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Lecture14Extras - Maxwell Model(better for stress...

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Unformatted text preview: Maxwell Model (better for stress relaxation) dσ dε Hookeʼs law when applied stress =E dt dt varies in time: dε σ assumes viscous = forces are dt η newtonian; plug in to Hookeʼs law equation above. rearrange to get: 1 ∂σ σ ∂ε( t ) += E ∂t η ∂t if stress is constant, this predicts no creep! 29 Maxwell Model 1 ∂σ σ ∂ε( t ) += E ∂t η ∂t impose a condition of constant strain, integrate σ E ln = − t σ0 η rearrange: ⎡E⎤ ⎢− t ⎥ ⎣η⎦ ⎡ t⎤ ⎢− ⎥ ⎣ τt ⎦ σ = σ 0e = σ 0e 30 Maxwell Model ln σ E =− t σ0 η σ = σ 0e ⎡E⎤ ⎢− t ⎥ ⎣η⎦ = σ 0e ⎡ t⎤ ⎢− ⎥ ⎣ τt ⎦ 31 Voigt Model (better for creep) dε( t ) σ ( t ) = Eε( t ) + η d (t ) apply constant stress: dε( t ) ε( t ) σ 0 + '= d( t ) τt η ⎛ t ⎞⎤ ⎡ ⎜− ' ⎟ ⎜ ⎟ σ0 ⎢ ⎝ τ t ⎠⎥ ε( t ) = 1− e ⎥ E⎢ ⎣ ⎦ solution to linear differential equation: 32 ...
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Lecture14Extras - Maxwell Model(better for stress...

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