Research
My research interests lie in the study of mechanics of solid material systems, ranging from inert solids such as metals, alloys, polymers and hydro-gels to living and growing material such as biological tissues, using analytical and computational tools. My graduate research work so far has primarily been with Professor Tal Cohen and is largely centered around the mechanics of soft solids. My undergraduate research work was on ductile fracture in metals, with Professor Shyam Keralavarma at IIT Madras. Below are some of the topics I have worked on or am currently working on.
Understanding the mechanics of growth and morphogenesis through a large deformation swelling-growth theory
Understanding the growth of soft biological systems is crucial in a wide range of applications with extensive societal consequences, such as plastic and reconstructive surgery, curbing the growth of tumors and bacterial colonies, tissue engineering of functional vascular grafts, etc. Solid tumors account for more than 85% of cancer mortality, and bacterial biofilms account for a significant part of all human microbial infections. Mechanics plays a crucial role in determining how these systems grow and acquire their shape (morphogenesis). My PhD dissertation work was in elucidating the mechanics of growth and morphogenesis in such soft systems, starting from universal underlying mechanisms.
Growing biological systems are a mixture of fluid and solid components and increase their mass by intake of diffusing species such as fluids and nutrients (swelling) and subsequent conversion of some of the diffusing species into solid material (growth). Experiments indicate that these systems swell by large amounts and that the swelling and growth are intrinsically coupled, with the swelling being an important driver of growth. However, most existing theories for swelling coupled growth employ linear poroelasticity, which is limited to small swelling deformations, and employ phenomenological prescriptions for the dependence of growth rate on concentration of diffusing species and the stress-state in the system. In particular, the termination of growth is enforced through the prescription of a critical concentration of diffusing species and a homeostatic stress. In contrast, by developing a fully coupled swelling-growth theory that accounts for large swelling through nonlinear poroelasticity, we show that the emergent driving stress for growth automatically captures all the above phenomena.
Below we show the ability of theory to model volumetric growth behavior. Specifically, the thermodynamic driving stress is able to simultaneously capture the effect of both applied stresses and diffusion-consumption constraints without ad hoc phenomenological prescriptions. It is able to reproduce conventionally assumed dependence of growth rate on concentration of diffusing species - increasing with increasing concentration, saturation at high values and unfavorable growth below a critical value. At the top, we show the ability of the theory to model morphogenesis that arises under mechanical confinement. The theory also shows that the shape of dynamically growing inclusions evolves similar to fluid filled expansion of voids when the remodelling processes (such as cellular rearrangement and microstructural evolution) are much faster than the volumetric growth processes (such as cell division and extracellular matrix production), a result previously also predicted in our previous work (Li et al., 2022) where we proposed a growth law for shape evolution based on energy minimization.
Probing local nonlinear viscoelasticity in soft solids
Characterizing the local mechanical properties of soft and biological materials is crucial in understanding several fundamental biological processes and in several medical and engineering applications. Classical testing methods face several limitations when it comes to testing of very soft materials such as sample preparation in specific shapes, boundary effects, and inhomogeneous deformation, apart from usually being able to only capture the bulk material properties. Biological materials and tissues are often heterogeneous with spatially varying properties and also show significant differences in mechanical behaviour between in-vivo and ex-vivo conditions. Thus minimally invasive testing methods that can characterize mechanical properties locally and in-vivo are important. Needle based measurement techniques such as Cavitation Rheology and Volume Controlled Cavity Expansion (VCCE), allow for minimally invasive local mechanical testing, but have been limited to measuring the elastic (quasi-static/rate-independent) material properties. In this paper we published in the Journal of Mechanics and Physics of Solids, we propose several enhancements to the VCCE technique to adapt it for characterization of rate-dependent material response at low to medium stretch rates. We demonstrate that the enhanced method becomes essential for accurate characterization of even the quasistatic material properties, since the loading protocol used in previous studies can lead to viscoelastic stiffening and saturation of material response that would give an illusion of quasi-static conditions.
Shear shock evolution in incompressible soft solids
My Master's work at MIT was on nonlinear evolution of shear waves in soft solids. A material whose shear stress-strain response is linear would relay a shear wave loading waveform imparted on its surface unchanged (ignoring dissipation) at the constant linear elastic shear wavespeed. On the other hand, in materials with a nonlinear shear response, the shape of the velocity or shear strain profile of the shear wave can evolve as the wave is relayed through the material. For a shear stiffening material, this nonlinear effect would steepen the loading waveform as it is being relayed eventually resulting in a shock.
Steepening of loading wave into a shear shock (shock location indicated by red cross)
Spreading of loading wave in shear softening material
A recent study demonstrates that such shear shocks could be a primary damage mechanism in Traumatic brain injuries (TBI's) due to high local accelerations and gradients at the shock. Therefore understanding how these shocks form and how far they take to evolve based on the loading/impact on the material and the mechanical properties of the material itself, is of paramount importance in beginning to design protective structures or studying the phenomenon in laboratory settings. Most analyses of nonlinear shear waves have been by the acoustics community that employs reduced wave equations using assumptions of weak non-linearity or small deformations, assumptions that may not hold for realistic loading scenarios. In our paper in the Journal of Mechanics and Physics of Solids, we present the first closed form solutions that solve for the distance taken for shock formation while solving the fully nonlinear elastodynamics equations. We encapsulate our results in non-dimensional phase plots that determine regimes in which a shock can be realized. We use our results to study a simplistic version of the problem of shear impact of the human brain and find that our results are in agreement with the experiments of the brain study. It is also suggested that the non-dimensional maps can guide the design of protective structures by determining the combination of loading parameters, material dimensions, and elastic properties that can avoid shock formation.
Recently postdoctoral research fellow Harold Berjamin (NUI Galway) and I published a paper in a special issue of Wave Motion where we analyze the same shear shock problem while also accounting for viscosity in the solid. We demonstrate once again that usual small deformation approximations employed in acoustics lead to inaccurate predictions, emphasizing the need for large deformation frameworks in analyzing shear shock evolution for real world applications.
Observation of ultra-slow shockwaves in tunable magnetic lattice
Nonlinear evolution of shock-waves have been traditionally difficult to observe in experiments and the first report of observation of shear shock evolution for example was only in the 2000s. In this paper published in Physical Review Letters, we develop a simple tunable nonlinear system using an array of magnets that allows us to demonstrate easily observable nonlinear shockwave evolution in a table-top experimental setup. The discrete nature of the system allows us to draw parallels to atomic scale shock-wave modelling and we show that the magnetic lattice system preserves the key quantitative features of shock-wave propagation in an analogous Copper atomic lattice system. Thus the system can potentially serve as a "magnifying glass" to illuminate shock processes.
Nonlinear inclusion theory with application to confined growth and morphogenesis
The work of J.D. Eshelby in the late 50s revolutionized our understanding of the elastic stress and strain fields due to an ellipsoidal inclusion/inhomogeneity that undergoes a transformation of shape and size. However Eshelby's solutions are limited to the linear elastic regime. In this paper in the Journal of the Mechanics and Physics of Solids, we develop an approximate theory for the growth of inclusions embedded in a solid medium for large deformations. Furthermore , instead of an inert inclusion undergoing a prescribed transformation strain, we allow for active re-modelling of the inclusion with the goal of modelling growth of active biological systems embedded in a soft matrix. Our experimental model system involves growth and morphogenesis of Vibrio cholerae biofilms embedded in hydrogels. While growth models typically prescribe a constitutive evolution law that ties together the volumetric growth rate and evolution of stress-free shape, the biofilm system re-organizes at a much faster rate than it grows and the volumetric growth rate is constant. Thus we prescribe kinematically prescribe the volumetric growth and propose a free energy minimization based path for evolution of the growth shape of the inclusion. While confined growth and morphogenesis is important in several systems such as tumors and bacterial biofilms, an interesting practical example where mechanical confinement affects the growth shape is in confined growth of vegetables and fruits such as shown below for tomatoes. This work was recently covered by MIT News. In my PhD dissertation, I developed a more general and accurate description of the growth process wherein the growth rates and shape evolution is determined by thermodynamic driving stresses arising from underlying biochemical processes. In the limit where remodelling processes such as cell re-arrangement is much faster than the volumetric growth processes such as cell division and extra-cellular matrix production, the free energy minimization based results are qualitatively reproduced.
Interfacial cavitation
Cavitation has long been recognized as a crucial predictor, or precursor, to the ultimate failure of various materials, ranging from ductile metals to soft and biological materials. Traditionally, cavitation in solids is defined as an unstable expansion of a void or a defect within a material. The critical applied load needed to trigger this instability – the critical pressure – is a lengthscale independent material property and has been predicted by numerous theoretical studies for a breadth of constitutive models. While these studies usually assume that cavitation initiates from defects in the bulk of an otherwise homogeneous medium, an alternative and potentially more ubiquitous scenario can occur if the defects are found at interfaces between two distinct media within the body. Such interfaces are becoming increasingly common in modern materials with the use of multi-material composites and layer-by-layer additive manufacturing methods. However, a criterion to determine the threshold for interfacial failure, in analogy to the bulk cavitation limit, has not been previously reported (in neo-Hookean or rubbery solids). In our recently published paper in PNAS Nexus, we fill this gap. Our theoretical model captures a lengthscale independent limit for interfacial cavitation, and is complemented with experimental results. In this colloborative work, my role was in the numerical simulations and establishing the cavitation limit.
Thermo-chemo-mechanically coupled cavity dynamics
Understanding the dynamics induced by confined thermo-chemical processes occurring inside a solid medium is a fundamental problem of interest in several applications, such as hotspot formation and micro-explosions and laser-induced cavitation for energy focusing and material characterization. In this paper in the Proceedings of the Royal Society A, we study the motion of a cavity under thermo-chemical loading by coupling the mechanics of the solid with the thermodynamics and the chemical kinetics inside the cavity. The model is shown to capture interesting experimental observations such as multi-phase energy bursts. More generally the dimensionless response of the system for any heat power supply loading is studied numerically and analytical expressions for the mechanical response limits are derived.
Surface patterns generated in hydrogels
Another problem that has interested me during my spare time is the study of surface patterns generated during hydrogel swelling and de-swelling in different media. However this exploration was put on hold following the COVID-19 situation. Below are some cool patterns that arise during swelling de-swelling in water.
Undergraduate Work
My undergraduate research work was under the supervision of Professor Shyam Keralavarma. My most substantial work with him was in developing and validating an analytic criterion for void coalescence in anisotropic ductile materials . Our work was published in the International Journal of Plasticity. My undergraduate thesis can be found here. I've spelled out the crux of some of my undergraduate research projects below.
Analytical criterion for void coalescence in anisotropic porous ductile solids
The primary damage process in metals undergoing ductile fracture is the nucleation, growth and coalescence of microvoids. Voids nucleate from multiple sources and grow in isolation due to plastic deformation of surrounding material when stressed under loading. When a critical condition is reached neighboring voids interact through plastic strain localization in the inter-void ligament, a process called void coalescence. Void coalescence is followed by quick plastic collapse of the inter void ligament and crack growth. As a result prediction of onset of void coalescence is crucial in predictive modeling of ductile fracture. Experiments and micro mechanical studies have indicated a strong influence of plastic anisotropy on material ductility. Dr Keralavarma and I have developed an analytical criterion for predicting void coalescence in anisotropic porous ductile solids using micromechanics and have validated it for the case of transversely isotropic materials through numerical simulations run on Abaqus. We first validated our finite element model, developed in Abaqus to calculate the limit load for void coalescence, using existing void coalescence criteria for isotropic materials. Subsequently we introduced anisotropy and studied the trends in numerical coalescence stress by which time we had developed an analytical criterion using micromechanics which agreed very well with the numerical predictions. Our analysis leads to a closed form expression for the coalescence function analogous to the Gurson yield function in the pre-coalescence regime which is exciting.
Some images from the Abaqus simulations run for isotropic and transversely isotropic materials modelling the coalescence of spheroidal and cylindrical voids under pure axial loading relative to the voids have been shown. The symmetry is exploited by modeling only a quarter of the cylindrical unit cell. Figures are for certain chosen anisotropy ratios and parameter ratios related to the void and ligament sizes (Scale of the stress plots have not been shown).
Abaqus models for predicting coalescence stress. Plots of Von Mises stress.
(Images not to scale)Integrated Void growth-coalescence model for anisotropic solids
This was my B.Tech project as part of my Honors program. Anisotropic void growth models were integrated with our void coalescence criterion and implemented as a UMAT which was subsequently tested using void coalescence simulations. Complete details can be found under the UG thesis tab at the top.
User Material Subroutine (UMAT) modeling anisotropic plasticity with strain hardening
Rolled sheet metals exhibit planar anisotropy which is generally described by the Lankford coefficient. Motivated by the potential appication in metal forming applications, I developed a User Material subroutine in Abaqus for modeling materials exhibiting anisotropic plasticity, described by the Quadratic Hill yield criterion with isotropic power law hardening, under plane stress condition. The convex cutting plane algorithm was used to model incremental plasticity behaviour.