Studying Problems in Elasticity and Plasticity Using a QR Decomposition of the Deformation Gradient

Abstract

With the recent developments of QR kinematics and the associated constitutive model, in this thesis, we address some of the fundamental issues in QR kinematics and extend this method to study some problems in elasto-plasticity. In this framework, the matrix of the deformation gradient is decomposed into an orthogonal rotation R and an upper-triangular matrix U, called the Laplace stretch. The QR decomposition can be achieved using different techniques, of which a Gram-Schmidt procedure is most suitable for our application. A Gram-Schmidt procedure requires the specification of a particular coordinate direction and a specific coordinate plane, which includes this particular coordinate direction, given some coordinate systems of interest. Unfortunately, this coordinate direction and associated coordinate plane are not known a priori, because they require information from both the triad of base vectors and the deformation in question. This issue is resolved by introducing a strategy whereby that edge of a representative cube undergoing the least amount of transverse shear under a given deformation, and the adjoining coordinate plane that experiences the least amount of in-plane shear are selected. Next, a compatibility condition for the Laplace stretch is derived, whenever a right Cauchy-Green tensor C = F^T F is prescribed. Here, we choose the right Cauchy-Green tensor as our primary kinematic variable and show that a vanishing of the Riemann curvature tensor imposes restrictions on the spatial variations of certain elements of the Laplace stretch U. A natural extension of our work on compatibility is to study the incompatibility of a pertient space when the QR kinematics is applied to elastoplasticity. Using the property that the set of all upper-triangular matrices form a group under multiplication, Freed et al.(2019) proposed an elastic-plastic decomposition of Laplace stretch, i.e., U=U^e U^p. Using this decomposition, we study the geometric dislocation density tensor and Burgers vector. The geometric dislocation density tensor G is obtained using the classical argument of failure of a Burgers circuit in a suitable configuration k_p where the deformation of a body is solely due to the movement of dislocations. The geometric features of space k_p are explored and it has been shown that the derived geometric dislocation tensor is related to the torsion of k_p. The total dislocation density can be additively decomposed into the dislocation density due to plastic "straining" and a term representing the incompatibility of rotation field. The latter of which is physically similar to Nye's definition of dislocation density tensor. Based on this kinematics, a constitutive model has been developed for isotropic, elastic-plastic materials. A maximum rate of dissipation criterion has been used in deriving the constitutive equations as this criterion is valid for a wider class of materials. Two cases of plastic deformation -volume-preserving and dilatant-pressure dependent deformations have been considered. As illustration of the proposed model, the classical J_2 plasticity and Drucker-Prager model has been derived. The concept of plastic spin has also been investigated in this framework. It has been shown that the intermediate configuration k_p acts as a macroscopic manifestation of the material substructure. Expressions for a substructural spin and a material spin have been obtained using appropriate physical arguments based on this configuration. An internal state variable has been considered to represent the macroscopic manifestation of the microstructural properties. Considering the orientational properties of this internal variable with respect to the material substructure, an expression for the plastic spin has been obtained and its implication in the context of single crystal plasticity has been shown. Finally, this plastic spin has been incorporated into a constitutive model by means of an appropriate definition of the co-rotational rate of the internal state variable.