tailieunhanh - Introduction to Elasticity Part 15

Tham khảo tài liệu 'introduction to elasticity part 15', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Matrix and Index Notation David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge MA 02139 September 18 2000 A vector can be described by listing its components along the xyz cartesian axes for instance the displacement vector u can be denoted as Ux Uy Uz using letter subscripts to indicate the individual components. The subscripts can employ numerical indices as well with 1 2 and 3 indicating the x y and z directions the displacement vector can therefore be written equivalently as U1 U2 U3. A common and useful shorthand is simply to write the displacement vector as Uị where the i subscript is an index that is assumed to range over 1 2 3 or simply 1 and 2 if the problem is a two-dimensional one . This is called the range convention for index notation. Using the range convention the vector equation Uị a implies three separate scalar equations U1 a U2 a U3 a We will often find it convenient to denote a vector by listing its components in a vertical list enclosed in braces and this form will help us keep track of matrix-vector multiplications a bit more easily. We therefore have the following equivalent forms of vector notation Second-rank quantities such as stress strain moment of inertia and curvature can be denoted as 3x3 matrix arrays for instance the stress can be written using numerical indices as Ơ11 Ơ12 Ơ13 Ơ21 a22 Ơ23 31 Ơ32 Ơ33 Here the first subscript index denotes the row and the second the column. The indices also have a physical meaning for instance u23 indicates the stress on the 2 face the plane whose normal is in the 2 or y direction and acting in the 3 or z direction. To help distinguish them we ll use brackets for second-rank tensors and braces for vectors. Using the range convention for index notation the stress can also be written as ơịj where both the i and the j range from 1 to 3 this gives the nine components listed explicitly above. 1 Since the stress matrix is symmetric . ơịj ơji only