Khanh Chau Le

The averaging procedure developed for polycrystalline bars under axially symmetric tension or compression is extended to arbitrary macroscopically uniform stress and strain states. The mean resolved shear stress and the mean resolved elastic and plastic shear strains are defined as root-mean-square averages over all slip-system orientations. It is shown that, if the plastic slip rate on a system is proportional to the resolved shear stress acting on it, the stress deviator determines the direction of plastic flow. The result is a dislocation-mediated plasticity in which eight ordinary differential equations per material point replace the yield condition and the flow rule. With one set of identified material parameters it describes tension, compression and torsion of copper over a broad range of temperatures and strain rates.

This study utilizes the variational‐asymptotic method to establish a one‐dimensional theory for functionally graded rods characterized by general anisotropy from the three‐dimensional elasticity theory. A distinctive feature of this dimension reduction procedure is the numerical solution of dual cross‐sectional problems, which provide rigorous upper and lower bounds for the average transverse energy density. By employing the Prager–Synge identity, we derive an error estimate in the energetic norm to establish the asymptotic exactness of the model. This estimate is extended to the dynamic regime for low‐frequency vibrations. Furthermore, the dynamic validity of the theory is confirmed by comparing the one‐dimensional dispersion relations with exact analytical three‐dimensional solutions for wave propagation in composite rods. The results show that the developed one‐dimensional model captures the long‐wave asymptotic behavior of the three‐dimensional elastic body with high fidelity. Numerical benchmarks indicate that while the naive rod theory incurs errors up to in deflection predictions, the current VAM framework reduces this discrepancy to about , with log–log convergence studies confirming the theoretical accuracy.

We propose a phase field model of a macrocracked single crystal in which the crack and the geometrically necessary dislocations descend from a single energy functional. Energy minimization alone then decides dislocation nucleation, through an integral criterion evaluated in closed form along slip chords. The criterion yields a size effect inaccessible to point-wise strength conditions: a grain-size-dependent yield stress. With slip suppressed the model reproduces Griffith fracture; with fracture suppressed, the nucleation load measured by the full non-smooth solver agrees with the closed-form nucleation criterion to four percent. The coupled computations produce a two-stage response: at loads an order of magnitude below cleavage, dislocation bands emitted from the notch tip blunt and shield it, raising the initiation load; once the crack grows, the bands heal; a compact cluster of like-signed dislocations travels with the tip, its canceling partner walls pinned at the grain boundary, and the dissipated fracture resistance equals the elastic one. In the purely energetic, dissipationless limit, emission shields the crack but does not toughen it; toughening requires dissipation, incorporated in the sequel through the threshold resistance to dislocation motion.

This paper establishes a lower bound for the energy decay rate in piezoelectric cylinders. The bound incorporates material properties and geometric factors, including the cross-section’s Poincaré–Wirtinger and Korn constants. A detailed analysis of a circular cross-section cylinder yields a precise numerical lower bound, illustrating the practical application of this result.

This paper establishes a lower bound for the energy decay rate in piezoelectric cylinders. The bound incorporates material properties and geometric factors, including the cross-section's Poincaré-Wirtinger and Korn constants. A detailed analysis of a circular cross-section cylinder yields a precise numerical lower bound, illustrating the practical application of this result.

We construct a one-dimensional first-order theory for functionally graded elastic beams using the variational-asymptotic method. This approach ensures an asymptotically exact one-dimensional equations, allowing for the precise determination of effective stiffnesses in extension, bending, and torsion via numerical solutions of the dual variational problems on the cross-section. Our theory distinguishes itself by offering a rigorous error estimation based on the Prager-Synge identity, which highlights the limits of accuracy and applicability of the derived one-dimensional model for beams with continuously varying elastic moduli across the cross section.