Dielectric elastomer composites

The coupled electromechanical response of electroactive dielectric composites is examined in the setting of small deformation and moderate electric field. In this setting, the mechanical stress depends quadratically on the electric field through a combination of material electrostriction and Maxwell stress. It is rigorously shown that the macroscopic mechanical stress of the composite also depends quadratically on the macroscopic electric field. It is further demonstrated that the effective electromechanical coupling can be computed from the examination of the uncoupled electrostatic and elastic problems. The resulting expressions suggest that the effective electromechanical coupling may be very large for microstructures that lead to significant fluctuations of the electric field. This idea is explored through examples involving sequential laminates. It is demonstrated that the electromechanical coupling – the macroscopic strain induced in the composite through the application of a unit electric field – can be amplified by many orders of magnitude by either a combination of constituent materials with high contrast or by making a highly complex and polydisperse microstructure. These findings suggest a path forward for overcoming the main limitation hindering the development of electroactive polymers.     

A nonlinear field theory of deformable dielectrics

Two difficulties have long troubled the field theory of dielectric solids. First, when two electric charges are placed inside a dielectric solid, the force between them is not a measurable quantity. Second, when a dielectric solid deforms, the true electric field and true electric displacement are not work conjugates. These difficulties are circumvented in a new formulation of the theory in this paper. Imagine that each material particle in a dielectric is attached with a weight and a battery, and prescribe a field of virtual displacement and a field of virtual voltage. Associated with the virtual work done by the weights and inertia, define the nominal stress as the conjugate to the gradient of the virtual displacement. Associated with the virtual work done by the batteries, define the nominal electric displacement as the conjugate to the gradient of virtual voltage. The approach does not start with Newton's laws of mechanics and Maxwell-Faraday theory of electrostatics, but produces them as consequences. The definitions lead to familiar and decoupled field equations. Electromechanical coupling enters the theory through material laws. In the limiting case of a fluid dielectric, the theory recovers the Maxwell stress. The approach is developed for finite deformation, and is applicable to both elastic and inelastic dielectrics. As applications of the theory, we discuss material laws for elastic dielectrics, and study infinitesimal fields superimposed upon a given field, including phenomena such as vibration, wave propagation, and bifurcation.     

Fully coupled, multi-axial, symmetric constitutive laws for polycrystalline ferroelectric ceramics

In this paper, a general form for multi-axial constitutive laws for ferroelectric ceramics is constructed. The foundation of the theory is an assumed form for the Helmholtz free energy of the material. Switching surfaces and associated flow rules are postulated in a modified stress and electric field space such that a positive dissipation rate during switching is guaranteed. The resulting tangent moduli relating increments of stress and electric field to increments of strain and electric displacement are symmetric since changes in the linear elastic, dielectric and piezoelectric properties of the material are included in the switching surface. Finally, parameters of the model are determined for two uncoupled cases, namely non-remanent straining ferroelectrics and purely ferroelastic switching, and then for the fully coupled ferroelectric case.     

A mathematical basis for strain-gradient plasticity theory—Part I: Scalar plastic multiplier

Strain-gradient plasticity theories are reviewed in which some measure of the plastic strain rate is treated as an independent kinematic variable. Dislocation arguments are invoked in order to provide a physical basis for the hardening at interfaces. A phenomenological, flow theory version of gradient plasticity is constructed in which stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition. Plastic work is also done at internal interfaces and a yield surface is postulated for the work-conjugate stress quantities at the interface. Thereby, the theory has the potential to account for grain size effects in polycrystals. Both the bulk and interfacial stresses are taken to be dissipative in nature and due attention is paid to ensure that positive plastic work is done. It is shown that the mathematical structure of the elasto-plastic strain-gradient theory has similarities to conventional rigid-plasticity theory. Uniqueness and extremum principles are constructed for the solution of boundary value problems.     

On a boundary condition used in Virtual Fields methods

The Virtual Fields Method (VFM) and the Eigenfunction Virtual Fields Method (EVFM) are inverse techniques for estimating constitutive properties from full-field experimental data. In these, a set of virtual fields is used in the Principle of Virtual Work (PVW) to yield a system of algebraic equations for the unknown material parameters. In a typical experiment, one does not know the distribution of tractions over the external surface of the specimen, but the total force is generally measured. In order to still enable evaluation of the external virtual work integral that appears in PVW, in all the work to date on Virtual Fields methods, the virtual displacements are restricted to be uniform over the portion of the exterior surface where tractions are prescribed so that the external virtual work is simply the inner product of the known total force vector and the uniform value of the chosen virtual displacement vector. In this work, we show that this constraint can be relaxed to obtain a more flexible version of EVFM. The proposed modification is used to obtain orthotropic elastic constants from a simulated unnotched Iosipescu test, and is shown to yield tighter estimates than previously obtained wherein the boundary virtual displacements were constrained to be uniform. This approach, which is novel to Virtual Fields methods, allows us to include domains in the interior of the specimen and therefore, results in an EVFM formulation capable of dealing with material heterogeneity, missing data and discontinuities in specimen geometry.     

A new type of Maxwell stress in soft materials due to quantum mechanical-elasticity coupling

All dielectrics deform when subjected to an electric field. This behavior is attributed to the so-called Maxwell stress and the origins of this phenomenon can be traced to geometric deformation nonlinearities. In particular, the deformation is large when the dielectric is elastically soft (e.g. elastomer) and negligible for most “hard” materials. In this work, we develop a theoretical framework which shows that a striking analog of the electrostatic Maxwell stress also exists in the context of quantum mechanical-elasticity coupling. The newly derived quantum-elastic Maxwell stress is found to be significant for soft nanoscale structures (such as the DNA) and underscores a fresh perspective on the mechanics and physics of polarons. We discuss potential applications of the concept for soft nano-actuators and sensors and the relevance for the interpretation of opto-electronic properties.     

A second strain gradient elasticity theory with second velocity gradient inertia – Part II: Dynamic behavior

This paper is the sequel of a companion Part I paper devoted to the constitutive equations and to the quasi-static behavior of a second strain gradient material model with second velocity gradient inertia. In the present Part II paper, a multi-cell homogenization procedure (developed in the Part I paper) is applied to a nonhomogeneous body modelled as a simple material cell system, in conjunction with the principle of virtual work (PVW) for inertial actions (i.e. momenta and inertia forces), which at the macro-scale level takes on the typical format as for a second velocity gradient inertia material model. The latter (macro-scale) PVW is used to determine the equilibrium equations relating the (ordinary, double and triple) generalized momenta to the inertia forces. As a consequence of the surface effects, the latter inertia forces include (ordinary) inertia body forces within the bulk material, as well as (ordinary and double) inertia surface tractions on the boundary layer and (ordinary) inertia line tractions on the edge line rod; they all depend on the acceleration in a nonstandard way, but the classical laws are recovered in the case of no higher order inertia. The classical linear and angular momentum theorems are extended to the present context of second velocity gradient inertia, showing that the extended theorems—used in conjunction with the Cauchy traction theorem—lead to the local force and moment (stress symmetry) motion equations, just like for a classical continuum. A gradient elasticity theory is proposed, whereby the dynamic evolution problem for assigned initial and boundary conditions is shown to admit a Hamilton-type variational principle; the uniqueness of the solution is also discussed. A few simple applications to wave propagation and dispersion problems are presented. The paper indicates the correct way to describe the inertia forces in the presence of higher order inertia; it extends and improves previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137]. Overall conclusions are drawn at the end of the paper.     

A mathematical basis for strain-gradient plasticity theory. Part II: Tensorial plastic multiplier

A phenomenological, flow theory version of gradient plasticity for isotropic and anisotropic solids is constructed along the lines of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain-gradient plasticity. J. Mech. Phys. Solids 52, 1379-1406]. Both energetic and dissipative stresses are considered in order to develop a kinematic hardening theory, which in the absence of gradient terms reduces to conventional J₂ flow theory with kinematic hardening. The dissipative stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition with associated plastic flow. The theory includes interfacial terms: elastic energy is stored and plastic work is dissipated at internal interfaces, and a yield surface is postulated for the work-conjugate stress quantities at the interface. Uniqueness and extremum principles are constructed for the solution of boundary value problems, for both the rate-dependent and the rate-independent cases. In the absence of strain gradient and interface effects, the minimum principles reduce to the classical extremum principles for a kinematically hardening elasto-plastic solid. A rigid-hardening version of the theory is also stated and the resulting theory gives rise to an extension to the classical limit load theorems. This has particular appeal as previous trial fields for limit load analysis can be used to generate immediately size-dependent bounds on limit loads.     

Magnetic field and magneto elastic stress in an infinite plate containing an elliptical hole with an edge crack under uniform electric current

Two-dimensional solutions of the electric current, magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack under uniform electric current. Using a rational mapping function, the each solution is obtained as a closed form. The linear constitutive equation is used for the magnetic field and the stress analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate which raises a plane stress state for a thin plate and the deformation of the plate thickness. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, electric current, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that the stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solving the present magneto elastic stress problem, dislocation and rotation terms appear, which makes the present problem complicate. Solutions of the magneto elastic stress are nonlinear for the direction of electric current. Stresses in the direction of the plate thickness are caused and the solution is also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length and the electric current direction.     

Nonproportional loading effects on elastic-plastic behavior based on stress resultants for thin plates of strain hardening materials

A stress resultant constitutive law in rate form is constructed for power-law hardening materials. The change of plate thickness is considered in the constitutive law. The elastic-plastic behavior of a plate element based on the stress resultant constitutive law under uniaxial combined tension and bending is determined under a limited number of nonproportional and unloading paths. The results based on the stress resultant constitutive law and the through-the-thickness integration method are compared within the context of both the small-strain and finite deformation approaches. The results indicate that the selection of the normalized equivalent stress resultant and the corresponding work-conjugate normalized equivalent generalized strain is appropriate for describing the hardening behavior in the stress resultant space. However, the hardening rule in a power law form must be modified for low hardening materials at large plastic deformation when finite deformation effects are considered.     

Constitutive equations for an electroactive polymer

Ionic electroactive polymers can be used as sensors or actuators. For this purpose, a thin film of polyelectrolyte is saturated with a solvent and sandwiched between two platinum electrodes. The solvent causes a complete dissociation of the polymer and the release of small cations. The application of an electric field across the thickness results in the bending of the strip and vice versa. The material is modeled by a two-phase continuous medium. The solid phase, constituted by the polymer backbone inlaid with anions, is depicted as a deformable porous media. The liquid phase is composed of the free cations and the solvent (usually water). We used a coarse grain model. The conservation laws of this system have been established in a previous work. The entropy balance law and the thermodynamic relations are first written for each phase and then for the complete material using a statistical average technique and the material derivative concept. One deduces the entropy production. Identifying generalized forces and fluxes provides the constitutive equations of the whole system: the stress–strain relations which satisfy a Kelvin–Voigt model, generalized Fourier’s and Darcy’s laws and the Nernst–Planck equation.     

Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity

We propose an approach to the definition and analysis of material instabilities in rate-independent standard dissipative solids at finite strains based on finite-step-sized incremental energy minimization principles. The point of departure is a recently developed constitutive minimization principle for standard dissipative materials that optimizes a generalized incremental work function with respect to the internal variables. In an incremental setting at finite time steps this variational problem defines a quasi-hyperelastic stress potential. The existence of this potential allows to be recast a typical incremental boundary-value problem of quasi-static inelasticity into a principle of minimum incremental energy for standard dissipative solids. Mathematical existence theorems for sufficiently regular minimizers then induce a definition of the material stability of the inelastic material response in terms of the sequentially weakly lower semicontinuity of the incremental variational functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of the quasi-convexity or the rank-one convexity of the incremental stress potential. This global definition includes the classical local Hadamard condition but is more general. Furthermore, the variational setting opens up the possibility to analyze the post-critical development of deformation microstructures in non-stable inelastic materials based on energy relaxation methods. We outline minimization principles of quasi- and rank-one convexifications of incremental non-convex stress potentials for standard dissipative solids. The general concepts are applied to the analysis of evolving deformation microstructures in single-slip plasticity. For this canonical model problem, we outline details of the constitutive variational formulation and develop numerical and semi-analytical solution methods for a first-level rank-one convexification. A set of representative numerical investigations analyze the development of deformation microstructures in the form of rank-one laminates in single slip plasticity for homogeneous macro-deformation modes as well as inhomogeneous macroscopic boundary-value problems. The well-posedness of the relaxed variational formulation is indicated by an independence of typical finite element solutions on the mesh-size.     

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part II – the formulation of elastoplastic coupling at large strain

The two key phenomena occurring in the process of ceramic powder compaction are the progressive gain in cohesion and the increase of elastic stiffness, both related to the development of plastic deformation. The latter effect is an example of ‘elastoplastic coupling’, in which the plastic flow affects the elastic properties of the material, and has been so far considered only within the framework of small strain assumption (mainly to describe elastic degradation in rock-like materials), so that it remains completely unexplored for large strain. Therefore, a new finite strain generalization of elastoplastic coupling theory is given to describe the mechanical behaviour of materials evolving from a granular to a dense state.The correct account of elastoplastic coupling and of the specific characteristics of materials evolving from a loose to a dense state (for instance, nonlinear – or linear – dependence of the elastic part of the deformation on the forming pressure in the granular – or dense – state) makes the use of existing large strain formulations awkward, if even possible. Therefore, first, we have resorted to a very general setting allowing general transformations between work-conjugate stress and strain measures; second, we have introduced the multiplicative decomposition of the deformation gradient and, third, employing isotropy and hyperelasticity of elastic response, we have obtained a relation between the Biot stress and its ‘total’ and ‘plastic’ work-conjugate strain measure. This is a key result, since it allows an immediate achievement of the rate elastoplastic constitutive equations. Knowing the general form of these equations, all the specific laws governing the behaviour of ceramic powders are finally introduced as generalizations of the small strain counterparts given in Part I of this paper.     

Thermodynamic restrictions and wave features of a non-linear Maxwell model

A non-linear rate-type constitutive equation, established by Rajagopal, provides a generalization of the Maxwell fluid. This note embodies such a constitutive equation within the scheme of materials with internal variables thus allowing also for solids with both dissipative and thermoelastic mechanisms. The compatibility with the second law of thermodynamics, expressed by the Clausius–Duhem inequality, is examined and the restrictions on the evolution equations are determined. Next the propagation condition of discontinuity waves is derived, for shock waves and acceleration waves, by regarding the body as a definite conductor. Infinitesimal shock waves and acceleration waves show similar effects. The effective acoustic tensor proves to be the sum of a thermoelastic tensor and a tensor arising from the rate-type equation.     

Conservation laws of an electro-active polymer

Ionic electro-active polymer is an active material consisting in a polyelectrolyte (for example Nafion). Such material is usually used as thin film sandwiched between two platinum electrodes. The polymer undergoes large bending motions when an electric field is applied across the thickness. Conversely, a voltage can be detected between both electrodes when the polymer is suddenly bent. The solvent-saturated polymer is fully dissociated, releasing cations of small size. We used a continuous medium approach. The material is modelled by the coexistence of two phases; it can be considered as a porous medium where the deformable solid phase is the polymer backbone with fixed anions; the electrolyte phase is made of a solvent (usually water) with free cations. The microscale conservation laws of mass, linear momentum and energy and the Maxwell’s equations are first written for each phase. The physical quantities linked to the interfaces are deduced. The use of an average technique applied to the two-phase medium finally leads to an Eulerian formulation of the conservation laws of the complete material. Macroscale equations relative to each phase provide exchanges through the interfaces. An analysis of the balance equations of kinetic, potential and internal energy highlights the phenomena responsible of the conversion of one kind of energy into another, especially the dissipative ones : viscous frictions and Joule effect.     

Magneto-elastic stress in a thin infinite plate with an elliptical hole under uniform magnetic field

Two-dimensional magnetic field and magneto-elastic stress solutions are presented for a magnetic material of a thin infinite plate with an elliptical hole under uniform magnetic field. The linear constitutive equation is used for the magnetic field and the stress analyses. The magneto-elastic stress is analyzed using Maxwell stress since only Maxwell stress is caused as a body force according to the electro magneto theory. Except the approximation of the plane stress state in which the plate is thin, no further assumption is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress is completely satisfied without any linear assumptions on the boundary. First, magnetic field and stress for soft ferromagnetic material is analyzed and then those for paramagnetic and diamagnetic materials are analyzed. It is stated that the stress components are the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields are different each other in the plates. If the analysis of magnetic field of paramagnetic materials is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material. Shear deflection as well as stress in the direction of the plate thickness arises and the solutions are also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived.     

Continuous symmetries and constitutive laws of dissipative materials within a thermodynamic framework of relaxation: Part I: Formal aspects

An analysis of the continuous symmetries of the constitutive laws of inelastic materials written within a thermodynamical framework of relaxation is performed. This framework relies on the generalization of Gibb’s relationship outside the equilibrium of a uniform system, and the use of the fluctuation theory to model the material dissipation due to its internal microstructure change [Cunat, C., 2001. The DNLR approach and relaxation phenomena. Part I – Historical account and DNLR formalism. Mech. Time-depend. Mater. 5, 39–65]. The approach leads to a viscoelastic like formulation for small deformations, and changes gradually for finite strains towards elastoviscoplasticity (with or without damage) via a dependence of characteristic times with the loading path, in a way similar to the endochronic approach developed by Valanis [Valanis, K.C., 1975. On the fundations of the endochronic theory of viscoplasticity. Arch. Mech. 27, 857–868]. The present thermodynamic framework has been previously applied to elastoviscoplastic materials under cyclic and non-proportional loadings [Dieng, L., Abdul-Latif, A., Haboussi, M., Cunat, C., 2005b. Cyclic plasticity modeling with the distribution of non-linear relaxations approach. Int. J. Plasticity 21, 353–379]. The constitutive laws split into the state laws relating intensive variables (thermodynamics forces) to extensive-like variables, and the complementary evolution laws of the internal variables associated to the dissipative mechanisms. An interpretation of a non-equilibrium thermodynamic approach of irreversible processes in terms of an extremum principle is proposed, associated to a Lagrangian functional. It is shown that one possible choice for the Lagrangian kernel is the material derivative of the internal energy density, augmented by a complementary term that accounts for the evolution laws of the internal variables. Interpreting the material behavior during the non-equilibrium evolution as the Euler–Lagrange equations of the resulting action integral, a differential condition expressing both the local and variational symmetries encapsulated into the Lagrangian formulation is formulated. It is further shown that both symmetry conditions are fully equivalent along the optimal path corresponding to the satisfaction of the constitutive laws. In terms of both practical and methodological aspects, the predictive nature of the symmetry analysis is highlighted, as a systematic tool for the exploitation of the constitutive response. Its performance and utility are exemplified by the construction of a time–temperature equivalence principle for a dry viscous polymer (PA66); the calculated shift factor is shown to well agree with the empirical shift factor given by Williams–Landel–Ferry (WLF) expression. A systematic interpretation of the calculated symmetry groups of the constitutive laws in terms of master curves for various plastic and viscoplastic materials shall be presented in a forthcoming contribution.     

The fundamental role of nonlocal and local balance laws of material forces in finite elastoplasticity and damage mechanics

In this paper the fundamental role of independent balance laws of material forces acting on dislocations and microdefects is shown. They enable a thermodynamically consistent formulation of dissipative deformation processes of continua with dislocation motion and defect evolution in the material space on meso- and microlevel.The balance laws of material forces together with the classical balance laws of physical forces and couples, first and second laws of thermodynamics for physical and material space and general constitutive equations are the basis to develop a thermodynamically consistent framework of nonlocal finite elastoplasticity and brittle and ductile damage.It is shown that a weakly-nonlocal formulation of the balance laws of material forces leads to gradient theories, where local theories are obtained, if all gradient contributions are assumed to be small. In this case the local balance laws of material forces together with the constitutive equations represent evolution laws of the material forces. In the classical approach of internal variables they are assumed from the outset with the result that there is a large number of different propositions in the literature.The well-known splitting test of a circular cylinder of concrete is simulated numerically, where the process of deformation in the physical space and defect and plastic evolution in the material space is represented.     

Identification of orthotropic elastic constants using the Eigenfunction Virtual Fields Method

The Virtual Fields Method (VFM – Pierron and Grediac, 2012), an inverse method based on the principle of virtual work (PVW), is being increasingly used to estimate mechanical properties of materials from full-field deformations obtained from techniques such as Digital Image Correlation, moiré and speckle interferometry and grid methods. By making specific choices for virtual fields (VFs) in PVW, one obtains a system of algebraic equations, which is then solved for the unknown material constants. Recently, a new variant of VFM, known as the Eigenfunction Virtual Fields Method (EVFM) has been proposed (Subramanian, 2013). In EVFM, principal components of the measured (i.e. true) strain fields are used to systematically generate VFs. We extend EVFM to orthotropic elastic materials in this work, and estimate the relevant material parameters from full-field strain data generated from a finite-element model of an unnotched Iosipescu test. Varying levels of Gaussian white noise are added to the synthetic strain data to evaluate the sensitivity of EVFM to input noise. It is observed that for low to moderate noise, the material properties estimated by the proposed method are relatively insensitive to noise. However, when noise levels are high, the proposed method yields large variance in some of the computed properties when compared to the state-of-the-art optimized piecewise continuous VFM (Toussaint et al., 2006; Pierron and Grediac, 2012). Some of the large variance in properties estimated from noisy data using EVFM is traced to the sensitivity of the third dominant eigenfunction and modifications to the proposed method to address this issue are suggested.