#### apl. Prof. Dr.-Ing. Joachim Willi Otto Bluhm

Institute of Mechanics

University of Duisburg-Essen

##### Contact

- joachim[dot]bluhm[at]uni-due[dot]de
- 0201/183 2660
- personal website

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**Theoretical formulation and computational aspects of a two-scale homogenization scheme combining the TPM and FE2 method for poro-elastic fluid-saturated porous media**

Ricken, T. and Schröder, J. and Bluhm, J. and Bartel, F.*International Journal of Solids and Structures*241 (2022)The focus of this investigation lies on the development of a two-scale homogenization scheme for poro-elastic fluid-saturated porous media. For this purpose, the general concepts of the Theory of Porous Media (TPM) are combined with the FE2 method. After an introduction of the basics of TPM, the weak forms for the macroscopic and the microscopic scale will be formulated and the averaged macroscopic tangent moduli considering the microscale will be derived. Additionally, the formulation of lower level boundary conditions, which refer to the quantities that will be transmitted from the macro- to the microscale, in strict compliance with the Hill–Mandel homogeneity condition, is derived. Finally, a numerical example will be presented, pointing out the gained features of the methodology. © 2021view abstract 10.1016/j.ijsolstr.2021.111412 **Modeling of freezing processes of ice floes within the framework of the TPM**

Schwarz, A. and Bluhm, J. and Schröder, J.*Acta Mechanica*231 (2020)Sea ice is floating ice which is formed by the freezing of ocean water in the polar regions of the Earth, i.e., the Arctic and the Antarctic. Thus, a closed smooth ice surface can be formed consisting of these ice configurations. In recent years, the simulation of sea ice evolution, especially for the use in climate models, became more important, see, for instance, Danilov et al. (Geosci Model Dev 8:1747–1761, 2015) and the references therein. In the present paper, a coupled macroscopic model based on the Theory of Porous Media is introduced in view of the finite element simulation of the coalescence of ice floes due to freezing in calm sea and weather conditions. Attention is paid to the description of the temperature development, the determination of energy, enthalpy, specific heat and mass exchange between water and ice as well as volume deformations due to ice formation during freezing. The main idea is based on a theoretically motivated evolution equation for the phase transition of ice and water, which guarantees the thermodynamical consistency. Numerical examples show that the simplified model is indeed capable of simulating the temperature development and energetic effects during phase change. © 2020, Springer-Verlag GmbH Austria, part of Springer Nature.view abstract 10.1007/s00707-020-02686-8 **Elasto-plastic phase-field model of hydraulic fracture in saturated binary porous media**

Pise, M. and Bluhm, J. and Schröder, J.*International Journal for Multiscale Computational Engineering*17 (2019)In many fields of engineering, especially in geo sciences and rock mechanics, the theoretical and numerical modeling of hydraulic fracturing of porous materials plays an important role. Hydraulic fracturing is a well-known technology in which porous materials are fractured by a pressurized liquid. The process involves the pressure injection of a fracking fluid (primarily water, often enriched with filling materials and thickening agents) and accompanied by crack nucle-ation and propagation, as well as mass transport. This article presents a macroscopic model based on the Theory of Porous Media (TPM). For simplification, an incompressible binary model consisting of the solid and liquid phases is used. The development of the damage of the elastic-plastic solid phase is controlled by an evolution equation, which corresponds to known diffusive phase-field models within a continuum mechanical framework. A numerical example shows that the simplified model is indeed capable of simulating hydraulic fracturing of porous media. © 2019 by Begell House, Inc.view abstract 10.1615/IntJMultCompEng.2019027066 **Electro-chemical aspects of IPMCs within the framework of the theory of porous media**

Leichsenring, P. and Serdas, S. and Wallmersperger, T. and Bluhm, J. and Schröder, J.*Smart Materials and Structures*26 (2017)Ionomeric polymer-metal composites (IPMCs) consist of an ionomer with bound anionic groups and mobile counterions. They are plated with noble impermeable metal cover layers. By application of an electric voltage, a transport of the mobile ions towards the respective electrode occurs. Due to local electrostatic and ionic forces, a local deformation of the IPMC can be observed. Therefore IPMCs are promising candidates for electrochemical transducers. In the present research, the chemo-electro-mechanical behavior of IPMCs is described within the framework of the theory of porous media. First, the field equations are derived with respect to the second law of thermodynamics. Second, a reduced set of equations for the chemo-electric behavior is formulated and discretized by applying the finite element method. In the numerical investigations a parametric study of the time and space dependent behavior is carried out in order to quantify the influence of different material compositions. Based on this study, the characteristic response of IPMC to the application of an electric voltage can be predicted. Concluding, the obtained computational framework is an excellent tool for the design of electrochemical transducers. © 2017 IOP Publishing Ltd.view abstract 10.1088/1361-665X/aa590e **Continuum mechanical description of an extrinsic and autonomous self-healing material based on the theory of porous media**

Specht, S. and Bluhm, J. and Schröder, J.*Advances in Polymer Science*273 (2016)Polymers and polymeric composites are used in many engineering applications, but these materials can spontaneously lose structural integrity as a result of microdamage caused by stress peaks during service. This internal microdamage is hard to detect and nearly impossible to repair. To extend the lifetime of such materials and save maintenance costs, self-healing mechanisms can be applied that are able to repair internal microdamage during the usual service load. This can be realized, for example, by incorporating microcapsules filled with monomer and dispersed catalysts into the polymeric matrix material. If a crack occurs, the monomer flows into the crack, reacts with the catalysts, and closes the crack. This contribution focuses on the development of a thermodynamically consistent constitutive model that is able to describe the damage and healing behavior of a microcapsule-based self-healing material. The material under investigation is an epoxy matrix with microencapsulated dicyclopentadiene healing agents and dispersed Grubbs’ catalysts. The simulation of such a multiphase material is numerically very expensive if the microstructure is to be completely resolved. To overcome this, a homogenization technique can be applied to decrease the computational costs of the simulation. Here, the theoretical framework is based on the theory of porous media, which is a macroscopic continuum mechanical homogenization approach. The developed five-phase model consists of solid matrix material with dispersed catalysts, liquid healing agents, solidified healed material, and gas phase. A discontinuous damage model is used for the description of the damage behavior, and healing is simulated by a phase transition between the liquid-like healing agents and the solidified healed material. Applicability of the developed model is shown by means of numerical simulations of the global damage and healing behavior of a tapered double cantilever beam, as well as simulations of the flow behavior of the healing agents at the microscale. © Springer International Publishing Switzerland 2016.view abstract 10.1007/12_2015_338 **Theoretical framework of modeling of ionic EAPs within the Theory of Porous Media**

Bluhm, J. and Serdas, S. and Schröder, J.*Archive of Applied Mechanics*86 (2016)A thermo-electromechanical formulation for the description of ionic electroactive polymers is derived within the framework of the Theory of Porous Media. The model consists of an electrically charged porous solid saturated with an ionic solution. The saturated porous medium is assumed to be incompressible. Different constituents following different kinematic paths are considered such as solid, fluid, anions, cations and free charges. The electromechanical and the electrodynamic field equations are discussed. Based on the second law of thermodynamics, a consistent model is developed. With respect to the closure problem of the model, the needed constitutive relations and evolution equations are presented. © 2016, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00419-015-1110-8 **Modeling of self-healing effects in polymeric composites**

Bluhm, J. and Specht, S. and Schröder, J.*Archive of Applied Mechanics*85 (2015)Polymers and polymer composites are used in many engineering applications, but they can loose a high rate of stiffness and strength due to internal micro cracks/damages during their lifetime cycle. These damages are very hard to detect and nearly impossible to repair. To avoid failure due to such damages, a self-healing system is considered where microencapsulated healing agents and catalysts are embedded in the polymer matrix. For the numerical simulation of such a self-healing material, a thermodynamically consistent multiphase model, based on the Theory of Porous Media, is developed in this contribution. The different phases of the model are the solid matrix material with embedded catalysts, the liquid healing agents, the solid healed material and the gas phase, which represents the volume fraction of the micro cracks in the model. For the description of the healing mechanism, a mass exchange between the liquid healing agents and the solid healed material, in consideration of the change of the aggregate state, is introduced, which depends on the local concentration of catalysts in the polymer matrix. The applicability of the developed model is shown by means of numerical test simulations of a tapered double cantilever beam. © 2014, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00419-014-0946-7 **Concentration driven phase transitions in multiphase porous media with application to methane oxidation in landfill cover layers**

Ricken, T. and Sindern, A. and Bluhm, J. and Widmann, R. and Denecke, M. and Gehrke, T. and Schmidt, T.C.*ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik*94 (2014)This study focuses on a formulation within the theory of porus media for continuum multicomponent modeling of bacterial driven methane oxidation in a porous landfill cover layer which consists of a porous solid matrix (soil and bacteria) saturated by a liquid (water) and gas phase. The solid, liquid, and gas phases are considered as immiscible constituents occupying spatially their individual volume fraction. However, the gas phase is composed of three components, namely methane (CH4), oxygen (O2), and carbon dioxide (CO2). A thermodynamically consistent constitutive framework is derived by evaluating the entropy inequality on the basis of Coleman and Noll [8], which results in constitutive relations for the constituent stress and pressure states, interaction forces, and mass exchanges. For the final set of process variables of the derived finite element calculation concept we consider the displacement of the solid matrix, the partial hydrostatic gas pressure and osmotic concentration pressures. For simplicity, we assume a constant water pressure and isothermal conditions. The theoretical formulations are implemented in the finite element code FEAP by Taylor [29]. A new set of experimental batch tests has been created that considers the model parameter dependencies on the process variables; these tests are used to evaluate the nonlinear model parameter set. After presenting the framework developed for the finite element calculation concept, including the representation of the governing weak formulations, we examine representative numerical examples. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/zamm.201200198 **Energetic effects during phase transition under freezing-thawing load in porous media - a continuum multiphase description and FE-simulation**

Bluhm, J. and Bloßfeld, W.M. and Ricken, T.*ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik*94 (2014)In civil engineering, the frost durability of partly liquid saturated porous media under freezing and thawing conditions is a point of great discussion. Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. The volume increase in space and time corresponds to the moving freezing front inside the porous solid. In this paper, a macroscopic model based on the Theory of Porous Media (TPM) is presented which describes energetic effects of freezing and thawing processes. For simplification a ternary model consisting of the phases solid, ice and liquid is used. Attention is paid to the description of the temperature development, the determination of energy, enthalpy and mass supply as well as volume deformations due to ice formation during a freezing and thawing cycle. For the detection of energetic effects regarding the characterization and control of phase transition of water and ice, a physically motivated evolution equation for the mass exchange between ice and liquid is presented. Comparing experimental data with numerical examples shows that the simplified model is indeed capable of simulating the temperature development and energetic effects during phase change. In civil engineering, the frost durability of partly liquid saturated porous media under freezing/thawing conditions is a point of great discussion. Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. The volume increase corresponds to the moving freezing front inside the porous solid. A macroscopic model based on the Theory of Porous Media (TPM) is presented which describes energetic effects of freezing/thawing processes. To simplify a ternary model consisting of the phases solid, ice and liquid is used which describes the temperature development, the determination of energy, enthalpy and mass supply as well as volume deformations due to ice formation during a freezing and thawing cycle. Energetic effects regarding the phase transition are modelled by a physically motivated evolution equation for the mass exchange between ice and liquid. Comparing experimental data with numerical examples shows that the simplified model is capable of simulating the temperature development and energetic effects during phase change. Copyright © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/zamm.201200154 **Ice formation in porous media**

Bluhm, J. and Ricken, T. and Bloßfeld, M.*Lecture Notes in Applied and Computational Mechanics*59 LNACM (2011)Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. The volume increase in space and time corresponds to the moving freezing front inside the porous solid. In this contribution, a macroscopic model based on the Theory of Porous Media (TPM) is presented toward the description of freezing and thawing processes in saturated porous media. Therefore, a quadruple model consisting of the constituents solid, ice, liquid and gas is used. Attention is paid to the description of capillary suction, liquid- and gas pressure on the surrounding surfaces, volume deformations due to ice formation, temperature distribution as well as influence of heat of fusion under thermal loading. For detection of energetic effects regarding the control of phase transition of water and ice, a physically motivated evolution equation for the mass exchange based on the local divergence of the heat flux is used. Numerical examples are presented to the applications of the model. © 2011 Springer-Verlag Berlin Heidelberg.view abstract 10.1007/978-3-642-22738-7_8 **Modeling fluid saturated porous media under frost attack**

Ricken, T. and Bluhm, J.*GAMM Mitteilungen*33 (2010)Freezing and thawing are important processes in civil engineering. On the one hand frost damage of porous building materials like road pavements and concrete in regions with periodical freezing is well known. On the other hand, artificial freezing techniques are widely used, e.g. for tunneling in non-cohesive soils and other underground constructions as well as for the protection of excavation and compartmentalization of contaminated tracts. Ice formation in porous media results from a coupled heat and mass transport and is accompanied by the ice expansion. The volume increase in space and time is assigned to the moving freezing front inside the porous solid. In this paper, a macroscopic ternary model is presented within the framework of the Theory of Porous Media (TPM) in view of the description of phase transition. For the mass exchange between ice and water an evolution equation based on the local balance of the heat flux vector is used. Examples illustrate the application of the model for saturated porous solids under thermal loading. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/gamm.201010004 **Remodeling and growth of living tissue: A multiphase theory**

Ricken, T. and Bluhm, J.*Archive of Applied Mechanics*80 (2010)A continuum triphase model (i.e., a solid filled with fluid containing nutrients) based on the theory of porous media (TPM) is proposed for the phenomenological description of growth and remodeling phenomena in isotropic and transversely isotropic biological tissues. In this study, particular attention is paid on the description of the mass exchange during the stress-strain- and/or nutrient-driven phase transition of the nutrient phase to the solid phase. In order to define thermodynamically consistent constitutive relations, the entropy inequality of the mixture is evaluated in analogy to Coleman and Noll (Arch Ration Mech Anal 13:167-178, 1963). Thereby, the choose of independent process variables is motivated by the fact that the resulting phenomenological description derives both a physical interpretability and a comprehensive description of the coupled processes. Based on the developed thermodynamical restrictions constitutive relations for stress, mass supply and permeability are proposed. The resulting system of equation is implemented into a mixed finite element scheme. Thus, we obtain a coupled calculation concept to determine the solid motion, inner pressure as well as the solid, fluid and nutrient volume fractions. © 2009 Springer-Verlag.view abstract 10.1007/s00419-009-0383-1

#### elasticity and plasticity theory

#### finite element method

#### mechanical properties

#### phase transformations

#### porous materials