scroll identifier for mobile
main-content

Due to their high stiffness and strength and their good processing properties short fibre reinforced thermoplastics are well-established construction materials.

Up to now, simulation of engineering parts consisting of short fibre reinforced thermoplastics has often been based on macroscopic phenomenological models, but deformations, damage and failure of composite materials strongly depend on their microstructure. The typical modes of failure of short fibre thermoplastics enriched with glass fibres are matrix failure, rupture of fibres and delamination, and pure macroscopic consideration is not sufficient to predict those effects. The typical predictive phenomenological models are complex and only available for very special failures. A quantitative prediction on how failure will change depending on the content and orientation of the fibres is generally not possible, and the direct involvement of the above effects in a numerical simulation requires multi-scale modelling.

One the one hand, this makes it possible to take into account the properties of the matrix material and the fibre material, the microstructure of the composite in terms of fibre content, fibre orientation and shape as well as the properties of the interface between fibres and matrix. On the other hand, the multi-scale approach links these local properties to the global behaviour and forms the basis for the dimensioning and design of engineering components. Furthermore, multi-scale numerical simulations are required to allow efficient solution of the models when investigating three-dimensional problems of dimensioning engineering parts.

Bringing together mathematical modelling, materials mechanics, numerical methods and experimental engineering, this book provides a unique overview of multi-scale modelling approaches, multi-scale simulations and experimental investigations of short fibre reinforced thermoplastics. The first chapters focus on two principal subjects: the mathematical and mechanical models governing composite properties and damage description. The subsequent chapters present numerical algorithms based on the Finite Element Method and the Boundary Element Method, both of which make explicit use of the composite’s microstructure. Further, the results of the numerical simulations are shown and compared to experimental results.

Lastly, the book investigates deformation and failure of composite materials experimentally, explaining the applied methods and presenting the results for different volume fractions of fibres.

This book is a valuable resource for applied mathematics, theoretical and experimental mechanical engineers as well as engineers in industry dealing with modelling and simulation of short fibre reinforced composites.

### Chapter 1. Multi-Scale Methods in Simulation—A Path to a Better Understanding of the Behaviour of StructuresMulti-scalemethod

Efficient and accurate simulation methods are key for all development processes in all industries. Without simulation all influences of design changes, material choices, load situations, etc., on the performances of parts, sub-systems and systems, need to be confirmed by time consuming and expensive test procedures. On the other hand, including these influences into the design process by simulation together with efficient optimisation strategies allows to balance different performances, while allowing cost-efficient and ecological manufacturing as well as keeping a low weight of the complete system at the same time. To gain accurate results in a simulation, it is not sufficient to just model the geometry correctly; a main challenge is to model the material behaviour correctly. State of the art for metal based structures is to use material data that have been gained in experiments. Especially for fatigue performances, the expected lifetime fatigue material data is derived from databases or so called material laws. This approach does not take into account any manufacturing influences, even though these can locally lead to much improved behaviour. Therefore these parts are often still over-designed with respect to fatigue. For composite materials this approach is not even valid anymore as the behaviour strongly depends on the manufacturing process. Those influences can be analysed if one looks closer, i.e. at a different scale. The behaviour at the smaller scale then defines the material behaviour at the global scale. This process may even be recursive: even smaller scales are needed for the behaviour of intermediate scales—even down to atomistic levels. It is clear that such approaches can lead to tremendous computational cost. Therefore it is a key need to keep the process efficient, on one hand by analysing the methodologies on each scale, on the other hand by intelligent choice of the best method at each location.
Michael Hack

### Chapter 2. Indicators for the Adaptive Choice of Multi-Scale Solvers Based on Configurational Mechanics

For heterogeneous multi-scale methods,different analytical and numerical homogenisation methods can be applied on the micro-level, where the computational domain is a representative volume element (RVE). Several numerical homogenisation algorithms which are based on boundary element approaches, pseudo spectral discretizations, or finite element schemes are available for RVEs. However, each of these methods is only appropriate in subdomains of the macro-scale domain (component), e.g. in low-stress or highly stressed component regions. Therefore, indicators for the adaptive choice of solvers on the micro-scale are helpful. The proposed indicators make use of ideas from configurational mechanics.First of all, configurational forces are introduced as indicators. Then the multi-scale approach for configurational forces is explained and illustrated with an example. Afterwards the application of the configurational forces as an indicator for a refined homogenisation method is demonstrated. The last section is devoted to the scalability of heterogeneous multi-scale computations on parallel computers. A parallel finite element code is used for the macro-scale, and a PYTHON interface for the coupling with the different micro-scale solvers is described.
Ralf Müller, Charlotte Kuhn, Markus Klassen, Heiko Andrä, Sarah Staub

### Chapter 3. Modelling of Geometrical Microstructures and Mechanical Behaviour of Constituents

In addition to the macroscopic component geometry, a morphological microstructure model and material models for all individual phases of the material are required as input data to apply multi-scale methods. However, the advantage is that complicated mechanical coupon tests on the composite material can be avoided. This chapter explains the computation of morphological and material parameters on the example of short glass fibre reinforced polymers. The fibre orientation is the most important geometrical micro-structural parameter which has to be computed from µCT scans, whereas other micro-structural parameters (e.g. fibre length distribution and diameter) are a priori known. State-of-the-art methods for estimating local fibre orientations based on 3D image data are used to determine this essential microstructure feature depending on the sample position w.r.t. the flow front. After that the generation of virtual microstructures with the same morphological parameters as the µCT scans is considered. In the second part of this chapter, the identification of the material parameters is described for the polymer polybutylene terephthalate (PBT). All necessary parameters of a rate-independent elastoplastic model with damage are computed from cyclic tensile tests with increasing load amplitudes. Finally, the validation of the morphological and material models are illustrated by using an FFT-accelerated pseudo-spectral method as micro-scale solver.
Heiko Andrä, Dascha Dobrovolskij, Katja Schladitz, Sarah Staub, Ralf Müller

### Chapter 4. Parallel Inelastic Heterogeneous Multi-Scale Simulations

We recall the heterogeneous multi-scale method for elasticity and its extension to inelasticity within a two-scale energetic approach, where the fine-scale material properties are evaluated in Representative Volume Elements. These RVEs are located at Gauß points of a coarse finite element mesh. Within this $$\text {FE}^2$$ method the displacement is approximated on a coarse-scale, and depending on the strain at the Gauß points in every RVE a periodic micro-fluctuation and the internal variables describing the material history in this RVE are computed. Together, this defines the global energy and the dissipation functional, both depending on coarse-scale displacements as well as on fluctuations and internal variables on the micro-scale. Here we introduce a parallel realisation of this method which allows the computation of 3D micro-structures with fine resolution. It is based on the parallel representation of the RVE with distributed internal variables associated to each Gauß points, and a parallel multigrid solution method in the nonlinear computation of the micro-fluctuations and for the up-scaling of the algorithmic tangent within the incremental loading steps of the macro-problem. The efficiency of the method is demonstrated for a simple damage model combined with elasto-plasticity describing a PBT matrix material with glass fibre inclusions. For this investigation we use the material models in J. Spahn (Ph.D. thesis Kaiserslautern 2015) and the software developed by R. Shirazi Nejad (Ph.D. thesis Karlsruhe 2017).

### Chapter 5. Fast Boundary Element Methods for Composite Materials

In this chapter, we construct numerical solutions to the problems in the field of solid mechanics by combining the Boundary Element Method (BEM) with interpolation by means of radial basis functions. The main task is to find an approximation to a particular solution of the corresponding elliptic system of partial differential equations. To construct the approximation, the differential operator is applied to a vector of radial basis functions. The resulting vectors are linearly combined to interpolate the function on the right-hand side. The solvability of the interpolation problem is established. Additionally, stability and accuracy estimates for the method are given. A fast numerical method for the solution of the interpolation problem is proposed. These theoretical results are then illustrated on several numerical examples related to the Lamé system.
Richards Grzhibovskis, Christian Michel, Sergej Rjasanow

### Chapter 6. Experimental Studies

The theoretical results presented in the previous chapters are based on experimental investigations. Therefore, in this chapter, the experimental characterisation of the short fibre-reinforced composite is presented. The performed experiments increase in complexity, starting from uniaxial tensile tests at different strain rates up to multiaxial tests like true biaxial tests and the Nakajima test which introduce stress and strain states closer to applicational load cases. At least the results of the Nakajima tests can be regarded as verification experiments for the developed and implemented models. Effects like elasto-plasticity, damage and anisotropy are investigated in detail. The inhomogeneous strain states, which can be observed already in the uniaxial tests due to localisation phenomena in combination with damage, are evaluated using a three-dimensional optical strain measurement on the surface of the specimens. The underlying principles of digital image correlation are explained in detail.
Céline Röhrig, Stefan Diebels