This contribution presents a homogenization approach for the computationally efficient modeling of two-dimensional solid bodies that interact through structured surfaces. Instead of resolving the detailed microscale geometry, the method introduces a finite-thickness surrogate layer that captures the essential mechanical response of the structured contact zone. The proposed formulation is based on the description of microscale quantities, such as (normal) gap and relative sliding velocity, which are transferred to the macroscale using internal volumetric variables. The elastic behavior of the surrogate layer is identified through a mean-field homogenization approach that accounts for the non-linear dependency of stress response on the relative slip. As a result, the anisotropic and history-dependent behavior of the interface can be captured within the model. The proposed method is validated using two benchmark problems: a vertical stack of two structured blocks and a knurled interference fit. In both cases, the finite-thickness layer accurately reproduces the mechanical response of the fully resolved model, i.e., stick–slip transitions, hysteresis, and partial interface separation. The good agreement with the reference solutions, combined with a significant reduction in computational cost, demonstrates the potential of the method for efficient multi-scale interface modeling.

Hypodermic needles for the injection of fluids generally have annular cross-sections. The preference for round shapes is, besides manufacturing reasons, due to the ideal ratio between the cross-section circumference and area, which leads to optimal flow properties. However, when additional parameters are considered in the design, other forms can prove to be better: the healing of the wound induced by the needle is highly individual and effects such as skin tension can make non-round shapes more suitable. When reimagining the shape of hypodermic needles, relevant factors such as the buckling of the needle during piercing/injection, and the fluid flow through the needle must be considered. The current work demonstrates an approach for the identification of different geometries within the framework of multi-objective optimization. Conflicting aspects of microfluidic flow, buckling under piercing/injection conditions, as well as (simplified) tissue damage suggest the notion of Pareto-optimal solutions. Equipped with more advanced models, the proposed take on hypodermic needles can lead to a better standard geometry and even concepts of ad-hoc manufacturing of individually optimal geometries through additive methods in the context of personalized medicine.

A core task of industrial digital twins is the identification of parameters from measured process data. Inspired by such an industrial application, we consider a cantilever beam with Hookean material behavior subjected to a transverse follower force and focus on the inverse problem of determining Young’s modulus from known (measured) bending force and bending angle. Within the framework of Reissner’s beam theory, accounting for shear deformation, moderate strains, and large displacements, the relationship between applied force, bending angle and Young’s modulus is formulated. The inverse problem is discussed for the shear rigid Kirchhoff and the shear deformable Reissner formulation, including their corresponding theories for small displacements. While for the Kirchhoff case an analytical inverse solution is possible, the Reissner formulation requires numerical solution methods. The proposed identification strategies are verified through finite element simulations and a parameter study is conducted to quantify the accuracy across a broad range of geometric configurations. The comparison between the Reissner model, the Kirchhoff model and their linearizations highlights the influence of shear and geometric assumptions on the achievable identification accuracy. The outcome of this work is a verified solution framework for the identification of the Young’s modulus together with a clear characterization of the several variants of beam theory.

Following a research strategy that dates back to Ernst Mach, we study similarities and analogies that may exist for linear elastic and viscoelastic bodies when dynamically loaded by forces and eigenstrains. Particularly, we treat equivalence of displacements in two problems with different sets of loadings and possibly different constitutive behavior. Special interest is given to the presence of singular waves. In the present first part of our treatise, we deal with generalized body force analogies, while displacement control, i.e. shape control and displacement tracking will be treated in the forthcoming second part of our study. Besides a review on previous work, with an emphasis on unification of works of our group, some novel results in the form of generalizations are presented, and perspectives for prospective future work are given.

Shaft-hub connections are an important feature of many machines, with knurled interference fits (KIFs) being a novel connection method. While conventional shaft-hub connections use either friction or form closure, KIFs represent a combination offering the advantages of both concepts. To investigate the behavior of such a connection, a computational model of a knurled interference fit is developed. Assuming rotational symmetry, modeling a periodic unit cell of the setup is sufficient. This assumption does not only greatly alleviate the demands on computing power, but also allows to analyze the hysteretic behavior of the connection in a phenomenological way. Parameter studies indicate changes in the transmissible loads and stiffness of the connection when varying geometric dimensions such as tooth height and angle, material and contact characteristics or applied loads. Notably, the hysteresis behavior of KIFs differs significantly from conventional connections like interference fits with smooth shaft-hub interfaces, particularly due to coupling between the radial and circumferential directions due to the tooth angle. Moreover, the simulations demonstrate that a certain stick-slip behavior may also occur in frictionless settings as a result of the geometry.

The finite element simulation of thin viscoelastic membranes is a challenging problem. On the one hand, geometric dimensions of the membrane are adverse to classical finite element formulations, which require the thickness of the structure to be resolved. On the other hand, both material and geometric nonlinearities need to be considered in the large-strain regime. In the present paper, a Kirchhoff–Love shell formulation including thickness deformation is deduced for incompressible viscoelastic membranes at finite strain. For this formulation, a mixed shell element with low regularity is proposed to discretize a viscoelastic shell formulation based on a multiplicative split of the deformation gradient. The formulation is tested against full three-dimensional formulations in a variety of examples. Excellent accordance of results in quasi-static as well as dynamic simulations is observed.

The modeling and simulation of thin dielectric viscoelastic structures experiencing large deformations presents significant challenges. In this work, we introduce a shell formulation that captures the essential kinematic and constitutive features of these structures under transient electric loading. The formulation includes elastic and viscous membrane strains, curvature, as well as thickness deformation and the variation of the electric field through the thickness. A thermodynamically consistent model is developed and discretized using low-regularity shell elements within a variational framework. The proposed method is validated through several computational examples. The results demonstrate a high level of accuracy in capturing the transient large-strain behavior of dielectric elastomer shells even for very coarse discretizations.

In this article the bending behavior of sandwich plates with a Functionally Graded (FG) core with Boltzmann-type viscoelastic behavior integrated with piezoelectric layers at the top and bottom surfaces, is reported under electrothermomechanical loading within the framework of three-dimensional elasticity theory. For various boundary conditions, differential quadrature method (DQM) is used to obtain a semi-analytical solution along the in-plane coordinate axes. The present formulation is validated by comparing numerical results on bending, and stress with those published in the literature. A complete parametric study is performed to reveal the effect of different parameters such as length to thickness ratio, temperature gradient, mechanical loading, piezo-to-core thickness ratio, relaxation time constant, and applied voltage on the bending behavior of sandwich rectangular plates. The results show that an increase in the relaxation time leads to a decrease in the deflection of the plate; Also due to the applied voltage, the deflection curve along-the-thickness is nonlinear for the FG viscoelastic core; but it was linear in both piezoelectric layers.

So-called electro-active polymers are not only capable of undergoing very large deformations, but they also exhibit different electromechanical coupling effects due to their dielectric or electrostrictive characteristics. In the present paper the sub-class of dielectric elastomers is studied; in particular, we focus on thin dielectric elastomer shells, which undergo large deformations under an applied electric field. We develop a framework for the modeling and subsequent efficient simulation of thin structures made from incompressible dielectric elastomers. A thermodynamically consistent phenomenological continuum model based on the free energy density is adapted imposing the kinematic assumptions of Kirchhoff-Love shells in combination with including the thickness deformation and the electric field as independent unknown fields. To make the formulation accessible to finite element simulations, existing non-linear elastic shell mixed finite elements are extended to the present hyperelastic electromechanically coupled formulation for dielectric elastomer shells. Computational results of the shell theory are compared to three-dimensional results validating the proposed modeling and numerical simulation methodology, and showing high accuracy already for coarse finite element discretizations of the shell.

We discuss nonlinear modeling of ferroelectric plates as electro-elastic material surfaces assuming the plate as a two-dimensional continuum with five mechanical degrees of freedom for each material point.Constitutive coupling by means of electrostriction, ferroelectric hysteresis and piezoelectricity are accounted for.For that sake, the augmented free energy is additively decomposed into an elastic part, a dielectric part and an augmentation energy.The elastic part involves an additive decomposition of the plate strain measures into elastic and electrical parts.Here, the electrical parts account for electrostriction and piezoelectricity, where electrostriction is assumed to depend quadratically on the polarization.For the dielectric part, we introduce the internal energy as a function of the polarization and an internal polarization, which allows us to accurately capture nonlinearities, such as saturation.Then, we compute the free energy applying a Legendre transformation, such that the voltage enters the formulation.The augmentation energy accounts for the contribution from vacuum.Finally, we introduce a so-called dissipation function, by means of which irreversible polarization due to domain switching is accounted for.Numerical results computed with the plate model are compared to results using a three-dimensional formulation.

The three-dimensional bending behavior of a viscoelastic functionally graded material (FGM) layer embedded between piezoelectric layers and subjected to an electric field as well as a uniform transverse pressure is studied. An analytical solution is computed for a simply supported viscoelastic smart FGM plate using the state space technique along the thickness direction and a Fourier expansion along the in-plane coordinates. The governing differential equations in the time domain are transformed to the Laplace domain, solved in the Laplace domain, and transformed back to the time domain using the inverse Laplace transform. In the present study, the relaxation modulus of the FGM layer is assumed in the form of a Prony series, and it varies according to the power law in the thickness direction. The validity of the proposed approach is assessed by comparison of the numerical results of the approach with those of published works in the literature. The effects of geometric dimensions, electromechanical loads, and relaxation time constant on the behavior of the smart plate are investigated.

Ferroic materials are characterized by the presence of electric/magnetic dipoles that can be irreversibly re-oriented under sufficiently strong loading.Upon poling, an initially random distribution of dipoles on the microscopic domain scale results in a non-vanishing state of remanent polarization/magnetization on the macroscopic level.The remanent switching of dipoles constitutes the "smartness" in materials and their usage both as sensors and as actuators.Once being poled, material properties are typically assumed to be constant and uni-directional in engineering problems.To extend the operational range of ferroic transducers and open perspectives for novel applications, we seek for an accurate understanding of the evolution of the polarization/magnetization, which requires both physical and geometric non-linearities to be included in the modeling.To describe irreversible changes of the remanent state, we transfer phenomenological concepts and algorithms of associative elasto-plasticity to the field of electro-magneto-mechanics. To describe the dissipative response in a thermodynamically consistent manner, the principle of maximum dissipation is adopted.As in elasto-plasticity, constitutive equations for dissipative internal forces that drive the evolution of the remanent polarization/magnetization then follow as associated flow rules.To simplify the algorithmic treatment, we introduce the notion of dissipation functions, by which the constrained optimization problem is converted into an unconstrained problem, which can be efficiently solved by standard means.The vectors of remanent polarization/magnetization enter the model as additional internal variables.The constitutive response is governed by thermodynamic potentials, i.e., the free energy and dissipation functions.We derive a (mixed) variational formulation, identify appropriate function spaces for the dependent fields involved, and introduce a finite-element discretization.Representative numerical examples demonstrate the efficacy of the framework in electro-magneto-mechanically coupled problems.

Due to the high importance of viscoelastic materials in modern industrial applications, besides the intensive popularity of piezoelectric smart structures, analyzing their thermoelastic response in extreme temperature conditions inevitably becomes very important. Accordingly, this research explores the thermoviscoelastic response of sandwich plates made of a functionally-graded Boltzmann viscoelastic core and two surrounding piezoelectric face-layers subjected to electrothermal load in the platform of three-dimensional elasticity theory. The relaxation modulus of the FG viscoelastic layer across the thickness follows the power law model. the plate's governing equations are expressed in the Laplace domain to handle mathematical complications corresponding to the sandwich plate with a viscoelastic core. Then, the state-space method, combined with Fourier expansion, is utilized to extract the plate response precisely. Finally, the obtained solution is converted to the time domain using the inverse Laplace technique. Verification of the present formulation is compared with those reported in the published papers. Finally, the influences of plate dimension, temperature gradient, and relaxation time constant on the bending response of the above-mentioned sandwich plate are examined. As an interesting finding, it is revealed that increasing the length-to-thickness ratio leads to a decrease in deflections and an increase in stresses.

The present paper is concerned with static shape control of sub-domains of material bodies. Eigenstrains or their corresponding actuation stresses are used as the actuation applied only in the sub-domain. Two problems are of particular interest, strain tracking and displacement tracking of the sub-domain. Exact solutions for the two tracking problems are derived and numerical results computed with the Finite Element method are presented for a plane stress problem for validation.

A one-dimensional (1D) analytic example for dynamic displacement tracking in linear viscoelastic solids is presented. Displacement tracking is achieved by actuation stresses that are produced by eigenstrains. Our 1D example deals with a viscoelastic half-space under the action of a suddenly applied tensile surface traction. The surface traction induces a uni-axial shock wave that travels into the half-space. Our tracking goal is to add to the applied surface traction a transient spatial distribution of actuation stresses such that the total displacement of the viscoelastic half-space coincides with the shock wave produced by the surface traction in a purely elastic half-space. We particularly consider a half-space made of a viscoelastic Maxwell-type material. Analytic solutions to this tracking problem are derived by means of the symbolic computer code MAPLE. The 1D solution presented below exemplifies a formal 3D solution derived earlier by the present authors for linear viscoelastic solids that are described by Boltzmann hereditary laws. In the latter formal solution, no reference was made to shock waves. Our present solution demonstrates its validity also in the presence of singular wave fronts. Moreover, in our example, we show that, as was also indicated in our earlier work, the actuation stress can be split into two parts, one of them producing no stresses, and the other no displacements in two properly enlarged problems.

The present contribution is devoted to a special equivalence problem concerning the linear dynamic theories of elasticity and viscoelasticity. We seek for distributions of eigenstrain-induced actuation stresses acting upon a force loaded linear elastic body, such that the resulting displacements are equal to the displacements of a viscoelastic body that is loaded by the same set of forces, but in the absence of actuating stresses. Special emphasis is given to the presence of propagating singular waves. A three-dimensional solution for this equivalence problem is presented first. The presented special solution yields equivalence of stresses in both problems, so that a complete analogy is obtained. The solution then is exemplified for the one-dimensional case of a uni-axial shock wave propagating in a semi-infinite half-space made of a viscoelastic material of the Maxwell-type.