| dc.description.abstract | Postulates in solid mechanics can be broadly classified into three groups describing the geometry of a configuration, mechanical equilibrium and thermodynamics of the deformation process. The geometry defined on a configuration provides us with tools to analyse deformation. The strain tensor in three dimensional elasticity and the curvature tensor encountered in a Kirchhoff type shell theory are consequences of the geometric hypotheses. The
first part of this thesis (Chapters 2-4 to wit) uses a class of affine connections to study the deformation of elastic continua, shells and continua with point defects which are representatives of theories with flat, extrinsic
and intrinsic geometries. As a first case, we study non-linear elasticity whose geometry is Euclidean. The mainstay of our approach is to treat quantities defined on the co-tangent bundles of reference and deformed configurations as
primal. Such a treatment invites compatibility equations so that the base space (configurations of the elastic body) can be realised as a subset of an Euclidean space; Cartan's method of moving frames and the associated structure equations establish this compatibility. The geometric understanding of stress as a co-vector valued 2-form fits squarely within this program. We also show that for a hyperelastic solid, a relationship akin to the Doyle-Eriksen formula may be
written for the co-vector part of the stress 2-form. Using this kinetic and kinematic understanding, we rewrite the Hu-Washizu (HW) functional in terms of frames and differential forms. We also show that the compatibility of
deformation, constitutive rules and equations of equilibrium are obtainable as Euler-Lagrange equations of the HW functional. Following the same spirit, we reformulate the kinematics of Kirchhoff shells using the theory of moving
frames. This reformulation permits us to treat the deformation and geometry of the shell as two equally important but distinct entities. The structure equations which represent the familiar torsion and curvature free conditions (of
the ambient space) are used to combine deformation and geometry in a compatible manner. From such a point of view, Kirchhoff type theories have non-classical features which are similar to the equations of defect mechanics (theory of
dislocations and disclinations). Using the proposed framework, we solve a simple boundary value problem and thus demonstrate the importance of moving frames.
We then study the thermo-mechanics of a solid body with point defects using Weyl geometry. Here, we assume the geometries of reference and deformed configurations to be of the Weyl type. The Weyl one-form is introduced as an
additional degree of freedom that determines ratios of lengths between different tangent spaces. This one-form prevents the metric to be compatible with the connection (in the Riemannian sense). We exploit this incompatibility to
characterize metrical defects in the material body. When such a defective body undergoes temperature changes, additional incompatibilities appear and interact with the defects. This interaction is modelled using the Weyl transform, which keeps the Weyl connection invariant whilst changing the non-metricity of the configuration. An immediate consequence of adopting the Weyl connection for a configuration is that the critical points of the stored energy functional are shifted. We relate this change in the equilibrium point to the residual stresses developed in the body due to point defects. In order to relate stress and strain in our non-Euclidean setting, use is made of the Doyle-Ericksen formula, which is interpreted as a relation between the intrinsic geometry of the body and the stresses developed. Thus the Cauchy stress is postulated to be conjugate to the Weyl transformed metric tensor of the deformed configuration. The evolution equation for the Weyl one-form and temperature are arrived using the laws of thermodynamics. Using this model, the self-stress generated by a point defect is calculated and compared with linear elastic solution. We also obtain conditions on the defect distribution (Weyl one-form) that render a thermo-mechanical deformation stress-free. Using this condition, specific stress-free deformation profiles for a class of prescribed temperature changes are computed.
In the second part, (Chapters 6 \& 7), we discuss a phase-field approach to delamination. Pseudo-ductility encountered in laminated ceramic composites is our first focus. By pseudo-ductility, we mean the engineered ductility of ceramic laminated composites by a strategic placement of weak interfaces. The phase-field based brittle fracture model has the advantage of seamlessly modelling the growth of cracks at the interface and lamina. Using a finite element implementation of this brittle fracture model, we compute the response of laminated ceramics composites and study the influence of geometric and material properties. Important geometric properties studied in this work are lamina thickness, lamina thickness scheme and interface geometry. Material properties like Young's modulus and critical energy release rate of the interface are also studied. From these studies, the importance of weak interfaces and key interface material properties which influence the pseudo-ductility of ceramic laminated composites are established. We than apply the phase-field modelling to study delamination in orthotropic laminated composites. Here, we understand the crack phase-field as an internal variable with a thermodynamic origin. We use this modelling approach to simulate delamination in mode I, mode II and another such problem with multiple initial notches. The present approach is able to reproduce nearly all the features of the experimental load displacement curves, allowing only for small deviations in the softening regime. Numerical results also show forth a superior performance of the proposed method over existing approaches based on a cohesive law. | en_US |
