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This book provides an overview of multiscale approaches and homogenization procedures as well as damage evaluation and crack initiation, and addresses recent advances in the analysis and discretization of heterogeneous materials. It also highlights the state of the art in this research area with respect to different computational methods, software development and applications to engineering structures.

The first part focuses on defects in composite materials including their numerical and experimental investigations; elastic as well as elastoplastic constitutive models are considered, where the modeling has been performed at macro- and micro levels. The second part is devoted to novel computational schemes applied on different scales and discusses the validation of numerical results. The third part discusses gradient enhanced modeling, in particular quasi-brittle and ductile damage, using the gradient enhanced approach. The final part addresses thermoplasticity, solid-liquid mixtures and ferroelectric models. The contents are based on the international workshop “Multiscale Modeling of Heterogeneous Structures” (MUMO 2016), held in Dubrovnik, Croatia in September 2016.

### Erratum to: Efficient Multiscale FE-FFT-Based Modeling and Simulation of Macroscopic Deformation Processes with Non-linear Heterogeneous Microstructures

Julian Kochmann, Lisa Ehle, Stephan Wulfinghoff, Joachim Mayer, Bob Svendsen, Stefanie Reese

### Evolution of Failure Mechanisms in Composite Shell Structures Using Different Models

Modelling of structures on different scales has been a popular subject in the past. Within such a strategy the structural behaviour is modeled on a macro-level, describing the structure itself, whereas the material behaviour is modeled on a micro-level. Here typically RVEs are used. The proper choice of boundary conditions for the RVE is a difficult task in case of shell structures. Here, results have been presented for homogeneous and layered structures for composite materials in (Gruttmann & Wagner, Int J Num Meth Eng 94:1233–1254, 2013) [10]. In the present paper we discuss the influence of material nonlinear behaviour, especially the damage behaviour of fiber reinforced polymers, within the above described setting in comparison to other modeling techniques.
Werner Wagner, Friedrich Gruttmann

### Micro-Macro Modelling of Metallic Composites

This contribution describes a scale bridging approach for modelling pressure independent elastoplastic unidirectional metallic composite materials by making use of an anisotropic elastoplastic constitutive model. The material under investigation is tungsten fiber reinforced copper (W/Cu) composite. To identify the yield surface of the composite, a finite element model of a repeating unit cell (RUC) is set-up (micro-model). Through virtual experiments, the yield surface of the composite is identified. An anisotropic elastoplastic constitutive model based on the identified yield surface, which makes use of the concept of structural tensors, is developed. This material model serves as the material model for macro computations. To ensure a good agreement between constitutive model and RUC during plastic evolution, multiple hardening functions are employed. The parameters of the constitutive model are identified and the constitutive model is validated against the response of the RUC.
Rex Bedzra, Stefanie Reese, Jaan-Willem Simon

### Comparison of Mechanical Tests for the Identification of Composite Defects Using Full-Field Measurements and the Modified Constitutive Relation Error

Composite parts manufactured in large batches always present defects. These may not influence the behavior of the structure or might on the contrary be seriously detrimental to the performance of the component. In the first case, their presence is negligible, in the other case it is fundamental to be aware of their presence to foresee countermeasures. In this framework, being able to localize and estimate the intensities of flaws is extremely interesting. In this article, we present an approach based on the Modified Constitutive Relation Error to characterize defects, employing as input the displacements field measured from simple static and dynamic tests. The identification capabilities from tensile, bending, vibration and compression tests are compared using pseudo-experimental results as input data; then the identification is shown on a real case for buckling experiments to show the potential of the method.
E. Barbarella, O. Allix, F. Daghia, E. Jansen, R. Rolfes

### Snap-Through of Bistable Configurations Generated from Variable Stiffness Composites

Structures made of variable stiffness (VS) composites possess a rich design space for bistable configurations that demonstrate different values of curvatures and out-of-plane displacements. In this study, several VS composites are investigated which can yield cylindrical bistable shapes similar to those generated from unsymmetrical cross-plies. Such configurations have been found favorable as a component for certain morphing applications. A semi-analytical model based on the Rayleigh-Ritz approach is presented to calculate the thermally induced multistable shapes as well as the snap-through forces particularly taking into account the curvilinear paths of VS composites. A nonlinear finite element analysis is performed to check the accuracy of the semi-analytical method. Cylindrical shapes generated from VS laminates and the corresponding straight fiber cross-ply laminates are analyzed and compared. The snap-through forces are subsequently calculated and compared for different VS laminates and the straight fiber cross-ply. It is observed that certain VS composites have a significant reduction of snap-through forces but with a marginal difference in out-of-plane displacement as compared to the corresponding straight fiber cross-ply laminate.
Ayan Haldar, José Reinoso, Eelco Jansen, Raimund Rolfes

### Invariant-Based Finite Strain Anisotropic Material Model for Fiber-Reinforced Composites

Short fibre reinforced plastic (SFRP) materials are intensively used in several engineering sectors due to their excellent mechanical properties and production rates. In this investigation, an invariant-based transversely isotropic elasto-plastic model for finite strain applications and its corresponding numerical treatment are presented. The current model is based on the multiplicative decomposition of the deformation gradient. The main characteristic of the formulation is the mathematical realization of the incompressibility assumption with regard to the plastic behaviour in anisotropic finite strain setting. The proposed model is complying with thermodynamic restrictions and allows robust reliable numerical simulations. The accuracy of the model is verified by comparison against experimental data, showing a very satisfactory level of agreement.
Aamir Dean, José Reinoso, Shahab Sahraee, Benedikt Daum, Raimund Rolfes

### Unified Approach to Sensitivity Analysis Based Automation of Multi-scale Modelling

Use of different kinds of multi-scale methods is limited with specifications of the problem to be solved. Standard two-level finite element homogenization approach $$\text {FE}^2$$ is appropriate for problems with weakly coupled scales. If the difference between two scales is finite, or in the region of high gradients the $$\text {FE}^2$$ multi-scale approach fails, then some sort of domain decomposition method can be applied. Our motivation was to create computational environment, where the multi-scale code is automatically derived and various types of multi-scale approaches can be freely mixed. The described approach uses an advanced feature of software tools AceGen and AceFEM, that is automatic generation of the finite element codes for analytical first and second order sensitivity analysis with respect to prescribed essential boundary conditions as a unifying factor. The automatic-differentiation-based formulation (ADB) enables unification and automation of various multi-scale approaches for an arbitrary nonlinear, time dependent, coupled problem (e.g. general finite strain plasticity).
N. Zupan, J. Korelc

### Efficient Multiscale FE-FFT-Based Modeling and Simulation of Macroscopic Deformation Processes with Non-linear Heterogeneous Microstructures

The purpose of this work is the prediction of micromechanical fields and the overall material behavior of heterogeneous materials using an efficient and robust two-scale FE-FFT-based computational approach. The macroscopic boundary value problem is solved using the finite element (FE) method. The constitutively dependent quantities such as the stress tensor are determined by the solution of the local boundary value problem. The latter is represented by a periodic unit cell attached to each macroscopic integration point. The local algorithmic formulation is based on fast Fourier transforms (FFT), fixed-point and Newton-Krylov subspace methods (e.g. conjugate gradients). The handshake between both scales is defined through the Hill-Mandel condition. In order to ensure accurate results for the local fields as well as feasible overall computation times, an efficient solution strategy for two-scale full-field simulations is employed. As an example, the local and effective mechanical behavior of ferrit-perlit annealed elasto-viscoplastic 42CrMo4 steel is studied for three-point-bending tests. For simplicity, attention is restricted to the geometrically linear case and quasi-static processes.
Julian Kochmann, Lisa Ehle, Stephan Wulfinghoff, Joachim Mayer, Bob Svendsen, Stefanie Reese

### Experimental-Numerical Validation Framework for Micromechanical Simulations

A combined experimental-numerical framework is presented in order to validate computations at the microscale. It is illustrated for a flat specimen with two holes, which is made of cast iron and imaged via in situ synchrotron laminography at micrometer resolution during a tensile test. The region in the reconstructed volume between the two holes is analyzed via Digital Volume Correlation (DVC) to measure displacement fields. Finite Element (FE) simulations, whose mesh is made consistent with the studied material microstructure, are driven by measured Dirichlet boundary conditions. Damage levels and gray level residuals for DVC measurements and FE simulations are assessed for validation purposes.
Ante Buljac, Modesar Shakoor, Jan Neggers, Marc Bernacki, Pierre-Olivier Bouchard, Lukas Helfen, Thilo F. Morgeneyer, François Hild

### Stochastic Upscaling via Linear Bayesian Updating

In this work we present an upscaling technique for multi-scale computations based on a stochastic model calibration technique. We consider a coarse scale continuum material model described in the framework of generalised standard materials. The model parameters are considered uncertain in this approach, and are approximated using random variables. The update or calibration of these random variables is performed in a Bayesian framework where the information from a deterministic fine scale model computation is used as observation. The proposed approach is independent w.r.t. the choice of models on coarse and fine scales. Simple numerical examples are shown to demonstrate the ability of the proposed approach to calibrate coarse-scale elastic and inelastic material parameters.

### A Model Reduction Technique in Space and Time for Fatigue Simulation

The simulation of mechanical responses of structures subjected to cyclic loadings for a large number of cycles remains a challenge. The goal herein is to develop an innovative computational scheme for fatigue computations involving non-linear mechanical behaviour of materials, described by internal variables. The focus is on the Large Time Increment (LATIN) method coupled with a model reduction technique, the Proper Generalized Decomposition (PGD). Moreover, a multi-time scale approach is proposed for the simulation of fatigue involving large number of cycles. The quantities of interest are calculated only at particular pre-defined cycles called the “nodal cycles” and a suitable interpolation is used to estimate their evolution at the intermediate cycles. The proposed framework is exemplified for a structure subjected to cyclic loading, where damage is considered to be isotropic and micro-defect closure effects are taken into account. The combination of these techniques reduce the numerical cost drastically and allows to create virtual S-N curves for large number of cycles.
Mainak Bhattacharyya, Amélie Fau, Udo Nackenhorst, David Néron, Pierre Ladevèze

### Finite and Virtual Element Formulations for Large Strain Anisotropic Material with Inextensive Fibers

Anisotropic material with inextensive or nearly inextensible fibers introduce constraints in the mathematical formulations of the underlying differential equations from mechanics. This is always the case when fibers with high stiffness in a certain direction are present and a relatively weak matrix material is supporting these fibers. In numerical solution schemes like the finite element method or the virtual element method the presence of constraints—in this case associated to a possible fiber inextensibility compared to a matrix—lead to so called locking-phenomena. This can be overcome by special interpolation schemes as has been discussed extensively for volume constraints like incompressibility as well as contact constraints. For anisotropic material behaviour the most severe case is related to inextensible fibers. In this paper a mixed method is developed for finite elements and virtual elements that can handle anisotropic materials with inextensive and nearly inextensive fibers. For this purpose a classical ansatz, known from the modeling of volume constraint is adopted leading stable elements that can be used in the finite strain regime.
P. Wriggers, B. Hudobivnik, J. Schröder

### A Micromorphic Damage-Plasticity Model to Counteract Mesh Dependence in Finite Element Simulations Involving Material Softening

A gradient-extended damage-plasticity material model is presented which belongs to the class of micromophic media as proposed by Forest (J Eng Mech 135:117–131, 2009) [17]. A ‘two-surface’ formulation is utilized in which damage and plasticity are treated as independent but strongly coupled dissipative phenomena. To this end, separate yield and damage criteria as well as loading/unloading conditions are introduced. The model is thermodynamically consistent and accounts for both nonlinear kinematic and isotropic hardening as well as damage hardening. Various theoretical and numerical aspects of the formulation are discussed. Emphasis is also put on a procedure to enforce stress constraints at the local integration point level which provides, for instance, the basis for a straightforward integration of 3D gradient-extended material models into beam or shell elements or for their usage in 2D plane stress computations. A structural example problem illustrates the merits of the model and its ability to deliver mesh-independent results in coupled damage-plasticity finite element simulations.
Tim Brepols, Stephan Wulfinghoff, Stefanie Reese

### Modeling of Material Deformation Responses Using Gradient Elasticity Theory

Realistic description of material deformation responses demands more accurate modeling at both macroscopic and microscopic scales. Multiscale techniques employing several homogenization schemes are mostly used, in which a transition between nonlocal and local continuum formulations has been performed. Therein the transition of state variables is not defined fully consistently. In the present contribution a novel multiscale approach is proposed, where the same nonlocal theories at both scales are coupled, and discretisation is performed only by means of the $$C^{1}$$ finite element based on the strain gradient theory. The advantage of the new computational procedure is discussed in comparison with the approach using a local concept at microlevel. Employing the strain gradient continuum theory, a damage model for quasi-brittle materials is proposed and embedded into the $$C^{1}$$ continuity triangular finite element. The softening response of homogeneous materials under assumption of isotropic damage law is considered. The regularization superiority over the conventional implicit gradient enhancement procedure is demonstrated.
Jurica Sorić, Tomislav Lesičar, Filip Putar, Zdenko Tonković

### 3D Dynamic Crack Propagation by the Extended Finite Element Method and a Gradient-Enhanced Damage Model

A combined continuous-discontinuous approach to fracture is presented to model crack propagation under dynamic loading. A gradient-enhanced damage model is used to evaluate degradation of the material ahead of the crack. This type of model avoids mesh dependency and pathological effects of local damage models. Discrete cracks are reflected by means of extended finite elements (XFEM) and level sets. For the transition between damage and discrete fracture a damage based criterion is utilized. A discrete crack propagates if a critical damage value at the crack front is reached. The propagation direction is also determined through the damage field. Finally a dynamic mode II crack propagation example is simulated to show the capabilities and robustness of the employed approach.
M. Pezeshki, S. Loehnert, P. Wriggers, P. A. Guidault, E. Baranger

### A 3D Magnetostrictive Preisach Model for the Simulation of Magneto-Electric Composites on Multiple Scales

In this contribution we derive a three dimensional ferroelectric Preisach model based on an orientation distribution function. Therefore, the classical scalar Preisach model is modified and applied on the individual orientations, which are uniformly distributed in the three dimensional space. This model is used to simulate the behavior of magneto-electric (ME) composites. Such effective multiferroic materials combine two or more ferroic characteristics and can exhibit a coupling between electric polarization and magnetization. Since most of the single-phase ME materials exhibit a weak magneto-electric coupling at low temperatures, two-phase ME composites produce an ME coupling at room temperature. The basic idea for the manufacturing of ME composites is to use the interaction of the ferroelectric and magnetostrictive phases in order to generate strain-induced ME properties. However, in contrast to single-phase multiferroics, the ME coefficient of composites significantly depends on the microscopic morphology and the electro- as well as magneto-mechanical properties of the individual constituents. Therefore, we implemented the 3D Preisach model into the FE$$^2$$-method in order to depict the realistic ferroelectric behavior and directly incorporate the microstructure by the consideration of underlying representative volume elements.
J. Schröder, M. Labusch

### A Multiscale Framework for Thermoplasticity

The chapter describes a homogenization procedure for thermoplasticity problems. The proposed model is suitable for the finite strain regime and supports a very wide class of plasticity models. The methodology starts from the thermodynamically consistent thermoelastic framework already described in the literature. The latter framework is now extended to account for inelastic deformations. The problem is separated by means of the isothermal split into a mechanical and a thermal step, both at the macroscale and the microscale. As demonstrated in an example, the method does provide a way to successfully homogenize microscale variables as well as tangent operators. Finally, limitations of the approach are pointed out.
Marko Čanađija, Neven Munjas

### A Method of Numerical Viscosity Measurement for Solid-Liquid Mixture

We present a space-time homogenization procedure for multiscale modeling of solid-liquid mixture. The derived mathematical model enables us to set up two separate governing equations at both macro- and micro-scales. The fluid in the macroscopic governing equation is teated as an equivalent homogeneous medium with average or homogenized viscosity and is regarded as an incompressible Newtonian fluid, whose motion is assumed to be governed by the Navier-Stokes equations. The microscopic equations of motion governing the coupling phenomenon of the fluid and solid particles in a certain local domain and are solved to determine the microscopic flow fields under adequate boundary and loading conditions. Then the macrosopic viscosity is determined as the quantity averaged over the microscopic domain and within a certain time interval. The numerical viscosity measurement (NVM) can be realized by this space-time homogenization procedure. A set of NVMs is presented to demonstrate that the solid-liquid mixture considered in this study possibly exhibits a macroscopic flow characteristics of a special type of non-Newtonian fluids.
Reika Nomura, Kenjiro Terada, Shinsuke Takase, Shuji Moriguchi

### Numerical Simulation of Hydrogen Embrittlement at the Example of a Cracked Pipeline

A continuum model for numerical simulation of hydrogen induced embrittlement of pipeline material is discussed within this work. For that, a transient hydrogen model considering trapping is coupled with an elasto-plastic material model considering von Mises yielding. The hydrogen enhanced plasticity (HELP) mechanism is assumed to be active within this problem statement and is realized by a hydrogen dependent reduction of the yield strength. An iterative numerical solution scheme is applied to solve the coupled problem. At the example of a pipeline with a blunted crack, the influence of hydrogen is investigated. A localized plastic zone is observed for high hydrogen concentrations, in line with the inherent phenomena of the HELP mechanism. However, when applying hydrogen boundary conditions which are considered to be realistic for an existing natural gas pipeline, no pronounced effect of hydrogen based on reducing the yield strength could be observed. Nevertheless, this numerical results do not imply a judgment if the HELP mechanism in general could be the prevalent mechanism for failure.
Milena Möhle, Udo Nackenhorst , Olivier Allix