick-walled tubes subjected first to a pure azimuthal shear deformation, followed by combined axial extension and radial inflation.KeywordsDeformation GradientStrain Energy FunctionPrincipal StretchAxial StretchIncrease Wall ThicknessThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This manuscript presents an analysis of the thermal and magnetic properties of particles moving in two dimensions under the influence of an external magnetic field and Aharanov-Bohm flux, considering an energy-dependent Hellmann potential. The energy eigenvalue is obtained using the asymptotic iteration method. Next, we model the system as a canonical ensemble and derive the partition function to acquire analytical expressions for the Helmholtz free energy, entropy, mean energy, and specific heat. We observe that an increase in the external magnetic field and AB flux generally increases the energy eigenvalues, with a stronger effect at lower flux values, while energy eigenvalues decrease with an increase in the screening parameter. Additionally, Helmholtz free energy and entropy functions are sensitive to temperature, particularly in the low-temperature regime, while specific heat exhibits a Schottky anomaly. We also compute the magnetisation and susceptibility functions, finding that both magnetisation and susceptibility increase with the external magnetic field strength and the AB flux.

The contribution is concerned with a thermodynamic consistent constitutive model for magnetostrictive materials and ferroelectric ceramics. It captures the nonlinear phenomenological behavior which is described by hysteresis effects. Magnetostrictive alloys and ferroelectric ceramics belong to the multifunctional materials. In recent years these materials have become widely–used in actor and sensor applications. They characterize an inherent coupling between deformation and magnetic or electric field. Due to the similarities of the coupled differential equations a uniform approach is applied for both phenomena. The presented three–dimensional material model is thermodynamically motivated. It is based on the definition of a specific free energy function and a switching criterion. Furthermore an additive split of strain and the magnetic or electric field in a reversible and an irreversible part is suggested. The irreversible quantities serve as internal variables, which is analog to plasticity theory. A one–to–one–relation between the two internal variables provides conservation of volume for the irreversible strains. The presented material model can approximate the ferromagnetic or ferroelectric hysteresis curve and the related butterfly hysteresis. Furthermore an extended approach for ferrimagnetic behavior, which occurs in magnetostrictive materials, is presented. Some numerical simulations demonstrate the capability of the presented model. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In current scenario, research in smart material deformations introduces a new classical continuum mechanics-based deformation approach that have enough potential to make a bold move from the conventional deformation approaches to a newer one. In line with that, the present work is concerned with the finite deformation modeling of a dielectric cylindrical actuator (DCA) with compliant electrodes. Herein, a common usage of DCA is to realize simultaneous axial and radial displacement with an electric field application across the thickness of the cylinder. Accordingly, we first presented an unified classical continuum mechanics-based approach to model the finite deformation of an incompressible isotropic electro-elastic cylindrical actuator under an applied electric field. Next, we formulate the constitutive relationships following the second law of thermodynamics with an amended form of energy function. This amended energy function successfully resolved the difficulty in the physical interpretation of Maxwell stress tensor exists in the literature under large deformations. In addition, we also propose a new energy density function for a class of an incompressible isotropic electro-elastic material. Further, we obtain a finite electro-elastic deformation model for a dielectric cylindrical actuator (DCA). Finally, we validated the obtained electro-elastic deformation model with their corresponding experimental data existing in the literature.

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.