### Homogenization-Based Constitutive Models for  Magnetorheological Elastomers at Finite Strain (NSF DMS- 0708271)

Magnetorheological elastomers (MREs) are multi-phase,multi-functional material systems consisting of magnetically "hard" or "soft" particles embeddedin an elastomeric matrix phase. Because of the magnetic interactions between the particles,these materials are magnetostrictive and their mechanical response can be modified smoothlyand reversibly in real time. Conversely, the presence of strain in these materials can be detectedby induced changes in the overall magnetization. Although "macroscopic" (continuum mechanics)approaches for MREs have been in existence since the 1950s, more "microscopic" theories withtruly predictive capabilities are much more recent and so far have been restricted to the linear(infinitesimal strain) regime. The investigator develops nonlinear homogenization techniques and applies them to generate constitutive models for magnetorheological elastomers that are valid in the finite-strain regime. This builds on earlier work by the investigator for reinforced elastomers and uses extensions of variational "linear comparison" homogenization techniques that have been developed previously. At the theoretical level, the models account for: (i) the strongly nonlinear response of the constituents, including nonlinear ferrolectric behavior for the particles, as well as nonlinear mechanical response for the elastomeric matrix, (ii) microstructural information, such as particle shape and orientation, as well as their spatial and orientational distribution, (iii) coupled magnetoelastic constitutive behavior, and (iv) finite deformations. At the applicationslevel, the methodology is used to optimally select the constituent properties (e.g., magnetically hard vs. soft particles, magnetically isotropic vs. anisotropic particles) and the microstructural variables (e.g., particle shape and concentration, aligned distribution of orientations vs. random orientations, etc.) to enhance the magnetostrictive and sensing capabilities of these materials.  Magnetorheological elastomers (MREs) are composite materials with "smart" or "intelligent" properties. As a consequence of these special properties, MREs hold great promise for use as sensors and actuators in many industrial applications, including the automotive, electronics, and robotic industries. In addition, they are lightweight, inexpensive and easily processed into a myriad of shapes. However, in order to achieve their full potential, a better mathematical understanding is necessary of their complex -- highly coupled and nonlinear -- macroscopic behavior, especially in the large-deformation regime. To aid in this process, the investigator develops a homogenization-based approach to accurately model the macroscopic behavior of MREs, incorporating the dependence on the magnetic properties of the particles, their initial distribution and orientation (microstructure), as well as the evolution of this microstructure under large deformation. Finally, because of their actuation properties MREs provide an artificial analogue of human muscle, and the project has implications for other types of smart materials, including electroactive polymers (EAPs), which could find applications in biotechnology and other areas of Federal strategic interest.

### Finite-Strain, Constitutive Models for Semi-Crystalline  Polymers  (NSF CMMI- 0654063)

PI proposes to develop finite-strain constitutive models for semi-crystalline polymers, such as polyethylene and polypropylene. These are multi-phase material systems consisting of a "soft" phase that provides elasticity to the system in intimate contact with a "hard" phase that is

responsible for plastic deformation at large strains. They exhibit structure on two different length scales, ranging from the nanometer scale, at which the constituent polymer chains order themselves into "lamellar" structures, to the micrometer scale, where either randomly oriented

"lamellar grains," or "spherulitic grains" are observed. Because these polymers are among the most commonly used in many applications, including in the packaging, fluid distribution, automotive and toy industries, there is great interest in manipulating their structure to optimize

their mechanical response. Improvements in their mechanical properties are expected to have a positive impact on the environment. The methodology to be developed will make use of variational "linear comparison" homogenization techniques, and will account for: (i) the strongly nonlinear response of the constituent phases, (ii) the dual microstructure at the "grain" and "polycrystal" levels, (iii) coupled elasto-viscoplastic constitutive behavior, and (iv) finite deformations. The work will involve close collaborations with the experimental group of Jean-Yves Cavaille at the INSA de Lyon (France). Efforts will be made to recruit a top-level minority graduate student, who will benefit from the PI's on-going international collaborations.

### Modélisation mécanique multi-échelle de polymères nanostructures (Agence Nationale de la Recherche NANOMECA-2005)

L’objectif de ce projet est la caractérisation et la modélisation des propriétés mécaniques de polymères nanostructurés. Dans cette étude nous nous concentrerons principalement sur des matériaux présentant, à l’échelle nanomètrique, une structure lamellaire (empilement de différentes phases), au sein de laquelle on aura fait varier la densité de molécules liens entre phases: les copolymères à blocs et les polymères semi-cristallins en sont des parfaits exemples.

La modélisation micromécanique retenue est basée sur les techniques d’homogénéisation. Une des difficultés réside dans la connaissance du comportement de chaque phase d’un empilement et de leur couplage mécanique. Pour accéder à ces données (par exemple module des phases) on utilisera des calculs de dynamique moléculaire et une détermination par méthode inverse à par partir de la modélisation micromécanique. La modélisation sera ensuite entreprise à l’échelle mésoscopique (à l’échelle de la sphérolite dans le cas des polymères semi-cristallins, et à l’échelle des microdomaines dans le cas des copolymères à blocs) et à l’échelle macroscopique, les estimations issues d’une première étape d’homogénéisation étant injectées comme paramètres d’entrées de la seconde étape.

Parallèlement, plusieurs techniques expérimentales sont mise en œuvre pour mesurer les déformations locales à l’échelle des empilements. La comparaison des différents résultats doit permettre d’affiner progressivement chaque échelle de modélisation jusqu’à ce que l’ensemble des résultats soient cohérents et prédictifs. Dans une première étape, la modélisation sera centrée sur des déformations suffisamment faibles pour la microstructure reste stable. On cherchera alors à déterminer les propriétés viscoélastiques du matériau jusqu’à la limite d’écoulement. Puis à plus long terme les plus grandes déformations seront explorées. On cherchera alors à comprendre la genèse des microstructures étirées pour les semi cristallins ainsi que l’évolution de la microstructure pour les copolymères à bloc. On prendra alors en compte ces évolutions structurales pour la modélisation du comportement mécanique. L’aspect hétérogène et anisotrope de cette microstructure, ainsi que son évolution au cours de la déformation, sera également pris en compte.

### Multiscale modeling of nanostructured triblock copolymers (DOE DE-FG02-04ER46110)

In this project, structure-property relations will be developed for styrenic thermoplastic elastomers (TPEs). These are multiphase polymeric materials which consist of a soft'' phase (e.g., polybutadiene) giving rise to the rubbery nature of the materials, and  a hard'' crystalline or glassy phase (e.g., polystyrene) yielding increased stiffness and enhanced large-deformation properties. More precisely, these materials are ABA-{\em triblock} copolymers that derive  their superior properties from a self-assembly process  where the hard blocks act as anchoring points for the soft blocks  in a way somewhat  analogous to cross-linking in a vulcanized rubber. Because the self-assembly process takes place at the level of molecules, these materials develop a domain'' structure at the nanometer scale, or {\em nanostructure}. However, under typical processing conditions, they also develop a granular'' structure at the micron level, or {\em microstructure}, which is similar to that of metal polycrystals.  Therefore, TPEs exhibit structure a two different length scales and there is growing experimental evidence that this dual structure greatly affects the overall response of macroscopic samples. Because TPEs now constitute a multibillion dollar industry, and  are expected to replace traditional vulcanates in many applications, there is great interest in manipulating this structure to optimize the mechanical response of these materials. In spite of numerous applications that transcend many different technical areas of critical importance of the nation ranging from the automotive to the biotech industries, there has been relatively

little modeling and simulation work done to date on these materials and that is one of the main motivations for the proposed work.

### US-France Cooperative Research: Field Fluctuations, Microstructure Evolution and Coupled Phenomena in Random Heterogeneous Materials (NSF OISE- 0231867)

This three-year award for US-France collaboration in materials engineering involves researchers at the University of Pennsylvania, California Institute of Technology, University of Michigan, the French National Center for Scientific Research, Ecole Polytechnique and Ecole Normale Superieure. Pedro Ponte Castaneda in the US and Pierre Suquet in France lead a cooperative project on computation and measurement of nonlinear behavior of model composite materials. They will study problems of homogenization, including nonlinear properties, field fluctuations, microstructure evolution and coupled physical phenomena in different materials systems. The research is motived by several applications of technological questions. These include creep and forming operations of metal polycrystal systems, studies of porous elastomers, high-temperature behavior of metal-matrix composites and smart/active materials (including shape-memory alloys and ferroelectrics). The goal is to predict the behavior of macroscopic composite materials by investigating the components. The US investigators have theoretical expertise. This is complemented by the French investigators' experimental and numerical expertise. This award represents the US side of parallel proposals to the NSF and the CNRS. NSF will cover travel funds and living expenses for the US investigator and his students. The CNRS will support visits by French researchers and students to the United States.

### Nonlinear Homogenization and Applications to Porous and Nematic Elastomers (NSF DMS- 0204617)

This project proposes to develop and apply homogenization techniques for estimating the macroscopic behavior of heterogeneous hyperelastic material systems that are characterized by nonconvex energy functions. Two prototypical examples will be analyzed in detail: porous elastomers and "polydomain" liquid crystal elastomers (LCEs). The first is an example of a two-phase composite with one void phase and one elastic phase. These materials are used extensively in various industries for their insulation and shock absorption properties. The second is an example of a polycrystalline aggregate involving single-crystal grains made of a liquid-crystal elastomeric phase. They are materials that exhibit "soft'' modes of deformation, are capable of being actuated by temperature and light inputs, and also exhibit the remarkable property of becoming optically transparent at sufficiently large stretches. In particular, we propose to develop simple estimates of the Hashin-Shtrikman and self-consistent type for these materials, which have already been found to be extremely useful in other contexts. However, because of the finite deformations involved in elastomeric systems, it will be necessary to also characterize the evolution of the microstructure (e.g., porosity, texture, void or grain shape and orientation) in these systems and its implications on the overall behavior. Because of the non-convexity of the relevant energy functions, the possible development of instabilities must also be taken into account, especially because such unstable modes may be useful in the design of devices. This program of research will involve an exciting combination of several tools in mathematical analysis, including calculus of variations, convex analysis, differential equations, and optimization, and is likely to impact our understanding of constitutive theory of complex materials in general. This proposal is concerned with heterogeneous material systems that can undergo large elastic (recoverable) deformations. Examples of these material systems include, among others, carbon-black-filled elastomers, polymeric foams, liquid crystalline elastomers, block copolymers and skeletal muscle tissue. Two essential features characterize their mechanical response: 1) they can undergo large elastic (recoverable) deformations; and 2) they exhibit non-unique behavior (micro-buckling and other instabilities). In addition, most importantly from the applications point of view, the behavior of many of these material systems can be controlled by external fields (temperature, electric, magnetic, chemical inputs). Because of their remarkable properties, these materials, usually appearing in the form of composites or polycrystalline aggregates, will continue to provide the vehicles for many technological innovations, ranging from rubber tires, more than a century ago, to light-activated switches and artificial muscles, today. Because of their highly nonlinear properties, the characterization of these material systems is also mathematically challenging. In particular, even though much progress has been made in recent years in developing rigorous homogenization frameworks for these material systems, much work remains to be done in terms of developing "constructive" mathematical tools to estimate the constitutive behavior of specific systems within this class.

### Macroscopic Behavior and Field Fluctuations in Random Heterogeneous Materials: Theory and Applications (NSF CMMI-0201454)

This project is concerned with heterogeneous material systems with nonlinear constitutive behavior and complex, random microstructures that evolve in time. Because of their scientific and technological importance, the focus will be on viscoplastic systems including porous and other composite materials, as well as polycrystals. There are three principal themes that will be investigated in the context of multi-scale modeling of these material systems: (i) macroscopic, or effective behavior, (ii) field fluctuations, and (iii) microstructure evolution. Close interactions with experimentalists and numerical analysts will ensure the practical relevance of the work, as well as the development of numerical tools for eventual industrial use. The effective behavior serves to characterize the average response of heterogeneous materials at a sufficiently large length scales, and can be estimated by means of suitable homogenization techniques. Although homogenization estimates are already available for composites and polycrystals, it is proposed here to make use of the "second-order'" method that has been developed recently by the PI. This method has been found to deliver accurate estimates for some model, two-dimensional problems and it is proposed to apply it to model the effective behavior of three-dimensional viscoplastic composites and polycrystals using experimentally measured microstructural information. This "second-order'" method has the further advantage that it automatically delivers estimates for the "second moments" of the field fluctuations. The field fluctuations can be used to measure the strain and stress heterogeneities in the constituent phases of a composite, or grains in a polycrystal. Such information can be useful to model the development of twinning and other instabilities in polycrystals, as well as incipient failure in composites. The field fluctuations can also be used to generate more accurate (and smoother) predictions for texture evolution. Analogous investigations can also be carried out in the context of porous materials, where the microstructure (pore size, shape, orientation and distribution) is also known to evolve during a typical deformation process, such as extrusion or hot forging. Parts of this work will be carried out in close collaboration with three C.N.R.S. laboratories in France. Complementary numerical and experimental investigations will be carried out in these laboratories, which will be supported by the CNRS in France. A separate proposal will be submitted to the Division of International Programs for additional travel expenses under the NSF/CNRS cooperative scheme.

### Micro-Macro Estimates For Low-Symmetry Polycrystals (NSF CMS-9972234)

The computation of the effective or average mechanical response of polycrystalline aggregates is a classical problem dating back to the early work of Taylor. Recently, we have been extending the "variational" and "second-order" homogenization procedures to generate estimates for the effective behavior of viscoplastic polycrystals. Preliminary results for a model two-dimensional  nonlinear polycrystal are very encouraging, especially for low symmetry, high-anisotropy crystal systems (e.g., HCP materials). For more details, refer to deBotton and Ponte Castañeda (1995), Ponte Castañeda and Nebozhyn (1997) (see below for full references). Project funded by NSF.

Nonconvex Homogenization and Applications (NSF DMS-9971958)

This award will support research on a challenging class of mathematical problems in the field of nonlinear homogenization. Specifically, the project is concerned with the study of the effective or "homogenized" behavior of finite elastic composites such as elastic foams or particle-reinforced rubber. These are heterogeneous material systems that are characterized by nonlinear and in fact nonconvex local energy functions. The key idea is to make use of available homogenization estimates for suitably chosen "linear comparison composites" to generate corresponding estimates for the nonlinear composites. This is an idea that has been pioneered with success by the PI in the context of nonlinear problems with convex energy functions (e.g., plasticity, nonlinear conductivity, etc.). By suitably enlarging the class of "trial" linear comparison media, a generalized version of the method will be developed that will be suitable for nonconvex problems. The objective of this research is thus to develop mathematical methods for estimating the macroscopic behavior of nonlinear elastic composite materials that are subjected to finite deformations. The work will be directly relevant to a range of technologically important material systems, from the commonplace, such as elastomeric foams, to the more sophisticated, such as shape-memory alloys. In particular, the work will focus on models for particle-reinforced elastomers (e.g., carbon-black filled rubbers) and porous elastic solids (e.g., polymeric foams). As a result, effective estimates for the response of such materials will be produced that are easy to use by the engineering community and that can be compared to experimental studies.

### Heterogeneous Materials With Evolving Microstructure: Constitutive Modeling and Applications to  Localization in Manufacturing Processes (NSF CMS- 9622714)

This project will study the evolution of microstructural changes during finite deformations of heterogeneous materials. Internal variables will be used to characterize the current state of the microstructure. These internal variables will be governed by incremental relations. Special attention will be given to shear loadings which frequently are neglected in earlier models. Several problems in the area of metal forming and ductile fracture will be analyzed. These models are expected to be applicable to porous and particle reinforced aluminum.

Shear localization in metals plays an important role in restricting the potential applications of several manufacturing processes, such as hot forging and extrusion. This difficulty is compounded by the presence of microstructural defects that may enhance, or delay the onset of localization during processing. This project is concerned with the theoretical development of constitutive models for porous materials, accounting for the evolution of the microstructure, that will be useful in assessing the effect of porosity on localization.  For more details, refer to Ponte Castañeda and Zaidman (1994) and Kailasam and Ponte Castañeda (1998) (see below for full references). Project funded by NSF and AFOSR.

Porosity evolution in uniaxial compaction of a porous metal. Comparison of the predictions (continuous line) of the anisotropic model of Ponte Castañeda and Zaidman (1994) with the experimental results (dark circles) of Haghi (1992; Ph.D. Thesis, Mechanical Engineering, M.I.T) and the predictions of the Gurson model (dotted line). Note that the anisotropic model, which captures the development of anisotropy due to the change in shape of the pores, is in much better agreement with the experimental results than the isotropic model of Gurson. Reprinted from M. Kailasam (1998; Ph.D. Thesis, Mechanical Engineering and Applied Mechanics, University of Pennsylvania.)

Axisymmetric extrusion of porous metal. (a) Contour maps of anisotropy (i.e., pore shape); (b) Contour maps of distribution of the anisotropy axes (i.e., orientation of the pores). Reprinted from M. Kailasam (1998; Ph.D. Thesis, Mechanical Engineering and Applied Mechanics, University of Pennsylvania.)

Effective flow stress of model two-dimensional polycrystal as function of grain anisotropy for a value of the strain-rate sensitivity parameter equal to 0.1. Comparison of "variational" and "second-order" estimates with Taylor and Sachs bounds and predictions based on other schemes. Note that the classical incremental and secant schemes violate the recently established upper bound of Kohn and Little (1998; to appear in SIAM Journal on Applied Mathematics), while the "variational" and "second-order" estimates satisfy the bound. Reprinted from Bornert and Ponte Castañeda (1998).

### Nonlinear Composite Materials

The tremendously successful studies of the effective or average properties of composite materials have mostly dealt so far with linear constitutive behaviors. However, more often than not nonlinear effects are critical in the understanding of the effective properties of real engineering materials. This project addresses nonlinear constitutive and kinematical effects as observed in low-temperature plasticity and high-temperature creep of metals, as well as in the large deformation of polymers. For details, refer to the recent review article by Ponte Castañeda and Suquet (1998) "Nonlinear Composites." Advances in Applied Mechanics 34, 171-302. Project funded by NSF and ONR.

Yield surfaces for fiber-reinforced composites as functions of the fiber-to-matrix flow stress ratio. Comparison of predictions of variational procedure of Ponte Castañeda (1991) (continuous lines) with FFT numerical simulations of Moulinec and Suquet (1995) (points). In this figure, S33 denotes the macroscopic stress in the direction of the fibers and S11 the corresponding shear stress  transverse to the fibers. The fiber volume fractions is 50%. Reprinted from P. Ponte Castañeda and M. Zaidman  (1996) "On the finite deformation of nonlinear composite materials. Part I. Instantaneous effective potentials." International Journal of Solids and Structures 33, 1271-1286.

Maps of accumulated plastic strain in the transverse plane according to the numerical simulations for the fiber-reinforced composites of Figure 1. (a) Pure shear transverse to the fibers; (b) and (c) Combined in-plane shear and axial stress; (d) Pure uniaxial stress along the fibers. (Note that state (b) corresponds to the stress at vertex of the yield surface in Figure 1.) Reprinted from Moulinec and Suquet  (1995) "An FFT-based numerical method for computing mechanical properties of composite materials from images of their microstructure." In Microstructure-Property Interactions in Composite Materials (R. Pyrz, ed.) Kluwer, Dordrecht, The Netherlands, pp. 235-246.