hw2solution - HW#2-SOLUTION Stress Strength...

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HW#2-SOLUTION: Stress & Strength Nano-Indentation October 6, 2003 MIT 1.033/1.57 Fall 2003 Instructor: Franz-Josef ULM 1 Statically Admissible Stress Fields A statically admissible stress field is a stress field which satisfies ( i ) the force boundary conditions, ( ii ) the stress vector continuity condition on any surface in the material; ( iii ) the symmetry of the stress tensor; ( iv ) the momentum balance. 1.1 Boundary Conditions For the nanoindentation test, the boundary conditions are: In 1 : Frictionless contact: t = e r : σ rz = 0 on z = 0; r < r 0 ; n = e z : T ( n = e z ) · t = 0 (1) t = e θ : σ θz = 0 Force boundary condition: on z = 0; r < r 0 ; n = e z : N d = F e z σ · n da = σ · e z da (2) 2 0 2 0 A = πr A = πr From a combination of (1) and (2), it follows: σ rz A = 0 on z = 0; r r 0 ; n = e z : = 0 (3) σ θz A σ zz A = H 1 where σ ij A σ ij da stands for the stress average of σ ij over the surface = 2 0 A = πr A = πr 0 2 , and H = F/A is the micro-hardness measured in the nanoindentation test. Note that it cannot a priori be concluded that σ zz = F/A , since σ zz may not be constant over the contact area. In 2 : σ rz = 0 on z = 0; r > r 0 ; n = e z : T ( n = e z ) = 0 σ θz = 0 (4) σ zz = 0 A
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�� October 6, 2003 page 2 1.2 Continuity of Stress Vector On the interface between domain 1 and 2 : σ (1) = σ (2) rr rr : T (1) ( n = e r ) + T (2) ( z > 0; r = r 0 ; n = e r n = e r ) = 0 σ (1) = σ (2) (5) θr θr σ (1) = σ (2) zr zr 1.3 Form of the Stress Tensor Given the rotational symmetry of the problem, σ θr = 0 in . The stress tensor, therefore, is of the diagonal form: in Ω : σ = σ rr e r e r + σ θθ e θ e θ + σ zz e z e z
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