Page 43 - 《应用声学》2024年第1期

第 43 卷 第 1 期 寻天雨等： 基于四水听器的充水弹性管声速测量方法 39

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附录A

T
在柱坐标系下，轴对称情况下管壁固体材料中的位移 u = (u r, u θ, u z) 和应力分量 σ rz、σ rr 为
∂Φ ∂Φ ∂Φ ∂Φ e r ∂ (rΨ θ) e z ∂ (rΨ θ)
u = ∇Φ + ∇ × Ψ = e r + e z + ∇ × (Ψ θe θ) = e r + e z − +
∂r ∂z ∂r ∂z r ∂z r ∂r
( ) ( )
∂Φ ∂Ψ θ ∂Φ Ψ θ ∂Ψ θ
= − e r + + + e z,
∂r ∂z ∂z r ∂r
( )
∂u r ∂u z
σ rz = µε rz = µ + ,
∂z ∂r
( ) ( )
∂u r u r 1 ∂u θ ∂u z ∂u r ∂u r u r ∂u z ∂u r
σ rr = λ (ε rr + ε θθ + ε zz) + 2µε rr = λ + + + + 2µ = λ + + + 2µ .
∂r r r ∂θ ∂z ∂r ∂r r ∂z ∂r
其中，ε rz、ε rr、ε θθ、ε zz 表示应变，在柱坐标系下，它们与固体中位移的关系为
∂u r ∂u z ∂u r u r 1 ∂u θ ∂u z
ε rz = + , ε rr = , ε θθ = + , ε zz = .
∂z ∂r ∂r r r ∂θ ∂z
附录B

5 × 5 的矩阵 Q 的所有元素为

k r
Q 11 = Q 21 = Q 31 = 0, Q 41 = J 0(k rb), Q 51 = J 1(k rb),
ρω 2
Q 12 = − 2jk zk lrJ 1(k lra), Q 13 = Q 12| J(·)→N(·) ,
( 2 ) 2
k tr 2 1 3k tr k tr k tr
Q 14 = J 0(k tra) + k z − − J 1(k tra) − J 2(k tra) + J 3(k tra),
2a a 2 4 2a 4
Q 15 = Q 14| J(·)→N(·) ,
Q 22 = Q 12| a→b, Q 23 = Q 22| J(·)→N(·) ,
Q 24 = Q 14| a→b, Q 25 = Q 24| J(·)→N(·) ,
( 2 2 ) 2
Q 32 = − λ (ω/c l) + µk lr J 0(k lra) + µk lr J 2(k lra),
Q 33 = Q 32| J(·)→N(·) , Q 34 = −µjk zk tr (J 0(k tra) − J 2(k tra)) , Q 35 = Q 34| J(·)→N(·) ,
Q 42 = Q 32| a→b, Q 43 = Q 42| J(·)→N(·) , Q 44 = Q 34| a→b, Q 45 = Q 44| J(·)→N(·) ,
Q 52 = − k lrJ 1(k lrb), Q 53 = Q 52| J(·)→N(·) , Q 54 = −jk zJ 1(k trb), Q 55 = Q 54| J(·)→N(·) .