文档介绍：Extensional Flow and Extensional Viscosity 拉伸及剪切粘度的定义
1. Contraction Flow (injection, extrusion. etc.)
2. Fiber spinning 剪切
3. Film blowing
4. Blow Forming
5. Thermo forming (Vacuum forming, Pressure 拉伸
forming, etc. )
6. pumping, filling, blending, spreading, ink-jets
Industrial processing conditions always have some
form of extensional deformation
L1 − L0 ∆L
应变: ε= = L0初始长度; L1最终长度
L0 L0
∆L
应变速率:ε& =
L0∆t
l F 真实拉伸应变和拉伸应变速率的表示法
δ e = ⋅= λ⋅ f Hencky
l0 A0
样品尺寸沿受力方向上均匀
Rate of Deformation Tensor for Simple Extensional Flow Stress-Strain Relation for Extensional Flow
∂υσ= − p δ+ ηε⇒
ε= 3 or υ= ε x ij ij & ij
&33 ∂x3 3 &33 3 ε is the extensional rate
& σ 11 σ 12 σ 13 1 0 0 2 ε& 0 0
x σσσ= − p 0 1 0 + η 0 −ε 0
3 21 22 23 &
σ 31 σ 32 σ 33 0 0 1 0 0 −ε&
∂υ1
∂υε= ∂x or υ= ε x σ 11 = − p + 2ηε& σ+ σ+ σσ
ε= 2 or υ= ε x &11 1 1 &11 1 11 22 33 11
&22 ∂x 2 2 &22 2 p = - = −
σ= σ= − p −ηε 3 3
x 22 33 &
2 x1
σ11
2ε 0 0 ∴σ11 = + 2ηε& ⇒σ11 = 3ηε&
&11 3
∴ε= 0 2ε 0
&ij &22 We define a new material function the Extensional (or Elongational) Viscosity:
0 0 2ε&33
σ11
For simple extensional flow, due to conservation of mass: ηE (ε&) =
ε&
2ε 0 0 At very low extension (stretch) rates:
ε&11 = ε& &
ε∴ε& ij = 0 −ε& 0
ε= ε= −& ηE = 3ηo
&22 &33