tailieunhanh - Introduction to Elasticity Part 6

Tham khảo tài liệu 'introduction to elasticity part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Constitutive Equations David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge MA 02139 October 4 2000 Introduction The modules on kinematics Module 8 equilibrium Module 9 and tensor transformations Module 10 contain concepts vital to Mechanics of Materials but they do not provide insight on the role of the material itself. The kinematic equations relate strains to displacement gradients and the equilibrium equations relate stress to the applied tractions on loaded boundaries and also govern the relations among stress gradients within the material. In three dimensions there are six kinematic equations and three equilibrum equations for a total of nine. However there are fifteen variables three displacements six strains and six stresses. We need six more equations and these are provided by the material s consitutive relations six expressions relating the stresses to the strains. These are a sort of mechanical equation of state and describe how the material is constituted mechanically. With these constitutive relations the vital role of the material is reasserted The elastic constants that appear in this module are material properties subject to control by processing and microstructural modification as outlined in Module 2. This is an important tool for the engineer and points up the necessity of considering design of the material as well as with the material. Isotropic elastic materials In the general case of a linear relation between components of the strain and stress tensors we might propose a statement of the form ij Sijkl kl where the Sijki is a fourth-rank tensor. This constitutes a sequence of nine equations since each component of ij is a linear combination of all the components of ơịj. For instance 23 S2311 ƠÌ1 S2312 ƠÌ2 S2333 33 Based on each of the indices of Sijki taking on values from 1 to 3 we might expect a total of 81 independent components in S. However both ij and ơịj are symmetric with six .