Tham khảo tài liệu 'applied structural mechanics fundamentals of elasticity part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 76 4 State of strain 0 alx 0 aly I 2 2 kalx 0 a x 0 aly I 2ka z 0 J 2 2 kalx 2kaly 2kaJz Ũ -------- 0 alx - c arbitrary aiy alx - 1 2 c . Thus it yields the eigenvector with arbitrary c f R. By analogy the reader may calculate the two other eigenvectors and obtain r 1 1 1 a2 c a3 c l 2 l i 2 l- ữ with arbitrary c F IR. c According to the volume dilatation is equal to the first invariant h t Sy k k 3k 5k which may be alternatively calculated as e J k 2 ự 13 k 2 - 13 k 5 k Exercise A-4-4 The strain - displacement relation for the normal strain 22 and the material law of an isotropic infinite disk with an elliptical hole are to be determined by using the elliptical hyperbolical coordinates from Exercise A-2-3. Solution The physical components of the strain tensor tensor of second order are first calculated according to y ap pp - Inserting 5 and 6 from A-2-3 yields the following for 22 Exercise A-4-4 77 Y Y n p22 p22 - L V-L L 22 E22 22 V B 6 - Ỵ22 J V2Í2 Ị yg yg AF2 T 1 27 7 yg _È_ âẸ2 tÊ 1 1 or . 1 . 2r2 V2. -sinh2 c sinh Ẹ 4- sin u 1 - o c2 z. . 2 r1 . -JrV1 2. -sin2Ẹ c sình Ẹ sin Ệ V t ị sinh2 Ệ1 c2 sinh21 1 4- sin2 2 .2J1-Vil . Ỉ . .21 . . 2ỒV sin Ệ sin 2 ỉ V 4- sinh sin Ẹ z A F 1 nrl - .2 J . . 2 -2 .-1 2 t sinh 2 5 sinh 4- sin 5 v u - 1 t2211 . 1 2. -- -sin2Ệ sinh sin V Xi c sinh2 sin2Ệ2 1 sinh 2 g1 2 - sinh2 sin2 2 __ 1 s22 - c 1__ c sinh2 sin2Ẹ2 1 v _ _ 2 sinh2 sin2 Ẹ2 ò Í 2 sinh2 1 4- sin2 2 3 2 The material law for the state of plane stress without consideration of temperature terms reads according to yaP - Da 7ỐT 3a with apy6 2E 1 j Soj jg-yi 3b Insertion yields 2 E f 0t y 56 oí E cự3 -y6 We now obtain the material law for Y22 yap 22 2E 26 t Ẽ2ỐÍ With the covariant metric components y6 and ẸƠ 0 for p ũ sea -- g for p Ơ 78 5 Constitutive laws finally follows V22 V _ 1 11 1 V. _ _ _ . V _ 1 22 6226111 T 2E 822622 822622 ES22822P 22 V . 11 Egt V V 2E g 1 6 Es T _ 1 J j-iA1 . _2r2 2 22 Ec sinh sin 22 111 T -vt . 4
