tailieunhanh - Finite Element Analysis - Thermomechanics of Solids Part 3

Tham khảo tài liệu 'finite element analysis - thermomechanics of solids part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 3 Introduction to Variational and Numerical Methods INTRODUCTION TO VARIATIONAL METHODS Let u x be a vector-valued function of position vector x and consider a vectorvalued function F u x u x x in which u x du dx. Furthermore let v x be a function such that v x 0 when u x 0 and v x 0 when u x 0 but which is otherwise arbitrary. The differential dF measures how much F changes if x changes. The variation dF measures how much F changes if u and u change at fixed x. Following Ewing we introduce the vector-valued function ộ e F as follows Ewing 1985 o e F F u x ev x u x ev x x - F u x u x x The variation dF is defined by __ f dod dF el I I de J e v 7 with x fixed. Elementary manipulation demonstrates that dF dFev trf ev l du vdu J in which 4 ev ev j. If F u then dF du ev. If F u then dF d u 3m ev . This suggests the form dF -dF du trf du i. du vdu J The variational operator exhibits five important properties 1. d . commutes with linear differential operators and integrals. For example if S denotes a prescribed contour of integration d dS d j dS 43 2003 by CRC CRC Press LLC 44 Finite Element Analysis Thermomechanics of Solids 2. 8 f vanishes when its argument f is prescribed. 3. 8 . satisfies the same operational rules as d . . For example if the scalars q and r are both subject to variation then 8 qr q8 r 8 q r. 4. Iff is a prescribed function of scalar x and if u x is subject to variation then 8 fu f8u. 5. Other than for number 2 the variation is arbitrary. For example for two vectors v and w vTd w 0 implies that v and w are orthogonal to each other. However vT8w implies that v 0 since only the zero vector can be orthogonal to an arbitrary vector. As a simple example Figure depicts a rod of length L cross-sectional area A and elastic modulus E. At x 0 the rod is built in while at x L the tensile force P is applied. Inertia is neglected. The governing equations are in terms of displacement u stress S and linear strain E .