Karlsruhe Institute of Technology Institute of Engineering Mechanics University of Paderborn Chair of Engineering Mechanics 27 th International Workshop Research in Mechanics of Composites Bad

Conventional macro mechanical models and closed form estimates are in many cases not sufficient to appropriately predict the stiffness and strength of composite materials. Composite failure occurs as a result of complex microstructural damage mechanisms, which arise simultaneously with other nonlinear effects in the microstructure. In this contribution we propose an approach, where nonlinear material effects caused by progressive damage behavior are captured directly on a finer scale. The microstructural constituents are spatially resolved and modeled explicitly. A periodic boundary value problem on a volume element of the microstructure is solved in the numerical homogenization procedure. Because of the highly complex microstructure a fine regular grid is used instead of a boundary aligned mesh to avoid the mesh generation. The periodic boundary value problem is reformulated into a volume integral equation of the Lippmann-Schwinger type and solved efficiently using Fast Fourier Transforms (FFT). In the work at hand, a glass fiber reinforced composite is considered and a ductile damage model is applied for the polymer matrix. The numerical method is validated with experimental data for the considered thermoplastic composite material. References [1] J. Spahn, H. Andrä, M. Kabel, R. Müller: A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms. Comp. Meth. Mech. Engrg., 268, 871–883 (2014). A new internal variables homogenization scheme for linear viscoelastic materials based on an exact Eshelby interaction law Stéphane Berbenni, Florence Dinzart, and Hafid Sabar 1 Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux, UMR CNRS 7239, Université de Lorraine, 57045 Metz, France 2 Laboratoire de Mécanique Biomécanique Polymères Structures, Ecole Nationale Ingénieurs de Metz, 57078 Metz, France Abstract. A new time-incremental internal variables homogenization scheme for Maxwellian linear viscoelastic heterogeneous materials is proposed. This scheme is based on the exact solution of the ellipsoidal Eshelby inclusion problem obtained in the time domain. In contrast with current existing methods, the effective behavior as well as the evolution laws of the averaged stresses per phase are solved incrementally in the time domain without need to analytical or numerical inverse Laplace-Carson transforms. This is made through a time-differential equation in addition to the more classic strain rate concentration equation. In addition, the new derived interaction law for the Eshelby inclusion problem is provided in a compact matrix form. It is proved that this is an exact formulation for an arbitrary anisotropic ellipsoidal Maxwellian inclusion embedded in an isotropic Maxwellian matrix. In order to show the interest of the present approach, the results are reported and discussed with a Mori-Tanaka homogenization scheme for two-phase composites in comparisons with other exact or approximate methods. A new time-incremental internal variables homogenization scheme for Maxwellian linear viscoelastic heterogeneous materials is proposed. This scheme is based on the exact solution of the ellipsoidal Eshelby inclusion problem obtained in the time domain. In contrast with current existing methods, the effective behavior as well as the evolution laws of the averaged stresses per phase are solved incrementally in the time domain without need to analytical or numerical inverse Laplace-Carson transforms. This is made through a time-differential equation in addition to the more classic strain rate concentration equation. In addition, the new derived interaction law for the Eshelby inclusion problem is provided in a compact matrix form. It is proved that this is an exact formulation for an arbitrary anisotropic ellipsoidal Maxwellian inclusion embedded in an isotropic Maxwellian matrix. In order to show the interest of the present approach, the results are reported and discussed with a Mori-Tanaka homogenization scheme for two-phase composites in comparisons with other exact or approximate methods. Comparison of finite volume and finite element methods for the prediction of process induced residual stresses during resin transfer molding Alexander Bernath and Frank Henning Institute of Vehicle System Technology, Chair of Lightweight Technology, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany Abstract. Resin transfer molding (RTM) shows huge potential for the economical large batch production of structural parts made of high performance composite materials. However, during manufacturing residual stresses arise due to temperature driven expansion or contraction and chemical induced shrinkage of the matrix material. As a consequence, final parts show significant geometrical deviations (warpage) and therefore often violate tolerance requirements. Strategies exist which aim to reduce warpage but proper application usually requires extensive experimental knowledge. Therefore, reliable prediction of deviations by using simulation methods is one promising approach for reducing development times and risks while at the same time avoiding the need for comprehensive experimental studies. Up to now, available simulation tools, which are able to solve this type of problem, are solely based on the finite element method (FEM). This work aims on using the finite volume method (FVM) for this purpose as it offers similar accuracy but higher memory efficiency compared to the FEM and shows good scaling in case of parallel computing [1,2]. Moreover, various open source libraries exist which enable researchers to easily tailor the software to their specific needs [3]. This approach has been followed in this work. Mathematical models were implemented in order to describe the material behavior of both, the matrix itself and the resulting composite. The same has been done for the FEM in order to compare these two discretization techniques. For this purpose both have been applied to generic geometries. References [1] N.A. Fallah, C. Bailey, M. Cross, G.A. Taylor: Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis, Applied Mathematical Modelling, 24, 439–455 (2000). [2] H. Jasak, H.G. Weller: Application of the finite volume method and unstructured meshes to linear elasticity, International Journal for Numerical Methods in Engineering, 48, 267–287 (2000). [3] P. Cardiff, A. Karac, A. Ivankovic: A Large Strain Finite Volume Method for Orthotropic Bodies with GeneralMaterial Orientations,ComputerMethods in Applied Mechanics and Engineering, 268, 318–335 (2013). Resin transfer molding (RTM) shows huge potential for the economical large batch production of structural parts made of high performance composite materials. However, during manufacturing residual stresses arise due to temperature driven expansion or contraction and chemical induced shrinkage of the matrix material. As a consequence, final parts show significant geometrical deviations (warpage) and therefore often violate tolerance requirements. Strategies exist which aim to reduce warpage but proper application usually requires extensive experimental knowledge. Therefore, reliable prediction of deviations by using simulation methods is one promising approach for reducing development times and risks while at the same time avoiding the need for comprehensive experimental studies. Up to now, available simulation tools, which are able to solve this type of problem, are solely based on the finite element method (FEM). This work aims on using the finite volume method (FVM) for this purpose as it offers similar accuracy but higher memory efficiency compared to the FEM and shows good scaling in case of parallel computing [1,2]. Moreover, various open source libraries exist which enable researchers to easily tailor the software to their specific needs [3]. This approach has been followed in this work. Mathematical models were implemented in order to describe the material behavior of both, the matrix itself and the resulting composite. The same has been done for the FEM in order to compare these two discretization techniques. For this purpose both have been applied to generic geometries. References [1] N.A. Fallah, C. Bailey, M. Cross, G.A. Taylor: Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis, Applied Mathematical Modelling, 24, 439–455 (2000). [2] H. Jasak, H.G. Weller: Application of the finite volume method and unstructured meshes to linear elasticity, International Journal for Numerical Methods in Engineering, 48, 267–287 (2000). [3] P. Cardiff, A. Karac, A. Ivankovic: A Large Strain Finite Volume Method for Orthotropic Bodies with GeneralMaterial Orientations,ComputerMethods in Applied Mechanics and Engineering, 268, 318–335 (2013). μCT-based characterization of the fibrous microstructure in Al(OH)3-filled SMC Benjamin Bertram and Kay André Weidenmann Institute for Applied Materials – WK, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany Abstract. The sheet molding compound (SMC) process is applied in the automotive industry to form a discontinuous fiber-reinforced polymer. The complex microstructure of SMC materials encompasses pores, filler particles and fibers, whose random orientation is caused by the flow and disaggregation of fiber bundles during molding and is likely affected by the large variety of SMC process parameters in use. Micro X-ray computed tomography (μCT) is widely used to investigate 3D-fiber orientation distributions; however, CaCO3 filler particles inhibit μCT because it has a linear X-ray attenuation that is similarly high as that of glass fibers.Al(OH)3 is therefore investigated as an alternative filler, which allows for tomography in absorption-mode lab-based μCT systems. Homomorphic filtering and a cylindrical μCT specimen shape are used to further optimize the fiber-matrix-contrast and to compute orientation distributions encoded as 4th order orientation tensors introduced by Advani and Tucker [1], which can be directly used for micro-mechanical modeling. The fiber volume content is a further important input parameter for the micromechanical modeling of the effective modulus. An incineration method is adapted to the specific oxidation of the Al(OH)3-filled matrix. Incineration is time-consuming and limited in spatial resolution, but can be used as a ground truth for the regression of the fiber content from 2D-xray image features, allowing for a non-destructive application to sections of the SMC plate. The tensile stiffness and strength of the Al(OH)3-filled SMC are similar to those of the corresponding CaCO3-filled versions, while facilitating the X-ray-based analysis of the fibrous microstructure and volume content. References [1] S. G. Advani, C. L. T. Iii: The Use of Tensors to Describe and Predict Fiber Orientation in Short Fiber Composites, J. Rheol., 31,8 751-784 (1987). The sheet molding compound (SMC) process is applied in the automotive industry to form a discontinuous fiber-reinforced polymer. The complex microstructure of SMC materials encompasses pores, filler particles and fibers, whose random orientation is caused by the flow and disaggregation of fiber bundles during molding and is likely affected by the large variety of SMC process parameters in use. Micro X-ray computed tomography (μCT) is widely used to investigate 3D-fiber orientation distributions; however, CaCO3 filler particles inhibit μCT because it has a linear X-ray attenuation that is similarly high as that of glass fibers.Al(OH)3 is therefore investigated as an alternative filler, which allows for tomography in absorption-mode lab-based μCT systems. Homomorphic filtering and a cylindrical μCT specimen shape are used to further optimize the fiber-matrix-contrast and to compute orientation distributions encoded as 4th order orientation tensors introduced by Advani and Tucker [1], which can be directly used for micro-mechanical modeling. The fiber volume content is a further important input parameter for the micromechanical modeling of the effective modulus. An incineration method is adapted to the specific oxidation of the Al(OH)3-filled matrix. Incineration is time-consuming and limited in spatial resolution, but can be used as a ground truth for the regression of the fiber content from 2D-xray image features, allowing for a non-destructive application to sections of the SMC plate. The tensile stiffness and strength of the Al(OH)3-filled SMC are similar to those of the corresponding CaCO3-filled versions, while facilitating the X-ray-based analysis of the fibrous microstructure and volume content. References [1] S. G. Advani, C. L. T. Iii: The Use of Tensors to Describe and Predict Fiber Orientation in Short Fiber Composites, J. Rheol., 31,8 751-784 (1987). Micro structure based modeling of temperature dependent stiffness of SMC Barthel Brylka, Viktor Müller, Thomas Böhlke, Martin Hohberg, and Frank Henning 1 Institute of Engineering Mechanics, Chair for Continuum Mechanics Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany 2 Institute of Vehicle System Technology (FAST), Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany Abstract. In many applications, sheet moulding compounds (SMC) are used if class A surface is required. Especially the possibility of forming complex structures leads to an increase of applications in the last few years. Compression moulding as manufacturing process for SMC leads to manufacturing related anisotropic as well as inhomogeneous material properties. These material properties like stiffness, thermal extension, damage and fracture are strongly anisotropic as well as temperature dependent and related to the local fiber orientation distribution which is influenced by flow of the melt. Dimensioning strategies need to take, therefore, the manufacturing process into account by simulation of the compression moulding process. Especially in automotive applications, material models for these inhomogeneous and anisotropic as well as temperature and strain-rate dependent materials are needed. For this reason, homogenization schemes are used based on the micro structure information from mould filling simulation as well as computer tomography, to predict the effective anisotropic, temperature dependent stiffness of SMC. The temperature dependent stiffness of the resin-filler composite matrix as well as the effective SMC has been measured by dynamic mechanical analysis (DMA). A comparison of the effective stiffness predicted by Mori-Tanaka approximation [1] based on mould filling simulation results as well as computer tomography will be presented. These results will be compared to the effective anisotropic stiffness and their scatter investigated in DMAmeasurements. References [1] Y. Benveniste: A new approach to the application of Mori-Tanaka’s theory in composite materials, Mech. Mat., 6, 147–157 (1987). [2] Z. Jendli, F. Meraghni, J. Fitoussi, D. Baptiste: Multi-scales modelling of dynamic behaviour for discontinuous fibre SMC composites, Comp. Sci. Technol., 69, 97–103 (2009). In many applications, sheet moulding compounds (SMC) are used if class A surface is required. Especially the possibility of forming complex structures leads to an increase of applications in the last few years. Compression moulding as manufacturing process for SMC leads to manufacturing related anisotropic as well as inhomogeneous material properties. These material properties like stiffness, thermal extension, damage and fracture are strongly anisotropic as well as temperature dependent and related to the local fiber orientation distribution which is influenced by flow of the melt. Dimensioning strategies need to take, therefore, the manufacturing process into account by simulation of the compression moulding process. Especially in automotive applications, material models for these inhomogeneous and anisotropic as well as temperature and strain-rate dependent materials are needed. For this reason, homogenization schemes are used based on the micro structure information from mould filling simulation as well as computer tomography, to predict the effective anisotropic, temperature dependent stiffness of SMC. The temperature dependent stiffness of the resin-filler composite matrix as well as the effective SMC has been measured by dynamic mechanical analysis (DMA). A comparison of the effective stiffness predicted by Mori-Tanaka approximation [1] based on mould filling simulation results as well as computer tomography will be presented. These results will be compared to the effective anisotropic stiffness and their scatter investigated in DMAmeasurements. References [1] Y. Benveniste: A new approach to the application of Mori-Tanaka’s theory in composite materials, Mech. Mat., 6, 147–157 (1987). [2] Z. Jendli, F. Meraghni, J. Fitoussi, D. Baptiste: Multi-scales modelling of dynamic behaviour for discontinuous fibre SMC composites, Comp. Sci. Technol., 69, 97–103 (2009). Coupling of mold flow simulations with two-scale structural mechanical simulations for long fiber reinforced thermoplastics Fabian Buck, Barthel Brylka, Viktor Müller, Timo Müller, Andrew N. Hrymak, Frank Henning, and Thomas Böhlke 1 Institute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany 2 Institute of Vehicle Systems Technology, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany 3 Department of Chemical & Biochemical Engineering, University of Western Ontario, London, Canada Abstract. Due to their great weight-related strength and stiffness characteristic values and their versatile usability for complex molding, long fiber reinforced thermoplastics (LFT) become more important in industrial applications. For the calculation of the mechanical properties, which are mainly influenced by the manufacturing induced fiber orientation, the entire simulation process including flow and fiber orientation analysis [1], mapping of the obtained fiber orientation information, homogenization and FE structural simulations has to be investigated. Examination parts are compression molded polypropylene plates with 30 wt.-% glass fibers as reinforcement. The filling of the mold during compression is examined with the commercial software Moldflow and the obtained fiber orientation is validated with μCT data of selected samples. To calculate the effective stiffness properties based on simulated and measured μCT fiber orientation data three different homogenization schemes are used and compared: a two step scheme, a Mori-Tanaka approach and the self-consistent method [2]. The numerical results of FE calculations are validated with dynamic mechanical analysis (DMA) results of tensile and three point bending tests with static and dynamic load at different positions of the LFT plates. The obtained results identify that the combined methods allow a prediction of the essential mechanical properties and make clear that exact knowledge of the fiber orientation gradients induced by the different charge positions is essential for the entire simulation process. References [1] J. H. Phelps, C. L. Tucker III: An anisotropic rotary diffusion model for fiber orientation in shortand long-fiber thermoplastics, Journal of Non-Newtonian Fluid Mechanics, 156(3), 165–176 (2009). [2] V. Müller, T. Böhlke, F. Dillenberger, S. Kolling: Homogenization of Elastic Properties of Short Fiber Reinforced Composites Based on Discrete Microstructure Data, PAMM, 13(1), 269–270 (2013). Due to their great weight-related strength and stiffness characteristic values and their versatile usability for complex molding, long fiber reinforced thermoplastics (LFT) become more important in industrial applications. For the calculation of the mechanical properties, which are mainly influenced by the manufacturing induced fiber orientation, the entire simulation process including flow and fiber orientation analysis [1], mapping of the obtained fiber orientation information, homogenization and FE structural simulations has to be investigated. Examination parts are compression molded polypropylene plates with 30 wt.-% glass fibers as reinforcement. The filling of the mold during compression is examined with the commercial software Moldflow and the obtained fiber orientation is validated with μCT data of selected samples. To calculate the effective stiffness properties based on simulated and measured μCT fiber orientation data three different homogenization schemes are used and compared: a two step scheme, a Mori-Tanaka approach and the self-consistent method [2]. The numerical results of FE calculations are validated with dynamic mechanical analysis (DMA) results of tensile and three point bending tests with static and dynamic load at different positions of the LFT plates. The obtained results identify that the combined methods allow a prediction of the essential mechanical properties and make clear that exact knowledge of the fiber orientation gradients induced by the different charge positions is essential for the entire simulation process. References [1] J. H. Phelps, C. L. Tucker III: An anisotropic rotary diffusion model for fiber orientation in shortand long-fiber thermoplastics, Journal of Non-Newtonian Fluid Mechanics, 156(3), 165–176 (2009). [2] V. Müller, T. Böhlke, F. Dillenberger, S. Kolling: Homogenization of Elastic Properties of Short Fiber Reinforced Composites Based on Discrete Microstructure Data, PAMM, 13(1), 269–270 (2013). Material properties and model parameters, necessary for the analysis of static, cyclic, dynamic stress states and impact Ralf Cuntze Markt Indersdorf, retired from industry, linked to Carbon Composites e.V., Augsburg Abstract. Industry looks for robust and reliable analysis procedures in order to replace the expensive ’Make and Test Method’ as far as reasonable. Virtual tests shall reduce the amount of physical tests. In this context: Structural Design Development can be only effective if realistic input information is available for Design Dimensioning and for Manufacturing, as well. Reliable material properties are an essential key to achieve this effectiveness accompanied by the necessary high fidelity analyses. Topics here are high performance laminates and associated structural properties. The talk is structured according to: 1. Design dimensioning in structural design, some definitions 2. Modeling of materials (elasticity, strength) and analysis 3. Material properties (matrix, fiber, interphase, composite) 4. Special material properties and model parameters 5. Test methods and material sheets 6. Design verification and certification 7. Conclusions with Annex Industry looks for robust and reliable analysis procedures in order to replace the expensive ’Make and Test Method’ as far as reasonable. Virtual tests shall reduce the amount of physical tests. In this context: Structural Design Development can be only effective if realistic input information is available for Design Dimensioning and for Manufacturing, as well. Reliable material properties are an essential key to achieve this effectiveness accompanied by the necessary high fidelity analyses. Topics here are high performance laminates and associated structural properties. The talk is structured according to: 1. Design dimensioning in structural design, some definitions 2. Modeling of materials (elasticity, strength) and analysis 3. Material properties (matrix, fiber, interphase, composite) 4. Special material properties and model parameters 5. Test methods and material sheets 6. Design verification and certification 7. Conclusions with Annex Uncertainty quantification for linear elastic bodies with two fluctuating parameters Alex Dridger, Ismail Caylak, and Rolf MahnkenChair of Engineering Mechanics,University of Paderborn, Paderborn, Germany Abstract. Nowadays the uncertainty quantification becomes an important factorin many physics and engineering applications and should be taken into account.Stochastic partial differential equations (SPDEs) are commonly used to treat these kindof issues.In this paper, we study a numerical method to solve a linear elastic systemwith stochastic coefficients. We introduce the strong form of the SPDE and thecorresponding weak form. To solve this problem it is necessary to discretizethe equation in the spatial and the stochastic area. The spatial discretization isperformed by ordinary finite element methods whereas the stochastic discretizationuses WIENER’S polynomial chaos [2] to expand the coefficients in deterministic andstochastic parts. Similar to the spatial discretization there are shape fuctions (so-calledHermite polynomials [1],[2]) for the stochastic discretization.We consider a material behavior with two parameters. Two different approaches arepresented: The first one presupposes the knowledge of the distribution of the randomvariables modulus of elasticity E and the shear modulus G. The second one assumesthe distribution of Poisson’s ratio ν instead of the distribution of G to be known.Computational approaches involving polynomial chaos are used to expand thesevariables. Therefore, GALERKIN projection [1] can be applied to reduce the stochasticPDE into a system of deterministic PDEs.This work considers normally distributed random variables. Thus the number ofstochastic dimensions is equal to the number of independent input parameters.Subsequently, the resulting equation system is solved iteratively. Finally, our methodis applied in a numerical example for a plate with a ring hole.References[1] H. G. Matthies and A. Keese. Galerkin methods for linear and nonlinear ellipticstochastic partial differential equations. Comput. Methods Appl. Mech. Engrg.,194:1295-1331, 2005.[2] R. G. Ghanem and P. D. Spanos. Stochastic Finite Elements: A Spectral Approach.Springer-Verlag, New York, 1991. Nowadays the uncertainty quantification becomes an important factorin many physics and engineering applications and should be taken into account.Stochastic partial differential equations (SPDEs) are commonly used to treat these kindof issues.In this paper, we study a numerical method to solve a linear elastic systemwith stochastic coefficients. We introduce the strong form of the SPDE and thecorresponding weak form. To solve this problem it is necessary to discretizethe equation in the spatial and the stochastic area. The spatial discretization isperformed by ordinary finite element methods whereas the stochastic discretizationuses WIENER’S polynomial chaos [2] to expand the coefficients in deterministic andstochastic parts. Similar to the spatial discretization there are shape fuctions (so-calledHermite polynomials [1],[2]) for the stochastic discretization.We consider a material behavior with two parameters. Two different approaches arepresented: The first one presupposes the knowledge of the distribution of the randomvariables modulus of elasticity E and the shear modulus G. The second one assumesthe distribution of Poisson’s ratio ν instead of the distribution of G to be known.Computational approaches involving polynomial chaos are used to expand thesevariables. Therefore, GALERKIN projection [1] can be applied to reduce the stochasticPDE into a system of deterministic PDEs.This work considers normally distributed random variables. Thus the number ofstochastic dimensions is equal to the number of independent input parameters.Subsequently, the resulting equation system is solved iteratively. Finally, our methodis applied in a numerical example for a plate with a ring hole.References[1] H. G. Matthies and A. Keese. Galerkin methods for linear and nonlinear ellipticstochastic partial differential equations. Comput. Methods Appl. Mech. Engrg.,194:1295-1331, 2005.[2] R. G. Ghanem and P. D. Spanos. Stochastic Finite Elements: A Spectral Approach.Springer-Verlag, New York, 1991. A phase-field approach for lower bainitic transformationconsidering carbide formation Martin Düsing and Rolf MahnkenChair of Engineering Mechanics (LTM),University of Paderborn, Paderborn, Germany Abstract. The phase-field method is a widely used tool to model the materialbehaviour on amesoscale [1]. Especially for steel there aremany approaches descridingthe different transformations using this method [2]. Yet there are few phase-fieldmodels for the bainitic transformation [3], because it is one of most complextransformations in steel. Bainite consists of carbides, bainitic ferrite and may haveresidual austenite. In recent reports the formation of carbides has not been considered.A phase field model to simulate the transformation of lower banite including carbondiffusion und carbide formation has been developed. To model the evolution ofthe carbides it is nescessary to simulate the diffusion of the carbon. Therefore themodel which is based on a classical phase-field approach, is coupled with a viscousCahn-Hilliard eqaution to simulate the typical coarsening of the carbon concentration.During the isothermal process a sheaf of bainitic ferrite grows. The carbon startscoarsening because the bainitic ferrite can only contain a fraction of the carbon whichwas stored in the austenite. At the peaks of the carbon concentration carbides areexcreted. The simulations show the described growth characteristics of the lowerbanite transformation including carbide formation successfully.References[1] I. Steinbach: Phase-field models in materials science, Modelling Simul. Mater. Sci.Eng., 17, (2009).[2] A. Yamanaka, T. Takaki, Y. Tomita: Phase-Field Simulation of Austenite to FerriteTransformation and Widmanstätten Ferrite Formation in Fe-C Alloy, MaterialsTransactions, 47(11) 2725–2731 (2006).[3] T.T. Arif, R.S. Qin: A phase-field model for bainitic transformation, ComputationalMaterials Science, 77, 230–235 (2013). The phase-field method is a widely used tool to model the materialbehaviour on amesoscale [1]. Especially for steel there aremany approaches descridingthe different transformations using this method [2]. Yet there are few phase-fieldmodels for the bainitic transformation [3], because it is one of most complextransformations in steel. Bainite consists of carbides, bainitic ferrite and may haveresidual austenite. In recent reports the formation of carbides has not been considered.A phase field model to simulate the transformation of lower banite including carbondiffusion und carbide formation has been developed. To model the evolution ofthe carbides it is nescessary to simulate the diffusion of the carbon. Therefore themodel which is based on a classical phase-field approach, is coupled with a viscousCahn-Hilliard eqaution to simulate the typical coarsening of the carbon concentration.During the isothermal process a sheaf of bainitic ferrite grows. The carbon startscoarsening because the bainitic ferrite can only contain a fraction of the carbon whichwas stored in the austenite. At the peaks of the carbon concentration carbides areexcreted. The simulations show the described growth characteristics of the lowerbanite transformation including carbide formation successfully.References[1] I. Steinbach: Phase-field models in materials science, Modelling Simul. Mater. Sci.Eng., 17, (2009).[2] A. Yamanaka, T. Takaki, Y. Tomita: Phase-Field Simulation of Austenite to FerriteTransformation and Widmanstätten Ferrite Formation in Fe-C Alloy, MaterialsTransactions, 47(11) 2725–2731 (2006).[3] T.T. Arif, R.S. Qin: A phase-field model for bainitic transformation, ComputationalMaterials Science, 77, 230–235 (2013). Use of high resolution CT for fiber based materials Hermann Finckh, Florian Fritz, and Goetz T. GresserGerman Institutes of Textile and Fiber Research (DITF), Institute of Textile Technology and Process Engineering(ITV), Denkendorf, Germany Abstract. Fiber-reinforced plastics (FRP) have outstanding properties for lightweightconstruction as they possess very high mass specific strength and stiffness. For along time numerical methods are applied in automotive, aerospace etc. to computemechanical properties using idealizations. With increasing efficiency of computermore detailed and precise simulation models have been used, also allowing complexmaterial models. Main focus in FRP is laid on considering correct fiber orientation,fiber distribution and fabric structure to optimize fabric layout. The quality of theFRP part strongly depends on the individual stages of the production processes.In the Resin Transfer Molding (RTM) process the reinforcement fabrics are drapedinto a mold, the tool is closed and resin is injected at one or several injectionpoints to infiltrate the fabric. The draping of the fabric into the mold results inlocal reorientation of the fibers. Additionally, by closing the tool the fiber volumecontent is changed locally. This has a strong influence on the infiltration process asflow resistance is changed by compression of the fibers resulting in a variation ofresin velocity and therefore changing the flow direction. Because of many unknownparameters, process simulation gains importance. Simulation of textile processes usingmicro/meso models have been a key research topic at the ITV, e.g. braiding, weaving,warping, knitting [1]. Now numerical simulation is supplemented with state of theart μ-computer tomography (CT) which proved to be a very valuable experimentaltechnique. μ-CT is not only used for quality assurance (imperfections, air pockets) anddamage detection (cracks) but for generating detailed simulation models as well as thevalidation of numerical simulations. For CFD simulation (e.g. RTM infiltration) theinput data "local permeability" of fabrics can be computed on voxel-data with latestanalysis-software. This presentation gives an overview of detailed models gainedby process simulation and the novel possibilities with high-resolution computertomography for more precise numerical simulations of FRP.References[1] H. Finckh: Numerische Simulation der mechanischen Eigenschaften textilerFlächengebilde Gewebeherstellung. In: 3. LS-DYNA Anwenderforum 2004, 15 p.,Germany, Bamberg, 2004 Fiber-reinforced plastics (FRP) have outstanding properties for lightweightconstruction as they possess very high mass specific strength and stiffness. For along time numerical methods are applied in automotive, aerospace etc. to computemechanical properties using idealizations. With increasing efficiency of computermore detailed and precise simulation models have been used, also allowing complexmaterial models. Main focus in FRP is laid on considering correct fiber orientation,fiber distribution and fabric structure to optimize fabric layout. The quality of theFRP part strongly depends on the individual stages of the production processes.In the Resin Transfer Molding (RTM) process the reinforcement fabrics are drapedinto a mold, the tool is closed and resin is injected at one or several injectionpoints to infiltrate the fabric. The draping of the fabric into the mold results inlocal reorientation of the fibers. Additionally, by closing the tool the fiber volumecontent is changed locally. This has a strong influence on the infiltration process asflow resistance is changed by compression of the fibers resulting in a variation ofresin velocity and therefore changing the flow direction. Because of many unknownparameters, process simulation gains importance. Simulation of textile processes usingmicro/meso models have been a key research topic at the ITV, e.g. braiding, weaving,warping, knitting [1]. Now numerical simulation is supplemented with state of theart μ-computer tomography (CT) which proved to be a very valuable experimentaltechnique. μ-CT is not only used for quality assurance (imperfections, air pockets) anddamage detection (cracks) but for generating detailed simulation models as well as thevalidation of numerical simulations. For CFD simulation (e.g. RTM infiltration) theinput data "local permeability" of fabrics can be computed on voxel-data with latestanalysis-software. This presentation gives an overview of detailed models gainedby process simulation and the novel possibilities with high-resolution computertomography for more precise numerical simulations of FRP.References[1] H. Finckh: Numerische Simulation der mechanischen Eigenschaften textilerFlächengebilde Gewebeherstellung. In: 3. LS-DYNA Anwenderforum 2004, 15 p.,Germany, Bamberg, 2004 FFT-based homogenization of elasticity at largedeformations Matthias Kabel, Thomas Böhlke and Matti Schneider1 Department of Flow and Material SimulationFraunhofer ITWM, Kaiserslautern, Germany2 Institute of Engineering Mechanics,Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany3 Department of Lightweight Structures and Polymer TechnologyChemnitz University of Technology, Chemnitz, Germany Abstract. We will present a Newton-Krylov solver for the FFT-based homogenizationmethod of Moulinec and Suquet [4] for elasticity problems at large deformations [2].Compared to earlier solvers, cf. [3] and [1], which rely upon gradient descent, ourmethod is significantly faster, reducing the computational times by factors of 20 to100 for some test examples.Exploiting the special structure of Green’s operator, we deduce an algorithmic variantreducing the memory requirements by 40%. For validation we present results forboth simple microstructures with analytically known solution fields and complexmicrostructures of glass fiber reinforced polymer structures (GFRP).References[1] P. Eisenlohr, M. Diehl, R.A. Lebensohn, and F. Roters: A spectral method solution tocrystal elastoviscoplasticity at finite strains, International Journal of Plasticity, 46(0),37–53 (20013)[2] M. Kabel, T. Böhlke, and M. Schneider: Efficient fixed point and Newton-Krylovsolver for FFT-based homogenization of elasticity at large deformations Computa-tional Mechanics, accepted[3] N. Lahellec, J. C. Michel, H. Moulinec, and P. Suquet: Analysis of inhomogeneousmaterials at large strains using fast fourier transforms. In: IUTAM Symposium onComputationalMechanics of SolidMaterials at Large Strains (ed. C.Miehe), pp. 247–258,Springer Netherlands, 2003[4] H.Moulinec and P. Suquet. A numerical method for computing the overall responseof nonlinear composites with complex microstructure. Computer Methods in AppliedMechanics and Engineering, 157(12),69 – 94 (1998) We will present a Newton-Krylov solver for the FFT-based homogenizationmethod of Moulinec and Suquet [4] for elasticity problems at large deformations [2].Compared to earlier solvers, cf. [3] and [1], which rely upon gradient descent, ourmethod is significantly faster, reducing the computational times by factors of 20 to100 for some test examples.Exploiting the special structure of Green’s operator, we deduce an algorithmic variantreducing the memory requirements by 40%. For validation we present results forboth simple microstructures with analytically known solution fields and complexmicrostructures of glass fiber reinforced polymer structures (GFRP).References[1] P. Eisenlohr, M. Diehl, R.A. Lebensohn, and F. Roters: A spectral method solution tocrystal elastoviscoplasticity at finite strains, International Journal of Plasticity, 46(0),37–53 (20013)[2] M. Kabel, T. Böhlke, and M. Schneider: Efficient fixed point and Newton-Krylovsolver for FFT-based homogenization of elasticity at large deformations Computa-tional Mechanics, accepted[3] N. Lahellec, J. C. Michel, H. Moulinec, and P. Suquet: Analysis of inhomogeneousmaterials at large strains using fast fourier transforms. In: IUTAM Symposium onComputationalMechanics of SolidMaterials at Large Strains (ed. C.Miehe), pp. 247–258,Springer Netherlands, 2003[4] H.Moulinec and P. Suquet. A numerical method for computing the overall responseof nonlinear composites with complex microstructure. Computer Methods in AppliedMechanics and Engineering, 157(12),69 – 94 (1998) Comparison of hyperelastic micromorphic, micropolarand microstretch continua Constitutive models andcomputation Thorben Leismann and Rolf MahnkenChair of Engineering Mechanics (LTM),University of Paderborn, Paderborn, Germany Abstract. Micromorphic continua are equipped with additional degrees of freedomin comparison to the classical continuum, representing micro deformations of thematerial points of a body. Therefore, they are able to account for material size-effectsand to regularize the boundary value problem, when localization phenomena arise.Micromorphic continua allow for arbitrary micro deformations, while the special casesmicropolar continuum and microstretch continuum only allow for micro rotationand microstretch respectively, see [1]. Only the micropolar case has been coveredextensively in the literature, e.g. [2].One goal of this presentation is to make the transition from a full micromorphiccontinuum, as presented in [3], to a micropolar or microstretch continuum, by varyingthe constitutive equations. To this end two different possibilities are presented forhyperelasticity with large deformations. This leads to four different material models,which are then compared using simple numerical examples.Another goal is to present a constitutive model encompassing the micromorphic,micropolar and microstretch continuum as special cases and arbitrary mixtures ofmicropolar and microstretch parts, enabling the representation of versatile materialbehaviour.References[1] A. C. Eringen: Microcontinuum Field Theories. Springer, 1999[2] W. Ehlers, W. Volk: On theoretical and numerical methods in the theory of porousmedia based on polar and non-polar elasto-plastic solid materials, Int. J. SolidsStructures, 35, 4597–4617 (1998).[3] C. B. Hirschberger, E. Kuhl and P. Steinmann: On deformational and cofigurationalmechanics of micromorphic hyperelasticity Theory and computation, Comput.Methods Appl. Mech. Engrg., 196, 4027–4044 (2007). Micromorphic continua are equipped with additional degrees of freedomin comparison to the classical continuum, representing micro deformations of thematerial points of a body. Therefore, they are able to account for material size-effectsand to regularize the boundary value problem, when localization phenomena arise.Micromorphic continua allow for arbitrary micro deformations, while the special casesmicropolar continuum and microstretch continuum only allow for micro rotationand microstretch respectively, see [1]. Only the micropolar case has been coveredextensively in the literature, e.g. [2].One goal of this presentation is to make the transition from a full micromorphiccontinuum, as presented in [3], to a micropolar or microstretch continuum, by varyingthe constitutive equations. To this end two different possibilities are presented forhyperelasticity with large deformations. This leads to four different material models,which are then compared using simple numerical examples.Another goal is to present a constitutive model encompassing the micromorphic,micropolar and microstretch continuum as special cases and arbitrary mixtures ofmicropolar and microstretch parts, enabling the representation of versatile materialbehaviour.References[1] A. C. Eringen: Microcontinuum Field Theories. Springer, 1999[2] W. Ehlers, W. Volk: On theoretical and numerical methods in the theory of porousmedia based on polar and non-polar elasto-plastic solid materials, Int. J. SolidsStructures, 35, 4597–4617 (1998).[3] C. B. Hirschberger, E. Kuhl and P. Steinmann: On deformational and cofigurationalmechanics of micromorphic hyperelasticity Theory and computation, Comput.Methods Appl. Mech. Engrg., 196, 4027–4044 (2007). Thermodynamic modeling and simulation ofsemicrystalline polymers Alexander Lion and Michael JohlitzInstitute of Mechanics,University of the Federal Armed Forces Munich, Neubiberg, Germany Abstract. In order to simulate production processes or the long-term behaviourof technical components which are made of semi-crystalline polymers, constitutivemodels describing the crystallisation behaviour in combination with the glasstransition are needed. These polymers exhibit amorphous and crystalline regionswhich are coupled by an interface layer, i.e. by the rigid amorphous fraction. The massfractions of the phases are essentially determined by the temperature process that thematerial has experienced.To represent the thermomechanical behaviour of such polymers, the constitutivemodel for the specific free energy is based on a hybrid formulation as proposedin: the volumetric part of the total free energy is described by an energy functionof the Gibbs type whereas the isochoric part is modelled by an energy functionof Helmholtz type. In this contribution, the Gibbs free energy per unit mass of asemicrystalline polymer is assumed to be the sum of five terms. The first term isthe chemical potential of the thermoviscoelastic glass-forming amorphous phase, thesecond term is that of the thermoelastic crystalline phase and the third contributionis the temperature-dependent chemical potential of the rigid amorphous fraction.The two remaining contributions are the enthalpy and the entropy of mixing. Thedegree of crystallinity is related to the mass fraction of the crystalline phase and isan internal state variable of the model, the mass fraction of the rigid amorphousfraction is assumed to be proportional to the mass of the crystalline phase and themodel for the entropy of mixing is developed. When the model for the specific Gibbsfree energy is inserted into the Clausius-Duhem inequality, evolution equations forthe degree of crystallinity and for the second internal variable describing the glasstransition are derived in combination with an algebraic equation for the temperature-and crystallization-induced changes in the specific volume.The properties of the theory are demonstrated by a series of simulations. Theconstitutive model is able to represent the characteristic behaviour of the isobaricspecific heat of semi-crystalline polymers during heating and cooling. The pronounceddependence of the exothermic crystallisation peak on the temperature rate is describedas well as endothermic melting. The equilibrium solution of the constitutive modeldelivers an expression in closed form for the equilibrium degree of crystallisationwhich depends on both the pressure and the temperature. In order to simulate production processes or the long-term behaviourof technical components which are made of semi-crystalline polymers, constitutivemodels describing the crystallisation behaviour in combination with the glasstransition are needed. These polymers exhibit amorphous and crystalline regionswhich are coupled by an interface layer, i.e. by the rigid amorphous fraction. The massfractions of the phases are essentially determined by the temperature process that thematerial has experienced.To represent the thermomechanical behaviour of such polymers, the constitutivemodel for the specific free energy is based on a hybrid formulation as proposedin: the volumetric part of the total free energy is described by an energy functionof the Gibbs type whereas the isochoric part is modelled by an energy functionof Helmholtz type. In this contribution, the Gibbs free energy per unit mass of asemicrystalline polymer is assumed to be the sum of five terms. The first term isthe chemical potential of the thermoviscoelastic glass-forming amorphous phase, thesecond term is that of the thermoelastic crystalline phase and the third contributionis the temperature-dependent chemical potential of the rigid amorphous fraction.The two remaining contributions are the enthalpy and the entropy of mixing. Thedegree of crystallinity is related to the mass fraction of the crystalline phase and isan internal state variable of the model, the mass fraction of the rigid amorphousfraction is assumed to be proportional to the mass of the crystalline phase and themodel for the entropy of mixing is developed. When the model for the specific Gibbsfree energy is inserted into the Clausius-Duhem inequality, evolution equations forthe degree of crystallinity and for the second internal variable describing the glasstransition are derived in combination with an algebraic equation for the temperature-and crystallization-induced changes in the specific volume.The properties of the theory are demonstrated by a series of simulations. Theconstitutive model is able to represent the characteristic behaviour of the isobaricspecific heat of semi-crystalline polymers during heating and cooling. The pronounceddependence of the exothermic crystallisation peak on the temperature rate is describedas well as endothermic melting. The equilibrium solution of the constitutive modeldelivers an expression in closed form for the equilibrium degree of crystallisationwhich depends on both the pressure and the temperature. Effective viscosity of fiber suspensions Felix Ospald and Matti Schneider1 Research Group Numerical Mathematics (Partial Differential Equations),TU Chemnitz, Chemnitz, Germany2 Department of Lightweight Structures and Polymer Technology,TU Chemnitz, Chemnitz, Germany Abstract. Exact knowledge of the effective viscosity of a fiber suspension is ofparamount importance for accurate simulations of injection molding. Due to its stronganisotropy, the full viscosity tensor is very difficult to determine by measurementsalone. Therefore, we make use of micromechanical simulations. In this talk we discussa finite difference discretization on a Cartesian staggered grid, whose solution can beaccelerated by use of Fast Fourier Transform. Our method enjoys similar benefits as themethod proposed by Moulinec and Suquet [1, 2]. In particular, our approach permitsa matrix-free implementation and can handle several hundred million unknownson a single workstation. To deal with the infinite viscosities of the fibers, we workwith a dual scheme acting on the fluidities (i.e. the inverse of the viscosities) of theconstituents. The numerical results are compared to existing numerical solutions. Wewill further discuss some aspects of the implementation, parallelization and scalabilityof the method.AcknowledgementsFelix Ospald and Matti Schneider gratefully acknowledge financial support by theGerman Research Foundation (DFG), Federal Cluster of Excellence EXC 1075 “MERGETechnologies for Multifunctional Lightweight Structures”.References[1] H. Moulinec, P. Suquet: A fast numerical method for computing the linear andnonlinear mechanical properties of composites, Comptes rendus de l’Académie dessciences. Séries II, Mécanique, physique, chimie, astronomie, 318(11), 1417-1423 (1994).[2] H. Moulinec, P. Suquet: A numerical method for computing the overall response ofnonlinear composites with complexmicrostructure,Comp.Meth. Appl. Mech. Engng.,157, 69-94 (1998). Exact knowledge of the effective viscosity of a fiber suspension is ofparamount importance for accurate simulations of injection molding. Due to its stronganisotropy, the full viscosity tensor is very difficult to determine by measurementsalone. Therefore, we make use of micromechanical simulations. In this talk we discussa finite difference discretization on a Cartesian staggered grid, whose solution can beaccelerated by use of Fast Fourier Transform. Our method enjoys similar benefits as themethod proposed by Moulinec and Suquet [1, 2]. In particular, our approach permitsa matrix-free implementation and can handle several hundred million unknownson a single workstation. To deal with the infinite viscosities of the fibers, we workwith a dual scheme acting on the fluidities (i.e. the inverse of the viscosities) of theconstituents. The numerical results are compared to existing numerical solutions. Wewill further discuss some aspects of the implementation, parallelization and scalabilityof the method.AcknowledgementsFelix Ospald and Matti Schneider gratefully acknowledge financial support by theGerman Research Foundation (DFG), Federal Cluster of Excellence EXC 1075 “MERGETechnologies for Multifunctional Lightweight Structures”.References[1] H. Moulinec, P. Suquet: A fast numerical method for computing the linear andnonlinear mechanical properties of composites, Comptes rendus de l’Académie dessciences. Séries II, Mécanique, physique, chimie, astronomie, 318(11), 1417-1423 (1994).[2] H. Moulinec, P. Suquet: A numerical method for computing the overall response ofnonlinear composites with complexmicrostructure,Comp.Meth. Appl. Mech. Engng.,157, 69-94 (1998). CT-based FE simulations of composite materials:Possibilities, trends, and applications Dieter H. PahrInstitute of Lightweight Design and Structural Biomechanics,Vienna University of Technology, Vienna, Austria Abstract. 3D imaging systems are widely used in the field of biomedical engineeringto obtain patient-specific simulation models. Similar methodologies are gettingpopular in the field of composite engineering.Three-dimensional image data build the basis of CT-based simulation models. Usually,they are taken by computer tomography but any kind of imaging technique is possiblewhich yields a 3D array of gray-values. The image resolution i.e. the size of a 3D pixel(so called voxel) determines the possible FE model types [1]. High resolution microFE models showing a detailed microscopic architecture. In general such models needspecialized FE solver on powerful computer clusters [2]. Low resolution homogenizedFEmodels are based on amaterial mapping (gray-value to elasticity) and can be solvedon standard PCs [2]. Materials or structures can be analysed by using such models.Typical tasks are material characterization (homogenization, material laws), materialcalibration (multi-scale gray-value mapping functions), and structural multi-scaleanalysis. For practical applications automated software tools are needed for an efficientgeneration, analysis, and results evaluation [3].In future, 3D images based simulation models will change the classical designphilosophy from a standardized modelling concept to a component specific modellingconcept. The talk will highlight possibilities, trends, and applications. Methodologiesfrom the biomechanical field will be presented and it will be shown how they could beapplied in the field of composite engineering.References[1] D. H. Pahr and P. K. Zysset: From high-resolution CT Data to Finite ElementModels: Development of an Integrated Modular Framework, Comp Meth BiomechBiomed Eng, 12, 45–57 (2009).[2] D. H. Pahr and P. K. Zysset: A comparison of enhanced continuum FE with microFE models of human vertebral bodies. J Biomech, 42, 455–462 (2009).[3] D. H. Pahr: Medtool User Manual www.dr-pahr.at. 3D imaging systems are widely used in the field of biomedical engineeringto obtain patient-specific simulation models. Similar methodologies are gettingpopular in the field of composite engineering.Three-dimensional image data build the basis of CT-based simulation models. Usually,they are taken by computer tomography but any kind of imaging technique is possiblewhich yields a 3D array of gray-values. The image resolution i.e. the size of a 3D pixel(so called voxel) determines the possible FE model types [1]. High resolution microFE models showing a detailed microscopic architecture. In general such models needspecialized FE solver on powerful computer clusters [2]. Low resolution homogenizedFEmodels are based on amaterial mapping (gray-value to elasticity) and can be solvedon standard PCs [2]. Materials or structures can be analysed by using such models.Typical tasks are material characterization (homogenization, material laws), materialcalibration (multi-scale gray-value mapping functions), and structural multi-scaleanalysis. For practical applications automated software tools are needed for an efficientgeneration, analysis, and results evaluation [3].In future, 3D images based simulation models will change the classical designphilosophy from a standardized modelling concept to a component specific modellingconcept. The talk will highlight possibilities, trends, and applications. Methodologiesfrom the biomechanical field will be presented and it will be shown how they could beapplied in the field of composite engineering.References[1] D. H. Pahr and P. K. Zysset: From high-resolution CT Data to Finite ElementModels: Development of an Integrated Modular Framework, Comp Meth BiomechBiomed Eng, 12, 45–57 (2009).[2] D. H. Pahr and P. K. Zysset: A comparison of enhanced continuum FE with microFE models of human vertebral bodies. J Biomech, 42, 455–462 (2009).[3] D. H. Pahr: Medtool User Manual www.dr-pahr.at. Computational homogenization of elasticityon a staggered grid Matti Schneider, Felix Ospald, and Matthias Kabel1 Department of Lightweight Structures and Polymer Technology,TU Chemnitz, Chemnitz, Germany2 Research Group Numerical Mathematics (Partial Differential Equations),TU Chemnitz, Chemnitz, Germany3 Department of Flow and Material Simulation,Fraunhofer ITWM, Kaiserslautern, Germay Abstract. We propose to discretize the problem of elastic homogenization by finitedifferences on a staggered grid, and introduce fast and robust solvers.Our method shares some properties with the FFT-based homogenization technique ofMoulinec and Suquet [1], which has received widespread attention recently due to itsrobustness and computational speed. These similarities includes the use of FFT, andthe resulting performing solvers.The staggered grid discretization however, offers three crucial improvements.First, solutions obtained by our method are completely devoid of the spuriousoscillations characterizing solutions obtained by Moulinec-Suquet’s discretization.Second, the iteration number of our solvers are bounded independently of the meshspacing and the contrast. In particular, our solvers converge for three-dimensionalporous structures, which cannot be handled by Moulinec-Suquet’s method.Third, the finite difference discretization allows for algorithmic variants withlower memory consumption. More precisely, it is possible to reduce the memoryconsumption of the Moulinec-Suquet algorithms by 50%.Applications include the computation of effective elastic properties of fiber reinforcedporous structures and plastification of long fiber thermoplastics.AcknowledgementsFelix Ospald and Matti Schneider gratefully acknowledge financial support by theGerman Research Foundation (DFG), Federal Cluster of Excellence EXC 1075 ’MERGETechnologies for Multifunctional Lightweight Structures’. We propose to discretize the problem of elastic homogenization by finitedifferences on a staggered grid, and introduce fast and robust solvers.Our method shares some properties with the FFT-based homogenization technique ofMoulinec and Suquet [1], which has received widespread attention recently due to itsrobustness and computational speed. These similarities includes the use of FFT, andthe resulting performing solvers.The staggered grid discretization however, offers three crucial improvements.First, solutions obtained by our method are completely devoid of the spuriousoscillations characterizing solutions obtained by Moulinec-Suquet’s discretization.Second, the iteration number of our solvers are bounded independently of the meshspacing and the contrast. In particular, our solvers converge for three-dimensionalporous structures, which cannot be handled by Moulinec-Suquet’s method.Third, the finite difference discretization allows for algorithmic variants withlower memory consumption. More precisely, it is possible to reduce the memoryconsumption of the Moulinec-Suquet algorithms by 50%.Applications include the computation of effective elastic properties of fiber reinforcedporous structures and plastification of long fiber thermoplastics.AcknowledgementsFelix Ospald and Matti Schneider gratefully acknowledge financial support by theGerman Research Foundation (DFG), Federal Cluster of Excellence EXC 1075 ’MERGETechnologies for Multifunctional Lightweight Structures’. References[1] H. Moulinec, P. Suquet. A fast numerical method for computing the linear andnonlinear mechanical properties of composites, Comptes rendus de l’Académie dessciences. Série II, Mécanique, physique, chimie, astronomie, 318(11), 1417-1423 (1994). Micro-mechanical modeling of long fiber reinforcedthermoplastics using local fiber orientation distributionfrom mold filling simulation Lukas Schulenberg, Thomas Seelig, and Dong-Zhi Sun1 Fraunhofer Institute for Mechanics of Materials (IWM), Freiburg, Germany2 Institute of Mechanics, Karlsruhe Institute of Technology (KIT),Karlsruhe, Germany Abstract. Long fiber reinforced thermoplastics (LFT) combine the advantages ofshort-fiber reinforcements and those of infinitely long fibers which results in excellentmaterial properties, e.g. high strength and toughness. These are important quantitiesfor automotive crash applications and for that reason it is necessary to correctly predictthe material behavior in numerical crash simulations.In this study, the spatial heterogeneity of LFT made of polypropylene with 30 wt%glass fibers has been analyzed in plate shaped structures. Tensile tests on specimenstaken from different positions in the plate have been performed, showing significantvariations of the measured local stiffness with position.The approach by Mori and Tanaka [1] has been used to micro-mechanicallyapproximate the composite stiffness in the linear elastic range. The method, originallysuitable only for aligned fiber configurations, is extended to account for the local fiberorientation distribution (FOD) with the approach of Advani and Tucker [2]. The latteris provided from mold filling simulations performed at the Fraunhofer Institute forIndustrial Mathematics (ITWM).The homogenized material model for LFT has been implemented as a user routine in acommercial finite element code. Two different methods to homogenize the orientationdistribution have been analyzed. Strengths and weaknesses of the implementedmethods have been elaborated especially under the consideration of a feasible CPUtime. The gained knowledge has been used to introduce a set of parameters which areable to adjust the material behavior. Finally, experimental tests were reproduced bynumerical simulations.References[1] T. Mori, K. Tanaka: Average Stress in the Matrix and Average Elastic Energy ofMaterials with Misfitting Inclusions. Acta Metallurgica 21, 571-574 (1973).[2] S. Advani, C. Tucker: The Use of Tensors to Describe and Predict Fiber Orientationin Short Fiber Composites. Journal of Rheology 31 (8), 751-784 (1987). Long fiber reinforced thermoplastics (LFT) combine the advantages ofshort-fiber reinforcements and those of infinitely long fibers which results in excellentmaterial properties, e.g. high strength and toughness. These are important quantitiesfor automotive crash applications and for that reason it is necessary to correctly predictthe material behavior in numerical crash simulations.In this study, the spatial heterogeneity of LFT made of polypropylene with 30 wt%glass fibers has been analyzed in plate shaped structures. Tensile tests on specimenstaken from different positions in the plate have been performed, showing significantvariations of the measured local stiffness with position.The approach by Mori and Tanaka [1] has been used to micro-mechanicallyapproximate the composite stiffness in the linear elastic range. The method, originallysuitable only for aligned fiber configurations, is extended to account for the local fiberorientation distribution (FOD) with the approach of Advani and Tucker [2]. The latteris provided from mold filling simulations performed at the Fraunhofer Institute forIndustrial Mathematics (ITWM).The homogenized material model for LFT has been implemented as a user routine in acommercial finite element code. Two different methods to homogenize the orientationdistribution have been analyzed. Strengths and weaknesses of the implementedmethods have been elaborated especially under the consideration of a feasible CPUtime. The gained knowledge has been used to introduce a set of parameters which areable to adjust the material behavior. Finally, experimental tests were reproduced bynumerical simulations.References[1] T. Mori, K. Tanaka: Average Stress in the Matrix and Average Elastic Energy ofMaterials with Misfitting Inclusions. Acta Metallurgica 21, 571-574 (1973).[2] S. Advani, C. Tucker: The Use of Tensors to Describe and Predict Fiber Orientationin Short Fiber Composites. Journal of Rheology 31 (8), 751-784 (1987). Multi-scale modeling of damage in textile composites Jaan-W. Simon, Brett Bednarcyk, Bertram Stier, and Stefanie Reese1 RWTH Aachen University, Institute of Applied Mechanics, Aachen, Germany2 NASA Glenn Research Center, Cleveland, OH, U.S.A. Abstract. Textile composites have become very popular in industrial applicationsdue to their ease of manufacture, damage tolerance, and relatively low cost.However, physics-based modeling of the mechanical behavior of textile compositesis challenging because additional geometric complexities are introduced, which causesignificant local stress and strain concentrations. These internal concentrations areprimary drivers of nonlinearity, damage, and failure within textile composites, andthus must be captured in order for the models to be predictive.The macroscale approach to modeling textile-reinforced composites treats thecomposite as an effective, homogenizedmaterial, with properties typically determinedexperimentally. This approach is very computationally efficient and can be sufficientin the linear elastic regime, but, because it does not explicitly consider the complexmicrostructural geometry of the composite, it cannot be considered predictive. Incontrast, the mesoscale approach to modeling textile composites explicitly considersthe internal geometry of the reinforcing tows, and thus their interaction, and the effectsof their curved paths can be modeled. The tows, which are themselves anisotropic,are treated as effective (homogenized) materials, requiring use of anisotropic materialmodels to capture their behavior. The microscale approach goes one level lower,modeling the individual filaments that constitute the textile composite tows.This paper will compare mesoand micro-scale approaches to modeling thedeformation and damage of textile-reinforced polymer matrix composites. For themeso-scale approach, the woven composite architecture will be modeled using thefinite element method, and an anisotropic damage model for the tows will beemployed to capture the local nonlinear behavior. This same global finite elementmodel will be used in a micro-scale approach, but an embedded semi-analyticalmicromechanics model will be employed to represent the tows. The homogenizedtow behavior, as predicted by this micromechanics model, will be passed to the finiteelement model at the tow integration points. Finally, a third, more computationallyefficient, micro-scale approach will be examined, wherein both the global architectureof the woven composite and the tows are modeled using the aforementionedsemi-analytical micromechanics model, without the use of the finite element method.The goal will be the comparison and evaluation of these three approaches to modelingtextile-reinforced composites in terms of accuracy, efficiency, and utility. Textile composites have become very popular in industrial applicationsdue to their ease of manufacture, damage tolerance, and relatively low cost.However, physics-based modeling of the mechanical behavior of textile compositesis challenging because additional geometric complexities are introduced, which causesignificant local stress and strain concentrations. These internal concentrations areprimary drivers of nonlinearity, damage, and failure within textile composites, andthus must be captured in order for the models to be predictive.The macroscale approach to modeling textile-reinforced composites treats thecomposite as an effective, homogenizedmaterial, with properties typically determinedexperimentally. This approach is very computationally efficient and can be sufficientin the linear elastic regime, but, because it does not explicitly consider the complexmicrostructural geometry of the composite, it cannot be considered predictive. Incontrast, the mesoscale approach to modeling textile composites explicitly considersthe internal geometry of the reinforcing tows, and thus their interaction, and the effectsof their curved paths can be modeled. The tows, which are themselves anisotropic,are treated as effective (homogenized) materials, requiring use of anisotropic materialmodels to capture their behavior. The microscale approach goes one level lower,modeling the individual filaments that constitute the textile composite tows.This paper will compare mesoand micro-scale approaches to modeling thedeformation and damage of textile-reinforced polymer matrix composites. For themeso-scale approach, the woven composite architecture will be modeled using thefinite element method, and an anisotropic damage model for the tows will beemployed to capture the local nonlinear behavior. This same global finite elementmodel will be used in a micro-scale approach, but an embedded semi-analyticalmicromechanics model will be employed to represent the tows. The homogenizedtow behavior, as predicted by this micromechanics model, will be passed to the finiteelement model at the tow integration points. Finally, a third, more computationallyefficient, micro-scale approach will be examined, wherein both the global architectureof the woven composite and the tows are modeled using the aforementionedsemi-analytical micromechanics model, without the use of the finite element method.The goal will be the comparison and evaluation of these three approaches to modelingtextile-reinforced composites in terms of accuracy, efficiency, and utility. Microstructure and yield strength modeling in a severelydeformed Al2.5Cu1.5Mg alloy Morgan Tort1,2, Patrick Trimby, Gang Sha, Yuri Lapusta, and Kenong Xia1 Department of Mechanical Engineering, University of Melbourne, Victoria 3010, Australia2 French Institute Advanced Mechanics, IFMA, LAMI, F-63175 Aubiere, France3 Australian Key Centre for Microscopy & Microanalysis and School of Aerospace, Mechanical & MechatronicEngineering, University of Sydney, NSW 2006, Australia4 Chinese Center of Excellence in Atom Probe Tomography, Gleiter Institute of Nanoscience, Nanjing Universityof Science and Technology, Jiangsu, 210094 China Abstract. Establishing microstructure/property relationships is a fundamental taskfor the design of new materials. In the present work, the study focuses on oneternary Al-Cu-Mg alloy, which is the base system of the 2xxx aluminium series,including 2024 and 2014 alloys. A yield strength model for undeformed andseverely deformed Al-Cu-Mg based alloys is proposed, taking into account theintrinsic strengthening, cluster/GPB zones strengthening, S phase strengthening, grainboundary strengthening, solid solution hardening and dislocation hardening. A setof 8 different microstructures, corresponding to different states of precipitation anddeformation via equal channel angular pressing (ECAP), was fully characterized usingcutting edge characterization techniques. In particular, atom probe tomography (APT)was used to resolve and quantify the clusters constituted of only several atoms andtransmission Kikuchi diffraction (TKD) was employed to resolve grains as small as10 nm. It is shown that a very good agreement is found between the experimentalyield strength coming from tensile testing and the yield strength calculated usingthe microstructure’s parameters in the model developed. The model is also used toexamine the major contributors to the yield strength. Establishing microstructure/property relationships is a fundamental taskfor the design of new materials. In the present work, the study focuses on oneternary Al-Cu-Mg alloy, which is the base system of the 2xxx aluminium series,including 2024 and 2014 alloys. A yield strength model for undeformed andseverely deformed Al-Cu-Mg based alloys is proposed, taking into account theintrinsic strengthening, cluster/GPB zones strengthening, S phase strengthening, grainboundary strengthening, solid solution hardening and dislocation hardening. A setof 8 different microstructures, corresponding to different states of precipitation anddeformation via equal channel angular pressing (ECAP), was fully characterized usingcutting edge characterization techniques. In particular, atom probe tomography (APT)was used to resolve and quantify the clusters constituted of only several atoms andtransmission Kikuchi diffraction (TKD) was employed to resolve grains as small as10 nm. It is shown that a very good agreement is found between the experimentalyield strength coming from tensile testing and the yield strength calculated usingthe microstructure’s parameters in the model developed. The model is also used toexamine the major contributors to the yield strength. From FFT-based homogenization to guaranteed boundson effective linear properties Jaroslav Vondřejc, Jan Zeman, and Ivo Marek1 New Technologies for the Information Society, Faculty of Applied Sciences, University of West Bohemia, Plzeň,Czech Republic.2 Department of Mechanics and Department of Mathematics Faculty of Civil Engineering, Czech TechnicalUniversity in Prague, Prague 6, Czech Republic. Abstract. FFT-based homogenization algorithms belong to fast numerical methods forevaluating homogenized (effective) properties of periodic heterogeneous materials.Originally, the method was based on a solution of the Lippmann-Schwinger typeof an integral equation including the Green function for an auxiliary homogeneousproblem. A numerical solution proposed by Moulinec and Suquet [1] is based on theNeumann series expansion corresponding to a simple iteration procedure. We explainthe algorithm by the Galerkin method of corresponding variational formulation withan approximation space composed of trigonometric polynomials [2]. Techniques ofnumerical integrations are discussed and corresponding algorithms are supportedby convergence of their approximate solutions to the solution of weak formulation[2]. Moreover, the primal and the dual variational formulations, according to Dvořák[3], allow for evaluating arbitrary accurate guaranteed bounds on the homogenizedcoefficients [4]. All theoretical results are confirmed with numerical examples.Acknowledgement. This work has been supported by projectEXLIZ-CZ.1.07/2.3.00/30.0013 which is co-financed by the European Social Fundand the national budget of the Czech Republic and by the Czech Science Foundationthrough project No. P105/12/0331.References[1] H. Moulinec, P. Suquet: A fast numerical method for computing the linear andnonlinear mechanical properties of composites. Comptes rendus de l’Académie dessciences. Série II, Mécanique, physique, chimie, astronomie, 318(11):1417-1423, 1994.[2] Vondřejc, J. Zeman, I. Marek: An FFT-based Galerkin method for homogenizationof periodic media. Computers and Mathematics with Applications, 68(3):156-173, 2014.[3] J. Dvořák: Optimization of Composite Materials. Master’s thesis, CharlesUniversity, 1993.[4] Vondřejc, J. Zeman, I. Marek: Guaranteed bounds on homogenized properties byFFT-based Galerkin method. arXiv:1404.3614, 2014. FFT-based homogenization algorithms belong to fast numerical methods forevaluating homogenized (effective) properties of periodic heterogeneous materials.Originally, the method was based on a solution of the Lippmann-Schwinger typeof an integral equation including the Green function for an auxiliary homogeneousproblem. A numerical solution proposed by Moulinec and Suquet [1] is based on theNeumann series expansion corresponding to a simple iteration procedure. We explainthe algorithm by the Galerkin method of corresponding variational formulation withan approximation space composed of trigonometric polynomials [2]. Techniques ofnumerical integrations are discussed and corresponding algorithms are supportedby convergence of their approximate solutions to the solution of weak formulation[2]. Moreover, the primal and the dual variational formulations, according to Dvořák[3], allow for evaluating arbitrary accurate guaranteed bounds on the homogenizedcoefficients [4]. All theoretical results are confirmed with numerical examples.Acknowledgement. This work has been supported by projectEXLIZ-CZ.1.07/2.3.00/30.0013 which is co-financed by the European Social Fundand the national budget of the Czech Republic and by the Czech Science Foundationthrough project No. P105/12/0331.References[1] H. Moulinec, P. Suquet: A fast numerical method for computing the linear andnonlinear mechanical properties of composites. Comptes rendus de l’Académie dessciences. Série II, Mécanique, physique, chimie, astronomie, 318(11):1417-1423, 1994.[2] Vondřejc, J. Zeman, I. Marek: An FFT-based Galerkin method for homogenizationof periodic media. Computers and Mathematics with Applications, 68(3):156-173, 2014.[3] J. Dvořák: Optimization of Composite Materials. Master’s thesis, CharlesUniversity, 1993.[4] Vondřejc, J. Zeman, I. Marek: Guaranteed bounds on homogenized properties byFFT-based Galerkin method. arXiv:1404.3614, 2014. Participants Title,NameInstitutionE-Mail-Address Priv.-Doz.Dr.HeikoAndräFraunhoferInstituteforIndustrialMathematicsITWM Kaiserslautern,[email protected] Dr.StéphaneBerbenniLaboratoired’ÉtudedesMicrostructuresetdeMécaniquedesMatériaux UniversitédeLorraine Metz,[email protected] Dipl.-Ing.AlexanderBernathInstituteofVehicleSystemTechnology KarlsruheInstituteofTechnology(KIT) Karlsruhe,[email protected] M.Sc.RóbertBertótiInstituteofEngineeringMechanics KarlsruheInstituteofTechnology(KIT) Karlsruhe,[email protected] Dipl.-Inform.BenjaminBertramInstituteforAppliedMaterialsIAM-WK KarlsruheInstituteofTechnology(KIT) Karlsruhe,[email protected] Prof.Dr.-Ing.TomasBöhlkeInstituteofEngineeringMechanics KarlsruheInstituteofTechnology(KIT) Karlsruhe,[email protected]