Multiphase Material Model Research Articles

On the effects of a surrounding medium and phase split in coupled bone simulations

AbstractIn our previous contributions we established a multiscale, multiphase material model for the simulation of cancellous bone with the novel idea of including the full coupling of mechanical, electric and magnetic effects, which could be used for example, for the early detection of osteoporosis. While our calculations have already shown promising results, our previous approach lacks very important aspects, strongly limiting the applicability of our findings. In this paper we extend our base model by considering the effect of a surrounding medium on our bone specimen, using improved boundary conditions and differentiating between cortical bone, bone marrow and spongy bone to better reflect the physiological properties of bone. We show numerical results and compare our calculations to our previous modeling.

Phase‐field modeling of crack propagation based on multi‐crack order parameters considering mechanical jump conditions

AbstractThe phase field method is commonly used for the crack propagation modeling in modern material science, as they allow for an implicit tracking of the crack surface. However, most of these crack propagation models are for homogeneous materials, and there exist only a few approaches for heterogeneous systems. Recently, Schöller et al. [1] presented a novel phase‐field model for multiphase materials, e.g. composites, based on multi‐crack crack order parameters. Despite the quantitative advantages of the model, it is based on a simple scheme for the underlying homogenization problem. In this work, a more advanced homogenization scheme based on mechanical jump condition is applied to the model. Consideration of these jump conditions yields phase‐specific stresses and strains. Therefore, the mechanical driving force for crack propagation can be modeled as more independent of the elastic properties of other physical regions. Volume elements of a fiber reinforced polymer are used to demonstrate the limitations of the simple scheme, as well the improvement if considering mechanical jump conditions. Thereby, the contrast in the crack resistance of the two materials is varied. It is shown that the simple linear interpolation does not lead to reasonable crack paths for contrary contrasts of elastic modulus and crack resistance. Taking into account the mechanical jump conditions instead yields still reasonable results. For both the final crack paths and the stress‐strain curves of the system, the novel model is less sensitive to a change in fiber crack resistance. While the result of the simple scheme depend on the selected fiber crack resistance, although failure of the matrix is expected.

Mechanical performance of zirconia-silica bilayer coating on aluminum alloys with varying porosities: Deep learning and microstructure-based FEM

Despite significant efforts in multiscale modeling of multiphase materials, the effective mechanical response of the thin surface layers still remains elusive due to the complexity of heterogeneous microstructures and composition. This work presents a multiscale modeling approach for determining the effective elastic properties and mechanical behavior of heterogeneous materials such as thin multilayered oxides. This approach is specifically tailored for identifying the mechanical performance of porous solid oxides as the outermost layer. A generative adversarial network (GAN) is used to generate microstructure with varying porosities, complex pore shapes, and randomly distributed pores. The multiscale modeling framework is based on the Mechanics of Structure Genome (MSG) to concurrently derive micromechanics and structural analysis models from heterogeneous structures. We have evaluated the modeling approach based on the available experimental data in the literature. By resorting to the presented framework, not only the effective mechanical properties but also the global and local fields within the macro- and micro-structures, respectively, can be quantitatively predicted for a wide range of heterogeneous systems. A thorough investigation of the mechanical performance of aluminum alloys with deposited porous ZrO2 – SiO2 bilayer is presented to demonstrate the efficacy and applicability of this approach. Results show that the elastic properties of silica and zirconia are dramatically reduced when porosity exceeds 5% and 3%, respectively.

Micro-grinding Temperature Prediction Considering the Effects of Crystallographic Orientation and the Strain Induced by Phase Transformation

This paper proposes a physical-based model to predict the temperature in the micro-grinding of maraging steel 3J33b with the consideration of material microstructure and process parameters. In micro-grinding, the effects of crystallography on the grinding machinability become significant, since the depth of cut is of the same order as the grain size. In this research, the Taylor factor model for multi-phase materials is proposed to quantify the crystallographic orientation (CO) with respect to the cutting direction by examining the number and type of activated slip systems. Then, the flow stress model is developed, in which both the athermal stress resulted from the COs and the strain induced by the phase transformation are taken into account. On the basis of the flow stress model, the grinding forces are predicted followed by the calculation of the grinding heat. In the investigation, the triangular heat flux distribution and the reported energy partition model are applied in the calculation of workpiece temperature. Furthermore, the temperature model is validated by conducting an orthogonal-designed experiment, with the predictions of the maximum temperature in good agreement with the experimental data. Moreover, the predictive data is compared with the predictions resulted from the two other previously reported models. The results indicate that the proposed temperature model with considering the effect of CO and the phase transformation improved the prediction accuracy of the micro-grinding temperature.

Mechanical modelling of asphalt concrete using grid division

ABSTRACT In this paper, a simple method is introduced for the computational modelling of multiphase materials, and for the approximation of their mechanical response. The two-dimensional microstructures of six asphalt concrete specimens are selected; three of the specimens have ‘low’-, and three have ‘high’ aggregate fraction. A grid is used to divide the surface of each microstructure into square cells. The procedure of grid division is applied from 1 up to 100 divisions per side (i.e. up to 10,000 cells in the grid-divided specimen). To obtain an approximation of the mechanical response of the microstructure, the properties of the cells are estimated using three simple interpolation rules between the properties of the two phases, i.e. asphalt matrix and rock aggregates. It is found that the interpolation rules can yield reasonably representative results depending on the aggregate fraction of the microstructures and the number of divisions/cells in the grid. The grid-divided specimens allow approximating the overall mechanical response of the microstructures, and characteristics such as strain concentrations, overall deformations, and resulting force.

Force prediction in micro-grinding maraging steel 3J33b considering the crystallographic orientation and phase transformation

In micro-grinding, the effects of crystallography on grinding force become significant since the depth of cut is of the same order as the grain size. In this research, the Taylor factor model for multi-phase materials is proposed based on the previously reported Taylor factor model for monocrystalline material. Based on this model, the flow stress model is developed, which takes both the effect of CO on the athermal stress and the stress induced by the phase transformation into account. Based on the flow stress model, the predictive model of chip formation force is proposed by adapting parallel-sided shear zone theory. The rubbing force is modeled by applying Waldorf’s worn tool theory. Furthermore, the plowing force is predicted based on previously reported model by the authors. Subsequently, a comprehensive model of the micro-grinding force is proposed by considering mechanical-thermal loading, the effects of crystallography, and phase transformation. Finally, the model is validated by conducting an orthogonal-designed experiment with the result proving that the prediction of the model is capable to capture the magnitude and trend of the experimental data. Moreover, the proposed analysis are compared with the predictions of two other previously reported models with the result, indicating that the model that considers the effect of CO and the phase transformation improves the accuracy of the micro-grinding force.

A Model for Diffusion and Immobilization of Lithium in SiOC Nanocomposite Anodes

In order to simulate the diffusion of Li ions in SiOC nanocomposites, we developed a reaction–diffusion model for multiphase materials. This model extends existing models for single-phase diffusion through consideration of the ion transport across material interfaces. In each phase, this model regards mobile and immobilized ions together with the irreversible trapping process. The behavior of material interfaces is incorporated using a Butler–Volmer reaction kinetics model. The model is verified using a simple two-phase benchmark on a square domain. Simulations of the coupled diffusion in a random microstructure show a stalling effect, whereby the immobilization process effectively stops the diffusion of mobile ions during the first stages of intercalation.

Size dependent and micromechanical modeling of strain gradient-based nanoparticle composite plates with surface elasticity

The bending and free vibration behaviors of strain gradient-based Kirchhoff plates made of nanoparticle composites are investigated with the incorporation of the plate's surface effects. The modified strain gradient theory is used to capture the first-order deformation of the material structure while the impacts of the surface energy and surface tension are represented according to Gurtin and Murdoch surface elasticity theory. The plate is made of a composite material with nanocrystalline matrix and nanoparticles reinforcement. A size-dependent micromechanical model for multi-phase materials is developed to estimate the elastic moduli of the composite plate. This model has the merit to account for the effects of the surface energy of the inhomogeneities inside the composite structure. In addition, the impacts of the interface, grain size of the nanocrystalline matrix, porosity, and size of the reinforcing particles are incorporated into the micromechanical model. A simply supported composite plate consisting of nanocrystalline-silicon as the matrix and Al2O3-nanoparticles as the reinforcement is considered to study the influences of the size of the Si-grains, Al2O3-particles, voids, strain gradient deformation, and plate surface energy and tension on the deflection and natural frequencies of the plate. The results show that a decrease in the plate's thickness is accompanied with a decrease in the plate's deflection and an increase in the plate's natural frequency. Furthermore, the plate's strain gradient and surface tension contribute to its stiffness through a hardening effect while the interface and porosities inside the material structure affect the plate's stiffness with a softening mechanism. The plate's surface energy could stiffen or soften the plate depending on the values of the surface Lame constants. The surface energy of the inhomogeneities inside the material structure may increase or decrease the plate's stiffness, deflection, and natural frequencies depending on the inhomogeneities' sizes, their bulk material parameters, their surface material parameters, and their volume fractions.

Microstructure-based modelling of multiphase materials and complex structures

Micromechanical approaches are frequently employed to monitor local and global field quantities and their evolution under varying mechanical and/or thermal loading scenarios. In this contribution, an overview on important methods is given that are currently used to gain insight into the deformational and failure behaviour of multiphase materials and complex structures. First, techniques to represent material microstructures are reviewed. It is common to either digitise images of real microstructures or generate virtual 2D or 3D microstructures using automated procedures (e.g. Voronoi tessellation) for grain generation and colouring algorithms for phase assignment. While the former method allows to capture exactly all features of the microstructure at hand with respect to its morphological and topological features, the latter method opens up the possibility for parametric studies with respect to the influence of individual microstructure features on the local and global stress and strain response. Several applications of these approaches are presented, comprising low and high strain behaviour of multiphase steels, failure and fracture behaviour of multiphase materials and the evolution of surface roughening of the aluminium top metallisation of semiconductor devices.

Modeling of Self‐Healing phenomena in a Polymer matrix within the Framework of the Theory of Porous Media

AbstractIn this work, a multiphase material model will be presented in order to simulate self‐healing effects in a polymer. The material model is developed using the Theory of Porous Media (TPM). In order to describe the damage behavior of the polymer, the well known (1 − D)‐approach is included, and for the description of the healing, a phase change between the liquid like healing agents and the solid healed material is considered. This phase change depends on the amount of catalysts inside the polymer matrix. The applicability of the developed model, in order to describe the time dependency of the healing mechanism, will be shown by a virtual experiment of a self‐healing tapered double cantilever beam (TDCB). (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Finite Element Based Micro-mechanical Modeling of Interphase in Filler Reinforced Elastomers

Characteristic properties of elastomers can be tailored by embedding them with filler particles. Along with enhancing the overall properties of the composite, filler particles also induce some inelastic effects. In this paper, a finite element computational model is used to study the effect of microstructure morphology in filled elastomers, on its macroscopic large deformation behavior. A multiphase material model that accounts for the hypothesis of shift in glass transition temperature in the vicinity of the filler particle is developed to simulate the interphase between the fillers and the matrix. It also accounts for the breakdown and re-aggregation of filler networks under cyclic loading. Examples at the microstructural level, demonstrating the dynamics of the interphase using the developed multiphase model have been successfully simulated. The obtained results are in good qualitative agreement with the Mullins effect. Therefore, computational experiments using this methodology enable the prediction of the experimentally observed softening behavior in filled elastomers based on its microstructure evolution.

Prediction of microstructure in a Cosserat continuum using relaxed energies

AbstractWe develop a nonlinear incompressible multiphase material model in a Cosserat continuum with microstructure. The free energy of the material is enriched with an interaction potential taking into account the intergranular kinematics at the continuum scale. As a result the total energy becomes non‐convex, thus giving rise to the development of microstructural phases. To guarantee the existence of minimizers an exact quasi‐convex envelope of the corresponding energy functional is derived. As a result a three phase material energy appears, among them two of the phases are with microstructure in the translational motion (displacment field) and micromotion (microrotation field), whereas the third phase is without internal structure. The corresponding relaxed energy is then used for finding the minimizers of the two field minimization problem corresponding to a Cosserat continuum. Results from a numerical example predicting the development of microstructure in the material are presented. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Thermomechanical modelling strategies for multiphase steels

In multiphase steels, heat treatments such as quenching are usually applied to achieve a desired metallurgical composition to attain the expected mechanical properties. In these processes, residual thermal stresses arise during the cooling of the material which may induce a permanent deformation leading to dimensional instability. This deformation can be increased by the existence of phase transformations in the steel which should not be overlooked. In the current work, the thermomechanical modelling of multiphase materials is discussed. Firstly, a multiphase thermo-elasto-plastic-viscoplastic model is presented and applied to simulate several quenching heat treatments in a high carbon steel. The model uses the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation to describe the diffusion transformation and the Koistinen–Marburger model to characterise the diffusionless martensitic transformation in non-isothermal kinetics. This allows the observation of the evolution of the different steel phases (austenite, ferrite, pearlite, bainite and martinsite) during the cooling process. However, it is not possible to determine the residual stresses that arise in the intersection of the different phases. In the second part, the model that considers generalised multiphase transformation being compared with a multiphase homogenisation model for the case of a dual-phase steel during cooling and subsequent forming. The homogenisation micro-model operates over a periodic Representative Unit-Cell (RUC), detailing the heterogeneous material distribution due to the different metal phases. Therefore, it is possible to determine the residual stress fields in the intersection of the different phases. On the other hand, this model does not allow to reproduce the transformation process from austenite during cooling. Continuous cooling processes are studied in both parts. Following the heat treatment, tensile and shear test curves are presented and compared with experimental results for the second part.

Restrictions to the Energy Density in Energetic Models of Carbon Fiber Reinforced Carbon

In this contribution an energetic model for multi-phase materials is developed describing the influence of microstructure on different length scales as well as the evolution of phase changes. Restrictions on the energy functional are discussed. In such a non-convex framework, interfacial contributions serve for relaxing the total energy. Such models can be applied to describe the macroscopic material properties of carbon fiber reinforced carbon where phase transitions between regions of different texture of the carbon matrix are observed on a nanoscale as well as columnar microstructures on microscale.

Interface material failure modeled by the extended finite-element method and level sets

Multi-phase materials are characterized by the fact that they possess a specific heterogeneous microstructure. This feature often necessitates complex formulations in order to obtain realistic mechanical behavior with macroscopic material models. Furthermore, such complex models cause substantial difficulties when the corresponding material parameters need to be identified and their physical meaning interpreted. In multiscale analyses, usually the microstructure of a material point is resolved directly. Thus, the material behavior at the material point can be modeled in an elegant, natural and simple way. The present contribution aims at a detailed geometric modeling of multi-phase materials, as well as at a local mechanical modeling of material interfaces and interfacial failure. The geometry of the material distribution within the microstructure is described through level set functions. The mechanical modeling of material interfaces and interfacial cracks is accomplished by the extended finite-element method (X-FEM). The combination of both, the level set method and the X-FEM, allows one to model such internal features of the microstructure without the adaptation of the mesh. Thus, expensive meshing procedures can be avoided and considerable flexibility is obtained with respect to the variation of the material design. In the X-FEM, the finite-element approximation is enriched by appropriate functions through the concept of partition of unity. Remarkably, the level set method cannot only be applied to describe the geometry of the material interfaces and cracks, respectively, but also to support and even to develop the corresponding enrichment function.

Simulating Distortion and Residual Stresses in Carburized Thin Strips

This paper illustrates the application of a new multiphase material model for simulating distortion and residual stresses in carburized and quenched gear steels. Simulation is focused on thin, metallic strips that are heat treated to introduce a through-thickness carbon gradient. Because the material properties are strongly dependent on the carbon content, quenching causes significant transverse out-of-plane distortion. The material model accounts for a multiphase alloy structure where inelasticity in the individual phases is temperature and rate dependent. The model is fit to an extensive matrix of experimental data for low carbon steels (0.2–0.8 percent) whose transformation kinetics and mechanical response are similar to 4023 and 4620 alloys used in experiments. While residual stress data are limited, reasonable agreement with X-ray diffraction measurements was obtained. Comparisons of transverse deflections predicted numerically showed excellent agreement with those measured experimentally for all five thicknesses reported. Accurate transformation and lattice carburization strains are critical to correctly predict the sense and magnitude of these transverse distortions and in-plane residual stresses.

Numerical analysis of dynamic strain localisation in initially water saturated dense sand with a modified generalised plasticity model

Shear band dominated process in fully and partially saturated sand samples is simulated by means of dynamic strain localisation analysis together with a multiphase material model. The partially saturated medium is viewed as a multiphase continuum consisting of a solid skeleton and pores filled by water and air (vapour) which, once it appears, is presumed to remain at the constant value of cavitation pressure (isothermal monospecies approach). The governing equations are based on the general framework of averaging theories. A modified generalised plasticity constitutive model for partially saturated soils, developed from the general Pastor–Zienkiewicz sand model, has been implemented in a finite element code and used in the computational process. This model takes into account the effects of suction in the stiffness of the porous medium (solid skeleton) in partially saturated state. A case of strain localisation, which has been tested in laboratory observing cavitation of the pore water, is studied. Negative water pressures, which are of importance in localisation phenomena of initially fully saturated undrained samples of dilatant geomaterials, are obtained similarly to those observed experimentally.

Micromechanical modeling of the behavior of duplex stainless steels

The aim of the present paper is the micromechanical modeling of an aged duplex (austenite/ferrite) stainless steel. While the elastic properties of both phases are almost the same, the plastic mismatch between ferrite and austenite is high due to ferrite aging. Accounting for the complex micro and macrostructure of this material, three scales are investigated. The first one corresponds to each single phase. The second scale consists of a percolated (interwoven) network of both phases. The third scale is an aggregate of percolated bicrystals. For the first scale, each phase is modeled as a single crystal. In the second scale, F.E. simulations are performed on a unit cubic cell representing the percolated networks. A mean field model is then fitted in order to represent that bicrystal. This mean field model is used at the third scale to model the behavior of the aggregate of two-phase grains, using a model for multi-phase materials. At this scale, it becomes possible to represent the distribution of average ferrite stresses in each ferrite-austenite bicrystal. These variations are thought to be responsible for the heterogeneous damage nucleation observed in this material.

Strain localisation modelling and pore pressure in saturated sand samples

Dynamic strain localisation theory together with a multiphase material model is used to simulate shear band dominated processes in fully saturated sand samples. The fluid-saturated medium is viewed as multi-phase continuum consisting of a solid skeleton and pores filled by fluids. The governing equations are based on the general framework of averaging theories. A generalised plasticity constitutive model for fully saturated soils is adopted in the computational process. Both samples of medium-loose and dense sands are studied. Negative water pressures, which are important in localisation phenomena of fully saturated dilatant geomaterials, are obtained for dense sands, while positive water pressures result for medium-loose sands.

Stability analysis of the adhesive strength between substrate and a thin film

In the present work we show a simple way to describe the mechanical behaviour of an interface. In the recent history there was a lot of discussion about the modelling of multiphase materials. Generally, the interface is assumed as an ideal rigid boundary between matrix and inclusion or layer and substrate. In the following paper we will concentrate on the case of a layer-substrate interface. We will show a simple way to obtain a parameter specifying the interface from experimental results. This parameter is an essential part for the FE-calculation.