Axisymmetric bifurcations of thick spherical shells under inflation and compression

Incremental equilibrium equations and corresponding boundary conditions for an isotropic, hyperelastic and incompressible material are summarized and then specialized to a form suitable for the analysis of a spherical shell subject to an internal or an external pressure. A thick-walled spherical shell during inflation is analyzed using four different material models. Specifically, one and two terms in the Ogden energy formulation, the Gent model and an I1 formulation recently proposed by Lopez-Pamies. We investigate the existence of local pressure maxima and minima and the dependence of the corresponding stretches on the material model and on shell thickness. These results are then used to investigate axisymmetric bifurcations of the inflated shell. The analysis is extended to determine the behavior of a thick-walled spherical shell subject to an external pressure. We find that the results of the two terms Ogden formulation, the Gent and the Lopez-Pamies models are very similar, for the one term Ogden material we identify additional critical stretches, which have not been reported in the literature before

On Love-type waves in a finitely deformed magnetoelastic layered half-space

In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney–Rivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein–Gulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper

An energy–momentum time integration scheme based on a convex multi-variable framework for non-linear electro-elastodynamics

This paper introduces a new one-step second order accurate energy–momentum (EM) preserving time integrator for reversible electro-elastodynamics. The new scheme is shown to be extremely useful for the long-term simulation of electroactive polymers (EAPs) undergoing massive strains and/or electric fields. The paper presents the following main novelties. (1) The formulation of a new energy momentum time integrator scheme in the context of nonlinear electro-elastodynamics. (2) The consideration of well-posed ab initio convex multi-variable constitutive models. (3) Based on the use of alternative mixed variational principles, the paper introduces two different EM time integration strategies (one based on the Helmholtz’s and the other based on the internal energy). (4) The new time integrator relies on the definition of four discrete derivatives of the internal/Helmholtz energies representing the algorithmic counterparts of the work conjugates of the right Cauchy–Green deformation tensor, its co-factor, its determinant and the Lagrangian electric displacement field. (6) Proof of thermodynamic consistency and of second order accuracy with respect to time of the resulting algorithm is included. Finally, a series of numerical examples are included in order to demonstrate the robustness and conservation properties of the proposed scheme, specifically in the case of long-term simulations

Soft network composite materials with deterministic and bio-inspired designs

Hard and soft structural composites found in biology provide inspiration for the design of advanced synthetic materials. Many examples of bio-inspired hard materials can be found in the literature; far less attention has been devoted to soft systems. Here we introduce deterministic routes to low-modulus thin film materials with stress/strain responses that can be tailored precisely to match the non-linear properties of biological tissues, with application opportunities that range from soft biomedical devices to constructs for tissue engineering. The approach combines a low-modulus matrix with an open, stretchable network as a structural reinforcement that can yield classes of composites with a wide range of desired mechanical responses, including anisotropic, spatially heterogeneous, hierarchical and self-similar designs. Demonstrative application examples in thin, skin-mounted electrophysiological sensors with mechanics precisely matched to the human epidermis and in soft, hydrogel-based vehicles for triggered drug release suggest their broad potential uses in biomedical devices. © 2015 Macmillan Publishers Limited. All rights reservedopen7

Variational Foundations and Generalized Unified Theory of RVE-Based Multiscale Models

A unified variational theory is proposed for a general class of multiscale models based on the concept of Representative Volume Element. The entire theory lies on three fundamental principles: (1) kinematical admissibility, whereby the macro- and micro-scale kinematics are defined and linked in a physically meaningful way; (2) duality, through which the natures of the force- and stress-like quantities are uniquely identified as the duals (power-conjugates) of the adopted kinematical variables; and (3) the Principle of Multiscale Virtual Power, a generalization of the well-known Hill-Mandel Principle of Macrohomogeneity, from which equilibrium equations and homogenization relations for the force- and stress-like quantities are unequivocally obtained by straightforward variational arguments. The proposed theory provides a clear, logically-structured framework within which existing formulations can be rationally justified and new, more general multiscale models can be rigorously derived in well-defined steps. Its generality allows the treatment of problems involving phenomena as diverse as dynamics, higher order strain effects, material failure with kinematical discontinuities, fluid mechanics and coupled multi-physics. This is illustrated in a number of examples where a range of models is systematically derived by following the same steps. Due to the variational basis of the theory, the format in which derived models are presented is naturally well suited for discretization by finite element-based or related methods of numerical approximation. Numerical examples illustrate the use of resulting models, including a non-conventional failure-oriented model with discontinuous kinematics, in practical computations

Nonlinear constitutive modeling of electroelastic solids

In this chapter, the equations governing the mechanical behavior of electroelastic solids capable of finite deformations are summarized with particular reference to the development of constitutive equations describing the electromechanical interactions in soft dielectric materials. Following a brief summary of some background from continuum mechanics and nonlinear elasticity theory, the equations of electrostatics are given in Eulerian form and then re-cast in Lagrangian form. The electroelastic constitutive equations are based on the existence of a so-called total energy function, which may be regarded as a function of the Lagrangian form of either the electric field or electric displacement as the independent electric variable, together with the deformation gradient. For each form of the total energy function, corresponding expressions for the (total) nominal and Cauchy stress tensors are provided, both in full generality and for their isotropic specializations for unconstrained and incompressible materials. The general formulas are then applied to the basic problem of homogeneous deformation of a rectangular plate, and illustrated by the choice of a simple specific example of constitutive equation. As a further illustration, the theory is applied to the analysis of a nonhomogeneous deformation of a circular cylindrical tube subject to an axial force, a torsional moment, and a radial electric field generated by a potential difference between flexible electrodes covering its inner and outer curved surfaces

Universal relations in isotropic nonlinear magnetoelasticity

In this paper we first summarize the basic constitutive equations for (nonlinear) magnetoelastic solids capable of large deformations. Equivalent formulations are given using either the magnetic induction vector or the magnetic field vector as the independent magnetic variable in addition to the deformation gradient. The constitutive equations are then specialized to incompressible, isotropic magnetoelastic materials in order to determine universal relations. A universal relation, in this context, is an equation that relates the components of the stress tensor and the components of the magnetic field and/or the components of the magnetic induction that holds independently of the specific choice of constitutive law for the considered class or subclass of materials. As has been shown previously for the case in which the magnetic induction is the independent magnetic variable, in the general case there exists only one possible universal relation. We show that this is also the case if the magnetic field is taken as the independent variable and that the universal relations resulting from the two cases are equivalent. A number of special cases are found for certain specializations of the constitutive equations. These include some connections between the deformation, the magnetic field and magnetic induction that do not involve the components of the stress tensor. Universal relations are then examined for some representative homogeneous and inhomogeneous universal solutions