An adaptive approach for modeling a fiber-matrix composite with the FE2 method

In materials with a complicated microstructre [1], the macroscopic material behaviour is unknown. In this work a Fiber-Matrix composite is considered with elasto-plastic fibers. A homogenization of the microscale leads to the macroscopic material properties. In the present work, this is realized in the frame of a FE² formulation. It combines two nested finite element simulations. On the macroscale, the boundary value problem is modelled by finite elements, at each integration point a second finite element simulation on the microscale is employed to calculate the stress response and the material tangent modulus. One huge disadvantage of the approach is the high computational effort. Certainly, an accompanying homogenization is not necessary if the material behaves linear elastic. This motivates the present approach to deal with an adaptive scheme. An indicator, which makes use of the different boundary conditions (BC) of the BVP on microscale, is suggested. The homogenization with the Dirichlet BC overestimates the material tangent modulus whereas the Neumann BC underestimates the modulus [2]. The idea for an adaptive modeling is to use both of the BCs during the loading process of the macrostructure. Starting initially with the Neumann BC leads to an overestimation of the displacement response and thus the strain state of the boundary value problem on the macroscale. An accompanying homogenization is performed after the strain reaches a limit strain. Dirichlet BCs are employed for the accompanying homogenization. Some numerical examples demonstrate the capability of the presented method. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of Carbon Fiber Reinforced Plastics for Wave Propagation Analysis in Plates

The modeling of the carbon fiber reinforced plastic (CFRP) for finite element analysis of wave propagation in plates from this material is presented. The random distribution of fibers in the matrix material is considered in a microscale model in order to properly capture the experimentally observed “quasi-continuous mode conversion” (QCMC). (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Prediction of Macroscopic Material Failure

The aim of this contribution is the numerical determination of macroscopic material properties based on constitutive relationships characterising the microscale. A macroscopic failure criterion is computed using a three dimensional finite element formulation. The proposed finite element model implements the Strong Discontinuity Approach (SDA) in order to include the localised, fully nonlinear kinematics associated with the failure on the microscale. This numerical application exploits further the Enhanced–Assumed–Strain (EAS) concept to decompose additively the deformation gradient into a conforming part corresponding to a smooth deformation mapping and an enhanced part reflecting the final failure kinematics of the microscale. This finite element formulation is then used for the modelling of the microscale and for the discretisation of a representative volume element (RVE). The macroscopic material behaviour results from numerical computations of the RVE. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A hybrid element formulation for electromechanical problems

The difficulty in the modeling of ferroelectric materials is the coverage of the complicated interactions between electrical and mechanical quantities on the macroscale, which are caused by switching processes on the microscale. In the present work we present an electric hybrid element formulation where the stresses and the electric fields are derived by constitutive relations as presented in [1]. Therefore the displacements, the electric potential and the electric displacements are approximated by bilinear ansatz functions. Applying a static condensation procedure we obtain a modified finite element formulation governed by the degrees of freedoms associated to the displacements and the electric potential. The anisotropic material behavior is modeled within a coordinate-invariant formulation [6] for an assumed transversely isotropic material [4]. In this context a general return algorithm is applied to compute the remanent quantities at the actual timestep. Resulting hysteresis loops for the ferroelectric ceramics are presented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiscale damage simulation

Numerous materials show a softening behaviour at dynamic loading. The decrease of stress is caused by the evolution in the microscale in terms of areas where the local stiffness is reduced, e.g. due to micro-void growth. For a numerical treatment of this material behaviour, phenomenological damage approaches are used in daily engineering practice. For a better understanding of the micromechanical process of such phenomenological models, multiscale methods are becoming increasingly important. The physical quantities that are responsible for the microstructural evolution associated with the damage process are transferred into the numerical model. In this context, the method of configurational forces will be used to describe the geometrical changes of damaged areas. With the help of homogenization, macro- and microscale will be coupled. In consequence, each Gaussian point of the macroscale is modelled by an own microstructure (RVE), where the microscale evolves during the loading process according to observable damage phenomena. Hereby, we present the general case of hyperelastic materials at finite strains. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The heterogeneous multiscale method as a framework for coupling the finite element method and molecular dynamics

Hierarchical two-scale methods are computationally very powerful as there is no direct coupling between the macro- and microscale. Such schemes develop first a microscale model under macroscopic constraints, then the macroscopic constitutive laws are found by averaging over the microscale. The heterogeneous multiscale method (HMM) is a general top-down approach for the design of multiscale algorithms. While this method is mainly used for concurrent coupling schemes in the literature, the proposed methodology also applies to a hierarchical coupling. This contribution discusses a hierarchical two-scale setting based on the heterogeneous multi-scale method for quasi-static problems: the macroscale is treated by continuum mechanics and the finite element method and the microscale is treated by statistical mechanics and molecular dynamics. Our investigation focuses on an optimised coupling of solvers on the macro- and microscale which yields a significant decrease in computational time with no associated loss in accuracy. In particular, the number of time steps used for the molecular dynamics simulation is adjusted at each iteration of the macroscopic solver. A numerical example demonstrates the performance of the model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A computational two scaled model for the simulation of micro-heterogeneous low-alloyed TRIP steels

In transformation induced plasticity (TRIP) steel a diffusionless austenitic-martensitic phase transformation induced by plastic deformation can be observed, resulting in excellent macroscopic properties. In particular low-alloyed TRIP steels, which can be obtained at lower production costs than high-alloyed TRIP steel, combine this mechanism with a heterogeneous arrangement of different phases at the microscale, namely ferrite, bainite, and retained austenite. The macroscopic behavior is governed by a complex interaction of the phases at the micro-level and the inelastic phase transformation from retained austenite to martensite. A reliable model for low-alloyed TRIP steel should therefore account for these microstructural processes to achieve an accurate macroscopic prediction. To enable this, we focus on a multiscale method often referred to as FE² approach, see [6]. In order to obtain a reasonable representative volume element, a three-dimensional statistically similar representative volume element (SSRVE) [1] can be used. Thereby, also computational costs associated with FE² calculations can be significantly reduced at a comparable prediction quality. The material model used here to capture the above mentioned microstructural phase transformation is based on [3] which was proposed for high alloyed TRIP steels, see also e.g. [8]. Computations based on the proposed two-scale approach are presented here for a three dimensional boundary value problem to show the evolution of phase transformation at the microscale and its effects on the macroscopic properties. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the direct connection of rheological elements in nonlinear continuum mechanics

The structure of complicated phenomenological material models at finite strains is often exemplified with the help of rheological elements. Thereby, simple material behaviour, i.e. elasticity or viscous and plastic flow, are composes by components. In our approach, we directly apply this concept to obtain material models at finite strains. Towards this end, the thermodynamically consistent material behaviour of single elements is defined first. Subsequently, the elements are connected by evaluation of stress equilibria equations formulated on interconnecting configurations. The basic equations of this concept are presented using the example of nonlinear viscoelasticity of Maxwell type. The model results from a series connection of an elastic and a viscous element, whereas both are formulated in a thermodynamically consistent way within the framework on nonlinear continuum mechanics. Furthermore, an approach of numerical implementation using the stress equilibria is suggested. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains

In this contribution we analyze a generalization of the heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. The method was originally introduced by E and Engquist (Commun Math Sci 1(1):87–132, 2003) for homogenization problems in fixed domains. It is based on a standard finite element approach on the macroscale, where the stiffness matrix is computed by solving local cell problems on the microscale. A-posteriori error estimates are derived in L ²(Ω) by reformulating the problem into a discrete two-scale formulation (see also, Ohlberger in Multiscale Model Simul 4(1):88–114, 2005) and using duality methods afterwards. Numerical experiments are given in order to numerically evaluate the efficiency of the error estimate.     

Discrete-time analysis of the GI/G/1 system with Bernoulli retrials: An algorithmic approach

In this paper, we show that the discrete GI/G/1 system with Bernoulli retrials can be analyzed as a level-dependent QBD process with infinite blocks; these blocks are finite when both the inter-arrival and service times have finite supports. The resulting QBD has a special structure which makes it convenient to analyze by the Matrix-analytic method (MAM). By representing both the inter-arrival and service times using a Markov chain based approach we are able to use the tools for phase type distributions in our model. Secondly, the resulting phase type distributions have additional structures which we exploit in the development of the algorithmic approach. The final working model approximates the level-dependent Markov chain with a level independent Markov chain that has a large set of boundaries. This allows us to use the modified matrix-geometric method to analyze the problem. A key task is selecting the level at which this level independence should begin. A procedure for this selection process is presented and then the distribution of the number of jobs in the orbit is obtained. Numerical examples are presented to demonstrate how this method works.     

A convergent three‐level finite difference scheme for solving a dual‐phase‐lagging heat transport equation in spherical coordinates

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second‐order derivative of temperature with respect to time and a third‐order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in spherical coordinates and develop a three‐level finite difference scheme for solving the heat transport equation in a microsphere. It is shown that the scheme is convergent, which implies that the scheme is unconditionally stable. Results show that the numerical solution converges to the exact solution. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 60–71, 2004.     

A finite difference scheme for solving a nonlinear hyperbolic two-step model in a double-layered thin film exposed to ultrashort-pulsed lasers with nonlinear interfacial conditions

Hyperbolic two-step microscale heat transport equations have attracted attention in thermal analysis of thin metal films exposed to ultrashort-pulsed lasers. Exploration of temperature-dependent thermal properties is absolutely necessary to advance our fundamental understanding of microscale (ultrafast) heat transport. In this article, we develop a finite difference scheme, by obtaining an energy estimate, for solving the hyperbolic two-step model with temperature-dependent thermal properties in a double-layered microscale thin film with nonlinear interfacial conditions irradiated by ultrashort-pulsed lasers. The method is illustrated by investigating the heat transfer in a gold layer on a chromium layer.     

A Magneto-Visco-Elastic Model for Magnetorheological Elastomers

In this work we present the modular construction and FEM implementation of a material model for magnetorheological elastomers (MRE) that is firmly rooted in micromechanics and performs well against experimental data. Keeping in mind the composite nature at the microscale, we develop a constitutive formulation based on a multiplicative magneto-mechanical split of the deformation gradient. We consider both Lee- and Clifton-type right and left decompositions. In contrast to other recent formulations [6], this general approach allows the use of highly advanced micromechnically-based network models for polymers such as [4] and [5] in a modular format and to extend its application for coupled magnetomechanical response. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Material libraries for texturized thin metal sheets in elastic range

An approach for the simulation of the elastic material responses using stochastic generation of material samples and a subdomain-based FEM simulation is presented. Both the single-grain geometries and the lattice orientations are stochastically simulated. This simulations can be used to build a Material Library of responses which must be computed only once for a given material. Furthermore, the Library can be used in a fast two-scale simulation avoiding any detailed computation in the microscale. In this contribution, the basic ideas of the computational algorithm are presented and some results of the material library for Steel DC01 are given. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relaxed Incremental Variational Formulation for Damage in Fiber-Reinforced Materials

An incremental variational formulation for damage at finite strains is proposed based on the classical continuum damage mechanics. Since loss of convexity is obtained at some critical deformations a relaxed incremental stress potential is constructed which convexifies the original non-convex problem. The resulting model can be interpreted as the homogenization of a micro-heterogeneous material bifurcated into a strongly and weakly damaged phase at the microscale. A one-dimensional relaxed formulation is derived and based thereon, a model for fiber-reinforced materials is given. Finally, some numerical examples illustrate the performance of the model. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Calculation of Fiber Orientation in Three-Dimensional Arterial Walls

In this contribution an approach for the fiber reorientation in three-dimensional arterial walls is presented. In detail the load-bearing capacity of the tissue is increased by re orienting the fibers with respect to the principal stresses, cf. [1]. The improved fiber reorientation algorithm is combined with the polyconvex nonlinear anisotropic material model presented in [3]. The results of a three-dimensional finite element simulation, where the reorientation approach is applied to a short segment of a patient-specific arterial geometry, are presented. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiscale modelling and simulation of micro machining of titanium

The microscale morphology of micro machined component surfaces is directly connected to the heterogeneous microstructure. The deformation depends on the crystal structure, in case of the considered cp-titanium, the hcp crystal structure. In a first approach the crystal plastic deformation is modeled with isotropic hardening. A visco-plastic evolution law accounts for the rate dependency. The concept of configurational forces is used with the framework of crystal plasticity to model the cutting process of cp-titanium. The setting is implemented into the finite element method. The examples show the effect of the material heterogeneity on the deforamtion behavior and on the related configurational forces. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Configurational-Force-Driven Crack Propagation and Adaptive Mesh Refinement in Finite Inelasticity

We introduce a consistent variational framework for inelasticity at finite strains, yielding dual balances in physical and material space as the Euler equations. The formulation is employed for the simultaneous usage of configurational forces as both driving forces for crack propagation as well as h-adaptive mesh refinement. The theoretical basis builds upon a global balance of internal and external power, where the mechanical response is exclusively governed by two scalar functions, the free energy function and a dissipation potential. The resulting variational structure is exploited in the context of fracture mechanics and yields evolution equations for internal variables. In the discrete setting, we present a geometry model fully separated from the finite element mesh structure that represents structural changes of the material configuration due to crack propagation. Advanced meshing algorithms provide an optimal discretization at the crack tip. Local and global criteria are obtained via error estimators based on configurational forces being interpreted as indicators of an energetic misfit due to an insufficient discretization. The numerical handling is decomposed into a staggered algorithm scheme for the dual set of equilibrium equations in material and physical space and efficient mesh generation tools. Exemplary numerical examples are considered to illustrate the method and to underline the effects of inelastic material behaviour in the presented context. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase Field Approach for Martensitic Transformations and Crystal Plasticity

This work is motivated by cryogenic turning which allows end shape machining and simultaneously attaining a hardened surface due to deformation induced martensitic transformations. To study the process on the microscale, a multivariant phase field model for martensitic transformations in conjunction with a crystal plastic material model is introduced. The evolution of microstructure is assumed to follow a time-dependent Ginzburg-Landau equation. To solve the field equations the finite element method is used. Time integration is performed with Euler backward schemes, on the global level for the evolution equation of the phase field, and on the element level for the crystal plastic material law. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)