tailieunhanh - Introduction to Elasticity Part 5

Tham khảo tài liệu 'introduction to elasticity part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Example 1 To illustrate how volumetric strain is calculated consider a thin sheet of steel subjected to strains in its plane given by ex 3 ey -4 and Yxy 6 all in pin in . The sheet is not in plane strain since it can undergo a Poisson strain in the z direction given by ez -v ex ey 3 4 . The total state of strain can therefore be written as the matrix 3 6 0 e 6 -4 0 0 0 X 10-6 where the brackets on the e symbol emphasize that the matrix rather than pseudovector form of the strain is being used. The volumetric strain is 3 - 4 X 10-6 X 10-6 Engineers often refer to microinches of strain they really mean microinches per inch. In the case of volumetric strain the corresponding but awkward unit would be micro-cubic-inches per cubic inch. Finite strain The infinitesimal strain-displacement relations given by Eqns. are used in the vast majority of mechanical analyses but they do not describe stretching accurately when the displacement gradients become large. This often occurs when polymers especially elastomers are being considered. Large strains also occur during deformation processing operations such as stamping of steel automotive body panels. The kinematics of large displacement or strain can be complicated and subtle but the following section will outline a simple description of Lagrangian finite strain to illustrate some of the concepts involved. Consider two orthogonal lines OB and OA as shown in Fig. 4 originally of length dx and dy along the x-y axes where for convenience we set dx dy 1. After strain the endpoints of these lines move to new positions A1O1B1 as shown. We will describe these new positions using the coordinate scheme of the original x-y axes although we could also allow the new positions to define a new set of axes. In following the motion of the lines with respect to the original positions we are using the so-called Lagrangian viewpoint. We could alternately have used the final positions as our reference this is the .