hw2solution

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

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Unformatted text preview: 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 Ω 1 : — Frictionless contact: t = e r : σ rz = 0 on z = 0; r < r ; n = − e z : T ( n = − e z ) · t = 0 ↔ (1) t = e θ : σ θz = 0 — Force boundary condition: on z = 0; r < r ; n = − e z : N d = F e z ≡ σ · n da = − σ · e z da (2) 2 2 A = πr A = πr — From a combination of (1) and (2), it follows: ⎧ ⎫ ⎪ σ rz A = 0 ⎪ ⎨ ⎬ on z = 0; r ≤ r ; 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 A = πr A = πr 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 Ω 2 : ⎧ ⎫ ⎪ ⎨ σ rz = 0 ⎪ ⎬ on z = 0; r > r ; n = − e z : T ( n = − e z ) = 0 ⇒ σ θz = 0 (4) ⎪ ⎪ ⎩ ⎭ σ zz = 0 A October 6, 2003 page 2 1.2 Continuity of Stress Vector On the interface between domain Ω 1 and Ω 2 : ⎧ ⎫ σ (1) = σ (2) ⎪ ⎪ ⎨ rr rr ⎬ : T (1) ( n = e r ) + T (2) ( z > 0; r = r ; n = e r n = − e r ) = 0 ⇒ σ (1) = σ (2) (5) θr θr ⎪ ⎪ ⎭ ⎩ σ (1) = σ (2) zr zr 1.3 Form...
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This note was uploaded on 11/29/2011 for the course CIVIL 1.00 taught by Professor Georgekocur during the Spring '05 term at MIT.

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

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