3.044 MATERIALS PROCESSING LECTURE 18 Results from last time: Fluid Flow · the force of gravity is enough to deliver the metal into the mold · turbulent ﬂow → well mixed (good), but more aggressive on solid tube (bad) Heat Transfer · solidiﬁcation takes ∼ 13min, ∼ 17m Date: April 25th, 2012. 1 2 LECTURE 18 Can this process be continuous? Conservation of Volume: m3 = (0.0005)(1)(Vout ) (0.25)(1)(1.3) s m m = 25 Vout = 650 min hr ⇒ Such a high velocity is dangerous and requires an extremely long factory Therefore: The only path to fully continuous strip production is to cast even thinner ⇒ strip casting · How do you maintain throughput? – parallelize ⇒ multi-strand casting (requires huge capital investment) · New processes: mini-mills – record thin casting ∼ 2mm √ – s = 2γ αt 3.044 MATERIALS PROCESSING 3 Solid State Shape Forming: Hot solid material: must be high enough temperature to have viscous (ﬂuid like) ﬂow, but not so high that it melts Newtons law of viscosity τxy = −μ ∂vx ∂y Creep for small angles: γ ≈ ∂γ ∂ γ̇ = ∂t = ∂t ∂x = ∂y γ̇ = δx δy ≈ ∂x ∂y ∂ ∂y (vx ) ∂vx ∂y In conclusion: Newton’s law of viscosity for ﬂuids is same as the creep law for solids except for a factor of μ and the negative sign diﬀerence between solid and ﬂuid mechanics τxy = ± μγ̇ 4 LECTURE 18 m is deﬁned as strain rate sensitivity when m = 1, Newtonian ∼ ﬂuid like when m < 1, non-Newtonian Stability of Tensile Flow: Will necking to failure occur? How far will it stretch? σi Ai = σh Ah ⇓ stress ⇓ geometry relates to strain 3.044 MATERIALS PROCESSING General Power-Law Equation: σ = μΣ˙ m m ˙m i Ai = ˙h Ah Volume Conserved: V0 = V A0 L0 = AL True Strain: L = ln L0 A0 = ln A A = A0 exp(−) m ˙m i A0,i exp(−) = ˙h A0,h exp(−) 1 1 i h m m ˙i A0,i exp − = ˙h A0,h exp − m m d 1 di m1 i h h m A0,i exp − = A0,h exp − dt m dt m 1 i 1 h m m e− m = di A0,h e− m di A0,i 5

Lecture 18: Newtonian and Non-Newtonian Flow

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