# Research

The research activities of Kaushik Bhattacharya are at the intersection of Mechanics, Materials Science, and Applied Mathematics. Concepts in Mechanics and recent methods of Mathematics are used to generate ideas for the design, development, and directed discovery of new materials, the optimization of materials processing, and their exploitation in new applications. Virtually every material contains different features at different length scales and undergoes processes at a variety of time scales. For example, even the most straightforward piece of metal is typically made up of many crystallites (grains), which are made up of many atoms. This complexity is only compounded in sophisticated modern materials. Macroscopic applied loads and fields affect the microscopic structure; conversely, the microscopic structure affects the macroscopic behavior. Therefore, bridging length scales is a key theme, which is addressed in various materials and materials systems.

**Active materials like shape-memory alloys, ferroelectrics, soft active materials**

These materials are of inherent interest due to their unusual properties coupling mechanical, thermal, electrical, magnetic, optical, and other properties. Further, their characteristic microstructure makes them ideal for developing tools for multiscale modeling. While earlier activity studied hard active materials, recent activity considers soft materials like liquid crystal elastomers. The interest is understanding fundamental mechanics and physics that make endow these materials with unusual properties and then using this understanding to identify new materials, identify new modes of applications in soft robotics, implantable medical devices, reconfigurable devices, and structural damping, and to build efficient engineering models that accurately describe the complexity of these materials. Current work focuses on anomalous soft behavior of LCEs, combining controlling structural instabilities and actuation at a distance using light and other electromagnetic radiation.

**Fracture and related phenomena in heterogeneous media**

We seek to understand the process of fracture in a heterogeneous material where cracks can get pinned at interfaces, jump across regions, kink, and meander to avoid obstacles, and nucleate daughter cracks distally. We have developed an experimentally validated approach to quantifying the overall toughness against macroscopic failure and used this understanding to design materials with unusual fracture toughness. Recent work has developed a "fracture diode", where the resistance of a medium to a crack propagating from right to left can be significantly different from that to a crack propagating from left to right, and a composite medium that is both strong and tough. Current work focuses on the validation of computational models through in-situ experimental observations of crack propagation in three-dimensional heterogeneous specimens. Similar work concerns adhesion, friction, interface propagation. Mathematically, these involve the homogenization of free boundary or free discontinuity problems, and the effective behavior is dominated by extreme statistics.

**Optimal design of microstructure**

We study a variety of problems where we seek to find the micro and nanostructure that optimizes macroscopic performance. Recent examples include the optimal design of microstructure in energy conversation devices (battery, fuel cell), responsive materials for actuation, and meta-materials for multi-functionality and mechanical robustness. Current interest is in time-dependent phenomena including failure, and material synthesis.

**Mechanics of slender structures**

Slender structures -- where one or more dimensions are much smaller than the others as in strings, rods, membranes, plates, shells, etc.-- are common in nature and have been widely exploited in engineering. The slenderness and the resulting flexibility coupled with possible geometric constraints endows such structures with a rich range of mechanical responses. Thus the study of the mechanics of slender structures has been the subject of much research, though much of this literature relates to materials whose constitutive response is relatively simple. Our interest lies in slender structures made of materials like active materials whose constitutive behavior is highly nonlinear. Here, the geometric nonlinearity of the structure and the constitutive nonlinearity of the material can give rise to complex phenomena. Recent work has focussed on the robust actuation of thin sheets and the control of geometric structural instabilities. Current work focusses on obtaining periodic motion from steady sti

**Density functional theory for defects in crystalline solids**

Crystalline defects mediate numerous properties in metals and ceramics: vacancies affect creep and strength, dislocations provide plasticity, and interfaces affect strength and toughness. Remarkably they do so at extremely dilute concentrations since they directly link the quantum mechanics of the core to the macroscopic elastic and electrostatic fields. We study the structure of defects directly using density functional theory, a first-principles ab initio method. We have developed a sub-linear scaling method, MacroDFT, that provides accurate solutions to specimens involving millions of electrons. We use this method to study point and cluster defects, dislocation cores, twin boundaries, etc.

**Machine learning for multiscale modeling of material**

The macroscopic properties of materials that we observe and exploit in engineering applications result from complex interactions between physics at multiple lengths and time scales: electronic, atomistic, defects, domains, etc. Multiscale modeling seeks to understand these interactions by exploiting the inherent hierarchy where the behavior at a coarser scale regulates and averages the behavior at a finer scale. We seek to address the problems of scale transitions by exploiting machine learning to create efficient but accurate data-driven approximations. We formulate the problem of scale-transitions as input-output maps between infinite-dimensional spaces and develop methods to approximate them combining model reduction and neural networks. This results in methods that provide the accuracy of concurrent multiscale methods at the cost of traditional constitutive behavior. Recent work has demonstrated this approach on inelastic impact polycrystalline plasticity and density functional theory of crystalline solids. Current work focuses on identifying hidden variables that arise as a result of scale transitions.

Last updated: September 2021