Introduction: Philosophical Preliminaries

This chapter expounds some technical philosophical notions that figure prominently in the rest of the book. Among the phenomena elucidated are moral conflicts, strong permissibility versus weak permissibility, overtopping obligations versus non-overtopping obligations, weak absoluteness versus strong absoluteness, physical freedom versus deontic freedom, the Hohfeldian analysis of legal positions (claim-rights and duties, liberties and no-rights, powers and liabilities, and immunities and disabilities), and the distinction between causal relationships and constitutive relationships. Although all of these concepts and distinctions are of great significance in debates over the principle of freedom of expression, they also figure saliently in many other debates. Hence, the purpose of this introductory chapter is to provide the basic philosophical preliminaries that are needed for a rigorous engagement with the principle of freedom of expression. The chapter sets the stage for the subsequent chapters, where that principle is the focus of attention.

Nonlinear free vibration of size-dependent microbeams with nonlinear elasticity under various boundary conditions

In this study, nonlinear couple stress–strain constitutive relationships in the modified couple stress theory (MCST) are derived on the basis of previous classical stress–strain constitutive relationships of nonlinear elasticity materials. Hamilton's principle is employed to obtain higher-order nonlinear governing equations within the framework of the updated MCST, von Kármán geometric nonlinearity and Bernoulli–Euler beam theory. These mathematical formulations are solved numerically by the differential quadrature method together with an iterative algorithm to determine the nonlinear dynamic features of microbeams with four groups of boundary conditions. A detailed parametric study is conducted to analyze the influences of nonlinear elasticity properties on the nonlinear free vibration characteristics of the microbeams. Results show that these microbeams exhibiting nonlinear couple constitutive relationships have lower frequencies than their approximately simplified linear couple constitutive relationships. In addition, the frequencies of microbeams with nonlinear elasticity properties decrease as the vibration amplitude increases.