A homogenisation strategy for micromorphic continua based on particle mechanics

Materials with a granular microstructure frequently fail in narrow zones due to strain localisation. Examplarily, one may look at the shear-zone development in dry sand during bi- and triaxial loading, where grains in the shear-zone exhibit large displacements and rotations. Furthermore, localisation is also observed in materials, where the microstructure consists of grains and a binding material, such as for example metal-casting moulds. Here, sand grains are bound together via a polyurethan-based material and macroscopic material failure originates from the deformation and breakage of the binder material. Within a continuum-based modelling approach, these microstructural effects can be accounted for by the consideration of an additional microcontinuum at each material point of the macroscopic body. These extended continuum theories, such as the micromorphic continua and its micropolar and microstrain sub-formulations, assume a characteristic microcontinuum deformation on a lower scale and have been successfully applied in the field of granular media. Exemplarily, in the framework of a micropolar continua, it is possible to contact forces to stresses and couple stresses via an appropriate homogenisation technique. This method includes the introduction of a Representative Elementary Volume (REV) on the mesoscale situated between the particle and the continuum scale. In this contribution, a homogenisation strategy based on a particle-centre-based REV definition is presented that is generally valid for micromorphic and micropolar continua. Therefore, a grain-binder microstructure is investigated, where particle rotations contribute to the micropolar part, while binder deformations yield the additional macromorphic character. Numerical examples are given, where results from discrete-element simulations are locally averaged and show the individual activation of the microcontinuum characteristics in the localised zones. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional reconstruction of M. gastrocnemius contraction

There exist many different approaches investigating the contraction mechanisms of skeletal muscles. Thereby, the mechanical behavior, such as force generation in association with kinematic and microstructure, play an important role in modeling of muscle behavior. Besides the mechanical behaviour, the validation of muscle models requires the geometrical environment, too. The geometry of a muscle can be divided into macrostructure, existing of aponeurosis-tendon-complex (ATC) and muscle tissue (MT), as well as the fascicle architecture, representing the microstructure of the MT. In this study, the macrostructure of the isolated M. gastrocnemius was observed during isometric contraction by using three-dimensional optical measurement systems in combination with mechanical measurement techniques. The surface deformation was reconstructed at specific force and length relationships and further the muscle tissue, aponeurosis, and tendon were distinguished, building up a macroscopic geometrical dataset. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of bone elasticity based on tissue‐independent (‘universal’) phase properties

As candidates for tissue‐independent phase properties of cortical and trabecular bone we consider (i) hydroxyapatite, (ii) collagen, (iii) ultrastructural water and non‐collagenous proteins, and (iv) marrow (water) filling the Haversian canals and the intertrabecular space. From experiments reported in the literature, we assign stiffness properties to these phases (experimental set I). On the basis of these phase definitions, we develop, within the framework of continuum micromechanics, a two step homogenization procedure: (i) At a length scale of 100 – 200 nm, hydroxyapatite (HA) crystals build up a crystal foam ('polycrystal'), and water and non‐collagenous organic matter fill the intercrystalline space (homogenization step I); (ii) At the ultrastructural scale of mineralized tissues, i.e. 5 to 10 microns, collagen assemblies composed of collagen molecules are embedded into the crystal foam, acting mechanically as cylindrical templates. At an enlarged material scale of 5 to 10 mm, the second homogenization step also accommodates the micropore space as cylindrical pore inclusions (Haversian and Volkmann canals, inter‐trabecular space), that are suitable for both trabecular and cortical bone. The input of this micromechanical model are tissue‐specific volume fractions of HA, collagen, and of the micropore space. The output are tissue‐specific ultrastructural and microstructural (=macroscopic=apparent) elasticity tensors. A second independent experimental set (composition data and experimental stiffness values) is employed to validate the proposed model. We report a a good agreement between model predictions and experimentally determined macroscopic stiffness values. The validation suggests that hydroxyapatite, collagen, and water are tissue‐independent phases, which define, through their mechanical interaction, the elasticity of all bones, whether cortical or trabecular.     

A macroscopic consitutive model on induced anisotropy for polymers with weighting functions

The alignment of polymer chains is a well known microstructural evolution effect due to straining of polymers. This has a drastic influence on the macroscopic properties of the initially isotropic material, such as a pronounced strength in the loading direction of stretched films. Experiments on strain induced anisotropy at room temperature are analyzed by optical measurements. For modeling the effect of strain induced anisotropy a macroscopic constitutive model is presented. As a key idea, weighting functions are introduced to represent a strain-softening/hardening-effect to account for induced anisotropy. These functions represent the ratio between the total strain rate and a structural tensor. In this way, material parameters are used as a sum of weighted direction related quantities. In the finite element examples we simulate the cold-forming of amorphous thermoplastic films below the glass transition temperature subjected to different re-loading directions. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Passive muscle behaviour – experimental and numerical investigations

The material behaviour of skeletal muscles can be decomposed into two parts: an active part, describing the contractile mechanisms, and a passive one, characterising the passive components such as the connective tissue. Computational models are used to support the understanding of complex mechanism inside a muscle. In the present work, we focus on the three-dimensional passive tissue behaviour from the experimental as well as modelling point of view. Therefore, quasi-static experiments have been performed on specimens with regular geometry. By using a three-dimensional optical measurement system the shape of the specimens has been reconstructed at different deformation states. On the modelling side a hyperelastic model with transversal isotropic fibre orientation has been used to describe non-linear stress responses. The model has been validated by performing analyses for different fibre orientations. In summary, it figures out that the proposed modelling approach is able to reflect the experimental results in a satisfying manner. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiscale Modeling for the Simulation of Damage Processes at Refractory Materials under Thermal Shock

A brittle damage model based on multiscale considerations and homogenisation procedures is presented. Cell models are developed as RVE including different microstructural features. The material laws themselves are formulated on the continuum level. Local failure occurs if the damage variable reaches a critical value. For simple configurations of the microstructure, the relation between stress, strain und temperature is derived from analytical considerations. In order to properly model the thermo-mechanical coupling, the temperature-dependence of material constants is taken into account. Fracture and damage mechanical approaches are combined using different techniques. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the macroscopic material behaviour of textile reinforced concrete undertensile loading

This paper concerns with the development of the macroscopic material behaviour of textile reinforced concrete (TRC) using an analytical approach. Therefore the heterogeneous structure of TRC is modelled on the mesoscopic level. The overall material behaviour on the macroscopic level is obtained by means of the homogenisation technique. The analytical approach is based on the micro mechanical solution for a single inclusion according to Eshelby . In extension of this solution for multidirectional reinforced concrete an effective field approximation is used. This approach considers the interactions between the different orientated rovings and the micro cracks in an average sense. For the mechanical modelling of the bond behaviour between roving and matrix after initiating of the macro cracking a slip based bond model with a multiple linear shear stress-slip relation is used. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiphasic modelling of human brain tissue for intracranial drug-infusion studies

A direct intracranial infusion of a therapeutic solution into the extra-vascular space of human brain tissue is a promising medical application for the effective treatment of malignant brain tumours [1]. The advantage of this method, compared to an intra-vascular medication, is the targeted delivery with the circumvention of the blood-brain barrier (BBB), which prohibits the passing of therapeutic macro-molecules across the vascular walls into the brain parenchyma. The prediction of the resulting therapeutical distribution by a numerical simulation is challenging, since the spreading is affected by the complex nature of living brain tissue. For this purpose, a macroscopic continuum-mechanical model is established within the Theory of Porous Media (TPM), proceeding from a homogenisation of the underlying micro-structure [5]. The ternary four-component model consists of an elastically deformable solid skeleton (composed of tissue cells and vascular walls), which is perfused by two mobile but separated liquid phases, the blood and the overall interstitial fluid (treated as a real two-component mixture of the liquid solvent and the dissolved therapeutic solute). The strongly coupled solid-liquid-transport problem is simultaneously approximated in all primary unknowns using mixed finite elements (uppc-formulation) and consequently solved in a monolithic manner with an implicit time-integration scheme. This numerical investigation allows the computational study of several circumstances influencing the irregular distribution of infused drugs, as observed in clinical studies. Therefore, the microstructural perfusion characteristics in the extra-cellular space of the white-matter tracts are considered by a spatial diversification of the anisotropic permeability tensors, provided by Diffusion Tensor Imaging (DTI). Furthermore, Magnetic Resonance Angiography (MRA) enables the in vivo location of blood vessels within the brain tissue. Finally, the selection of appropriate material parameters has a crucial influence on the drug distribution profile and further occurring effects beyond. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Localisation in granular media: Particle approach,homogenisation and continuum modelling

The individual motion of grains in granular material has a strong influence on the macroscopic material behaviour, which is in particular the case for the phenomena of strain localisation in shear zones and justifies the need for techniques that incorporate a micro-macro transition. In this contribution, granular media are investigated in three steps. Firstly, a microscopic particle-based modelling is set up, where individual grains are considered as rigid uncrushable particles while their motion is obtained through Newton's equations of state. The inter-particle contact forces are thereby determined via constitutive contact-force formulations, which have to account for the envisaged material behaviour. The second step is the homogenisation of the obtained particle's displacements and contact forces through a particle-centre-based strategy towards continuum quantities. Therefore, Representative Elementary Volumes (REV) are introduced on the mesoscale and the specific construction of the REV boundary leads to the understanding of granular media as a micropolar continuum. Finally, in order to verify the homogenisation strategy, a continuum based micropolar model is applied to model localisation phenomena and a comparative study of the results is carried out in a qualitative way. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical homogenization of foam-like structures based on the FE2-approach

The computation of foam–like structures is still a topic of research. There are two basic approaches: the microscopic model where the foam–like structure is entirely resolved by a discretization (e.g. with Timoshenko beams) on a micro level, and the macroscopic approach which is based on a higher order continuum theory. A combination of both of them is the FE²-approach where the mechanical parameters of the macroscopic scale are obtained by solving a Dirichlet boundary value problem for a representative microstructure at each integration point. In this contribution, we present a two–dimensional geometrically nonlinear FE²-framework of first order (classical continuum theories on both scales) where the microstructures are discretized by continuum finite elements based on the p-version. The p-version elements have turned out to be highly efficient for many problems in structural mechanics. Further, a continuum–based approach affords two additional advantages: the formulation of geometrical and material nonlinearities is easier, and there is no problem when dealing with thicker beam–like structures. In our numerical example we will investigate a simple macroscopic shear test. Both the macroscopic load displacement behavior and the evolving anisotropy of the microstructures will be discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigations on an elastic micropolar continuum model for large deformations

The mechanical behaviour of cellular materials is dominated by the influence of the microtopology. If the characteristic microstructural length scale becomes comparable to the macroscopical length scale, classical homogenisation strategies fail and have to be replaced by extended approaches. In the present contribution, we present a micropolar continuum model for large deformations introducing the microrotation as an additional degree of freedom. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling internal erosion of cohesionless soils using a microstructural parameter

Internal erosion processes are of vital importance for the risk management of geotechnical structures as well as for the understanding of the macroscopic mechanical and hydraulic properties of the subsoil in various man-made constructions. Here, a 4-phase continuum model is presented and numerically applied to illustrative applications. The role of interfacial area and related microstructural parameters is addressed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bridging scales: An attempt to incorporate cellular responses within a three-dimensional FEM model of active muscle contraction

A mathematical model of the cellular responses of skeletal muscles has been integrated within a three-dimensional biomechanical Finite Element (FEM) model. The FEM model is based on a tri-cubic Hermite Finite Element discretisation of the governing equations of finite elasticity theory and a transversely isotropic constitutive law. To incorporate the cellular information, homogenised values of key physiological parameters, e.g. the pre- and post-power stroke concentration of crossbridge attachments, are computed at the Gauss points of the FEMintegration scheme. These values are then used to modify the stress tensor in such a way that it resembles the contractile response. The advantages of such an improved three-dimensional FEM model are far reaching. These models can be used, for example, to investigate and study local muscle contraction, muscle recruitment patterns, force generation, or fatigue response of skeletal muscles. As an illustrative example, one twitch of the tibialis anterior, in which 25% of the muscle fibres are excited by a nerve stimulus, is simulated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of Microstructure Evolution in Two‐Phase Materials using Generalized Driving Forces

In two‐phase materials, such as nickel base alloys or zirconia, the macroscopic material response is strongly influenced by the morphology of the microstructure. In nickel alloys microstructural rearrangements due to diffusion occur at elevated temperatures. A continuum mechanical model is presented that takes elastic inhomogeneity and eigenstrains together with an interface energy into account. The driving force for the diffusion process is identified and used to simulate morphology evolution and equilibrium shapes. The numerical simulation is done with a 3D Boundary Element Method applicable to anisotropic materials.     

Remarks on coupled multi-scale simulations and high performance computation

To describe material behaviour more precisely than nowadays, tomorrows simulation software will include multi-scale homogenisation techniques. State of the art are various scale bridging strategies, like FE-ODE, FE-Phasefield or FE²-Method. We are dealing with the FE²-Method, see [3], [4], which gives us the possibility to take the geometric, discrete micro-structure of a material into account. Furthermore, it allows to integrate enhanced continuum mechanical models. Here, we are researching on two-scale approach with poro-mechanical coupling based on the Theory of Porous Media (TPM), for more details see [1] or [2]. The framework is demanding and need a lot of computational effort. In order to receive industrial recognition, it is necessary to decrease the computation runtime. One way to go is definitely using high performance cluster. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of micro-stress evolution in dual phase polycrystals

Diffraction methods gain much attention in nondestructive residual stress analysis. While the determination of macroscopic residual stresses is of main interest, the presence of microscopic residual stresses arising from microstructural characteristics of the material can influence the analysis of the acquired data. The residual stress measurements by neutron diffraction on IN718 pancake forgings are analyzed in this work. We present a simple mechanical model supporting the hypothesis that the phase average of the microscopic residual stress accumulated during the forging process is anisotropic causing anisotropy of the macro stress free reference lattice parameter. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)