Exact solution for nonlinear static behaviors of functionally graded beams with porosities resting on elastic foundation using neutral surface concept

In this paper, an exact solution for nonlinear static behaviors of functionally graded (FG) beams with porosities resting on the elastic foundation is presented. The FG material properties with porosities are assumed to vary along the thickness of the beam, and two types of porosity distributions are considered. Actually, the geometrical middle surface of the FG beam selected in computations is very popular in the literature. By contrast, in this study, the physical neutral surface of the beam is utilized. Based on the Timoshenko beam theory, von Kármán nonlinear assumption, together with neutral surface concept, the nonlinear governing equations of the FG beam resting on the elastic foundation are derived. By using the physical neutral surface, the nonlinear governing equations have simple forms and can be solved directly. The exact solution for the problem with all immovable and moveable boundary conditions is conducted in detail. Some numerical investigations to show the effects of boundary conditions, material properties, length-to-thickness ratio, elastic foundation coefficients and several types of applied load on nonlinear static bending behaviors of the beam are given.

Maximum permissible inelastic deformations of hybrid rectangular plates during intense explosions

Для гибридных прямоугольных слоистых пластин из произвольных наборов изотропных конструкционных материалов с симметричной структурой расположения слоев относительно отчетной (нейтральной) поверхности получены аналитические формулы оценок допустимой остаточной повреждаемости при воздействии интенсивных нагрузок взрывного типа. Закрепления контура предусматривают любую комбинацию из жестких или шарнирных условий отдельных сторон прямоугольного контура. В основе рассмотренной модели использованы традиционные кинематические гипотезы Кирхгофа, деформационная теория А.А. Ильюшина, принцип слабейшего звена для используемых материалов и деформационный критерий предразрушения для всех используемых фазовых материалов. Использован наиболее распространенный в практике закон изменения во времени давления на фронте взрывной волны. For hybrid rectangular layered plates from arbitrary sets of isotropic structural materials with a symmetric structure of the arrangement of layers relative to the reporting (neutral) surface, analytical formulas for assessing the permissible residual damage under the action of intense explosive loads are obtained. Contour anchors provide for any combination of rigid or hinge conditions of individual sides of a rectangular contour. The considered model is based on the traditional kinematic hypotheses of Kirchhoff, the deformation theory A.A. Ilyushin, the principle of the weakest link for the materials used and the deformation criterion for pre-fracture for all used phase materials. The most widespread law of time variation of pressure at the front of a blast wave is used.

Exact solution by dynamic stiffness method for the natural vibration of porous functionally graded plate considering neutral surface

This paper presents the formulation of dynamic stiffness matrix for the natural vibration analysis of porous power-law functionally graded Levy-type plate. In the process of formulating the dynamic stiffness matrix, Kirchhoff-Love plate theory in tandem with the notion of neutral surface has been taken on board. The developed dynamic stiffness matrix, a transcendental function of frequency, has been solved through the Wittrick–Williams algorithm. Hamilton’s principle is used to obtain the equation of motion and associated natural boundary conditions of porous power-law functionally graded plate. The variation across the thickness of the functionally graded plate’s material properties follows the power-law function. During the fabrication process, the microvoids and pores develop in functionally graded material plates. Three types of porosity distributions are considered in this article: even, uneven, and logarithmic. The eigenvalues computed by the dynamic stiffness matrix using Wittrick–Williams algorithm for isotropic, power-law functionally graded, and porous power-law functionally graded plate are juxtaposed with previously referred results, and good agreement is found. The significance of various parameters of plate vis-à-vis aspect ratio ( L/b), boundary conditions, volume fraction index ( p), porosity parameter ( e), and porosity distribution on the eigenvalues of the porous power-law functionally graded plate is examined. The effect of material density ratio and Young’s modulus ratio on the natural vibration of porous power-law functionally graded plate is also explained in this article. The results also prove that the method provided in the present work is highly accurate and computationally efficient and could be confidently used as a reference for further study of porous functionally graded material plate.

Dynamic Stiffness Formulation for Out-of-Plane Natural Vibration of Elastically Supported Functionally Graded Plates

In this paper, the natural vibration characteristics of elastically supported functionally graded material plate are investigated using the dynamic stiffness method (DSM). Power-law functionally graded (P-FG) plate, the material properties of which vary smoothly along the thickness direction following the power-law function, that has been used for the analysis. Classical plate theory and Hamilton’s principle are used for deriving the governing differential equation of motion and associated edge conditions for P-FG plate supported by elastic foundation. During the formulation of dynamic stiffness (DS) matrix, the concepts of rotary inertia and neutral surface are implemented. Wittrick–Williams (W-W) algorithm is used as a solving technique for the DS matrix to compute eigenvalues. The results thus obtained by DSM for the isotropic, P-FG plate, and the P-FG plate with elastic foundation compare well with published results that are based on different analytical and numerical methods. The comparisons indicate that this approach is very accurate. Furthermore, results are provided for elastically supported P-FG plate under four different considerations in order to see the differences in frequencies with the inclusion or exclusion of neutral surface and/or rotary inertia. It is noticed that the inclusion of rotary inertia and neutral surface influences the eigenvalues of P-FG plate, and that cannot be discounted. The study also examines the influence of plate geometry, material gradient index, edge conditions, and elastic foundation modulus on the natural frequency of P-FG plate.

Synthesis of Dendritic Mesoporous Organosilica Nanoparticles under a Mild Acidic Condition with Homogeneous Wall Structure and Near-Neutral Surface

Dendritic mesoporous organosilica nanoparticles were synthesized under a mild acidic condition (pH 6.2), featuring a molecularly homogeneous wall structure and an unusual near-neutral charged surface, consequently enabling reduced protein fouling...