Summary A recent unsymmetric 4‐node, 8‐DOF plane element US‐ATFQ4, which exhibits excellent precision and distortion‐resistance for linear elastic problems, is extended to geometric nonlinear analysis. Since the original linear element US‐ATFQ4 contains the analytical solutions for plane pure bending, how to modify such formulae into incremental forms for nonlinear applications and design an appropriate updated algorithm become the key of the whole job. First, the analytical trial functions should be updated at each iterative step in the framework of updated Lagrangian formulation that takes the configuration at the beginning of an incremental step as the reference configuration during that step. Second, an appropriate stress update algorithm in which the Cauchy stresses are updated by the Hughes‐Winget method is adopted to estimate current stress fields. Numerical examples show that the new nonlinear element US‐ATFQ4 also possesses amazing performance for geometric nonlinear analysis, no matter whether regular or distorted meshes are used. It again demonstrates the advantages of the unsymmetric finite element method with analytical trial functions.

No abstract

No abstract

No abstract

No abstract

An improved Legendre polynomial series approach (AILPSA) is presented to investigate the circumferential thermoelastic Lamb wave in a fractional order orthotropic cylindrical plate. In the AILPSA, the analytical integration is developed based on the orthogonality and recursive properties of the Legendre polynomial to simplify the integral computation involved in the solving progress. As a consequence, the computational efficiency is improved significantly. Results are compared with those from the previous article to confirm the validity of the introduced method. Using the AILPSA, guided wave characteristics in various fractional order orthotropic cylindrical plates are investigated by solving the eigenvalues and eigenvectors of the system of algebraic equations. The influences of fractional order and radius‐thickness ratio on dispersion curve and displacement, temperature amplitudes are illustrated. The results show that the fractional order almost has no effect on the phase velocities of quasi‐elastic wave modes, but has notable effect on their attenuations. A smaller fractional order means a smaller temperature amplitude for the quasi‐elastic wave mode, but means a larger temperature amplitude for the thermal wave mode. A smaller radius thickness ratio indicates a smaller attenuation and a larger temperature amplitude.

Functionally graded piezoelectric-piezomagnetic material (FGPPM), with a gradual variation of the material properties in the desired direction(s), can improve the conversion of energy among mechanical, electric, and magnetic fields. Full dispersion relations and wave mode shapes are vital to understanding dynamic behaviors of structures made of FGPPM. In this paper, an analytic method based on polynomial expansions is proposed to investigate the complex-valued dispersion and the evanescent Lamb wave in FGPPM plates. Comparisons with other related studies are conducted to validate the correctness of the presented method. Characteristics of the guided wave, including propagating modes and evanescent modes, in various FGPPM plates are studied, and three-dimensional full dispersion and attenuation curves are plotted to gain a deeper insight into the nature of the evanescent wave. The influences of the gradient variation on the dispersion and the magneto-electromechanical coupling factor are illustrated. The displacement amplitude and electric potential and magnetic potential distributions are also discussed in detail. The obtained numerical results could be useful to design and optimize different sensors and transducers made of smart piezoelectric and piezomagnetic materials with high performance by adjusting the gradient property.

In this paper, we study monotonicity formulas of eigenvalues and entropies along the rescaled List's extended Ricci flow. We derive some monotonicity formulas of eigenvalues of Laplacian which generalize those of Li in [8] and Cao-Hou-Ling in [3]. Moreover, we also consider monotonicity formulas of F k -functional which can be seen as a generalized F-functional corresponding with steady Ricci breathers, and W k -functional which generalizes W-functional corresponding with expanding Ricci breathers.2000 Mathematics Subject Classification. 58C40, 53C44.