# The macroscopic effect of the microstructure on strength: Strong composites.

The theory of composite materials has been extensively studied for the past 100 years involving some of the most distinguished scientists including Poisson, Faraday, Maxwell, Rayleigh, and Einstein. The preponderance of the theory addresses the prediction of the overall effective thermal, electric, and elastic properties as functions of the underlying microstructure and properties of the constituent materials. However equally compelling and more important for high performance applications are delicate questions related to the strength of a composite body. Understanding strength requires a complete understanding of how local thermal, electric, and elastic field fluctuations depend on the applied load, underlying microsturcture and physical properties of the constituents.

### Project 1. The strongest thermo-elastic composite and optimal lower bounds on the local stress inside thermoelastic composites

Together with Y. Chen we are interested in understanding the effects of residual stress and strain inside random elastic composites (Chen and Lipton, 2010).
Here the problem of finding optimal lower bounds on the maximum hydrostatic stress seen inside pre-stressed
configurations of two phase elastic materials with fixed volume fractions is considered.
To get started we have considered the case of a prescribed isotropic residual stress inside composites made from two
elastically isotropic materials. The composite is subjected to a constant external hydrostatic stress and we seek to
discover lower bounds on the maximum hydrostatic stress inside the composite given that we know only the volume fraction
of the two component materials. The goal is to find a tight lower bound on the maximum stress that is realized by some
special configuration of the two materials. Here the bounds on the local field properties are expressed in terms of the
applied mechanical stress, the prestress, volume fractions, elastic properties, and the coefficient of thermal expansion of
each of the materials. This project was carried out together with Yue Chen, see (Chen and Lipton, 2010). A complete characterization of optimal lower
bounds and extremal microgeometries that realize them were obtained. **In this way we have identified the strongest
statistically isotropic composite microgeometries.**

*This work is supported by NSF Grant DMS-0807265 and the Air Force Research Office through award FA9550-08-1-0095*

### Project 2. The strongest composites, strength domains, and optimal lower bounds on the local stress field in random media

Together with Bacim Alali we are interested in understanding the effects of stress inside random elastic composites (Alali and Lipton, 2009). Many structures are hierarchical in nature and are made up of substructures distributed across several length scales. Examples include aircraft wings made from fiber reinforced laminates, bridges made from steel reinforced concrete, and naturally occurring structures like bone. The applied load can be greatly amplified by the local microstructure and can result in local stress and strain concentrations, see for example (Kelly and Macmillan, 1986). The presence of large local stress and strain often precedes the appearance of nonlinear phenomena such as fracture and yielding. Thus it is crucial to quantify load transfer between length scales when considering failure initiation inside multi-scale heterogeneous media. In this paroject we seek to quantify load transfer between length scales when the substructure or microstructure is known only in a statistical sense. We provide new tools for teasing out relationships that connect the local field response to applied uniform loads. These relationships provide explicit criteria on the applied loads that are necessary for failure initiation inside statistically defined heterogeneous media.

The results presented in this project provide new quantitative tools for the study of failure initiation inside random heterogeneous media.
For a given realization of the random medium, the theory of failure initiation posits that failure is initiated when certain rotational invariants
of the local elastic stress exceed threshold values. An example is an elastic--perfectly plastic material. Here the material deforms
elastically up to some threshold value and then yields undergoing plastic, or irreversible deformation.
Typical stress invariants used to describe failure include the local hydrostatic stress component $\sigma^H$ which measures the hydrostatic force
acting inside the material and the Von Mises equivalent stress $\sigma^V$ which measures the local shearing forces acting inside a
material. Various combinations of these two invariants are considered in the strength of composites
literature.
To fix ideas we introduce the macroscopic strength domain associated with the local Von Mises stress $\sigma^V$ for
two phase statistically homogeneous random elastic media. Here we suppose that only the volume fractions $\theta_1$ and $\theta_2$
of the two elastic materials are known.
The macroscopic strength domain $K^{Safe}$ is defined to be the set of applied constant stresses $\overline{\sigma}$ such that $\sigma^V$ lies
below the failure threshold inside each component material almost surely for every microstructure realization of the random medium with prescribed
volume fractions $\theta_1$ and $\theta_2$. An upper bound on the macroscopic strength domain is defined to be the set $\overline{K}$ of constant
stresses such that if $\overline{\sigma}$ lies outside $\overline{K}$ then $\sigma^V$ has attained the threshold on some subset inside one of the
component materials for every microstructure composed of materials one and two with prescribed volume fractions $\theta_1$ and $\theta_2$, so
\begin{eqnarray}
K^{Safe}\subset \overline{K}.
\label{setupper}
\end{eqnarray}
We apply the lower bounds on local fields to obtain explicit, tight upper bounds on the macroscopic strength domains
for statistically homogeneous random media.
These results provide new optimal upper bounds on the strength domain for typical examples of * elastic--perfectly plastic*
random heterogeneous media.

We develop methods for finding the first optimal lower bounds on the local stress for a ladder of progressively more general
sets of imposed macroscopic stress. As we progress to more general load cases we will apply additional hypotheses on the shear and
bulk moduli of the constituent materials. In this section we provide lower bounds for the following applied macroscopic
load cases: 1) lower bounds on the full local stress for imposed hydrostatic stresses, 2) lower bounds on the full local
stress inside the material with larger shear modulus for elastic problems with imposed shear stresses,
3) lower bounds on the full local stress for $\mu_1=\mu_2$, that are seen to be optimal for a special class of imposed macroscopic stresses,
4) lower bounds on the local Von Mises equivalent stress that are optimal for a similar special class of imposed macroscopic stress fields,
and 5) lower bounds on the hydrostatic and deviatoric components of the local stress for the full set of imposed macroscopic stresses
subject to the hypotheses $\mu_1=\mu_2$ or $\kappa_1=\kappa_2$ respectively. The composite geometries attaining the lower bounds are the **strongest composites
for the prescribed class of loadings**.

We close pointing out that there is a substantial mathematical literature and associated theory for characterizing the
strength domains of heterogeneous media made from *rigid-perfectly plastic* materials. Unlike an elastic-perfectly plastic material a
rigid-perfectly plastic material does not deform until yield occurs. For rigid--perfectly plastic materials the
local stress satisfies only the equilibrium equation ${\rm div}\sigma=0$ until the yield limit is reached.
This is distinct from the elastic-perfectly plastic model where the stress also satisfies a constitutive law relating it to
the local elastic strain.

Ealier results identifing strong composites for hydrostatic loading and prescribed electric field are given in: R. Lipton. Optimal lower bounds on the hydrostatic stress amplification inside random two-phase elastic composites." Journal of the Mechanics and Physics of Solids, 53, 2005, pp. 2471--2481. R. Lipton. Optimal lower bounds on the dilatational strain inside random two-phase elastic composites subjected to hydrostatic loading. Mechanics of Materials, 38, 2006, pp. 833--839. R. Lipton. Configurations of nonlinear materials with electric fields that minimize L^p norms. Physica B, 338, 2003, pp. 48--53. R. Lipton. Optimal lower bounds on the electric-field concentration in composite media. Journal of Applied Physics, vol. 96, 2004, pp. 2821--2827.

*This work is supported by NSF Grant DMS-0807265 and the Air Force Research Office through award FA9550-08-1-0095*

### Project 3. Optimal design of composite structures for strength and stiffness: an inverse homogenization approach

In joint work with Michael Stuebner (R. Lipton and M. Stuebner, 2007) we introduce a rigorously based numerical method for compliance minimization problems in the presence of point wise stress constraints. The method is based on new multiscale quantities that measure the amplification of the local stress due to the microstructure. The design method is illustrated for two different kinds of problems. The first identifies suitably graded distributions of fibers inside shaft cross sections that impart sufficient overall stiffness while at the same time adequately control the amplitude of the local stress at each point. The second set of problems are carried out in the context of plane strain. Here we recover a novel class of designs made from locally layered media for minimum compliance subject to point wise stress constraints. The stress constrained designs place the more compliant material in the neighborhood of stress concentrators associated with abrupt changes in boundary loading and reentrant corners.This work has appeared in R. Lipton and M. Stuebner. Structural and Multidisciplinary Optimization, 33 (2007) pp. 351--362 and R. Lipton and M. Stuebner. The Quarterly Journal of Mechanics and Applied Mathematics, 59, 2006, pp. 139--161.

Ealier work for desigining electric field strength using composite media is given in in: R. Lipton and A. Velo. Optimal design of gradient fields with applications to electrostatics. Nonlinear Partial Differential Equations and Their Applications: College de France Seminare Volume XIV, D. Cioranescu and J.-L. Lions Editors. Studies in Applied Mathematics and its Applications, vol. 31, Elsevier, Amsterdam, 2002, pp. 509--532.

*This reasearch was supported by NSF through grant DMS- 0406374 and by the Air Force Office of Scientific Research, Air Force Materiel Command USAF, under grant numbers F49620-02-1-0041 and FA9550-05-1-0008.*

### Project 4. Representation Formulas for $L^\infty$ Norms of Weakly Convergent Sequences of Gradient Fields

Together with Tadele Mengesha we find representation formulas for the limits of $L^\infty$ norms of gradient fields associated with weakly convergent sequences of solutions to PDE with oscillatory coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory.

The representation formula provides the quantitative link between the gradient field measured at the length scale of the heteroegeneity and the macroscopic homogenized gradient field see (Lipton and Mengesha, 2012). This provides a means to quantify the effect of loading at the structural length scale on the amplification of gradient fields at the length scale of the microstructure. The analysis is carried out in the most general setting of G or H-convergence of elliptic operators. We identify a set of local representation formulas that in the fine phase limit provide lower bounds on the limit superior of the $L^\infty$ norms of gradient fields. The representation formulas are given by modulation functions relating gradients of homogenized fields to the $L^\infty$ norms of the local gradient inside a preselected domain of interest. For periodic homogenization these are precisely the $L^\infty$ norm of the correctors.

These results have appeared in R. Lipton and T. Mengesha. ESAIM: Mathematical Modelling and Numerical Analysis September 2012 46: pp. 1121--1146.

*This work is supported by NSF Grant DMS-0807265 and the Air Force Research Office through award FA9550-08-1-0095*

### Project 5. Correctors and field fluctuations for the $p^\epsilon(x)$-Laplacian with rough exponents

Together with Silvia Jimenez (S. Jimenez and R. Lipton, 2010) we have pursued a multi-scale analysis of the $L^p$ norms of gradients of potential fields inside multi-phase nonlinear power law composites. Here a new corrector theory was developed for mixtures of nonlinear power law materials with different exponents in their respective power laws. Earlier corrector theories focused on mixtures of nonlinear materials with the same power law exponent (Dal Maso and Defranceschi, 1990). The new corrector theory is applied to develop lower bounds on the $L^p$ norm of local gradient fields inside periodic fine phase mixtures of two or more nonlinear power law dielectric materials. These results provide the explicit link between local gradient fields at the length scale of the microgeometry that are generated by applied loads at macroscopic length scales.

This work has appeared in S. Jimenez and R. Lipton. J. Math. Anal. Appl. 372 (2010) pp. 448--469.*This project is supported by NSF Grant DMS-0807265 and the Air Force Research Office through award FA9550-08-1-0095.*

### Project 6. Stress constrained G-closure and relaxation of structural design problems

A relaxation for stress constrained optimal design problems is presented. It is accomplished by introducing the stress constrained G-closure. For a finite number of stress constraints an explicit characterization of the stress constrained G-closure is given. It is shown that the stress constrained G-closure is characterized by all G-limits together with their derivatives. A local representation of all G-limits and their derivatives are developed.

This work has appeared in R. Lipton. Stress constrained G closure and relaxation of structural design problems. Quarterly of Applied Mathematics. vol. LXII, 2004, pp. 295--321.*This work is supported by NSF Grant DMS-0072469 and AFSOR award F49620-99-1-1009.*

## References

- B. Alali and R. Lipton. Optimal lower bounds on local stress inside random media. SIAM Journal On Applied Mathematics 70, 2009, pp. 1260--1282.
- B. Alali and R. Lipton.
- G. Dal Maso and A. Defranceschi. Correctors for the homogenization of monotone operators. Differential and Integral Equations, 3:6 (1990) 1151--1166
- Y. Chen and R. Lipton Optimal lower bounds on the local stress inside random thermoelastic composites. Acta Mechanica. 2010, Volume 213, Issue 1-2, pp 97-109.
- S. Jimenez and R. Lipton. Correctors and field fluctuations for the p
_{ε}(x)-Laplacian with rough exponents. J. Math. Anal. Appl. 372 (2010) 448-469. - A. Kelly and N.H. Macmillan, 1986. Strong Solids. Monographs on the Physics and Chemistry of Materials. Clarendon Press, Oxford.
- R. Lipton. Stress constrained G closure and relaxation of structural design problems. Quarterly of Applied Mathematics. vol. LXII, 2004, pp. 295--321.
- R. Lipton and T. Mengesha. Representation formulas for L-infinity norms of weakly convergent sequencies of gradient fields in homogenization. Mathematical Modeling and Numerical Analysis (M2AN) 46 (2012) 1121-1146.
- R. Lipton and M. Stuebner. Optimal design of composite structures for strength and stiffness: an inverse homogenization approach. Structural and Multidisciplinary Optimization, 33, 2007, pp. 351--362.
- R. Lipton and M. Stuebner. Inverse homogenization and design of microstructure for pointwise stress control. The Quarterly Journal of Mechanics and Applied Mathematics, 59, 2006, pp. 139--161.