Bernd Kröplin

Coupled chemo-electro-mechanical formulation for ionic polymer gels––numerical and experimental investigations

Abstract Polyelectrolyte gels consist of a network of crosslinked polymers with attached electric charges and a liquid phase. Variations of the chemical milieu surrounding the gel or application of an external electric field lead to a change in the swelling ratio of the gels. This phenomenon may be used for contraction and relaxation of chemo-electro-mechanical actuators and in particular of artificial muscles. In this paper electrically stimulated polymer gels, placed in a solution bath, are investigated. For these gels, we present a volume- and surface-coupled chemo-electro-mechanical multi-field formulation. This formulation consists of a convection–diffusion equation describing the chemical field, a Poisson equation for the electric field and a mechanical field equation. The model is capable to describe the local swelling and deswelling of ionic polymer gels as well as the ion concentrations and the electric potential in the gel and in the solution. A chemo-electric simulation is performed for a gel fiber at a given electric field; the anionic and cationic ion concentrations as well as the electric potential inside and outside the gel are computed for a given number of bound anionic groups. The resulting increase in the concentration differences on the anode side of the anionic gel can be considered as an indicator for a higher swelling ratio of the gel fiber. Mechanical and electrical measurements on anionic polyelectrolyte gels are in good agreement with the numerical results. This demonstrates the validity of the employed numerical model.

THERMO-MECHANICAL BENDING OF FUNCTIONALLY GRADED PLATES

In this work the deformations of a simply supported, functionally graded, rectangular plate subjected to thermo-mechanical loadings are analysed, extending Unified Formulation by Carrera. The governing equations are derived from the Principle of Virtual Displacements accounting for the temperature as an external load only. The required temperature field is not assumed a priori, but determined separately by solving Fouriers equation. Numerical results for temperature, displacement and stress distributions are provided for different volume fractions of the metallic and ceramic constituent as well as for different plate thickness ratios. They correlate very well with three-dimensional solutions given in the literature.

Zig-Zag and interlaminar equilibria effects in large deflection and postbuckling analysis of multilayered plates

Concerning, higher-order shear deformation theories (HSDT) for composite plates, this article presents a numerical investigation on local characteristics (distribution along the plate thickness of in-plane displacement, in-plane normal stress, and transverse shear stress component;) in the large-deflection and postbuckling fields. The von Kdrmdn theory in conjunction with a recent mixed two-dimensional model, elsewhere denoted by the acronym RMZC (Reissner-Mindlin zigzag continuity) is employed. The RAfZC model assumes two independent fields along the plate thickness for displacements and transverse shear stresses, respectively; the displacement model describes the so-called zigzag form for the in-plane components, while the stress field fulfils interlaminar equilibria. Standard displacement formulation is enforced by employing variationalty consistent constitutive equations between stress and displacement unknowns. Finite-element-type approximations are introduced on the plate domain, leading to governin...

An Extension of Reissner Mixed Variational Theorem to Piezoelectric Laminates

This paper presents a technique to consistently extend the Reissner Mixed Variational Theorem (RMVT) to problems involving multiple fields. In particular, a new partially mixed formulation is presented for multilayered structures embedding piezoelectric layers. The formulation satisfies a priorithe interlaminar continuity of the transverse stresses. Following Reissner, the constitutive equations for the coupled linear electromechanical behavior associated to the partially mixed variational statement are derived from a modified internal energy function by means of partial Legendre transformations. As a result, the mechanical displacements, the electric potential, as well as the transverse stresses, can be independently assumed. The resulting three-field formulation has been applied to an established axiomatic modelling technique for multilayered structures. An implementation and assessment of a large number of laminate theories, including “layerwise” and “equivalent single layer” descriptions, is performed. Numerical results are given within a closed-form solution technique for two exemplary case studies, for which an exact three-dimensional solution is available in open literature.

Thermodynamical Modeling of the Electromechanical Behavior of Ionic Polymer Metal Composites

Ionomeric polymer transducers are a class of smart materials which exhibit electromechanical coupling when subjected to low voltage (<5 V) excitation. Generally these materials are soft actuators exhibiting large bending strains (>5%) but correspondingly low force output. The mechanisms producing electromechanical coupling have so far not been completely understood. It is clear from experimental and theoretical investigations that diffusion and migration of ionic species within the polymer are the main cause for electromechanical coupling. For this reason we have developed a thermodynamically based mechanical model — using chemo-electrical inputs — which is able to predict the mechanical output i.e., deformation, bending, etc. for a given applied voltage to the IPMC strip. The chemo-electrical transport model is capable of computing the charge density profile in space and time as well as the current flux for applied electric fields. Based upon thermodynamic laws, the mechanical model has been developed to describe the strain within the material. The mechanical stress in this model is accomplished by two terms of the charge density, a linear and a quadratic one. The linear term represents the volume displacement caused by the charge migration while the quadratic term stands for the electrostatic forces caused by charge imbalances in the material. In this paper, numerical investigations of the electromechanical model as well as displacement measurements have been performed. A comparison of numerical and experimental investigations shows a very good correlation. This confirms the quality and the validity of the developed model.

Discrete element simulation of transverse cracking during the pyrolysis of carbon fibre reinforced plastics to carbon/carbon composites

The fracture behavior of fiber-ceramics like C/C-SiC strongly depends on the initial damage arising during the production process. We study the transverse cracking of the 90{\deg} ply in [0/90]S cross-ply laminates due to the thermochemical degradation of the matrix material during the carbonization process by means of a discrete element method. The crack morphology strongly depends on the fiber-matrix interface properties, the transverse ply thickness as well as on the carbonization process itself. To model the 90{\deg} ply a two-dimensional triangular lattice of springs is constructed where nodes of the lattice represent fibers. Springs with random breaking thresholds model the disordered matrix material and interfaces. The spring-lattice is coupled by interface springs to two rigid bars which capture the two 0{\deg} plies or adjacent sublaminates in the model. Molecular dynamics simulation is used to follow the time evolution of the model system. It was found that under gradual heating of the specimen, after some distributed cracking, segmentation cracks occur in the 90{\deg} ply which then develop into a saturated state where the ply cannot support additional load. The dependence of the micro-structure of damage on the ply thickness and on the disorder in spring properties is also studied. Crack density and porosity of the system are monitored as a function of the temperature and compared to an analytic approach and experiments.

Modeling of Temperature-Sensitive Polyelectrolyte Gels by the Use of the Coupled Chemo-Electro-Mechanical Formulation

Polyelectrolyte gels show adaptive viscoelastic characteristics. In water-based solutions they have enormous swelling capabilities under the influence of different possible stimulation types, such as chemical, electrical, or thermal stimulation. Possible applications for these intelligent materials can be either actuators, e.g., when used as artificial muscles or chemo-electric energy converters, or sensors, e.g., for measuring ion concentrations or pH-values of the solutions. In order to represent this active behavior, a coupled multi-field model is applied. The chemo-electro-mechanical multi-field formulation considers changes of concentrations, electric potential, or strains due to varying (initial) conditions. In the present work, a fully coupled 3-field formulation for polyelectrolyte gels using the Finite Element Method is applied. This formulation consists of a chemical, electrical, and mechanical field equation. The chemical field is described by a partial differential equation (PDE) first order in time and second order in space and includes diffusional, migrational, and convectional effects. The electrical field is represented by the Poisson equation, an elliptical PDE of second order in space. Finally, the mechanical field is described by a PDE of first order derived from the conservation of momentum. Due to the slow processes in time, inertia terms are neglected. The mechanical field is coupled to the chemo-electrical field by a prescribed strain stemming from an osmotic pressure term. A large dependency between the applied temperature and the actual swelling degree of the gel has been proven in experiments. In the present research, the thermal stimulation is investigated using two alternative approaches: The first is a straight forward modeling by modifying the actual temperature in the osmotic pressure term. As this approach does not lead to promising results, as a second alternative temperature-dependent material parameters obtained from experimental measurements are applied. The calibration of the derived simulation results is performed with experimental results from the literature. The Finite Elements introduced for the coupled multi-field formulation contain degrees of freedom for concentrations, electric potential, and mechanical displacements. They are adopted and applied in the commercial software package ABAQUS as a user defined element. For the numerical solution, the Newton-Raphson method in conjunction with the implicit backward Euler time integration scheme is applied.

Coupled chemo-electro-mechanical simulation of polyelectrolyte gels as actuators and sensors

Polyelectrolyte gels are ductile elastic electroactive materials. They consist of a polymer network with charged groups and a liquid phase with mobile ions. Changing the chemical or electric conditions in the gel-surrounding solution leads to a change of the chemo-electro-mechanical state in the gel phase: diffusion and migration of ions and solvent between the gel and solution phases trigger the swelling or shrinkage of the polymer gel. In case of chemical stimulation (change of pH or salt concentration), a swelling ratio of up to 100% may be obtained. Due to this large swelling ratio the gels exhibit excellent actuatoric capabilities. In this paper, a polyelectrolyte gel placed in a solution bath is investigated. The actuatoric and sensoric capabilities are described by a chemo-electro-mechanical model. The chemical field is represented by a convection-migration-diffusion equation while the electric field is described by a quasi-static Laplace equation. For the mechanical field a partial differential equation of first order in time is applied. Inertia effects are neglected due to the relatively slow swelling/shrinkage process. On the one hand, the coupling between the chemo-electrical and the mechanical field is realised by the differential osmotic pressure stemming from the concentration differences between gel and solution. On the other hand, the mechanical deformation influences the concentration of the bound charged groups in the gel. The three fields are solved simultaneously by applying the Newton Raphson method using finite elements in space and finite differences in time. The developed model is applicable for both, hydrogel actuators and sensors. Numerical results of swelling and bending are given for chemically and electrically stimulated polymer gels. In this paper we show the differences between the chemo-electric and the fully coupled chemo-electro-mechanical formulation for polymer gels in different solution baths. The inverse (sensor-) effect is demonstrated by the influence of the mechanical deformation on the gel, which results in a change of the chemical and electrical unknowns in the gel. The validity of the employed numerical model is shown by a comparison of the obtained results with experimental measurements.

Multiscale Modeling of Polymer Gels—Chemo-Electric Model versus Discrete Element Model

Polyelectrolyte gels are a very attractive class of actuation materials with remarkable electronic and mechanical properties with a great similarity to biological contractile tissues. They consist of a polymer network with ionizable groups and a liquid phase with mobile ions. Absorption and delivery of solvent lead to a large change of volume. This mechanism can be triggered by chemical (change of salt concentration or pH of solution surrounding the gel), electrical, thermal or optical stimuli. Due to this capability, these gels can be used as actuators for technical applications, where large swelling and shrinkage is desired. In the present work chemically stimulated polymer gels in a solution bath are investigated. To adequately describe the different complicated phenomena occurring in these gels, they can be modeled on different scales. Therefore, models based on the statistical theory and porous media theory, as well as a coupled multi-field model and a discrete element formulation are derived and employed. A refinement of the different theories from global macroscopic to microscopic are presented in this paper: The statistical theory is a macroscopic theory capable of describing the global swelling or bending, e.g., of a gel film, while the general theory of porous media (TPM) is a macroscopic continuum theory which is based on the theory of mixtures extended by the concept of volume fractions. The TPM is a homogenized model, i.e., all geometrical and physical quantities can be seen as statistical averages of the real quantities. The presented chemo-electro-mechanical multi-field formulation is a mesoscopic theory. It is capable of giving the concentrations and the electric potential in the whole domain. Finally the (micromechanical) discrete element (DE) theory is employed. In this case, the continuum is represented by distributed particles with local interaction relations combined with balance equations for the chemical field. This method is predestined for problems involving large displacements and strains as well as discontinuities. In this paper, the coupled multi-field model and the discrete element model for chemical stimulation of a polymer gel film with and without domain deformation are employed. Based on these results, the presented formulations are compared and conclusions on their applicability in engineering practice are finally drawn.

Micromechanically based continuum damage mechanics material laws for fiber-reinforced ceramics

Abstract A micromechanically based continuum damage mechanics material law is developed for fiber-reinforced ceramics. Its parameters and internal variables are obtained from micromechanical models rather than from macroscopic experiments. Micromechanical models are derived for the damage and failure mechanisms observed in 2D C/C–SiC composite samples loaded by tension, shear and compression. In specimens subjected to tension, the early stage of damage is characterized by transverse cracking. Further load increase induces fiber failure in the longitudinal plies, what leads to fracture. Shear loading initiates cracks oriented at 0°/90° and 45° to the fiber axes, the latter causing fracture. Specimens loaded in compression exhibit catastrophic failure due to microbuckling of fiber bundles. Simulation results show the ability of the developed model to describe the observed failure mechanism in experiments.