### Homogenized model of composits and polycrystals

Even in civil engineering field, composites are the most useful materials,
because they can be properly and effectively
designed for a particular purpose.
In such a designing process of materials, it is needed
that the characteristics of the overall properties of composite materials
and/or polycrystalline materials can be estimated
by a simple tool but with some degrees of accuracy.
Numerical simulations may predict more or less good estimate
of such average behavior, but to that end one must use
a large number of finite elements,
and it may not be easy to use in designing process.
From this viewpoint, an analytical model such as the classical Voigt model
may be much more useful than complicated numerical analyses.

Although there exists may such models, but
we have proposed that a two-phase material is modeled by
an artifical material with three materials, but
one of which will be set to be zero as a limit.
A typical result of the Young modulus and the Poisson ratio
is shown in the right-hand figure.
Our predictions are drawn by solid curves, while
dashed curves indicate the pre-existing results.
Our estimate lies between a pair of the pre-existing predictions
which are considered to be one of the upper and lower bounds.
Furthermore, our results are very close to those by
Hill's self-consistent method indicated by `SC' in the figure.
A figure below shows comparison with the experimental data
of Young's modulus and Poisson's ratio of a composite.

The folloing figures show elasto-plastic
predictions by our method.

### Material model with two inelastic mechanisms

The figure below shows several patterns of development of
the localized deformation in the ground beneath a direct footing.

It is true that the foundation materials such as sands and clays
are not continua, but continuum models are frequently used
in predicting the global behavior of such geo-materials.
However, if one uses a modified model with a simple plasticity
based on the metal plasticity, the localized deformation patterns
shown above are not properly predicted in numerical simulations.
We here add a non-coaxial flow rule to a non-associate plasticity
in order to take some properties of non-continuum into account.
Furthermore, in order to stabilize material responses with
such a non-coaxiality, another inelastic mechanism of
micro slips is added.
Such a model can yield the results in the figure above, where
the materials have different internal friction angles
(the one to the right has bigger angles than that in the left).
As can be seen in this figure, patterns quite similar to those
observed in the small-scall and large-scale experiments are
obtained by numerical simulations with this material model.

animation (symmetric loading)
animation (unsymmetric loading)

Figures below show the development of such a localized deformation
in the sedimentary layer above an active fault.

### Simplified FEM of compsite materials

It is nice to have an analytical averaging scheme of composites
explained above, but more precise simulation may be needed
at some stages of material design process.
There is a very powerful tool called a `Homogenization Method'
based on the singular perturbation method and possibly with an FEM
for the actual calculations.
We here imitate the method without any mathematical proofs between
their micro and macro analyses, and have proposed a method where
an analytically averaged constitutive model
is employed at an integration point of one finite element.
A figure at the left shows probabilistic perturbations
of stress components due to a variation of the volume fractions
of inhomogeneity obtained by the stochastic FEM.
Two curves correspond to two different composites with
inhomogeneities of different aspect ratios.

Such estimates
become possible because the stiffness matrix, for example,
is given explicitly in terms of micro-structures such as
volume fraction and aspect ratio of inclusions of composites.
The figure at the right shows a result of optimization
with respect to the volume fractions of reinforcing fibers
in a simple beam composed of composite materials.
The objective function is specified by

namely, by the maximization of stiffness ** K **
with a given force

**in terms of the volume fractions of fibers. The resulting objective function is a convex function as is shown in the right figure, and convergence of optimization is achieved at the fourth step. Starting from initial value as 3% for the volume fractions, we obtained their optimum values as**

*F**f*=1.1%, and

_{1}*f*=3.9%.

_{2}## Graduation Theses (last 3 years)

- 2013
- Generalization and Improvement of Predictive Method of Biphase Composite Material Average Elasticity (SUZUKI, Takahiro)
- A Consideration on Performance of Panel Points based on Redundancy (TAKIMOTO, Koudai)
- 2014
- An approximate self-consistent estimate of anisotropic elasticity of composites (ARAI, Akitomo)
- Quantitative evaluation of redundancy multigirder bridges based on nonlinear finite element analysis (KUMAGAI, Hiroyuki)
- Fundamental Consideration on Semianalytic Approach for Shear Lag using Periodic Boundary Condition (NISHII, Daiki)
- 2015
- Choice of stress rate for hypoelasticity from the viewpoint of shear resistance characteristics (FUJIMOTO, Masaaki)

## Master Theses (last 3 years)

- 2013
- Characteristics of objective stress rates used in incremental elastoplastic constitutive law with hypoelasticity and Localization of deformation (ARAKAWA, Junpei)
- Numerical Evaluation of the Nonlinear Shear Properties of Composite Member (SETOGAWA, Atsushi)
- 2014
- Numerical study of interaction between extended floor deck for non-jointification of a bridge and backfill soil (AKIBA, Shota)
- Fundamental consideration on the redundancy of steel Langer bridges based on nonlinear finite element analysis (KAWAMURA, Kota)
- Evaluation of Shear Properties of Composite Members with Considering Friction and Bond of Material Interface (KUROSAWA, Akifumi)
- Numerical Evaluation of Load Capacity of a Steel Truss Bridge with Considering Dynamic Effect of Member Failure (TSUKADA, Kenichi)
- 2015
- Approximate self-consistent prediction of average elastoplastic behavior of composite with stress-induced anisotropy (SUZUKI, Takahiro)
- Quantitative evaluation of dynamic effect of member failure of the steel truss bridges on nonlinear redundancy analysis (TAKIMOTO, Koudai)