The present contribution proposes a thermodynamically consistent model for the simulation of the ductile damage. The model couples the phase field method of fracture to the Armstrong-Frederick plasticity model with kinematic hardening. The latter is particularly suitable for simulating the material behavior under a cyclic load. The model relies on the minimum principle of the dissipation potential. However, the application of this approach is challenging since potentials of coupled methods are defined in different spaces: The dissipation potential of the phase field model is expressed in terms of rates of internal variables, whereas the Armstrong-Frederick model proposes a formulation depending on thermodynamic forces. For this reason, a unique formulation requires the Legendre transformation of one of the potentials. The present work performs the transformation of the Armstrong-Frederick potential, such that final formulation is only expressed in the space of rates of internal variables. With the assumption for the free energy and the joint dissipation potential at hand, the derivation of evolution equations is straightforward. The application of the model is illustrated by selected numerical examples studying the material response for different load cases and sample geometries. The paper provides a comparison with the experimental results as well. © 2021 Elsevier Ltd

The interior of a eukaryotic cell is a highly complex composite material which consists of water, structural scaffoldings, organelles, and various biomolecular solutes. All these components serve as obstacles that impede the motion of vesicles. Hence, it is hypothesized that any alteration of the cytoskeletal network may directly impact or even disrupt the vesicle transport. A disruption of the vesicle-mediate cell transport is thought to contribute to several severe diseases and disorders, such as diabetes, Parkinson's and Alzheimer's disease, emphasizing the clinical relevance. To address the outlined objective, a multiscale finite element model of the diffusive vesicle transport is proposed on the basis of the concept of homogenization, owed to the complexity of the cytoskeletal network. In order to study the microscopic effects of specific nanoscopic actin filament network alterations onto the vesicle transport, a parametrized three-dimensional geometrical model of the actin filament network was generated on the basis of experimentally observed filament densities and network geometries in an adenocarcinomic human alveolar basal epithelial cell. Numerical analyzes of the obtained effective diffusion properties within two-dimensional sampling domains of the whole cell model revealed that the computed homogenized diffusion coefficients can be predicted statistically accurate by a simple two-parameter power law as soon as the inaccessible area fraction, due to the obstacle geometries and the finite size of the vesicles, is known. This relationship, in turn, leads to a massive reduction in computation time and allows to study the impact of a variety of different cytoskeletal alterations onto the vesicle transport. Hence, the numerical simulations predicted a 35% increase in transport time due to a uniformly distributed four-fold increase of the total filament amount. On the other hand, a hypothetically reduced expression of filament cross-linking proteins led to sparser filament networks and, thus, a speed up of the vesicle transport. © 2021 Public Library of Science. All rights reserved.

The present study focuses on the receptor driven endocytosis typical of viral entry into a cell. A locally increased density of receptors at the time of contact between the cell and the virus is necessary in this case. The virus is considered as a substrate with fixed receptors on its surface, whereas the receptors of the host cell are free to move over its membrane, allowing a local change in their concentration. In the contact zone the membrane inflects and forms an envelope around the virus. The created vesicle imports its cargo into the cell. This paper assumes the diffusion equation accompanied by boundary conditions requiring the conservation of binders to describe the process. Moreover, it introduces a condition defining the energy balance at the front of the adhesion zone. The latter yields the upper limit for the size of virus which can be engulfed by the cell membrane. The described moving boundary problem in terms of the binder density and the velocity of the adhesion front is well posed and numerically solved by using the finite difference method. The illustrative examples have been chosen to show the influence of the process parameters on the initiation and the duration of the process. © 2021 Elsevier Ltd

The present contribution focuses on the thermodynamically consistent mechanical modeling of the strain-induced crystallization in unfilled polymers. This phenomenon is of particular importance for the mechanical properties of polymers as well as for their manufacturing and the application. The model developed uses the principle of the minimum of dissipation potential and assumes two internal variables: the deformations due to crystallization and the regularity of the network. In addition to the dissipation potential necessary for the derivation of evolution equations, the well-established Arruda-Boyce model is chosen to depict the elastic behavior of the polymer. Two special features of the model are the evolution direction depending on the stress state and the distinction of crystallization during the loading and unloading phase. The model has been implemented into the finite element method and applied for numerical simulation of the growth and shrinkage of the crystal regions during a cyclic tension test for samples with different initial configurations. The concept enables the visualization of the microstructure evolution, yielding information that is still inaccessible by experimental techniques. © 2020 Elsevier Ltd

The present contribution deals with the thermomechanical modeling of the strain-induced crystallization in unfilled polymers. This phenomenon significantly influences mechanical and thermal properties of polymers and has to be taken into consideration when planning manufacturing processes as well as applications of the final product. In order to simultaneously capture both kinds of effects, the model proposed starts by introducing a triple decomposition of the deformation gradient and furthermore uses thermodynamic framework for material modeling based on the Coleman–Noll procedure and minimum principle of the dissipation potential, which requires suitable assumptions for the Helmholtz free energy and the dissipation potential. The chosen setup yields evolution equations which are able to simulate the formation and the degradation of crystalline regions accompanied by the temperature change during a cyclic tensile test. The boundary value problem corresponding to the described process includes the balance of linear momentum and balance of energy and serves as a basis for the numerical implementation within an FEM code. The paper closes with the numerical examples showing the microstructure evolution and temperature distribution for different material samples. © 2020 by the authors. Licensee MDPI, Basel, Switzerland.

The present contribution focuses on the application of the multiscale finite element method to the modeling of actin networks that are embedded in the cytosol. These cell components are of particular importance with regard to the cell response to external stimuli. The homogenization strategy chosen uses the Hill-Mandel macrohomogeneity condition for bridging 2 scales: the macroscopic scale that is related to the cell level and the microscopic scale related to the representative volume element. For the modeling of filaments, the Holzapfel-Ogden β-model is applied. It provides a relationship between the tensile force and the caused stretches, serves as the basis for the derivation of the stress and elasticity tensors, and enables a novel finite element implementation. The elements with the neo-Hookean constitutive law are applied for the simulation of the cytosol. The results presented corroborate the main advantage of the concept, namely, its flexibility with regard to the choice of the representative volume element as well as of macroscopic tests. The focus is particularly placed on the study of the filament orientation and of its influence on the effective behavior. Copyright © 2018 John Wiley & Sons, Ltd.

Selective laser melting process has already been developed for many metallic materials, including steel, aluminum, and titanium. The quasistatic properties of these materials have been found to be comparable or even better than their conventionally-manufactured counterparts; however, for their reliable applications in operational components, their fatigue behavior plays a critical role, which is dominated by several process-related features, like surface roughness, remnant porosity, microstructure, and residual stresses, which are controlled by the processing features, like imparted energy density to the material, its corresponding solidification behavior, the cooling rate in the process, as well as post-processing treatments. This study investigates the influence of these parameters on the cyclic deformation behavior of selective laser melted as well as hybrid-manufactured aluminum alloys. The corresponding microstructural features and porosity conditions are evaluated for developing correlations between the process conditions to microstructure, the deformation behavior, and the corresponding fatigue lives. From the numerical point of view, damage development with respect to process-induced cyclic deformation behavior is assessed by the phase-field method, which has been identified as an appropriate method for the determination of fatigue life at the respective applied stress levels. Fatigue strength of SLM-processed parts is found better than their cast counterparts, while hybridization has further increased fatigue strength. No effect of test frequency on the fatigue life could be established. © 2018 by the authors.

Our previous work proposes a micromechanical model for dissolution-precipitation creep, an elasto-viscoplastic process supposed to be one of the main reasons for the tectonic motion of earth plates in the subduction zone. While the model in its original form enables the simulation of polycrystals with a limited number of crystals, the topic of the present contribution is its extension to simulating structures on a much larger spatial scale. For this purpose, a homogenization technique known as the multiscale finite element method is used. Here, the behavior of a heterogeneous body is simulated by solving two boundary value problems: one related to the structural level and one related to the representative volume element. The coupling of scales is established by introducing the Hill macrohomogeneity condition requiring the equality of the macropower with the volume average of the micropower. The method allows the simulating of various tasks at both levels. The examples concerned with simulating the tension tests of a macroscopic plate with different types of the microstructure are presented. © 2016 by Begell House, Inc.

In contrast to previous approaches that consider dissolution-precipitation creep as a multi-stage process and only simulate its governing subprocess, the present model treats this phenomenon as a single continuous process. The applied strategy uses the minimum principle of the dissipation potential according to which a Lagrangian consisting of elastic power and dissipation is minimized. Here, the elastic part has a standard form while the assumption for dissipation stipulates the driving forces to be proportional to two kinds of velocities: The material-transport velocity and the boundary-motion velocity. A Lagrange term is included to impose mass conservation. Two ways of solution are proposed. The strong form of the problem is solved analytically for a simple case. The weak form of the problem is used for a finite-element implementation and for simulating more complex cases. © 2015 The Author(s) Published by the Royal Society. All rights reserved.

This contribution aims at achieving two important goals: First, it outlines a numerical inverse homogenization strategy able to recover material parameters of the microstructure by using results of macroscopic tests. Second, it considers parameter identification for viscoelastic heterogeneous materials, which is a step providing the basis for the further extension toward the general treatment of dissipative processes. The approach proposed couples the Levenberg–Marquardt method with the multiscale finite element method. In this combination, the former is a gradient-based optimization strategy used to minimize a merit function while the latter is a numerical homogenization technique needed to solve the direct problem. The specific example studied in the paper deals with the investigation of a composite consisting of a viscoelastic curing polymer and a nonlinear elastic material. It proposes a three-step procedure for the evaluation of its material parameters and discusses the accuracy and the uniqueness of the solution. © 2015, Springer-Verlag Berlin Heidelberg.

This contribution deals with the application of the inverse homogenization method to the determination of geometrical properties of cancellous bone. The approach represents a combination of an extended version of the Marquardt-Levenberg method with the multiscale finite element method. The former belongs to the group of gradient-based optimization strategies, while the latter is a numerical homogenization method, suitable for the modeling of materials with a highly heterogeneous microstructure. The extension of the Marquardt-Levenberg method is concerned with the selection strategy for distinguishing the global minimum from the plethora of local minima. Within the numerical examples, the bone is modeled as a biphasic viscoelastic medium and three different representative volume elements are taken into consideration. Different models enable the simulation of the bone either as a purely isotropic or as a transversally anisotropic medium. Main geometrical properties of trabeculae are determined from data on effective shear modulus but alternative schemes are also possible. © 2012 Springer-Verlag.

The model proposed in this paper is based on the fact that the reflection might have a significant contribution to the attenuation of the acoustic waves propagating through the cancellous bone. The numerical implementation of the mentioned effect is realized by the development of a new representative volume element that includes an infinitesimally thin 'transient' layer on the contact surface of the bone and the marrow. This layer serves to model the amplitude transformation of the incident wave by the transition through media with different acoustic impedances and to take into account the energy loss due to the reflection. The proposed representative volume element together with the multiscale finite element is used to simulate the wave propagation and to evaluate the attenuation coefficient for samples with different effective densities in the dependence of the applied excitation frequency. The obtained numerical values show a very good agreement with the experimental results. Moreover, the model enables the determination of the upper and the lower bound for the attenuation coefficient. © 2012 Springer-Verlag.

In this contribution the properties and application of the multiscale finite element program MSFEAP are presented. This code is developed on basis of coupling the homogenization theory with the finite element method. According to this concept, the investigation of an appropriately chosen representative volume element yields the material parameters needed for the simulation of a macroscopic body. The connection of scales is based on the principle of volume averaging and the Hill-Mandel macrohomogeneity condition. The latter leads to the determination of different types of boundary conditions for the representative volume element and in this way to the postulation of a well-posed problem at this level. The numerical examples presented in the contribution investigate the effective material behavior of microporous media. An isotropic and a transversally anisotropic microstructure are simulated by choosing an appropriate orientation and geometry of the representative volume element in each Gauss point. The results are verified by comparing them with Hashin-Shtrikman's analytic bounds. However, the chosen examples should be understood as simply an illustration of the program application, while its main feature is a modular structure suitable for further development. © 2012 by Begell House, Inc.

The article deals with the contribution of reflection effects to the attenuation properties of cancellous bone. The bone behaviour is simulated by the multiscale finite element method, a numerical homogenization approach, suitable for the modelling of heterogeneous material with a highly oscillatory microstructure. The focus is on the modelling of a novel type of the representative volume element, which apart from the solid framework filled with fluid marrow also includes an infinitesimally thin 'transition' layer at the contact of the phases. The mentioned layer is implemented in order to simulate the amplitude transformation of an incident wave and to take the loss of energy caused by the reflection into account. The given numerical examples consider the simulation of wave propagation through a sample while the excitation frequency is varied. The numerical values are compared with the results which are determined without considering the reflection, in order to point out the contribution of the newly introduced phenomenon. © 2012 Copyright Taylor and Francis Group, LLC.

The contribution considers the application of inverse analysis to the identification of the material parameters of nonlinear composites. For this purpose a combination of the Levenberg-Marquardt method with the multiscale finite element method is used. The first one belongs to the group of gradient-based optimization methods, and the latter is a numerical procedure for modeling heterogeneous materials which is applicable in the case when the ratio of characteristic sizes of the scales tends to zero. Emphasis is placed on the investigation of problems with an increasing number of unknown materials parameters, as well as on the manifestation of the ill-posedness of inverse problems. These effects first occurred in the case of three-phase materials. The illustrative examples are concerned with cases where such a combination of experimental data is used that effects of ill-posedness are alleviated and a unique solution is achieved. © 2012 by Begell House, Inc.

This paper provides a continuum mechanical model for the curing of polymers, including the incompressibility effects arising at the late stages of the process. For this purpose, the free energy density functional is split into a deviatoric and a volumetric part, and a multifield formulation is inserted. An integral formulation of the functional is used to depict the time-dependent material behavior. The model is also coupled with the multiscale finite element method, a numerical approach serving for the modeling of heterogeneous materials with a highly oscillatory microstructure. The effects of the proposed extensions are illustrated on the basis of several numerical examples concerned with the study of the influence of Poisson's ratio on the curing process and the behavior of the microheterogeneous polymers. © 2012 Elsevier Ltd. All rights reserved.

In many cases, the microstructure of composite materials is not known and cannot directly be accessed such that an inverse analysis is necessary for its investigation. This approach requires the implementation of two tools: an optimization method for the minimization of the error problem and a mechanical approach for the solution of the direct problem, i.e. the simulation of composite materials. Our particular choice deals with the combination of the Levenberg-Marquardt method with the multiscale finite element method. The numerical examples are concerned with the investigation of the elastic parameters for two-phase materials. Emphasis is placed on the discussion of convergence and sensitivity with respect to the initial guess. © 2012 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved.

This contribution deals with the modeling of viscoelastic and shrinkage effects accompanying the curing of polymers at multiple length scales. For the modeling of viscous effects, the deformation at the microlevel is decomposed into an elastic and a viscoelastic part, and a corresponding energy density consisting of equilibrium and non-equilibrium parts is proposed. The former is related to the total deformation; it has the form of a convolution integral and depends on the time evolution of the material parameters. The non-equilibrium part depends on the elastic part of deformations only. The material parameters are constant in time, thus an integral form is not necessary. In contrast to the viscous effects, the modeling of shrinkage effects does not require any further extension of the expression for the energy density, but an additional decomposition of the deformation into a shrinkage and a mechanical part. Since the material compressibility is taken into consideration, a multifield formulation is applied for the equilibrium as well as for the non-equilibrium energy part. As a final aspect, the paper includes a study of macroheterogenous polymers for whose modeling the multiscale finite element method is applied. Within this numerical approach, a macroscopic body is treated as a homogeneous body whose effective properties are evaluated on the basis of the simulations which are carried out at the level of the representative volume element. The application of the model proposed is illustrated on the basis of examples studying the influence of individual parameters on the stress state as well as the influence of the volume fraction of different phases at the microscale on the effective material behavior. © 2012 Elsevier Ltd. All rights reserved.