tailieunhanh - Elasticity Part 14

Tham khảo tài liệu 'elasticity part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Sadd Elasticity Final Proof 6 35pm page 379 EXAMPLE 14-4 Kelvin State with Unit Loads in Coordinate Directions-Cont d For the case with the force in the x direction that is the state Sx x we get the following fields 2C 1 - 2n ---sin f cos y mR V C 3 - 4n Uf R R cos f cos y v 2mR C 3 - 4n .a uy sin y 2mR ƠR -- 2 - sin f cos y _C 1 - 2n Ơ0 Sf --Ị2-sin f cos y _ _C 2v - 1 . tRf -R---cos f cos y _ C l- 2n Trs -R2-sin y T0f 0 uR Notice that for the Kelvin state the displacements are of order O 1 R while the stresses are O 1 R2 and that Ị Tnds P Ị R X Tnds 0 for any closed surface S enclosing the origin. Using the basic Kelvin problem many related singular states can be generated. For example define S0 x S a u a s a e a where a 1 2 3. Now if the state S0 is generated by the Papkovich potentials f x and ộ x then S0 is generated by f x f a Ca x c a ei Further define the Kelvin state Sa x j as that corresponding to a unit load applied in the xa direction at point j as shown in Figure 14-7. Note that Sa x j Sa x j . Also define the set of nine states Sab x by the relation S x S x or equivalently Sjab x Sa x1 x2 x3 -Sa x1 - Ỗfi1h x2 - ỗỊỊ2h x3 - ổ h h Sa x1 X2 X3 -Sa x1 X2 xs fy1h ÔỊtth h Continued Micromechanics Applications 379 TLFeBOOK Sadd Elasticity Final Proof 6 35pm page 380 EXAMPLE 14-5 Force Doublet Consider the case of two concentrated forces acting along a common line of action but in opposite directions as shown in Figure 14-8. The magnitude of each force is specified as 1 h where h is the spacing distance between the two forces. We then wish to take the limit as h 0 and this type of system is called a force doublet. Recall that this problem was first defined in Chapter 13 see Exercise 13-18. From our previous constructions the elastic state for this case is given by Saa x with no sum over a. This form matches the suggested solution scheme presented in Exercise 13-18. xa - direction FIGURE 14-8 Force doublet

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