# Overview

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"Mesoscale" refers to an intermediate length scale that spans the range between the nanoscale and the macroscale. This length scale aids in linking microscale behavior to macroscale results.

# Tutorials

## Phase Field Modeling

A broad spectrum of moving boundary problems can be successfully handled with the diffuse-interface or phase-ﬁeld models. Such models are constructed by assuming that the free energy of a non-uniform system F, depends on (among other variables) the phase-ﬁeld variable $\boldsymbol{\phi}$, and its gradient ${\bigtriangledown}{\phi}$. In the volume V:

F = $\int\limits_{V}g \left ( {\phi}, {\bigtriangledown} {\phi} \right )dV$        (1)

Mathematically, the phase-ﬁeld formulation gives rise to two types of problems. The non-conserved phase-ﬁeld variables, such as those in solidiﬁcation and melting problems, are often assumed to evolve according to the ‘‘ﬁrst-order relaxation”, whereby the rate of change of the phase variable, $\dot{\phi}$, is proportional to the variational derivative of the free energy, mediated by the kinetic mobility B:

$\dot{\phi}$ = $-\boldsymbol B{\boldsymbol{\delta}F}/\boldsymbol{\delta}\boldsymbol{\phi}$

Such a formulation leads to a 2nd order, diffusion-type partial differential equation (PDE) for $\boldsymbol{\phi}$. However, if the phase- ﬁeld variable is subject to a conservation law, e.g.

$\frac{d}{dt}\int\limits_{V}{\phi} dV = 0$          (2)

then the rate $\dot{\phi}$ is proportional to the divergence of the ﬂux, which in turn, is proportional to the gradient of a potential. This potential is equal to the variational derivative of the free energy:

$\dot{\phi}$= $-{\bigtriangledown}.$$\quad \left \lbrack -B {\bigtriangledown} \left ({\boldsymbol{\delta}F}/\boldsymbol{\delta}\boldsymbol{\phi} \right ) \right \rbrack$          (3)

Examples include diffusion-controlled solid-state phase transformations, and interfaces between immiscible ﬂuids . Typically, the 4th order evolution PDE (3) is coupled to a 2nd order PDEs (stress, heat, etc.). For example, in solid state phase transformations, the lattice continuity across the interfaces between mismatched phases produces elastic strains , so that the 2nd order elasticity PDEs are coupled to the phase ﬁeld Eq. (3). The problems in this class have been addressed in the past, with the numerical tools tailored for speciﬁc problems. However, a general numerical method for solving the coupled equations, applicable to a variety of geometries and boundary conditions, has remained a challenging problem. The ﬁnite difference method (FDM) was used by Cahn and Kobayashi for one-dimensional modeling of the rapid coarsening and buckling in coherently self-stressed thin plates. Johnson used FDM to study the growth of the intermediate phase in a thin ﬁlm diffusion couple. Leo et al.used a pseudo-spectral method for the phase-ﬁeld model of coarsening in a two dimensional, elastically stressed, binary alloys. Fourier transform techniques, and related Fourier-spectral methods , have been used to investigate the microstructure evolution (mostly coarsening) resulting from solid-state phase transformations. The drawbacks of such methods are seen when the problem requires modeling of irregular domains, nonlinear and history-dependent problems. A versatile method is needed, applicable to irregular domains, a variety of boundary conditions, and various forms of geometric and material nonlinearities. The ﬁnite element method (FEM) and the related mesh-free method are such methods. [1]

Matlab tool box for texture MTEX

# Mesoscale Research

## Metals

### Magnesium

The mesoscale, although used generically to mean many things that are "intermediate," means essentially crystal plasticity in the context of multiscale modeling of metals. Crystal plasticity essentially is a continuum theory, but has discrete quantities starting at the grain or crystal scale. Polycrystalline averaging starting from the grain scale could be compared to the polycrystalline internal state variable models that operate at the macroscale. It can reach down to the Dislocation Dynamics results to help determine the material constants for the work hardening rules. Two different crystal plasticity formulations are given here:

1. Polycrystal formulations for FCC, BCC, and HCP crystals
2. Visco Plasticity Self Consistent (VPSC) model: polycrystalline averaged quantities

Experiments for calibration (homogeneously applied stress and strain states) and validation (heterogeneous stress and strain states) are used to determine the material constants and particular microstructures. The microstructures of interest include the texture (grain/crystal orientation, stress-strain behavior, hardening response, and twinning response if needed). Once the material model is calibrated and validated at the mesoscale, the results for the texture, yield surface, and hardening evolutions can be used in the macroscale internal state variable model and structural scale simulations.

Model Calibration for both the CP 1.0 and VPSC 1.0 models have fitting algorithms in which the material constants can be determined:

• CPfit 1.0
• VPSCfit 1.0

The following single element finite element input decks should be used to verify the material point simulator determinations: one element compression for 500 crystals aluminum polycrystal(ABAQUS-Standard) one element tension for 500 crystals aluminum polycrystal(ABAQUS-Standard) one element simple shear for 500 crystals aluminum polycrystal (ABAQUS-Standard)

## Geomaterials

### Concrete

Analysis of concrete at the mesoscale is beneficial as the size and distribution of constituents becomes evident. Concrete is a composite material made up of aggregates surrounded by a matrix. Large aggregates are easily visible at the macroscale, but small aggregates, unhydrated cement grains, and voids are first visible at the mesoscale.

## References

1. Mohsen Asle Zaeem, Sinisa Dj. Mesarovic, Finite element method for conserved phase fields: Stress-mediated diffusional phase transformation Original Research Article Journal of Computational Physics, Volume 229, Issue 24, 10 December 2010, Pages 9135-9149

[2] S. Groh, E.B. Marin, M.F. Horstemeyer, et al., Multiscale modeling of the plasticity in an aluminum single crystal, International Journal of Plasticity, 25(2009), 1456-1473.

[3] E.B. Marin, P.R. Dawson, On modeling the elasto-viscoplastic response of metals using polycrystal plasticity, Computer Methods in Applied Mechanics and Engineering, 165(1998), 1-21.

[4] E.B. Marin, P.R. Dawson, J.T. Jenkins, Aggregate size effect on the predicted plastic response of hexagonal close-packed polycrystals, Modelling Simulation Materials Science Engineering, 3(1995), 845-864.

[5] F. Roters, Application of crystal plasticity FEM from single crystal to bulk polycrystal, Computational Materials Science, 32(2005), 509-517.

[6] F. Roters, P. Eisenlohr, L. Hantcherli, et al., Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications, Acta Materialia, 58(2010), 1152-1211.

[7] D. Peirce, R.J. Asaro, A. Needleman, Analysis of nonuniform and localized deformation in FCC single crystals, Acta Metallurgica, 30(1982), 1087-1119.

[8] D. Peirce, R.J. Asaro, A. Needleman, Material rate dependence and localized deformation in crystalline solids, Acta Metallurgica, 31(1983), 1951-1976.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

## Media in category "Mesoscale"

This category contains only the following file.