Two-scale FE–FFT- and phase-field-based computational modeling of bulk microstructural evolution and macroscopic material behavior

The overall, macroscopic constitutive behavior of most materials of technological importance such as fiber-reinforced composites or polycrystals is very much influenced by the underlying microstructure. The latter is usually complex and heterogeneous in nature, where each phase constituent is governed by non-linear constitutive relations. In order to capture such micro-structural characteristics, numerical two-scale methods are often used. The purpose of the current work is to provide an overview of state-of-the-art finite element (FE) and FFT-based two-scale computational modeling of microstructure evolution and macroscopic material behavior. Spahn et al. (Comput Methods Appl Mech Eng 268:871–883, 2014) were the first to introduce this kind of FE-FFT-based methodology, which has emerged as an efficient and accurate tool to model complex materials across the scales in the recent years.

The Wigner-Seitz unit cell (rhombus) for a honeycomb lattice fails to establish a k-vector in the 2D space, which is required for the Bloch electron dynamics. Phonon motion cannot be discussed in the triangular coordinates, either. In this paper, we propose a rectangular 4-atom unit cell model, which allows us to discuss the electron and phonon (wave packets) motion in the k-space. The present paper discusses the band structure of graphene based on the rectangular 4-atom unit cell model to establish an appropriate k-vector for the Bloch electron dynamics. To obtain the band energy of a Bloch electron in graphene, we extend the tight-binding calculations for the Wigner-Seitz (2-atom unit cell) model of Reich et al. (Physical Review B, 66, Article ID: 035412 (2002)) to the rectangular 4-atom unit cell model. It is shown that the graphene band structure based on the rectangular 4-atom unit cell model reveals the same band structure of the graphene based on the Wigner-Seitz 2-atom unit cell model; the π-band energy holds a linear dispersion (ε−k ) relations near the Fermi energy (crossing points of the valence and the conduction bands) in the first Brillouin zone of the rectangular reciprocal lattice. We then confirm the suitability of the proposed rectangular (orthogonal) unit cell model for graphene in order to establish a 2D k-vector responsible for the Bloch electron (wave packet) dynamics in graphene.

Ferromagnetic materials are characterized by a heterogeneous micro-structure that can be altered by external magnetic and mechanical stimuli. The understanding and the description of the micro-structure evolution is of particular importance for the design and the analysis of smart materials with magneto-mechanical coupling. The macroscopic response of the material results from complex magneto-mechanical interactions occurring on smaller length scales, which are driven by magnetization reorientation and associated magnetic domain wall motions. The aim of this work is to directly base the description of the macroscopic magneto-mechanical material behavior on the micro-magnetic domain evolution. This will be realized by the incorporation of a ferromagnetic phase-field formulation into a macroscopic Boltzmann continuum by the use of computational homogenization. The transition conditions between the two scales are obtained via rigorous exploitation of rate-type and incremental variational principles, which incorporate an extended version of the classical Hill---Mandel macro-homogeneity condition covering the phase field on the micro-scale. An efficient two-scale computational scenario is developed based on an operator splitting scheme that includes a predictor for the magnetization on the micro-scale. Two- and three-dimensional numerical simulations demonstrate the performance of the method. They investigate micro-magnetic domain evolution driven by macroscopic fields as well as the associated overall hysteretic response of ferromagnetic solids.

Material models for ductile damage, crack initiation, and crack growth are of high interest, e.g. for metal forming simulations. Empirical engineering approaches are often applied, but the numerical results are sensitive to the discretization if no method is utilized to prevent ill-posedness of the underlying boundary value problem due to strain softening. In order to face this issue, an empirical damage model is equipped with a gradient-enhancement which introduces an additional length scale parameter. Until the initiation of damage, the material is modeled with standard von Mises plasticity. Damage initiation is taken into account by an uncoupled failure indicator. After damage initiation, material degradation is assumed to be driven by a non-local quantity, which depends on plastic deformation and stress triaxiality. During damage evolution, the macroscopic material behavior becomes dependent on hydrostatic stress, which is motivated by well known void growth and coalescence mechanisms. A calibration strategy is developed to determine the parameters of strain hardening, damage initiation, and damage evolution as well as the internal length step-by-step. The proposed model is calibrated to experimental data of a pressure vessel steel. Reasonable predictions of smooth and notched tensile tests as well as a small punch test show the validity of the model for loadings from moderate to high stress triaxialities.

The current knowledge about fatigue behavior of concrete is still incomplete. This concerns especially the progress of fatigue which precedes the fatigue failure. Therefore, the process of fatigue itself under cyclic compressive loading was investigated in a systematic and comprehensive way. The aim of this investigation was to obtain a deeper insight and to provide a better understanding of the damage process occurring within the material during fatigue loading. Concrete cylinders were tested with a number of cycles to failure between 106 and 107. To investigate macroscopic and microscopic changes in the material, various methods of non-destructive and destructive testing were used. One main result was, in contrast to other authors, that the investigated changes in macroscopic material behavior could not be explained only by a development of micro-cracks. The results indicated rather, that the related changes in the fatigue behavior are mainly a result of viscous processes in the hardened cement paste, similar to the processes of creep. Based on the experimental results a description of the possible processes was derived which take place in the material structure during fatigue loading and also leads to the observed macroscopic changes in the material behavior. In this context, the results have shown that in case of isotropic material behavior a stiffness reduction related to a scalar value could not capture the damage effect on the stress-strain-relationship caused by fatigue.

Plastic deformation behaviour of Fe–Cu composites predicted by 3D finite element simulations

Damage observation in glass fiber reinforced composites via μ-tomography

Three-dimensional visualization and microstructure-based modeling of deformation in particle-reinforced composites

Microstructure evolution and semi-solid deformation behavior of an A356 aluminum alloy processed by strain induced melt activated method

The macroscopic behavior of shape memory materials is modeled within the framework of multifield theories. Two scalar fields and a second-order tensor field are used as descriptors of the relevant microstructural phenomena. In this way it is possible to allow for pseudoelasticity and shape memory effect, as well as low and high temperature reorientation of Martensitic variants. The general aspects of the theory are discussed paying special attention to the treatment of balance equations and to the exploitation of the constitutive structure, which is characterized by the prescription of a response function for the entropy production. An example of an explicit model is also given.

The Cosserat continuum theory or theory of polar materials is intended to reflect the effects of microscopic structure in the macroscopic behavior of materials. These theories are intimately related with the notion of nonlocality at a single point. In contrast with the mechanical nonlocal effects in these theories, in this paper, thermodynamical nonlocality is taken into consideration. The entropy production rate presented here is affected by the fluctuations of temperature, radiation and heat fulx vecter, which do not appear in the condition of local equilibrium. By averaging the microscopic quantities, macroscopic variables which express the nonequilibrium condition can be obtained. The first law of thermodynamics in this study has the same form as for a thermal simple body. However, the form of the second law is modified by the above fluctuations and the secondary moment of microvolume. Through these equations, we may expect to construct constitutive equations which may include the effects of strong nonequilibrium conditions.

The macroscopic behavior of materials in an EM field is determined by their values of permittivity e r and permeability μ r relative to free space which are generally greater than unity. For specially engineered metamaterials, it has been found possible to obtain other effective values for e r and μ r . Rafael has developed basic phenomenological models for simulating refraction, scattering and transmission effects of metamaterials in a plane wave and/or point source environment. These tools are building blocks for such metamaterial applications as lenses and cloaks, and include hardware design, ray tracing, full wave solutions, scattering and cloaking.

The Atomic Kinetic Monte Carlo (AKMC) method provides insights into the macroscopic behavior of materials through atomistic-level simulations and finds broad applications in materials science innovation. Improving simulation scale and performance remains a consistent focus in the development of parallel AKMC software. We port the AKMC software to GPU clusters. To alleviate the memory pressure in large-scale complex system simulations, we redesign the data layout and propose the Lattice Data Compression and Vacancy Data Decompression algorithms. Additionally, We propose a multi-level pipeline scheme combined with an on-demand communication forwarding and merging strategy to reduce data transfer and communication overhead. Compared to state-of-the-art KMC software, MISA-AKMC achieves a 10.41-fold improvement in computational throughput and a 52.07-fold expansion in simulation scale. We implement the first true micrometer-scale AKMC simulation involving 20 quadrillion atoms on GPU clusters. MISA-AKMC achieves 96.03% parallel efficiency in weak scaling and 85.29% in strong scaling on 16,000 GPUs.

Thermal expansion properties of textile reinforced composites with certain structure symmetries can be efficiently calculated by a size-limited unit cell. In this paper, a general approach is developed for the establishment of such a unit cell model. For the derivation of unit cell boundary conditions, three rules are summarized according to the displacement fields in translational, reflectional and 180° rotational symmetric structures under a uniform temperature change loading. The application scope of present unit cell method is clarified from the thermal and mechanical point of views. Three typical composites, i.e., unidirectional fiber reinforced composite, plain woven composite and multi-harness (4HS, 5HS, 6HS, 7HS and 8HS) satin woven composites are then studied, and four, three and two size-reducing unit cells are formulated, respectively. The thermal expansion behaviors of each composite are analyzed, and the effective thermal expansion coefficients are predicted. The influence of structure symmetries on the deformation pattern of unit cell models is clarified. The numerical models are validated by the identical results obtained from unit cells of different sizes and also by the results available in literatures. The approach developed in this paper can be applied to thermal expansion studies of any other composites with relevant structure symmetries.

In this work, a mathematical modeling is developed to estimate the ageing macroscopic response and effective behavior of viscoelastic composites. The modeling is developed directly in time domain by considering the Volterra tensorial integral product describing the local behavior and governing equations. The ageing dynamic Green’s tensor is introduced in order to solve the partial differential equations for the time dependent displacement and make the transition between local and macroscopic fields. The localization tensor making the scale transition is obtained through using the heterogeneity problem and the Mori-Tanaka mean field approach. Once this step is completed, averaging techniques are used to derive the effective ageing behavior of viscoelastic composites. To compute the macroscopic response and the effective behavior a numerical scheme is used. Numerical results are presented with respect to the shape and volume fraction of inclusions and for different ageing viscoelastic composites.