A Solid Shell Finite Element Formulation for Dielectric Elastomers

To analyse the behaviour of thin structures of dielectric elastomer (DE) material a solid shell finite element is presented. The main characteristics of DEs are a non‐linear hyper elastic behaviour, the quasi‐incompressibility, and the ability to transform electric energy into mechanical work. Applying a voltage to thin DE structures may produce large elongation strains of 120‐380%. These large strains, the efficient electro‐mechanical coupling, and the light weight make DEs very attractive for the usage in actuators. Thus, there is a need for detailed research. With respect to the electro‐mechanical coupling a constitutive model is presented. An electric stress tensor and a total stress tensor are introduced by considering the electrical body force and couple in the balance of linear momentum and angular momentum, respectively. The governing equations are derived and embedded in the solid shell formulation. The element formulation is based on a Hu‐Washizu mixed variational principle using six independent fields: displacements, electric potential, strains, electric field, mechanical stresses, and dielectric displacements. It allows large deformations and accounts for physical nonlinearities to capture two of the main characteristics of DEs. The shell element could be applied for the modelling of arbitrary curved thin structures. The ability of the present element formulation is demonstrated in several examples. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Thin piezoelectric laminates are used for a wide range of technical applications. A four‐node piezoelectric shell element is presented to analyse such structures effectively. In case of bending dominated problems incompatible approximation functions of the electrical and mechanical fields cause incorrect results. In order to overcome this problem the finite element formulation is based on a mixed variational principle implying six independent fields: displacements, electric potential, strains, electric field, mechanical stresses and dielectric displacements. This allows for an interpolation of the strains and the electric field in thickness direction independent of the bilinear interpolation functions. A piecewise quadratic approach for the shear strains in thickness direction and the corresponding electric field is proposed for arbitrarily layered shells. Regarding coupling of electrical and mechanical fields this yields to an appropriate balance of the approximation functions. Numerical examples show more precise results in contrast to standard elements with lowest order interpolation functions. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A continuum thermodynamics framework for the diffuse interface or phase field approach to domain evolution is presented. The theory distinguishes the fundamental balance laws which are universal (i.e. mechanical equilibrium, Gauss' law, and a new micro-force balance) from the constitutive laws, which describe the behavior of a specific material. A finite element formulation based on a virtual work statement is implemented using mechanical displacements, electrical potential, and polarization components (the vector order parameter) as nodal degrees of freedom. The finite element code is then applied to simulate the growth of a 180° domain needle through a parent domain.

Non-local behavior of two collinear mixed-mode limited-permeable cracks in a functionally graded piezoelectric medium

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

Dielectric elastomers (DE) have lately drawn attention for their use in soft actuators and electromechanical energy harvester. Excellent dielectric properties and high mechanical deforming capability are key requirements of DEs. Given that the output strain of DE system is proportional to the applied electric field squared (and vice versa), - high electric field is needed to generate large displacement. Similarly to any other insulation exposed to high electric field, DEs undergo electrical ageing. Furthermore, DEs undergo continuous mechanical deformation during its lifetime. Therefore, the lifetime of DE system can be shortened by the simultaneous action of electric and dynamical mechanical stress. This can lead to a faster overall ageing of DE system with respect to other common HV devices, that usually are mechanically-static. To extend the useful life of the system it is firstly important to know how fast the electromechanical degradation process is and what is the contribution of electrical and mechanical stresses. Secondly, assess its ageing stage and adjust its output accordingly, albeit with a lower effectiveness. Partial discharge testing has been used to assess the health of a silicon polymer elastomer typically used in dielectric elastomer generator. The mechanical ageing of a DE system has been qualitatively evaluated by monitor PD activity over different cycles of usage and reported in this paper.

A scaled thickness conditioning for solid- and solid-shell discretizations of thin-walled structures

Free vibration of simply supported functionally graded and layered magneto-electro-elastic plates by finite element method

A scaled boundary finite element method for bending analysis of fiber-reinforced piezoelectric laminated composite plates

In this paper, static bending analysis of functionally graded plates with piezoelectric layers has been carried out considering geometrical nonlinearity in different sets of mechanical and electrical loadings. Only the geometrical nonlinearity has been taken into account. The governing equations are obtained using potential energy and Hamilton’s principle. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect by using higher order elements. The present finite element used displacement and electric potential as nodal degrees of freedom. Results are presented for two constituent FGM plate under different mechanical boundary conditions. Numerical results for FGM plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response of the plate are also studied.

This paper presents a piezoelectric solid element that can be used for modeling thin sensors and actuators. The element has been extended from an eighteen-node assumed strain solid element that can alleviate locking in mechanical problems. For the fully coupled mechanical-piezoelectric formulation, the electric potential is regarded as a nodal degree of freedom in addition to three translations in the original element. Consequently, the induced electric potential can be calculated for a prescribed load and the actuation displacement computed for an input voltage. Since the assumed strain solid element can relieve locking, the proposed element can also be used to analyze the behavior of very thin actuators. In addition, a finite element code was developed based on the proposed formulation and typical numerical examples were solved for code validation. Parametric studies for the THUNDER actuator have been also conducted using the code, where specific combinations of materials used for the layers plus the curvature of THUNDER were both found to improve the actuation displacement.

In this paper, two efficient elements are proposed by utilizing hybrid Trefftz method for the analysis of thin plate bending. The triangular element, THT, and the quadrilateral element, QHT, which have 9 and 12 degrees of freedom, respectively. Two independent displacement fields are defined for internal and boundary of the elements. The internal field is selected in such a way that it satisfies the governing equation of the thin plate. Boundary field is dependent on the nodal degrees of freedom via boundary interpolation functions. For deriving boundary interpolation functions, element's edges are assumed to deform like a beam, and the related interpolation functions are used for the boundary fields. The solution accuracies of the famous and hard bench mark problems, such as circular and skew plates, prove the justification of suggested elements. Based on the various test problems, QHT in comparison to other four-sided elements, and THT in comparison to triangular ones, show better results and rapider convergence rate.

An efficient multiphysics solid shell based finite element approach for modeling thin sheet metal forming processes

A systematic development of a continuum model is presented, which is capable of describing the magneto-mechanical behavior of magnetic polymer gels commonly referred to as “ferrogels”. In the present research, ferrogels are treated as multicomponent, multiphase materials. They consist of a polymer network (P), fixed magnetic particles (f), mobile magnetic particles (m), and liquid (L). By considering ferrogels as multicomponent materials, interaction among constituents of ferrogels can be captured. This helps in understanding the process occurring inside ferrogels under the influence of external stimuli, such as magnetic fields. In our modeling approach, the field equations of ferrogels are derived within the framework of the theory of mixtures. The basic equations include Maxwell’s equations, balance of mass, linear momentum, angular momentum, energy, and entropy. In the framework of the theory of mixtures, balance relations are first presented at the constituent level also referred to as partial balance relations. By summing partial balance relations over all constituents and imposing the restrictions of theory of mixtures, balance relations of mixture (for the ferrogel) are obtained. In the current work the specific magnetization (magnetization per density) is considered as an evolving variable. It is demonstrated that balance of angular momentum is satisfied using the evolution equation of specific magnetization and constitutive laws. In the process of modeling, a suitable free energy function is introduced and thermodynamically consistent constitutive laws are formulated. Introducing certain assumptions, a reduced model of the ferrogel, a coupled magneto-mechanical formulation, is subsequently presented. The reduced model consists only of a polymer network (P) and fixed magnetic particles (f). It is concluded that the reduced model compares well to the existing ones in the literature. The magneto-mechanical problem based on the reduced model is solved in 2D using the finite element method. The only unknowns for the finite element method implementation are mechanical displacement and magnetic potential. Deformation of a ferrogel in a magnetic field is subsequently investigated. Elongation and contraction of a ferrogel are observed when a magnetic field is applied in the x- and y-directions, respectively. The numerical results were compared with existing experimental work in the literature. A good qualitative agreement was found between numerical and experimental results.

Static bending and forced vibration analyses of a piezoelectric semiconductor cylindrical shell within first-order shear deformation theory