Marcio A. A. Cavalcante

Transient Thermomechanical Analysis of a Layered Cylinder by the Parametric Finite-Volume Theory

The recently developed parametric finite-volume theory for functionally graded materials is employed to investigate the response of a layered cylinder under transient thermal loading that simulates a cyclic thermal shock durability test. The results reveal a potential for the occurance of two distinct failure modes that may be activated due to two different stress components reaching critical values during different portions of the thermal cycle at different locations. These are delamination of the ceramic top coat from the bond coat, and radial cracking of the top coat that potentially initiates at the outer surface subjected to concentrated transient thermal load. Steady-state analysis substantially underestimates the magnitude of the radial and hoop stresses and, moreover, does not predict the stress reversals during cooldown that likely initiate radial cracks at the outer surface. The fidelity with which local stress fields are captured provides a convincing evidence that the parametric finite-volume theory is an attractive alternative to the finite-element analysis for this class of problems.

Transient Finite-Volume Analysis of a Graded Cylindrical Shell Under Thermal Shock Loading

A major failure mechanism in thermal barrier coatings involves delamination of the ceramic top coat from the substrate, which is accompanied and/or aided by transverse cracks that initiate at the outer surface. Techniques to mitigate these failure modes include grading the transition region between the ceramic top coat and the metallic bond coat by gradually varying the content of the two phases. This concept is explored herein for a graded cylinder subjected to transient thermal cyclic loading, that simulates a thermal shock durability test, using the parametric finite-volume theory for functionally graded materials. Previous investigation into the transient response of a three-layer cylinder under the considered thermal shock loading revealed two potential failure modes, one of which was a direct result of transient effects. Herein, examination of average phase-level stress fields indicates that grading influences the initiation of delamination and potential radial cracking through redistribution of the radial and hoop stress components in the thermal coatings graded region upon rapid heating. The large hoop stress reversal responsible for radial crack initiation at the outer surface, however, is not reduced upon rapid cooling, pointing to the importance of accurately modeling transient effects in thermal shock testing as well as crack-growth management through grading.

Generalized FVDAM Theory for Periodic Materials Undergoing Finite Deformations—Part I: Framework

The recently constructed generalized finite-volume theory for two-dimensional linear elasticity problems on rectangular domains is further extended to make possible simulation of periodic materials with complex microstructures undergoing finite deformations. This is accomplished by embedding the generalized finite-volume theory with newly incorporated finite-deformation features into the 0th order homogenization framework, and introducing parametric mapping to enable efficient mimicking of complex microstructural details without artificial stress concentrations by stepwise approximation of curved surfaces separating adjacent phases. The higher-order displacement field representation within subvolumes of the discretized unit cell microstructure, expressed in terms of elasticity-based surface-averaged kinematic variables, substantially improves interfacial conformability and pointwise traction and nontraction stress continuity between adjacent subvolumes. These features enable application of much larger deformations in comparison with the standard finite-volume direct averaging micromechanics (FVDAM) theory developed for finite-deformation applications by minimizing interfacial interpenetrations through additional kinematic constraints. The theory is constructed in a manner which facilitates systematic specialization through reductions to lower-order versions with the 0th order corresponding to the standard FVDAM theory. Part I presents the theoretical framework. Comparison of predictions by the generalized FVDAM theory with its predecessor, analytical and finite-element results in Part II illustrates the proposed theorys superiority in applications involving very large deformations.

Generalized Finite-Volume Theory for Elastic Stress Analysis in Solid Mechanics—Part I: Framework

A generalized finite-volume theory is proposed for two-dimensional elasticity problems on rectangular domains. The generalization is based on a higher-order displacement field representation within individual subvolumes of a discretized analysis domain, in contrast with the second-order expansion employed in our standard theory. The higher-order displacement field is expressed in terms of elasticity-based surface-averaged kinematic variables, which are subsequently related to corresponding static variables through a local stiffness matrix derived in closed form. The novel manner of defining the surface-averaged kinematic and static variables is a key feature of the generalized finite-volume theory, which provides opportunities for further exploration. Satisfaction of subvolume equilibrium equations in an integral sense, a defining feature of finite-volume theories, provides the required additional equations for the local stiffness matrix construction. The theory is constructed in a manner which enables systematic specialization through reductions to lower-order versions. Part I presents the theoretical framework. Comparison of predictions by the generalized theory with its predecessor, analytical and finite-element results in Part Il illustrates substantial improvement in the satisfaction of interfacial continuity conditions at adjacent subvolume faces, producing smoother stress distributions and good interfacial conformability.

The Korteweg–de Vries equation on a metric star graph

We prove local well-posedness for the Cauchy problem associated with Korteweg–de Vries equation on a metric star graph with three semi-infinite edges given by one negative half-line and two positive half-lines attached to a common vertex, for two classes of boundary conditions. The results are obtained in the low regularity setting by using the Duhamel boundary forcing operator, in context of half-lines, introduced by Colliander and Kenig (Commun Partial Differ Equ 27(11/12): 2187–2266, 2002), and extended by Holmer (Commun Partial Differ Equ 31:1151–1190, 2006) and Cavalcante (Differ Integral Equ 30(7/8):521–554, 2017).

Parametric Finite‐Volume Theory for Functionally Graded Materials

A parametric formulation of the finite‐volume theory for functionally graded materials is presented based on a mapping of a square reference subcell onto a quadrilateral subcell in the actual discretized microstructure. This formulation significantly advances the capability and utility of the theory, enabling modeling of curved boundaries of functionally graded structural components, as well as inclusions employed for grading purposes, without the disadvantage of stress concentrations at the corners of rectangular subcells used in the standard version.