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EN0175-09

# EN0175-09 - EN0175 10 03 06 Stress in a solid Examples of...

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Unformatted text preview: EN0175 10 / 03 / 06 Stress in a solid Examples of common stress states: 1. Uniaxial stress 1 e v 2 e v 3 e v σ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = σ σ Principal stresses: σ σ = I , = = III II σ σ Principal directions: 1 e v , 2 e v , ( 3 e v 2 e v , 3 e v can be replaced by any direction in 2-3 plane) 2. Biaxial stress 3 e v Substrate 1 e v 2 e v Film ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = σ σ σ Principal stresses: σ σ σ = = II I , = III σ Principal directions: 1 e v , 2 e v , ( 3 e v 1 e v , 2 e v can be replaced by any direction in 1-2 plane) 1 EN0175 10 / 03 / 06 3. Hydrostatic stress 3 e v 1 e v 2 e v σ σ σ I σ σ σ σ σ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = Principal stresses: σ σ σ σ = = = III II I Principal directions: any direction (since there is no shear stress on any cut plane) 4. Pure shear 1 e v 2 e v 3 e v τ 1 2 2 1 e e e e v v v v ⊗ + ⊗ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = τ τ τ τ σ Principal stresses: 2 EN0175 10 / 03 / 06 3 2 1 3 2 1 3 2 1 = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − ⇒ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ n n n n n n n n n λ λ τ τ λ λ τ τ For nontrivial solution of , i n det = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − λ λ τ τ λ , which indicates ( ) τ τ λ τ λ λ − = ⇒ = − − , , 2 2 Following the convention III II I σ σ σ > > , we can write τ σ = I ,...
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EN0175-09 - EN0175 10 03 06 Stress in a solid Examples of...

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