The unifying theme of my research interests is to bridge the gap between solid mechanics and mathematics in problems that matter in real applications.

Nonlinear Mechanics of Defects and Differential Geometry. Defects control many of the interesting properties of solids. There has been an intense effort in understanding the role played by defects in mechanical and physical properties of solids in the computational physics, materials science, and mechanics communities. Historically, defects in solids were mathematically predicted by the Italian mathematician Vito Volterra (1907). It took almost another thirty years for an experimental observation of dislocations. G.I. Taylor, E. Orowan, and V.M. Polyani independently in 1934 presented the first theoretical analyses of the relevance of dislocations in plasticity of metals. Analysis of defects has been overwhelmingly restricted to linear elasticity in almost all the existing theoretical works. Very little is known about the stress fields and energetics of defects in nonlinear solids. In the 1950s Kondo (1955) and Bilby and his coworkers (1955) independently investigated the deep connections between non-Riemannian geometries and the mechanics of distributed defects. Their studies were mainly kinematic, and unfortunately, these geometric works remained mostly formal without any stress calculations for distributed defects.

My efforts in the past ten years have resulted in a fully nonlinear geometric theory that can be used to generate exact solutions for defects in nonlinear solids. These include dislocations, disclinations, point defects, and combinations of line and point defects that we call discombinations. In particular, I obtained (in collaboration with A. Goriely) the first exact solution for the stress field of distributed point defects in nonlinear solids since the corresponding work in linear elasticity by A.E.H. Love more than 90 years ago. I have also done work on clarifying the geometric meaning of Burgers’ vector in dislocation mechanics and the effect of material inhomogeneities and anisotropies on the stress fields and energetics of defects in nonlinear solids.

More recently, in collaboration with a former student (Fabio Sozio) I have been working on formulating a geometric field theory of dislocation dynamics and finite plasticity in single crystals. This is done by starting from the multiplicative decomposition of the deformation gradient into elastic and plastic parts. Cartan’s moving frames are used to describe the dislocated lattice structure via differential 1-forms. In this theory the primary fields are the dislocation fields, defined as collections of differential 2-forms. The defect content of the lattice structure is then determined by the superposition of the dislocation fields. All these differential forms constitute the internal variables of the system. The evolution equations for the internal variables are derived starting from the kinematics of the dislocation 2-forms, which is expressed using the notions of flow and of Lie derivative. This is then coupled with the rate of change of the lattice structure through Orowan’s equation. The governing equations are derived using a two-potential approach to a variational principle of the Lagrange-d’Alembert type. As in the nonlinear setting the lattice structure evolves in time, the dynamics of dislocations on slip systems is formulated by enforcing some constraints in the variational principle. Using the Lagrange multipliers associated with these constraints, one obtains the forces that the lattice exerts on the dislocation fields in order to keep them gliding on some given crystallographic planes. Moreover, this geometric formulation allows one to investigate the integrability—and hence the existence—of glide surfaces,  and how the glide motion is affected by it. Linearizing the governing equations of this nonlinear theory allows one to identify the nonlinear effects that do not appear in the linearized setting.

Differential Complexes of Nonlinear Elasticity and Mixed Finite Element Methods. Differential complexes are sequences of linear operators such that the image of each operator is a subset of the kernel of the next operator. The differential complex of linear elasticity introduced by Ekkehart Kröner (1959) contains information about some topological properties of elastic bodies. Arnold and Winther (2002) obtained compatible finite element spaces for the mixed formulation of linear elasticity by appropriately discretizing the linear elasticity complex such that the discrete complexes preserve all the topological information of the linear elasticity complex. By generalizing this approach, Arnold and his coworkers showed that it is also possible to obtain stable mixed methods for some linear operators associated to specific classes of differential complexes. In collaboration with a former PhD student we showed that the linear elastostatics complex on flat spaces is equivalent to the Calabi complex, which is a well-known complex in differential geometry. We introduced various complexes for nonlinear elastostatics for the first time in the literature. In particular, in terms of the deformation gradient and the first Piola-Kirchhoff stress tensor, we obtained a differential complex for nonlinear elastostatics that is isomorphic to the R3-valued de Rham complex. We have also worked on Hilbert complexes of nonlinear elasticity that have applications in structure-preserving mixed finite element methods.

It has been known for quite some time in the finite element literature that internal constraints, e.g., incompressibility constraint should be treated very carefully to avoid numerical artifacts and instabilities. One path for developing efficient and robust numerical schemes for incompressible elasticity is the use of mixed finite elements. In collaboration with two former PhD students we have introduced a new family of mixed finite elements (CSMFEs) for compressible and incompressible nonlinear elasticity. We formulated a four-field mixed formulation with the displacement, the displacement gradient, the first Piola-Kirchhoff stress, and a pressure-like field as the four independent unknowns. Using the Hilbert complexes of nonlinear elasticity we identify the solution spaces of the independent unknown fields. In particular, we define the displacement in H1, the displacement gradient in H(curl), the stress in H(div), and the pressure field in L2. The test spaces of the mixed formulations are chosen to be the same as their corresponding solution spaces. Next, in a conforming setting, we approximate the solution and the test spaces with some piecewise polynomial subspaces of them. This approach results in compatible-strain mixed finite element methods (CSFEMs) that can be considered structure preserving in the sense that the differential complex structure of nonlinear elasticity is preserved at the discrete level. In particular, both the Hadamard compatibility condition and the continuity of traction are satisfied at the discrete level independently of the refinement level of the mesh. By considering several numerical examples, we have demonstrated that CSFEMs are capable of capturing very large strains and accurately approximating stress and pressure fields. Using CSFEMs, we do not observe any numerical artifacts, e.g., checkerboarding of pressure, hourglass instability, or locking in our numerical examples. Moreover, CSFEMs provide an efficient framework for modeling heterogeneous solids.

Nonlinear Non-Euclidean Elasticity (Anelasticity): Nonlinear Analogues of Eshelby’s Inclusion Problem. The term “non-Euclidean solids” was coined by Henri Poincaré in 1902, and refers to mathematical objects that represent solids with distributed eigenstrains, and hence residual stresses. Anelasticity is the study of finite deformations of bodies that, in addition to elastic deformations, undergo non-elastic deformations or microstructural changes due to other physical, chemical, or biological processes, e.g., bulk growth and remodelling, accretion, swelling in gels, plasticity, thermal expansion/contraction, diffusion, etc. We refer to strains due to non-elastic deformations as anelastic strains or eigenstrains.

Classically, inclusions in an elastic body are pieces of elastic materials that have been inserted into the material. For instance, in the simplest case a spherical elastic ball is compressed or stretched to fit inside a given spherical shell. The problem is then to find the stress in the new ball and the deformation of both materials. In general, the problem of inclusions is to combine two different stress-free bodies and constrain them geometrically so that they create a new, possibly residually stressed, body. More generally, we can consider a single body and assume that the body undergoes a local change of volume described by general eigenstrains (prestrains), the particular shrink-fit problem corresponding to uniform dilatational eigenstrains. Physically these eigenstrains can be generated by thermal expansions, swelling, shrinking, growth, or any other anelastic effects. In the linearized setting, Eshelby (1957) calculated the stress field of an ellipsoidal inclusion with uniform eigenstrains using superposition. For the special class of harmonic materials there are some recent 2D solutions for inclusions. In the nonlinear case, there were no explicit three-dimensional analytic solutions for inclusions before my work in collaboration with A. Goriely. More specifically, we obtained the stress field of a ball made of an arbitrary incompressible isotropic nonlinear elastic solid with a spherically-symmetric eigenstrain distribution. In the case of an ellipsoidal inclusion in an infinite linearly elastic isotropic solid, Eshelby observed that stress inside the inclusion is uniform. As a special example, we considered a spherical inclusion at the center of the ball. We showed that for any incompressible isotropic nonlinear elastic solid with pure dilatational eigenstrain, the stress inside the inclusion is uniform and hydrostatic. Recently, I extended this analysis to arbitrary compressible solids, twist eigenstrains, finite eigenstrains in nonlinear elastic wedges and tori, and nonlinear anisotropic solids.

Universal Deformations in Nonlinear Elasticity and Anelasticity. For a given class of solids, it turns out that one cannot deform an elastic body to an arbitrary shape by only applying boundary tractions; most likely body forces are needed to maintain the desired deformation. Those deformations that can be maintained by only applying boundary tractions are called universal or controllable. The set of universal deformations explicitly depends on the class of materials. The systematic study of this problem began in the 1950s in the seminal woks of Jerry Ericksen. The anelastic analogue of Ericksen’s problem is: Determine both the deformations and the eigenstrains such that a solution to the anelastic problem exists for arbitrary strain-energy density functions. Anelasticity is described by finite eigenstrains. In a nonlinear solid, these eigenstrains can be modeled by a Riemannian material manifold whose metric depends on their distribution. In this framework, I showed that the natural generalization of the concept of homogeneous deformations is the notion of covariantly homogeneous deformations—deformations with covariantly constant deformation gradients. It was proved that these deformations are the only universal deformations and that they put severe restrictions on possible universal eigenstrains. It was shown that, in a simply-connected body, for any distribution of universal eigenstrains the material manifold is a symmetric Riemannian manifold and that in dimensions two and three the universal eigenstrains are zero-stress.

In the incompressible case my collaborators and I showed that i) all the known universal deformations are symmetric with respect to some Lie subgroups of the special Euclidean group, and ii) Ericksen’s problem can be extended to its anelastic version. We explicitly characterized the universal eigenstrains that share the symmetries present in the classical problem, and showed that in the presence of eigenstrains, the six known classical families of universal solutions merge into three distinct anelastic families, distinguished by their particular symmetry group. Some generic solutions of these families correspond to well-known cases of anelastic eigenstrains. Additionally, we showed that some of these families possess a branch of unconventional solutions, and demonstrate the unique features of these solutions and the equilibrium stress they generate.

Another related problem is the characterization of the universal displacements of linear elasticity: those displacement fields that can be maintained by applying boundary tractions in the absence of body forces for any linear elastic solid in a given anisotropy class. I showed that the universal displacements for compressible isotropic linear elastic solids are constant-divergence harmonic vector fields. It was also shown that any divergence-free displacement field is a universal displacement for incompressible linear elastic solids. Further, the universal displacements for all the anisotropy classes (triclinic, monoclinic, tetragonal, trigonal, orthotropic, transversely isotropic, and cubic) were characterized. As expected, universal displacements explicitly depend on the anisotropy class: the smaller the symmetry group, the smaller the space of universal displacements. In the extreme case of triclinic materials where the symmetry group only contains the identity and minus identity, the only possible universal displacements are linear homogeneous functions. Recently, I extended the study of universal deformations to anisotropic solids. Assuming that the material preferred directions respect the symmetries of the universal deformations, the universal material preferred directions for transversely isotropic, orthotropic, and monoclinic nonlinear solids were obtained. In another recent work I characterized the universal material inhomogeneities for both compressible and incompressible nonlinear isotropic solids.

Nonlinear Mechanics of Accretion. Mechanics of growing bodies has been a focus of intense research in recent years partly because of its emerging applications in biomechanics and additive manufacturing. Structures that are built by a process of accretion have been common in engineering for a long time, e.g., built up of concrete dams in successive layers, metal solidification, layer-by-layer gluing of composites, wound rolls, thermal and laser-based 3D printing, etc. There are many examples of accretion in Nature, e.g., volcanic and sedimentary rock formation, ice and snow cover build-up, formation of planetary objects, equilibrium shape configurations for rubble piles, the growth of biological tissues, etc. In a growing body mass is not conserved and the body after unloading may be residually stressed. There are two types of growth: bulk growth and surface growth (accretion). In bulk growth, in the language of continuum mechanics, material points are preserved; only the mass density and the natural (stress-free) configuration of the body evolve. Due to its similarity with finite plasticity the idea of the decomposition of the deformation gradient into elastic and growth parts has been borrowed from plasticity. There are many theoretical and computational works in the literature of (bulk) growth mechanics. In surface growth, instead, new material points are added to or are removed from the boundary of a deforming body, i.e., the set of material points is time dependent. Moreover, the relaxed (natural) configuration of the body explicitly depends not only on the surface growth characteristics (accretion flux, accretion velocity, etc.), but also on the history of loading and deformation during accretion. The mechanics of surface growth is much less developed mainly because of the complexities involved in modeling the kinematics of accretion and the intrinsic incompatible nature of accreting bodies.

Surface growth can be seen as the study of the formation of non-Euclidean solids through a continuous joining of infinitely many two-dimensional layers. In the past few years I have worked on formulating a geometric theory of the nonlinear mechanics of accreting bodies. In this formulation the stress-free (natural) state of an accreting body is a time-dependent Riemannian manifold with a time-independent metric that is non-flat, in general. The material metric, and consequently the Riemannian structure of a material manifold, depends on the growth velocity and the state of deformation of the body at the time of attachment of the new material points. In this accretion theory the Riemannian material structure is not known a priori; the material metric is calculated after solving the accretion boundary-value problem. Our geometric theory is capable of modeling very complex nonlinear accretion problems. We have demonstrated this using several numerical examples. For future work I plan on developing numerical schemes that can be used in the analysis of very complex accretion problems for additive-manufacturing applications.

Nonlinear and Linear Elastodynamic Transformation Cloaking. Invisibility has been a dream for centuries. Making objects invisible to electromagnetic waves, for example, is both theoretically and practically important and has been a subject of intense research in recent years. Pendry and co-workers (2006) and Leonhardt and co-workers (2006) independently showed the possibility of electromagnetic cloaking. This was later experimentally verified. The idea of transformation cloaking has been explored in many other fields, e.g., acoustics, optics, thermodynamics (design of thermal cloaks), diffusion, and elastodynamics (Milton, et al., 2006, Yavari and Golgoon, 2019, 2021). The least understood among these applications is elastodynamics. In my opinion, the main problem is that most of the existing works have formulated the problem of elastodynamic transformation cloaking incorrectly.

With a former PhD student (Ashkan Golgoon), we formulated the problems of nonlinear and linear elastodynamic transformation cloaking in a geometric framework. In this formulation a cloaking transformation is introduced that maps the boundary-value problem of an isotropic and homogeneous elastic body (virtual problem) to that of an anisotropic and inhomogeneous elastic body with a hole surrounded by a cloak that is to be designed (physical problem). The virtual body has a desired mechanical response while the physical body is designed to mimic the same response outside the cloak. We showed that nonlinear elastodynamic transformation cloaking is not possible while nonlinear elastostatic transformation cloaking may be possible for special deformations, e.g., radial deformations in a body with either a cylindrical or a spherical cavity. In the case of linear elastodynamic, in agreement with the previous observations in the literature, we showed that the elastic constants in the cloak are not fully symmetric; they do not possess the minor symmetries. We proved that elastodynamic transformation cloaking is not possible regardless of the shape of the hole and the cloak. We showed that linear elastodynamic transformation cloaking cannot be achieved for gradient elastic solids and generalized Cosserat solids either; similar to classical linear elasticity the balance of angular momentum is the obstruction to transformation cloaking.

Many thin three-dimensional bodies can be idealized as elastic plates, which are, in turn, suitable models for quite a few physical, engineering, and biological systems. Recently, in collaboration with a former student (Ashkan Golgoon) we formulated a mathematical framework to investigate the transformation cloaking problem in Kirchhoff-Love plates and elastic plates with both in-plane and out-of-plane displacements. It turns out that none of the existing works in the literature has properly formulated the transformation cloaking problem in elastic plates. One of the new findings of this work is a set of cloaking compatibility equations. The cloaking compatibility equations are a system of second-order nonlinear partial differential equations (PDEs), and the balance of linear and angular momenta for the (physical) plate lead to fourth-order and third-order nonlinear PDEs. It was shown that the cloaking compatibility equations and the boundary and continuity conditions are obstruction to exact transformation cloaking in the case of circular holes and a generic radial cloaking map.

These no-go theorems are significant and imply that the path forward for engineering applications of elastic cloaking is approximate cloaking. For future work I plan on formulating the elastostatic and elastodynamic cloaking as optimal design problems. Given a set of static or dynamic loadings the goal will be to design a cloak (with inhomogeneous and perhaps anisotropic elastic properties) that makes the response of the outside body as close as possible to that of the corresponding body without any holes or inhomogeneities.