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Introduction to Continuum Mechanics 3 Episode 8

Tham khảo tài liệu 'introduction to continuum mechanics 3 episode 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 266 Torsion of a Noncircular Cylinder 2 Fig. 5.12 5.14 Torsion of a Noncircular Cylinder For cross-sections other than circular the simple displacement field of Section 5.13 will not satisfy the tractionless lateral surface boundary condition see Example 5.13.4 . We will show that in order to satisfy this boundary condition the cross-sections will not remain plane. We begin by assuming a displacement field that still rotates each cross-section by a small angle Ớ but in addition there may be a displacement in the axial direction. This warping of the cross-sectional plane will be defined by 1 p x2 3 . Our displacement field now has the form W1 ip x2 - 3 W2 -X3Ớ X 3 2ớ 1 5.14.1 The associated nonzero strains and stresses are given by r12 T21 2 i i2 - x3ớ H í5-14-2 dr2 713 T31 2 4 E13 n x2 e n 5.14.2b ƠX3 The second and third equilibrium equations are still satisfied if Ỡ constant. However the first equilibrium equation requires that ểV ỀỊ. A 5.14.3 dxị dxị Therefore the displacement field of Eq. 5.14.1 will generate a possible state of stress if p satisfies Eq. 5.14.3 . Now we compute the traction on the lateral surface. Since the bar is The Elastic Solid 267 cylindrical the unit normal to the lateral surface has the form n 2e2 3e3 and the associated surface traction is given by t Tn ỊẰ Ở - 2x3 3 2 z d p rr 2 dx2 z dcp kn3 ei ft0 -n2x3 n3x2 4 V -n e1 i We require that the lateral surface be traction-free i.e. t 0 so that on the boundary the function p must satisfy the condition - v7- _ 5.14.4 - Vự -n ớ 2 3- 3 2 Equations 5.14.3 and 5.14.4 define a well-known boundary-value problem which is known to admit an exact solution for the function p. Here we will only consider the torsion of an elliptic cross-section by demonstrating that p 1x2X3 5.14.5 gives the correct solution. Taking A as a constant this choice of p obviously satisfy the equilibrium equation Eq. 5.14.3 . To check the boundary condition we begin by defining the elliptic boundary by the .