| dc.description.abstract | This thesis focuses on three areas of nonclassical continuum mechanics of solids. In the first part, we develop a few peridynamics (PD) models and solution strategies for discretized PD equations in the context of the mechanics of solids and structures. A strategy for removal of zero energy modes in PD correspondence models is presented in the second part of this thesis and a way of modelling solid continua with defects is outlined drawing upon analogies from gauge theory. In the last part, exploring conformal gauge symmetries in elastic solids, we show how several electromechanical and magnetomechanical phenomena can emerge solely from local conformal symmetry considerations of the Lagrangian.
We start with formulating a PD theory for thick linear elastic shells to model fracture and fragmentation in these structures. Effects of shear deformation and coupling between surface wryness with in-plane stress resultants and surface strain with moment resultants are considered. A few numerical simulations on thick plate, thick cylindrical shell and quasi-static fracture propagation on thick cylindrical shell are presented. Next, a reduced dimensional PD theory is developed for axisymmetric structures. Apart from reduction of computational burden, it eliminates stress singularity near the axis of symmetry due to the nonlocality in PD. We furnish a few numerical simulations on Taylor impact test with copper and steel specimens and compared them with experimental observations. After that, inelastic response of ceramics is investigated using phase field based PD theory to eliminate some of the limitations of Deshpande-Evans (DE) ceramics constitutive model. A macroscopic PD phase field based integro-differential damage evolution rule is used replacing DE crack growth law which removes possible mesh dependent solutions. We numerically solve a spherical cavity expansion problem using dimensional reduction and demonstrate evolution of damage and plastic fronts. Next, a general procedure for solving discretized PD continuum and atomic systems is presented using Hamilton-Jacobi theory and time-dependent perturbation techniques. Here, approximate analytical solutions of positions and momenta are obtained as functions of initial conditions and time with which separate analysis for each initial condition can be eliminated resulting in saving in computational time. A few simulations on linear discretized PD problems are furnished to demonstrate the efficacy of our method. We also solved graphene sheets under tension and shear loading using simplified Tersoff potential for given initial conditions. After that, flexoelectricity – an electromechanical coupling phenomenon is modeled in PD to investigate nanoscale fracture propagation in dielectrics. An analytical solution is presented for an infinite 3D body considering bond based case. Incorporating damage through phase field theory, we present a few numerical simulations on damage propagation in a flexoelectric plate.
In the second part of the thesis, we develop a sub-horizon based PD theory to eliminate zero-energy and other unphysical deformation modes from the correspondence framework of non-ordinary state based PD which requires only a minor alteration of the conventional PD correspondence equations and little additional computational demand. With this, one may study convergence of the solutions for a fixed horizon size with increasing particle density and obtain meaningful nonlocal solutions. We also outlined a way to model defective continua in this framework drawing upon analogies from a translation invariant gauge theory of solids.
In the last part, a conformal gauge theory of solids is laid out. We note that, if the pulled back metric of the current configuration (right Cauchy-Green tensor) is scaled with a constant, the volumetric part of Lagrange density changes but the isochoric part remains invariant. However, under a position dependent scaling, isochoric part loses its invariance. In order to restore invariance of the isochoric part, we introduce a 1-form compensating field and modify the definition of derivative to a gauge covariant one (minimal replacement). Noting close connection with Weyl geometry, we impose Weyl condition through the Lagrangian and for the evolution of 1-form, a minimal coupling is constructed. We obtain Euler-Lagrange equations from Hamilton’s principle and noticed a close similarity with flexoelectricity governing equations interpreting the exact part of 1-form with electric field and the anti-exact part with the polarization vector. Next, piezoelectricity and electrostriction phenomena are modeled through contraction of Weyl condition in various manners. We also modeled magnetomechanical phenomena applying Hodge decomposition theorem on the 1-form which leads to curl of a pseudo-vector field and a vector field. Identifying the pseudo-vector field with the magnetic potential and vector part with magnetization, flexomagnetism, piezomagnetism and magnetostriction phenomena are modeled. | en_US |
