Appendix A Symbols, Sign Convention, and Units
A.1 Common Symbols
Aij , Bij , Dij or Aijkl , Bijkl , Dijkl or A, B, D: extensional, coupling, bending stiffnesses ~ ~ ~ Aij , Bij , Dij : detailed in extended symbols 2
t t t Aij , Bij , Dij : detailed in extended symbols 2
h h h Aij , Bij , Dij : detailed in extended symbols 2
~t ~t Aij ,Bij : detailed in extended symbols 2 ~ ~ ~ A, B, D : detailed in extended symbols 2 A,B : material eigenvector matrices
ak ,bk : material eigenvectors
Cij or Cijkl or C: elastic stiffness
E E Cij or Cijkl : elastic stiffness at constant electric field c, d or ct ,dt : heat eigenvectors
ch ,dh : moisture eigenvectors
D j : electric displacement
Dij : bending stiffness (used in coupling problem) D : detailed in extended symbols 8 (used in 2D problem) E : Young’s modulus of isotropic materials
Ek : electric field, sometimes used as Young’s modulus of k-direction
(0) Ek : mid-plane electric field
(1) Ek : rate of electric field change
C. Hwu, Anisotropic Elastic Plates, DOI 10.1007/978-1-4419-5915-7, 589 C Springer Science+Business Media, LLC 2010
590 Appendix A
eij or ekij : piezoelectric stress tensors ˆ fi : prescribed point force
fα (zα ) : elasticity complex functions
f (z) : complex function vector
G: energy release rate, sometimes used as shear modulus θ θ G1( ),G3 ( ) : detailed in extended symbols 7
g(zt ) or gt (zt ) : thermal complex function
gh (zh ) : hygrocomplex function
gij or gijk : piezoelectric strain/voltage tensor
H : moisture content
H 0 : mid-plane moisture content
H * : rate of moisture content change
hi : heat flux (used in 2D problem)
h* : heat flux in surface normal direction (2D problem) I : unit matrix
I1,I2 ,I3 : detailed in extended symbols 3
i1,i 2 ,i3 : base vectors, detailed in extended symbols 3
i = −1 : pure imaginary number Im: imaginary part of a complex number
K I , K II , K III : stress intensity factors
t h *t *h Kij , Kij , Kij , Kij : detailed in extended symbols 2
t kij or kij : heat conduction coefficient
h kij : moisture diffusion coefficients
= T k (K II , K I , K III ) : vector of stress intensity factors, sometimes used as coefficient vector
M ij : bending moments (used in coupling problems) Appendix A 591
M : impedance matrix, detailed in extended symbols 8
M* : bimaterial matrix, detailed in extended symbols 8 ( mi : moisture transfer resultant (used in hygroproblem)
mˆ i : prescribed moisture transfer resultant (used in hygroproblems) or prescribed bending moment (used in coupling problems) ~ mi : surface resultant moments
mˆ : prescribed bending moment
mn ,m s : moment on the surface with normal n and s
Nij : stress resultants
N : fundamental elasticity matrix
N1, N2 , N3 : submatrices of fundamental elasticity matrix
Nm : fundamental elasticity matrix of mixed formalism
N(θ ) : detailed in extended symbols 6
Nˆ (θ ), Nˆ (θ,α) : detailed in extended symbols 6 ~ ~ N(θ ), N(θ,α) : detailed in extended symbols 6 ( N()θ : detailed in extended symbols 6
θ <−1> <−1> {N3 ( )} , N3 : detailed in extended symbols 6 n : normal direction, detailed in extended symbols 3
Qi : transverse shear force ( qi : heat flux resultant (used in coupling problem, same as hi in 2D problem)
qˆi : prescribed heat flux resultant
Qij or Q: elastic stiffness (used for lamina)
Q, R,T : matrices related to material properties
Qm , R m ,Tm : Q, R,T of mixed formalism r, θ : polar coordinate system Re: real part of a complex number ˆ Sij : reduced elastic compliances 592 Appendix A
Sij or S: elastic compliances
S, H, L: Barnett–Lothe tensors
S*, H*,L* : detailed in extended symbols 7 ~ ~ ~ S, H,L : detailed in extended symbols 7
s : tangential direction, detailed in extended symbols 3 T : temperature
T 0 : mid-plane temperature
T * : rate of temperature change
ti : surface traction ~ ti : surface resultant force t : surface traction vector
t n ,t s : traction on the surface with normal n and s
∞ ∞ t1 , t 2 : detailed in extended symbols 9 tr : trace of the matrix
U i : displacement (used in coupling problem)
ui : mid-plane displacement or displacement (2D problem) u : displacement vector
ud : generalized displacement vector (coupling problem) Δu : crack opening displacement
Vi : effective transverse shear force W : detailed in extended symbols 8
w : deflection, displacement in x3-direction
x1, x2 , x3 or x,y,z : rectangular coordinate system = + μ zα x1 α x2 : anisotropic complex variables = +τ zt x1 x2 : heat complex variables
zˆα : prescribed value of zα
α : thermal expansion coefficient of isotropic material Appendix A 593
α t ij : coefficients of thermal expansion
α h ij : coefficients of moisture expansion
h t *h *t αi ,αi ,αi ,αi : detailed in extended symbols 5 β i : negative slope in xi -direction (coupling problem) β ij : thermal moduli (used in 2D thermal problems) β σ ij : dielectric non-permittivities at constant stress (used in piezoelectric problems)
β : slope vector (used in coupling problem)
ββ,,,123 β β : vector of thermal moduli, detailed in extended symbols 5 (used in 2D problem) θ ~* γ, γ( ), γt , γ h , γi : detailed in extended symbols 5 ε ij : mid-plane strains or strains (2D problem)
∞ ∞ ε1 ,ε2 : detailed in extended symbols 9
ζ α : detailed in extended symbols 10 ˆ ζ α : prescribed value of ζ α
η or ηt : heat eigenvector
ηh : moisture eigenvector κ ij : curvature
μ or μα : material eigenvalue
μ(θ ) : detailed in extended symbols 4
μˆ(θ ), μˆ(θ,α) : detailed in extended symbols 4
μ~(θ ), μ~(θ,α) : detailed in extended symbols 4
ν : Poisson’s ratio ξ ij : strains (used in coupling problems)
ξ = (a,b)T : material eigenvector
ψ σ = ei : point on the circular boundary 594 Appendix A
σ ij : stresses τ τ or t : heat eigenvalue τ h : moisture eigenvalue
τ (θ ),τˆ(θ ),τˆ(θ,α) : detailed in extended symbols 5
ϕ : hole boundary parameter (used in coupling problem) φ i : stress function φ d : generalized stress function vector (coupling problem) φ : stress function vector
ψ : hole boundary parameter (used in 2D problem) ψ i : bending stress function ψ : bending stress function vector
ωε jk : dielectric permittivities at constant strains A.2 Extended Symbols
1. Superscripts, subscripts, etc.
•T : transpose of vector or matrix − • 1 : inverse of matrix •k : kth power of variable or function or matrix •(k ) • or k : variable or function or matrix of the kth material or sometimes used as the kth sub-matrix
•′ : differentiation of function • : complex conjugate
•ˆ : usually used as a prescribed value
< •α >: diagonal matrix whose component is varied according to its subscript α
2. Extension, coupling, bending stiffnesses, etc.: ~ ~ ~ t t t t *t ~t ~t Aij ,Bij ,Dij , Aij ,Bij ,Dij ,Kij ,Kij , Aij ,Bij ,Kt ,K h Appendix A 595
~ ~ ~ ~ = − −1 ~ = −1 ~ = −1 Aij , Bij , Dij : related to Aij , Bij , Dij by A A BD B, B BD , D D t t t h h h t h *t *h ~t ~t Aij , Bij , Dij , Aij , Bij , Dij , Kij , Kij , Kij , Kij , Aij , Bij , Kt , Kh : defined as follows h / 2 h / 2 h / 2 A = C dx , B = C x dx , D = C x 2 dx ijks ∫−h / 2 ijks 3 ijks ∫−h / 2 ijks 3 3 ijks ∫−h / 2 ijks 3 3 h / 2 h / 2 h / 2 At = C α t dx , Bt = C α t x dx , Dt = C α t x 2dx ij ∫−h / 2 ijks ks 3 ij ∫−h / 2 ijks ks 3 3 ij ∫−h / 2 ijks ks 3 3 h / 2 h / 2 h / 2 Ah = C α h dx , B h = C α h x dx , D h = C α h x 2 dx ij ∫−h / 2 ijks ks 3 ij ∫−h / 2 ijks ks 3 3 ij ∫−h / 2 ijks ks 3 3 h / 2 h / 2 h / 2 h / 2 K t = k t dx , K *t = k t x dx , K h = k h dx , K *h = k h x dx ij ∫−h / 2 ij 3 ij ∫−h / 2 ij 3 3 ij ∫−h / 2 ij 3 ij ∫−h / 2 ij 3 3 ~t = t − ~ t ~t = ~ t Aij Aij Bijkl Bkl , Bij Dijkl Bkl , = t τ −τ = h τ −τ Kt K22 ( t t ) / 2i, Kh K22 ( h h ) / 2i
3. Base vectors, unit matrix, and their extensions: i k ,s,n,I , I k
⎧1⎫ ⎧0⎫ ⎧0⎫ ⎧cosθ ⎫ ⎧− sinθ ⎫ = ⎪ ⎪ = ⎪ ⎪ = ⎪ ⎪ θ = ⎪ θ ⎪ θ = ⎪ θ ⎪ i1 ⎨0⎬ , i2 ⎨1⎬, i3 ⎨0⎬ , s( ) ⎨sin ⎬, n( ) ⎨ cos ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0⎭ ⎩0⎭ ⎩1⎭ ⎩ 0 ⎭ ⎩ 0 ⎭
⎡1 0 0⎤ ⎡1 0 0⎤ ⎡0 0 0⎤ ⎡0 0 0⎤ I = ⎢0 1 0⎥, I = ⎢0 0 0⎥ , I = ⎢0 1 0⎥ , I = ⎢0 0 0⎥ ⎢ ⎥ 1 ⎢ ⎥ 2 ⎢ ⎥ 3 ⎢ ⎥ ⎣⎢0 0 1⎦⎥ ⎣⎢0 0 0⎦⎥ ⎣⎢0 0 0⎦⎥ ⎣⎢0 0 1⎦⎥ Note: the dimensions of the above vectors or matrices may change depending on the problems, for example, sT = (cosθ, sinθ), nT = (−sinθ, cosθ) for inplane problem sT = (cosθ, sinθ, 0), nT = (−sinθ, cosθ, 0) for in-plane anti-plane coupling sT = (cosθ, sinθ, 0, 0), nT = (−sinθ, cosθ, 0, 0) for electromechanical problem θ where the angle is directed counterclockwise from the positive x1-axis to the direction of s 596 Appendix A
4. Material eigenvalue: μ, μ(θ ), μˆ(θ ), μ~(θ ), μˆ(θ,α), μ~(θ,α)
μcos θ− sinθ μ(θ)= μsin θ+cosθ 1 θ μˆ (θ,α) =cos(θ−α) +sin(θ−α)μ (α ), μ (θ, α) =μ (ω)dω % π ∫α = = = μ μ(0), μ ˆ (θ) μˆ(θ, 0), μ%(θ) μ%(θ, 0), −1 μˆ(θ)=μˆ(0,θ)=cosθ−μ(θ)sinθ
τ τ θ τ θ τ θ α ~* ~* 5. Heat eigenvalue and its extension: , ( ), ˆ( ), ˆ( , ) , β1,β2 ,β , γ1 , γ 2 , etc. Two-dimensional problems: τθcos− sin θ τθ()= τθsin+ cos θ τˆ(θ,α) = cos(θ −α) + sin(θ −α)τ (α), τ = τ (0), τˆ(θ ) = τˆ(θ,0) = cosθ +τ sinθ
⎧c⎫ ⎡0 N ⎤⎧β ⎫ ⎧γ (θ )⎫ ⎡0 N (θ )⎤⎧β (θ )⎫ η = , γ = − 2 1 , γ(θ ) = 1 = − 2 1 , ⎨ ⎬ ⎢ T ⎥⎨ ⎬ ⎨ θ ⎬ ⎢ T θ ⎥⎨ θ ⎬ ⎩d⎭ ⎣I N1 ⎦⎩β2 ⎭ ⎩γ 2 ( )⎭ ⎣I N1 ( )⎦⎩β2 ( )⎭ β β ⎧ 11 ⎫ ⎧ 12 ⎫ = ⎪β ⎪ = ⎪β ⎪ β1 ⎨ 21 ⎬, β2 ⎨ 22 ⎬ ⎪β ⎪ ⎪β ⎪ ⎩ 31 ⎭ ⎩ 32 ⎭ θ = θ θ = θ = β1( ) βs( ), β2 ( ) βn( ), β [β1 β2 β3 ]
1 2π − 1 2π − ~γ * = τˆ 1 (θ )γ (θ )dθ, ~γ * = τˆ 1 (θ )γ (θ )dθ 1 2π ∫0 1 2 2π ∫0 2
Coupled stretching–bending problems: τθcos−− sin θ τθ cos sin θ τθ()==th , τθ () , thτθ++ θ τθ θ thsincos sincos τθ=+ θ τ θ τθ =+ θ τ θ ˆˆtthh( ) cos sin , ( ) cos sin
c c η = ⎧ t ⎫ η = ⎧ h ⎫ t ⎨ ⎬, h ⎨ ⎬ ⎩dt ⎭ ⎩dh ⎭ 0 (N ) ⎧α~t ⎫ 0 (N ) ⎧α~ h ⎫ γ = − ⎡ m 2 ⎤⎪ 1 ⎪ γ = − ⎡ m 2 ⎤⎪ 1 ⎪ t It ⎢ T ⎥⎨ ⎬, h It ⎢ T ⎥⎨ ⎬ , I (N ) ~t I (N ) ~ h ⎣ m 1 ⎦⎩⎪α 2 ⎭⎪ ⎣ m 1 ⎦⎩⎪α 2 ⎭⎪ I I I 0 0 0 = ⎡ 1 2 ⎤ = ⎡ ⎤ = ⎡ ⎤ It ⎢ ⎥, I1 ⎢ ⎥, I 2 ⎢ ⎥ ⎣I 2 I1 ⎦ ⎣0 0⎦ ⎣0 I⎦ Appendix A 597
ttθθθ ⎪⎪⎧⎫γα11()⎡⎤0N(())m 2 ⎪⎪⎧⎫% () γ ()θ ==−I , tt⎨⎬tt⎢⎥θ T ⎨⎬ ⎪⎪γα()θθ⎣⎦IN(())m 1 ⎪⎪ () ⎩⎭22 ⎩⎭% hhθθθ ⎪⎪⎧⎫γα11()⎡⎤0N(())m 2 ⎪⎪⎧⎫% () γ ()θ ==−⎨⎬I ⎨⎬ hthhθθ⎢⎥IN(())θ T ⎩⎭⎪⎪γα22()⎣⎦m 1 ⎩⎭⎪⎪% ()
α~t (θ ) = cosθα~t + sinθα~t , α~t (θ ) = −sinθα~t + cosθα~t 1 1 2 2 1 2 ~h θ = θ~ h + θ~h ~t θ = − θ~h + θ~ h α1 ( ) cos α1 sin α2 , α2 ( ) sin α1 cos α2 ~ ~ ⎧α~t ⎫ ⎧ α~t ⎫ ⎧At ⎫ ⎧B t ⎫ ~t = ⎪ A1 ⎪ ~t = ⎪ A2 ⎪ ~t = ⎪ 1i ⎪ ~t = ⎪ 1i ⎪ α1 ⎨ ⎬, α2 ⎨ ⎬, α Ai ⎨ ~ ⎬, α Bi ⎨ ~ ⎬ ⎪~t ⎪ ⎪− ~t ⎪ t t ⎩α B2 ⎭ ⎩ α B1 ⎭ ⎩⎪A2i ⎭⎪ ⎩⎪B2i ⎭⎪ ~ ~ ⎧α~h ⎫ ⎧ α~ h ⎫ ⎧Ah ⎫ ⎧B h ⎫ ~h = ⎪ A1 ⎪ ~ h = ⎪ A2 ⎪ ~ h = ⎪ 1i ⎪ ~h = ⎪ 1i ⎪ α1 ⎨ ⎬, α2 ⎨ ⎬, α Ai ⎨ ~ ⎬, α Bi ⎨ ~ ⎬ ⎪~h ⎪ ⎪− ~ h ⎪ h h ⎩α B2 ⎭ ⎩ α B1 ⎭ ⎩⎪A2i ⎭⎪ ⎩⎪B2i ⎭⎪
tht h ⎪⎪⎧⎫ααAi ⎪⎪⎧⎫ Ai ⎪⎪⎧⎫ α Bi ⎪⎪⎧⎫ α Bi αααth==, , ** t = , α h = ii⎨⎬tht ⎨⎬ i ⎨⎬ i ⎨⎬ h ⎩⎭⎪⎪ααBi ⎩⎭⎪⎪ Bi ⎩⎭⎪⎪ α Di ⎩⎭⎪⎪ α Di tt t h ⎪⎪⎧⎫ABD11ii ⎪⎪⎧⎫ ⎪⎪⎧⎫ 1 i ⎪⎧⎫ A 1 i⎪ αααttt===, , , α h =, Ai⎨⎬tt Bi ⎨⎬ Di ⎨⎬ t Ai ⎨ h⎬ ⎩⎭⎪⎪ABD22ii ⎩⎭⎪⎪ ⎩⎭⎪⎪ 2 i ⎩⎪ A 2 i⎭⎪ hh ⎪⎪⎧⎫BD11ii ⎪⎪⎧⎫ ααhh===, , i 1,2 Bi⎨⎬hh Di ⎨⎬ ⎩⎭⎪⎪BD22ii ⎩⎭⎪⎪
1 π − 1 π − ~γ t* = τˆ 1 (ω)γ t (ω)dω, ~γ h* = τˆ 1 (ω)γ h (ω)dω, i = 1,2 i π ∫0 t i i π ∫0 h i
6. Elasticity matrix: θ ˆ θ ~ θ ˆ θ α ~ θ α ~ (k ) θ ~ (k ) θ ~ (k ) θ θ <−1> N, N( ), N( ), N( ), N( , ), N( , ) , N1 ( ), N2 ( ), N3 ( ) ,{N3 ( )}
Nξξ==μ , N ()θ ξξμ ()θ 1 θ NINNNˆ (θα , )=−+− cos( θ α ) sin( θ α ) ( α ), % ( θα , ) = ( ω )d ω π ∫α NN==(0), Nˆˆ (θθ ) N ( ,0), N%% ( θθ ) == N ( ,0), NN %% ( π ),
( −1 NN()θθ==ˆˆ () N (0,) θθθθ =− cos I sin N ()
~ θ ~ θ ~ θ ~ θ α ~ θ α ~ θ α ~ θ N1 ( ), N 2 ( ), N3 ( ) & N1( , ), N2 ( , ), N3 ( , ) : submatrices of N( ) & ~ N(θ,α) (((θθθ ( θ NNN123(), (), (): sub-matrices of N() 598 Appendix A
~ (k ) θ ~ (k ) θ ~ (k ) θ ~ θ ~ θ ~ θ N1 ( ), N2 ( ), N3 ( ) : N1 ( ) , N2 ( ) , N3 ( ) of kth material θ <−1> θ {N3 ( )} : sub-inverse of N3 ( ) defined in (5.21); <−1> N3 : sub-inverse of N3
7. Barnett–Lothe tensors and their extensions: * * * ~ ~ ~ θ θ S, H, L, S , H ,L ,S, H,L , G1( ),G3 ( )
ππ =−=TT11θθ == θθ SABIidid(2 ) N12 ( ) , H2AAN ( ) , ππ∫∫00
π =−T =−1 θθ L2BBid N3 () π ∫0
⎡ S H ⎤ ~ = ~ π = or N N( ) ⎢ T ⎥ ⎣− L S ⎦ S*, H*,L* : S, H, L defined for multi-materials, for example, * * + + ⎡ S H ⎤ 1 ⎡ S1 S2 H1 H2 ⎤ ⎢ ⎥ = ⎢ ⎥ , for bimaterials − * *T 2 − + T + T ⎣⎢ L S ⎦⎥ ⎣ (L1 L2 ) S1 S2 ⎦ 1 2π 1 2π 1 2π S* = N (ω)dω, H* = N (ω)dω, L* = − N (ω)dω , for 2π ∫0 1 2π ∫0 2 2π ∫0 3 general multi-materials ~ ~ ~ S, H,L : S, H and L defined for multi-materials: ~ ~ ⎡ S H ⎤⎡ S* H* ⎤ ⎢ ⎥⎢ ⎥ = −I . ~ ~T − * *T ⎣⎢− L S ⎦⎥⎣⎢ L S ⎦⎥ ⎡ S H ⎤ ~ 2 π = − ~ π = Note: N ( ) I, where N( ) ⎢ T ⎥ ⎣− L S ⎦ θ = T θ − θ −1 θ = − θ −1 G1( ) N 1 ( ) N3 ( )SL , G3 ( ) N3 ( )L
8. Impedance matrix, bimaterial matrix, and their extensions: M, M*, D, W =−=−−−11T −− 11 =+ − 1 MBAHSHHHSii i, MABLSLLLS−−−−−−111111==−=+iiiT MM*11=+=−− Mii(), ABABDW − 1 − − 1 =− 12 1122 =+−−11 = − 1 − − 1 DL12 L, WSLSL 1122 Appendix A 599
∞ ∞ ∞ ∞ 9. Uniform stresses and strains at infinity: İ1 ,İ2 ,t1 ᇫ, t 2 σ ∞ σ ∞ ε ∞ ε ∞ 11 ½ 21 ½ 11 ½ 21 ½ ∞ = °σ ∞ ° ∞ = °σ ∞ ° ∞ = ° ε ∞ ° ∞ = ° ε ∞ ° t1 ® 12 ¾ᇫ, t 2 ® 22 ¾, İ1 ® 12 ¾ᇫ, İ2 ® 22 ¾ °σ ∞ ° °σ ∞ ° ° ε ∞ ° ° ε ∞ ° ¯ 13 ¿ ¯ 23 ¿ ¯2 13 ¿ ¯2 23 ¿
10. Mapped variable:ζ α ,ψ ,θ, ρ Elliptical hole or inclusion: 2 2 2 2 zα + zα − a − b μα 1 1 ½ ζ = , z = (a − ibμ )ζ + (a + ibμ ) α − μ α ® α α α ζ ¾ a ib α 2 ¯ α ¿ ρ cosθ = −a sinψ , ρ sinθ = b cosψ , ρ = a2 sin 2 ψ + b2 cos2 ψ use ϕ to replace ψ in the stretching–bending problem
Polygon-like hole or inclusion: −1 k −k zα = a{(1− iμα c)ζ α + (1+ iμα c)ζ α + ε (1+ iμα )ζ α + ε (1− iμα )ζ α }/ 2 ρ cosθ = − a(sinψ + kε sin kψ ), ρ sinθ = a(c cosψ − kε cos kψ ) ρ 2 = a 2{k 2ε 2 + sin 2ψ + c2 cos2ψ + 2kε sinψ sin kψ − 2ckε cosψ coskψ }
Crack or line inclusion: 1 a 1 ζ = {z + z 2 − a2 }, z = ζ + ½ Į Į Į α ® α ζ ¾ a 2 ¯ α ¿
A.3 Sign Convention Since the positive directions of stresses and displacements and their related physical responses are defined according to their directions along the coordinate axes, to consider the sign conventions for these symbols two different right-handed Cartesian coordinate systems are shown in Figs. A.1 and A.2. The former contains five sub-figures showing the positive directions of several physical quantities based upon the coordinate system with upward positive x3 , whereas the latter is based upon the coordinate with downward positive x3 . These two figures show the positive directions of (a) the coordinate system (x1, x2 , x3 ) , ~ ~ ~ displacements (u1,u2 ,u3 ) , tractions (t1,t2 ,t3 ) , resultant forces (t1, t2 , t3 ) , moments ~ ~ ~ β β σ (mx ,my ) , resultant moments (m1,m2 ,m3 ) , and slopes ( x , y ); (b) the stresses ij ;
(c) the stress resultants Nij , bending moments M ij , transverse shear forces Qi , effective transverse shear forces Vi , corner forces tc , and transverse distributed load q; (d) the quantities in the Cartesian coordinate (x, y) and the tangent–normal β β β β coordinate (s,n) : (,uux y )– (,uus n ), ( x , y ) –( s , n ) , and (mx ,my ) –(ms ,mn ). 600 Appendix A
Note that in Figs. A.1 and A.2 the moment or slope is represented by a vector in the form of a double-headed arrow. The direction of the moment or slope is indicated by the right-hand rule – namely, using your right hand, let your fingers curl in the direction of the moment or slope, and then your thumb will point in the direction of the vector.
A.4 Units The International System of Units, abbreviated SI units, is a worldly accepted version of the metric system. In SI units for mechanics, mass in kilograms (kg), length in meters (m), and time in seconds (s) are selected as the base units, and force in newton (N) is derived from the preceding three by Newton’s second law. Thus, force (N) = mass (kg) × acceleration (m/s2) is one of the derived units used in mechanics. For the convenience of readers’ reference, Table A.1 shows some physical quantities and their SI units as well as SI symbols and relations used in this book. The SI unit prefixes used for the multiplication factors of the SI symbols are then shown in Table A.2.
Scaling Factor Table A.3 is an example of material properties of graphite/epoxy and left-hand quartz by using SI units. From this table we see that some values of the properties such as elastic stiffness constants are in the order above 9, while some others such as dielectric permittivity are in the order of −11. Thus, it is very possible that the fundamental elasticity matrix N constructed through (11.18) and (11.19) for piezoelectric materials will contain the components ranging from − 10 11 to 10 9 , which is numerical ill-condition. To avoid any trouble caused by the ill-conditioned matrix, a proper scale adjustment is necessary. For example, we may rewrite the constitutive relation (11.13)1 into σ/(/)(/),ECE=−E ε eEE ij00 ijkl kl kij k 0 (A.1) =+ε ωε Dejjklkljkk()(/), E00 EE
9 2 in which E 0 is a reference number, such as 10 N/m , used for scale adjustment. Equation (A.1) means that in numerical calculation the material ε ε E ω E ω properties C ijkl and jk are replaced by Cijkl /E0 and E 0 jk . By such σ replacement the output values of stresses and electric fields are ij / E 0 and
E k / E 0 , which should be multiplied by E 0 to return to their original units.
Appendix A 601
Elastic Materials
From the definition given in (3.48) we know that N1 is dimensionless and
N2 and N3 are, respectively, proportional to 1/E and E where E stands for the Young’s modulus. Thus, the fundamental elasticity matrix N will become ill-conditioned even for the pure elastic materials. Therefore, the scaling factor
E0 introduced in (A.1) to nondimensionalize the elastic constants Cijkl is suggested to be used in the computer programming for elastic materials, i.e., every
Cijkl will be divided by E0 in numerical calculation. To trace the effects of scaling factor, it is better to know the proportional relation with Young’s modulus for some commonly used matrices, which is shown in Table A.4. Take some examples from this table such as S, H, and L; we know that their values will become S, E 0H , and L / E0 when C ijkl is nondimensionalized by using
Cijkl / E 0. To return their original values, the latter two should be divided or multiplied by E0 .
602 Appendix A
x3,,,utt 333%
β x2,,,utt 222% xx,,mm% 2
m% 3 β ,m y y x1111,,,utt% m%1
(a) σ 33 σ 32 σ 31 σ 13 σ 12 σ σ 11 21 σ 22 σ 23 (b) N22 M 21
M 22 N12 M q 11 N12 M12 tc N11 Q2 ,V2 tc N11 M12 N12 Q1,V1 M11 N21 M 22 Q1,V1
N22 M 21 tc Q2 ,V2 tc
(c)
y,uy
n,un s,u β s x ,mx β s ,ms β ,m θ y y θ x,ux body(on the right hand side of the route) β n ,mn θ :counterclockwise
(d)
Fig. A.1 Sign convention for a right-handed Cartesian coordinate system with β σ upward positive x3 : (a) uttmmiii,,,% i ,% i , i; (b) ij ; (c) NMQVtqij, ij, i , i , c , ; (d) sn− coordinate.
Appendix A 603
β ,m m%1 yy x ,,,utt% β 1111 xx,,mm% 2
m% 3 x2,,,utt 222%
x3333,,,utt% (a)
σ 33 σ σ 32 31 σ σ 12 11 σ σ 13 21 σ 22 σ 23 (b) N22
M 21 N 21 M 22 N12
q M11 N11 tc t Q2 ,V2 M M12 c 12 N11 M11 Q1,V1 N M 12 22 N21 Q1,V1 M 21 N t tc 22 c Q2 ,V2
(c) β n ,mn
β y ,my θ x,ux β θ s ,ms θ β x ,mx n,un s,us
y,u y (d)
Fig. A.2 Sign convention for a right-handed Cartesian coordinate system with downward positive : (a) β ; (b) σ ; (c) x3 ui,ti,t%i,mi,m% i, i ij Nij , Mij ,Qi ,Vi ,tc , q; (d) sn− coordinate. 604 Appendix A
Table A.1 SI units Quantity SI unit SI Relation symbol (Base units) Length Meter m Mass Kilogram kg Time Second s Temperature Kelvin °K Electric current Ampere A (Derived units) Force Newton N N =⋅kg m/s2 Stress, pressure Pascal Pa Pa=N/m2 Energy, work Joule J J= N ⋅m (= volt ⋅coul) Power Watt W W=J/s Electric charge Coulomb C C= A⋅s (= joule/volt ) Electric potential Volt V V=J/C (= amp⋅ohm) Electric resistance Ohm Ω Ω = V/A Capacitance Farad F F=C/V Magnetic flux Weber Wb Wb = V⋅s Magnetic field Tesla T T=Wb/m2 (= N/(amp⋅ m)) Celsius Degree °C °C = °K- 273 temperature Electric field Volt/m Volt/m=N/coul 2 Electric Coul/m Coul/m2=N/ (m⋅ volt) displacement Permittivity Farad/m Farad/m=N/V2= C2/(m2 ⋅ N) Thermal Watt/ (m°K) W/ (m°K) = W/ (m°C) conductivity Thermal expansion 1/°K 1/ °K = 1/°C Thermal modulus N/m2 °K N/m2 °K = N/m2 °C
Table A.2 SI prefixes
Prefix Symbol Factor Pico p 10−12 Nano n 10−9 Micro μ 10−6 Milli m 10−3 Centi c 102 Kilo k 103 Mega M 106 Giga G 109 Tera T 1012 Appendix A 605
Table A.3 Material properties of graphite/epoxy and left-hand quartz
Left-hand Graphite/epoxy quartz E C11 (GPa) 183.71 86.74
E C12 (GPa) 4.63 6.99
E C13 (GPa) 4.22 11.91
E C22 (GPa) 11.29 86.74
E C23 (GPa) 3.24 11.91
E C33 (GPa) 11.27 107.2
E E C44 , C55 (GPa) 7.17 57.94
E C66 (GPa) 2.87 39.88 − E E − E C14 ,C24 , C56 (GPa) – 17.91
ε ω −12 11 (F/m) – 39.21×10
ε ω −12 22 (F/m) – 39.21×10 ε ω −12 33 (F/m) – 41.03×10
2 − − e11, e12 , e26 (C/m ) – 0.171 − 2 e25 ,e14 (C/m ) – 0.0406
Table A.4 Proportional relation with Young’s modulus of elastic materials ~ 1 ~ 1/E ~ E Fundamental elasticity N , N (θ ) N , N (θ ) N , N (θ ) matrices 1 1 2 2 3 3 Barnett–Lothe tensors S H L Green’s functions T* U* Influence matrices Yin Gin
Material eigenvector A~1/ E, B~ E matrices ⇒ AB−1~1/E, ABT~1, AAT~1/E, BBT~ E
Appendix B Hilbert Problem
B.1 Solution to the Hilbert Problem in Scalar Form The Hilbert problem is usually expressed in the form of scalar functions, like
+− F (s)−gF (s)=f (s), s on L except at the ends, (B.1) where g in general is a complex constant; L is the union of a finite number of arcs = L1, L2,..., Ln where the ends of the arcs Lk , k 1, 2,..., n, are ak and bk ; f ()s i s a complex function which satisfies the Holder conditions on L, i.e., for any two points ss12, on L, −≤−γ |()f sfscss12 ()|| 12 |, (B.2) where c and γ are positive constants and 0<γ ≤1; F()z i s a sectionally + − holomorphic function in the plane cut along L and F(s) and F(s) are, respectively, the limiting values of Fz() as zs→ from the left and right. The solution to (B.1) with g ≠ 1 has been shown in Muskhelishvili (1954) as χ ()z =+0 fsds() χ Fz()+ 0 () zpn () z, πχ∫L − (B.3a) 2()()issz0
where pn (z ) is an arbitrary polynomial with the degree not higher than n and χ 0 (z) is the basic Plemelj function satisfying the relation χχ+−= 00(sg ) ( s ), s on L , (B.3b) whose solution can be expressed as
n −−δδ 1 χδ(zzazb )=−∏ ( ) ( − )1 , = ln( g ). 0 jj π (B.3c) j=1 2 i
607 608 Appendix B
When g=1, the solution to (B.1) is
=+1()fsds Fz() pn (). z (B.4) 2πisz∫L −
B.2 Solution to the Hilbert Problem in Matrix Form The Hilbert problem in matrix form can be expressed as +− ψ ()sss+=Gψ ()tˆ (),s on L except at the ends, (B.5) where G is a complex constant matrix, L is the union of arcs, and tˆ()s is a given complex function vector. To find the solution ψ()z to problem (B.5), a similar approach as that of scalar form can be employed. First, we consider the homogeneous problem +− ψ ()ss+=Gψ ()0 , sL ∈ . (B.6)
With reference to (B.3c), a particular solution ψ0 to the problem (B.6) will be sought in the form n =−−−δδ −1 (B.7) ψλ0 ()zzazb∏ (jj ) ( ) , j=1 where δ is a complex constant and λ is a complex constant vector. The function ψ0 ()z is holomorphic in the entire plane cut along L, if a definite branch of this function is selected. It is readily verified by an investigation of the − − variation in the argument of z a j or z b j , when z describes a closed path beginning at a point s of the arc a jb j and leading, without intersecting L, from the left side of a jb j around the end a j to the right side of the arc or around the end b j , that +−= 2iπδ (B.8) ψψ00()se (). s
Hence, ψ0 ()z will satisfy the condition (B.6), provided πδ ().e2i IG+=λ 0 (B.9) If GMM= *1− * where MDW* =−i is the bimaterial matrix, the explicit solution for the eigenvalue δ has been given by Ting (1986) as 1 1 1 δ = + iε, δ = − iε, δ = , (B.10a) 1 2 2 2 3 2 where Appendix B 609
11+ β 1 − εβ=ln , =[]−tr(WD 12) 1/2, (B.10b) 21πβ− 2 and tr stands for the trace of the matrix. Thus, a particular solution ȥ0 ()z of the homogeneous problem has been found; it is given by (B.7) with δ and Ȝ determined by (B.9). Since there are three eigenvalues to (B.9), a linear combination of these particular solutions will still be one of the particular solutions, i.e., = (B.11a) ȥ000(zz) Xp() , where = (B.11b) X0 ()zzȁī () and
n −δ δ − = =< − α − α 1 > ȁ [Ȝ1Γ Ȝ 2 Ȝ 3 ], ī(z) ∏ (z a j ) (z b j ) . (B.11c) j=1 p 0 Γ is a coefficient vector. This particular solution does not vanish anywhere in − −1/ 2 − −1/ 2 the finite part of the plane and it is unbounded like | z a j | and | z b j | near the ends a j and bj , respectively. We now look for the most general solution to the homogeneous Hilbert = problem. For this purpose it will be noted that ȥ000()zzXp () , being a solution of the homogeneous problem, satisfies the condition +−+=∈ (B.12) XpGXp000()ss 00 () , sL, and hence, =− +−−1 (B.13) GX00()[ss X ()] By applying (B.13), (B.6) becomes +−+11−=∈ −−− (B.14a) [()]()[()](),X00ssȥ X ssȥ 0 ΓΓΓ sL Γ, or +−−= ∈ (B.14b) ȥȥ**()ss ()0 ,ΓΓΓ sL Γ,
−1 where ȥ* (z) denotes the sectionally holomorphic function [()]()X0 zzȥ . It follows from (B.14) that ȥ* (z) is holomorphic in the entire plane, except at the = ∞ point z , provided it is given suitable values on L. Further, since ȥ* (z) can only have a pole at infinity, it must, by the generalized Liouville theorem, be a polynomial. Thus, the most general solution of the homogeneous problem is given by 610 Appendix B
()h = (B.15) ψ ()zzzXp0 ()n (), where p n (z) is an arbitrary polynomial vector. If it is desired to obtain a solution which is also holomorphic at infinity, it must be assumed that the degree of the polynomial p n (z) does not exceed n. This follows from the behavior of X0 (z) at infinity as given in (B.11b) and (B.11c). Next consider the non-homogeneous problem. Using (B.13), the boundary condition (B.5) may be written as +−−= +−1 ˆ ∈ (B.16a) ψψ**(ss) () [Xt 0( sssL)] (), , where = −1 ψ*(z) [X0(z)] ψ(z). (B.16b) Each component of equation (B.16) is in the form of (B.1) with g = 1; hence, by (B.4) we have
11+− ψ()zz=+XXtXp () [ ()]()d ssszz1ˆ () (), (B.17) 2πisz00∫L − 0n where p n (z) is an arbitrary polynomial vector with the degree not higher than n. n.
B.3 Evaluation of a Line Integral in Scalar Form
L1
' + C C1 L1 1
− − L1 L1
'' ' L1 L1
C∞
Fig. B.1 Integration contour Consider the line integral
= h(s)ds j(z) + , (B.18) ∫L χ − 0 (s)(s z) χ where L is the union of a finite number of arcs LL12, ,..., Ln and 0 (z) is the basic Plemelj function. Suppose that h(s) is a polynomial, a situation which often occurs in practice. By residue theory, the integral around a closed contour C can be calculated as Appendix B 611
hsds() = 2,πηi (B.19) ∫C χ − 0 ()(ss z )
ηχ= where hz()/0 () z is the sum of residues of the poles of the integrand within + − ′ ′ ′′ = C. The closed contour C is the union of (Lk , Lk ,Ck ,Ck , Lk , Lk ), k 1,2,...,n and
C∞ (see Fig. B.1 for example with one arc L1 ). The summation of the integrals ′ ′′ along Lk and Lk vanishes since they have opposite directions and the integrand = across this line is continuous. If h(s) is bounded near the ends s ak and = ′ s bk , the integrals around the circles Ck and Ck can be proved to be zero ′ χ when the radii of the circles Ck and Ck tend to zero. Knowing that o (z) satisfying (B.3b), we have
h(s)ds = − h(s)ds + − (1 g) + . (B.20) ∫L + L χ − ∫L χ − k k o (s)(s z) k o (s)(s z)
Therefore, the integral around a closed contour C can now be written as
hsds()==−πη hsds () + hsds () 2(1)ig+ . ∫CLCχχχ−−−∫∫∞ 0 ()(ss z )o ()( ss z )0 ()( ss z ) (B.21)
iθ By replacing the contour of C∞ as ρe and letting ρ → ∞ , the line integral j(z) of (B.18) can now be evaluated by
iiθθ 2π ρρθ hsds()=− 1⎛⎞π hz () h ( e ) i e d + ⎜⎟2limi θθ . ∫∫L χχχρρ−−ρ→∞ 0 ii − o ()(ss z ) 1 g⎝⎠00 () z ( e )( e z ) (B.22) As an example, we now consider a straight arc L located on (−a,a) and = = − δ = h(s) h0 , g 1. From (B.3c), we have 1/ 2 and
χ = 1 0 (z) . (B.23) z 2 − a2
With the values given between (B.22) and (B.23), the line integral j(z) can be evaluated from (B.22) as
hods = − π − 2 − + i ho (z z a). (B.24) ∫L χ − o (s)(s z) 612 Appendix B
B.4 Evaluation of a Line Integral in Matrix Form Consider the line integral in matrix form
1 + − j(z) = [X (s)] 1h(s)ds, ³ − 0 (B.25) L s z
where L is the union of a finite number of arcs LL12, ,..., Ln and X0(z) is the matrix of basic Plemelj function satisfying the relation +−+=∈ (B.26) XGX000(sssL ) ( ) , . Suppose that h(s) is a polynomial and consider the closed contour shown in Fig. B.1. Following the procedure stated between (B.18) and (B.22), we can obtain
=−1 −−11 + j()z 2ʌissdsȘ [XIGh0 ()]( ) () , (B.27) ³ sz− C∞ where Ș is the sum of residues of the poles of the integrand within a closed contour C, i.e.,
1 −−11+=π [()](XIGh0 ssdsi )()2Ș . (B.28) ³C sz− The second term of (B.27) has the form
iș 2π ρ ie ρρiiθθ−−11+ θ lim [XIGh0 (ee )] ( ) ( )d , (B.29) ρ→∞ ³0 ρeziș −
where ρ is the radius of the contour C∞ . It can be shown that only terms independent of ρeiθ can contribute to the above integral. Then with a given function h(s), the integral j(z) can be explicitly evaluated from (B.27), (B.28), and (B.29). For example, consider an integral along a single line L = (−a,a) and let h(t)=h, where h is a given constant vector. The sum of the residues is =+−−11 Ș [()](XIGh0 z ). (B.30)
−1 To calculate the integral shown in (B.29), by (B.11c) we express īα (ζ ) for large |ζ | as
−1 δα −δα +1 −1 īα (ζ ) = (ȗ + a) (ȗ − a) = ζ + 2iaεα + O(ζ ), α =1,2,3. (B.31) With (B.31), (B.29) becomes Appendix B 613
2π ++<+z ρεiθ +> θ−−11 + lim() 1iθ LLeiaid 2α Λ (IG ) h ρ→∞ ∫0 ρe (B.32) −−11 =<+>22πεiz iaα Λ ()IG + g From (B.27), (B.30), and (B.32) we obtain the final result of j(z) as =−<+>+−−−111ε jX()z 2πiz {[0 ()] zia 2α Λ }(IG ) h . (B.33)
Appendix C Summary of Stroh Formalism
= = C.1 Two-Dimensional Problems: ui ui (x1, x2 ), i 1,2,3
Basic equations I. Anisotropic bodies under mechanical loadings (Section 3.1): ε = + σ = ε σ = = ij (ui, j u j,i ) / 2, ij Cijkl kl , ij, j 0, i, j, k,l 1,2,3. (C.1) = = Governing equations: Cijkluk,lj 0, i, j, k,l 1,2,3. (C.2) II. Anisotropic bodies under thermal loadings (Section 10.1): = − = hi kijT, j , hi,i 0, (C.3a)
ε = + σ = ε − β σ = = ij (ui, j u j,i )/ 2, ij Cijkl kl ijT, ij, j 0, i, j,k,l 1,2,3. (C.3b) = − β = = Governing equations: kijT,ij 0, Cijkluk,lj ijT, j 0, i, j, k,l 1,2,3. (C.4) III. Piezoelectric bodies under electromechanical loadings (Section 11.2): ε = (u + u )/ 2, σ = C E ε − e E , D = e ε + ω ε E , (C.5a) ij i, j j,i ij ijkl kl kij k j jkl kl jk k σ = = = ij, j 0, Di,i 0, i, j,k,l 1, 2, 3. (C.5b) = = = Governing equations: C pjqluq,lj 0, j,l 1, 2, 3, p, q 1,2,3,4 , (C.6a) C = C E , i, j,k,l =1,2,3; C = e , i, j,l =1,2,3, (C.6b) ijkl ijkl ij4l lij = = = −ω ε = C4 jkl e jkl , j,k,l 1,2,3; C4 j4l jl , j,l 1,2,3, (C.6c) = − σ = = u4, j E j , 4 j D j , j 1,2,3 . (C.6d) General solution I. Anisotropic bodies under mechanical loadings (Section 3.1): u = 2 Re{Af (z)}, φ = 2 Re{Bf (z)} , (C.7a) = = A [a1 a2 a3 ] , B [b1 b2 b3 ] , (C.7b) φ ⎧u1 ⎫ ⎧ 1 ⎫ ⎧ f1(z1) ⎫ = ⎪ ⎪ φ = ⎪φ ⎪ = ⎪ ⎪ = + μ α = u ⎨u2 ⎬, ⎨ 2 ⎬, f (z) ⎨ f2 (z2 )⎬ , zα x1 α x2 , 1,2,3 . (C.7c) ⎪ ⎪ ⎪φ ⎪ ⎪ ⎪ ⎩u3 ⎭ ⎩ 3 ⎭ ⎩ f3 (z3 )⎭ Anti-plane: u = 2 Re{}af (z) , φ = 2 Re {}bf (z) . (C.8) In-plane: u = 2 Re{Af (z)}, φ = 2 Re{Bf (z)} , (C.9a) = = A [a1 a2 ], B [b1 b2 ] , (C.9b) ⎧u ⎫ ⎧φ ⎫ ⎧ f (z ) ⎫ u = 1 , φ = 1 , f (z) = 1 1 . (C.9c) ⎨ ⎬ ⎨φ ⎬ ⎨ ⎬ ⎩u2 ⎭ ⎩ 2 ⎭ ⎩ f2 (z2 )⎭
615 616 Appendix C
II. Anisotropic bodies under thermal loadings (Section 10.1): T = 2Re{g′(z )}, h = −2Re{(k +τk )g′′(z )}, z = x +τx , t i i1 i2 t t 1 2 ( C.10a) = {}{}+ φ = + u 2 Re Af (z) cg(zt ) , 2 Re Bf (z) dg(zt ) , (C.10b) = = A [a1 a2 a3 ] , B [b1 b2 b3 ] , (C.10c) φ ⎧u1 ⎫ ⎧ 1 ⎫ ⎧ f1 (z1 ) ⎫ = ⎪ ⎪ φ = ⎪φ ⎪ = ⎪ ⎪ = + μ α = u ⎨u2 ⎬, ⎨ 2 ⎬, f (z) ⎨ f2 (z2 )⎬ , zα x1 α x2 , 1,2,3 . (C.10d) ⎪ ⎪ ⎪φ ⎪ ⎪ ⎪ ⎩u3 ⎭ ⎩ 3 ⎭ ⎩ f3 (z3 )⎭
III. Piezoelectric bodies under electromechanical loadings (Section 11.2): u = 2 Re{Af(z)}, φ = 2 Re{Bf (z)} , (C.11a) A = [a a a a ] , B = [b b b b ] , 1 2 3 4 1 2 3 4 (C.11b) φ ⎧u1 ⎫ ⎧ 1 ⎫ ⎧ f1(z1) ⎫ ⎪ ⎪ ⎪φ ⎪ ⎪ ⎪ ⎪u2 ⎪ ⎪ 2 ⎪ ⎪ f2 (z2 )⎪ u = , φ = , f (z) = , z = x + μ x , α = 1,2,3,4 . (C.11c) ⎨ ⎬ ⎨φ ⎬ ⎨ ⎬ α 1 α 2 ⎪u3 ⎪ ⎪ 3 ⎪ ⎪ f3 (z3 )⎪ ⎪ ⎪ ⎪φ ⎪ ⎪ ⎪ ⎩u4 ⎭ ⎩ 4 ⎭ ⎩ f4 (z4 )⎭
Material eigen-relation ⎡N N ⎤ a ξ = μξ = 1 2 ξ = ⎧ ⎫ = − −1 T = −1 = T N , N ⎢ T ⎥, ⎨ ⎬ , N1 T R , N 2 T N 2 , (C.12) ⎣N3 N1 ⎦ ⎩b⎭ = −1 T − = T N3 RT R Q N3 .
I. Anisotropic bodies under mechanical loadings (Section 3.1): = = = = Qik Ci1k1, Rik Ci1k 2 , Tik Ci2k 2 , i,k 1,2,3 , (C.13a)
⎡C11 C16 C15 ⎤ ⎡C16 C12 C14 ⎤ ⎡C66 C26 C46 ⎤ or Q = ⎢C C C ⎥, R = ⎢C C C ⎥, T = ⎢C C C ⎥. ⎢ 16 66 56 ⎥ ⎢ 66 26 46 ⎥ ⎢ 26 22 24 ⎥ (C.13b) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣C15 C56 C55 ⎦ ⎣C56 C25 C45 ⎦ ⎣C46 C24 C44 ⎦ II. Anisotropic bodies under thermal loadings (Section 10.1): τ 2 + + τ + = k22 (k12 k21) k11 0. (C.14a) c 0 N β (C.14b) = τ + = ⎧ ⎫ = −⎡ 2 ⎤⎧ 1 ⎫ Nη η γ , η ⎨ ⎬, γ ⎢ T ⎥⎨ ⎬ , β1,β2 : extended symbols 5, ⎩d⎭ ⎣I N1 ⎦⎩β2 ⎭ Q, R, T: same as Problem I, Eq. (C.13). Appendix C 617
III. Piezoelectric bodies under electromechanical loadings (Section 11.2): = = = = Qik Ci1k1, Rik Ci1k 2 , Tik Ci2k 2 , i,k 1,2,3,4, (C.15a) ⎡C E C E C E e ⎤ ⎡C E C E C E e ⎤ ⎡C E C E C E e ⎤ ⎢ 11 16 15 11 ⎥ ⎢ 16 12 14 21 ⎥ ⎢ 66 26 46 26 ⎥ ⎢C E C E C E e ⎥ ⎢C E C E C E e ⎥ ⎢C E C E C E e ⎥ (C.15b) or Q = 16 66 56 16 , R = 66 26 46 26 , T = 26 22 24 22 . ⎢ E E E ⎥ ⎢ E E E ⎥ ⎢ E E E ⎥ ⎢C15 C56 C55 e15 ⎥ ⎢C56 C25 C45 e25 ⎥ ⎢C46 C24 C44 e24 ⎥ ⎢ −ω ε ⎥ ⎢ −ω ε ⎥ ⎢ −ω ε ⎥ ⎣e11 e16 e15 11 ⎦ ⎣e16 e12 e14 12 ⎦ ⎣e26 e22 e24 22 ⎦
Generalized eigen relation Elasticity relation (Section 3.3): θ θ ⎡N1( ) N2 ( ) ⎤ ⎧a⎫ (C.16a) N(θ )ξ = μ(θ )ξ , N(θ) = , ξ = , ⎢ θ T θ ⎥ ⎨ ⎬ ⎣N3 ( ) N1 ( )⎦ ⎩b⎭ θ = − −1 θ T θ θ = −1 θ = T θ N1( ) T ( )R ( ), N 2 ( ) T ( ) N 2 ( ), (C.16b) θ = θ −1 θ T θ − θ N3 ( ) R( )T ( )R ( ) Q( ),
Q(θ ) = Q cos2 θ + (R + RT )sinθ cosθ + Tsin 2 θ, (C.16c) R(θ ) = R cos2 θ + (T − Q)sinθ cosθ − RT sin 2 θ, (C.16d) T(θ ) = T cos2 θ − (R + RT )sinθ cosθ + Qsin 2 θ. (C.16d) Thermal relation (Section 10.1): ⎧c⎫ ⎡0 N (θ )⎤⎧β (θ )⎫ (C.17a) N(θ )η = τ (θ )η +τˆ−1 (θ )γ(θ ), η = , γ(θ ) = − 2 1 , ⎨ ⎬ ⎢ T θ ⎥⎨ θ ⎬ ⎩d⎭ ⎣I N1 ( )⎦⎩β2 ( )⎭ +=+−+=+**T τθ θ θ ** Sc Hdii c γ%%12, Lc S d d γ , ˆ(),(),(),,ββ1212 γ%%γ : (C.17b) extended symbols 5.
Stresses (Sections 3.2 and 11.2)
Stresses in xi-coordinate system: σφσφ=−, = , i = 1,2,3 for Problems I&II, (C.18a) iiii1,22,1 σ=−φσ=φ = =σ =σ iiii1,2, 2,1, i1, 2,3, 4 for Problem III (D141, D242). (C.18b) Stresses in s-n coordinate system: (s: tangential direction, n: normal direction) t = φ , t = −φ , n ,s s ,n (C.19a) σ = nT φ , σ = −sT φ , σ = sT φ = −nT φ = σ , σ = iT φ , nn ,s ss ,n ns ,s ,n sn n3 3 ,s (C.19b) σ = − T φ s3 i3 ,n , = T φ = − T φ Dn i4 ,s , Ds i4 ,n . (C.19c) 618 Appendix C
Stresses in polar coordinate system: t = −φ / r, t = φ , r ,θ θ ,r (C.20a) σ = nT φ , σ = −sT φ / r, σ = sT φ = −nT φ / r, σ = iT φ , θθ ,r rr ,θ rθ ,r ,θ θ 3 3 ,r (C.20b) σ = − T φ r3 i3 ,θ / r, = T φ = − T φ Dθ i4 ,r , Dr i 4 ,θ / r, (C.20c)
= + β = C.2 Coupled Stretching–Bending Problems Ui ui (x1, x2 ) x3 i (x1, x2 ), U3 w(x1, x2 ) β = − β = − ξ = ε + κ = + = 1 w,1, 2 w,2 , ij ij x3 ij (U i, j U j,i )/ 2, i 1,2 .
Basic equations I. Laminates under mechanical loadings (Section 13.2): ε = + κ = β + β = ε + κ ij (ui, j u j,i ) / 2, ij ( i, j j,i ) / 2, Nij Aijkl kl Bijkl kl , (C.21a) = ε + κ M ij Bijkl kl Dijkl kl , = = = = Nij, j 0, M ij,ij 0, Qi M ij, j , i, j, k,l 1,2. (C.21b) Governing equations: + β = + β = β = β = Aijkl uk ,lj Bijkl k,lj 0, Bijkl uk ,lij Dijkl k,lij 0, 1,2 2,1, i,j,k,l 1,2 . (C.22) II. Laminates under hygrothermal loadings (Section 13.4): = 0 + * = 0 + * T T (x1, x2 ) x3T (x1, x2 ), H H (x1, x2 ) x3H (x1, x2 ), (C.23a) ( = − t 0 − *t * − t * ( = − h 0 − *h * − h * qi KijT, j Kij T, j Ki3T , mi Kij H , j Kij H , j Ki3 H , (C.23b) 1 ε = (u + u ) / 2, κ = (β + β ) / 2, (C.23c) ij i, j j,i ij 2 i, j j,i = ε + κ − t 0 − h 0 − t * − h * (C.23d) Nij Aijkl kl Bijkl kl AijT Aij H BijT Bij H , = ε + κ − t 0 − h 0 − t * − h * (C.23e) M ij Bijkl kl Dijkl kl BijT Bij H DijT Dij H , = = = ( = ( = = Nij, j 0, M ij,ij 0, Qi M ij, j , qi,i 0, mi,i 0, i, j,k,l 1,2. (C.23f)
Case 1: temperature and moisture content depend on x1 and x2 only Governing equations: ( = − t = ( = − h = qi,i KijT,ij 0, mi,i Kij H,ij 0, (C.24a) = + β − t − h = Nij, j Aijkluk,lj Bijkl k,lj AijT, j Aij H, j 0, (C.24b) = + β − t − h = = M ij,ij Bijkluk,lij Dijkl k,lij BijT,ij Bij H,ij 0, i, j,k,l 1,2. (C.24c) Appendix C 619
Case 2: temperature and moisture content depend on x3 only = 0 + * = 0 + * ( = − t * ( = − h * T T x3T , H H x3H , qi Ki3T , mi Ki3 H , (C.25a) = + β − t 0 − h 0 − t * − h * Nij Aijkluk,l Bijkl k,l AijT Aij H BijT Bij H , (C.25b)
= + β − t 0 − h 0 − t * − h * = M ij Bijkluk,l Dijkl k,l BijT Bij H DijT Dij H , Qi M ij, j . (C.25c) Governing equations: = + β = = + β = = Nij, j Aijkluk,lj Bijkl k,lj 0, M ij,ij Bijkluk,lij Dijkl k,lij 0, i, j,k,l 1,2 . (C.26)
III. Electro-Elastic Composite Laminates (Section 13.5): = (0) + (1) = (0) + (1) E1 E1 (x1, x2 ) x3E1 (x1, x2 ), E2 E2 (x1, x2 ) x3E2 (x1, x2 ), (C.27a) ε = + κ = β + β rs (ur,s us,r ) / 2, rs ( r,s s,r ) / 2, (C.27b) = ε + κ = ε + κ N pq Apqrsl rs Bpqrs rs , M pq Bpqrs rs Dpqrs rs , (C.27c) = = = = = N pj, j 0, M ij,ij 0, M 4 j, j 0, p, q, r, s 1,2,4; i,j 1,2. (C.27d) Governing equations: A u + B β = 0, B u + D β = 0, B u + D β = 0, (C.28a) pjrl r,lj pjrl r,lj ijrl r,lij ijrl r,lij 4 jrl r,lj 4 jrl r,lj β = β = = 1,2 2,1, p, r 1,2,4; i, j,l 1,2. (C.28b) General solution I. Laminates under mechanical loadings (Section 13.2): = φ = ud 2Re{Af (z)}, d 2Re{Bf (z)}, (C.29a) ⎧u⎫ ⎧φ ⎫ ⎧u ⎫ ⎧β ⎫ ⎧φ ⎫ ⎧ψ ⎫ (C.29b) u = , φ = , u = 1 , β = 1 , φ = 1 , ψ = 1 , d ⎨β⎬ d ⎨ψ⎬ ⎨ ⎬ ⎨β ⎬ ⎨φ ⎬ ⎨ψ ⎬ ⎩ ⎭ ⎩ ⎭ ⎩u2 ⎭ ⎩ 2 ⎭ ⎩ 2 ⎭ ⎩ 2 ⎭ = = (C.29c) A [a1 a2 a3 a4 ], B [b1 b2 b3 b4 ],
⎧ f1(z1) ⎫ ⎪ ⎪ f (z ) = ⎪ 2 2 ⎪ = + μ α = (C.29d) f (z) ⎨ ⎬, zα x1 α x2 , 1,2,3,4. ⎪ f3 (z3 )⎪ ⎪ ⎪ ⎩ f4 (z4 )⎭ 620 Appendix C
IIa. Laminates under hygrothermal loadings –
temperature and moisture content depend on x1 and x2 only (Section 13.4): = ′ = ′ T 2 Re{gt (zt )}, H 2 Re{gh (zh )}, (C.30a) ( = − t +τ t ′′ ( = − h +τ h ′′ qi 2 Re{(Ki1 t Ki2 )gt (zt )}, mi 2 Re{(Ki1 h Ki2 )gh (zh )}, (C.30b) = + + ud 2 Re{Af (z) ct gt (zt ) ch gh (zh )}, (C.30c) φ = + + d 2 Re{Bf (z) dt gt (zt ) dh gh (zh )},
⎧u⎫ ⎧φ ⎫ ⎧u ⎫ ⎧β ⎫ ⎧φ ⎫ ⎧ψ ⎫ u = , φ = , u = 1 , β = 1 , φ = 1 , ψ = 1 , d ⎨ ⎬ d ⎨ψ⎬ ⎨ ⎬ ⎨β ⎬ ⎨φ ⎬ ⎨ψ ⎬ ⎩β⎭ ⎩ ⎭ ⎩u2 ⎭ ⎩ 2 ⎭ ⎩ 2 ⎭ ⎩ 2 ⎭ (C.30d)
A,B,f (z) : same as Problem I, Eqs.(C.29c) and (C.29d). = +τ = +τ (C.30e) zt x1 t x2 , zh x1 h x2. IIb. Laminates under hygrothermal loadings –
temperature and moisture content depend on x3 only (Section 13.4) : = 0 + * = 0 + * ( = − t * ( = − h * T T x3T , H H x3 H , qi Ki3T , mi Ki3H , (C.31a) = φ = − ϑ + ϑ ud 2 Re{Af (z)}, d 2 Re{Bf (z)} x1 2 x2 1, (C.31b) ϑ = t 0 + h 0 + *t * + *h * = i αiT αi H αi T αi H , i 1,2. (C.31c) A,B,f (z) : same as Problem I, Eqs. (C.29c) and (C.29d). t h *t *h αi ,αi ,αi ,αi : extended symbols 5. III. Electro Elastic Composite Laminates (Section 13.5): = φ = u d 2 Re{Af (z)}, d 2 Re{Bf (z)}, (C.32a) f (z ) ⎧ 1 1 ⎫ ⎪ ⎪ f (z ) ⎧u ⎫ ⎧β ⎫ ⎧φ ⎫ ⎧ψ ⎫ ⎪ 2 2 ⎪ u φ 1 1 1 1 ⎪ f (z )⎪ = ⎧ ⎫ φ = ⎧ ⎫ = ⎪ ⎪ β = ⎪β ⎪ φ = ⎪φ ⎪ ψ = ⎪ψ ⎪ = ⎪ 3 3 ⎪ ud ⎨ ⎬, d ⎨ ⎬, u ⎨u2 ⎬, ⎨ 2 ⎬, ⎨ 2 ⎬, ⎨ 2 ⎬, f (z) ⎨ ⎬, (C.32b) β ψ f (z ) ⎩ ⎭ ⎩ ⎭ ⎪u ⎪ ⎪β ⎪ ⎪φ ⎪ ⎪ψ ⎪ ⎪ 4 4 ⎪ ⎩ 4 ⎭ ⎩ 4 ⎭ ⎩ 4 ⎭ ⎩ 4 ⎭ ⎪ f (z )⎪ ⎪ 5 5 ⎪ ⎩⎪ f6 (z6 )⎭⎪ = [][]= = + μ A a1 a2 a3 a4 a5 a6 , B b1 b2 b3 b4 b5 b6 , zk x1 k x2 , (C.32c) k = 1,2,3,4,5,6. Appendix C 621
Material eigen relation = μ = Nξ ξ , N It NmIt , (C.33a) N N (N ) (N ) I I I 0 0 0 = ⎡ 1 2 ⎤ = ⎡ m 1 m 2 ⎤ = ⎡ 1 2 ⎤ = ⎡ ⎤ = ⎡ ⎤ N ⎢ T ⎥, N m ⎢ T ⎥, I t ⎢ ⎥, I 1 ⎢ ⎥, I 2 ⎢ ⎥, ⎣N3 N1 ⎦ ⎣(Nm )3 (Nm )1 ⎦ ⎣I 2 I1 ⎦ ⎣0 0⎦ ⎣0 I⎦ (C.33b) ⎧a⎫ ξ = ⎨ ⎬, ⎩b⎭
= − −1 T = −1 = T = −1 T − = T (Nm )1 Tm R m , (Nm )2 Tm (Nm )2 , (Nm )3 R mTm R m Qm (Nm )3 . (C.33c)
I. Laminates under mechanical loadings (Section 13.2): ~ ⎡Q ~ R ~ ⎤ ⎡R ~ − Q ~ ⎤ ⎡ T~ − R ~ ⎤ (C.34a) = A B = A B = A B Qm ⎢ T ⎥, Rm ⎢ T T ⎥, Tm ⎢ ~ T ⎥, R ~ − T~ T~ R ~ − R ~ − Q ~ ⎣ B D ⎦ ⎣ B D ⎦ ⎣⎢ B D ⎦⎥ ~ ~ ~ ~ ~ ~ ======R B Bi2k1, Q X X i1k1, R X X i1k 2 , TX X i2k 2 , X A,B or D, i,k 1,2 , (C.34b) or ⎡ ~ ~ ~ ~ ⎤ A11 A16 B16 / 2 B12 ⎢ ~ ~ ~ ~ ⎥ A A B / 2 B (C.34c) Q = ⎢ 16 66 66 62 ⎥ , m ⎢ ~ ~ − ~ − ~ ⎥ B16 / 2 B66 / 2 D66 / 4 D26 / 2 ⎢ ~ ~ ~ ~ ⎥ B B − D / 2 − D ⎣⎢ 12 62 26 22 ⎦⎥ ⎡ ~ ~ − ~ − ~ ⎤ A16 A12 B11 B16 / 2 ⎢ ~ ~ ~ ~ ⎥ A A − B − B / 2 R = ⎢ 66 26 61 66 ⎥ , m ⎢ ~ ~ ~ ~ ⎥ B66 / 2 B26 / 2 D16 / 2 D66 / 4 ⎢ ~ ~ ~ ~ ⎥ ⎣⎢ B62 B22 D12 D26 / 2 ⎦⎥ ⎡ ~ ~ − ~ − ~ ⎤ A66 A26 B61 B66 / 2 ⎢ ~ ~ ~ ~ ⎥ − − (C.34d) = ⎢ A26 A22 B21 B26 / 2⎥ Tm ~ ~ ~ ~ . ⎢ − B − B − D − D / 2⎥ ⎢ 61 21 11 16 ⎥ − ~ − ~ − ~ − ~ ⎣⎢ B66 / 2 B26 / 2 D16 / 2 D66 / 4⎦⎥
II. Laminates under hygrothermal loadings (Section 13.4): t + τ t +τ 2 t = h + τ h +τ 2 h = K11 2 t K12 t K22 0, K11 2 h K12 h K22 0, (C.35a) ξ = μξ η = τ η + γ η = τ η + γ N , N t t t t , N h h h h , (C.35b) ⎡N N ⎤ a c c (C.35c) = 1 2 ξ = ⎧ ⎫ η = ⎧ t ⎫ η = ⎧ h ⎫ N ⎢ T ⎥, ⎨ ⎬, t ⎨ ⎬, h ⎨ ⎬, ⎣N3 N1 ⎦ ⎩b⎭ ⎩dt ⎭ ⎩dh ⎭ γ γ t , h : extended symbols 5.
Qm , R m ,Tm : same as Problem I, (C.34a), (C.34b), (C.34c), and (C.34d). 622 Appendix C
III. Electro Elastic Composite Laminates (Section 13.5): ~ ªQ ~ R ~ º ªR ~ − Q ~ º ª T~ − R ~ º (C.36a) = A B = A B = A B Qm « T », R m « T T », Tm « ~ T », R ~ − T~ T~ R ~ − R ~ − Q ~ ¬ B D ¼ ¬ B D ¼ ¬« B D ¼» ~ ~ ~ ~ ~ ~ ======R B Bp2r1, Q X X p1r1, R X X p1r2 , TX X p2r2 , X A, B or D, p,r 1,2,4, (C.36b) or ª ~ ~ ~ ~ ~ ~ º A11 A16 A17 B16 / 2 B12 B18 / 2 « ~ ~ ~ ~ ~ ~ » « A16 A66 A67 B66 / 2 B62 B68 / 2 » ~ ~ ~ ~ ~ ~ « A A A B / 2 B B / 2 » Q = « 17 67 77 76 72 78 » , m ~ ~ − ~ − ~ − ~ (C.36c) « B16 / 2 B66 / 2 B76 / 2 D66 / 4 D26 / 2 D68 / 4» ~ ~ ~ ~ ~ ~ « B B B − D / 2 − D − D / 2» « 12 62 72 26 22 28 » ~ ~ ~ − ~ − ~ − ~ ¬« B18 / 2 B68 / 2 B78 / 2 D68 / 4 D28 / 2 D88 / 4¼» ~ ~ ~ ~ ~ ~ ª A16 A12 A18 − B − B / 2 − B / 2º « 11 16 17 » ~ ~ ~ − ~ − ~ − ~ « A66 A26 A68 B61 B66 / 2 B67 / 2» ~ ~ ~ « ~ ~ ~ − B − B / 2 − B / 2» = A67 A27 A78 71 76 77 (C.36d) R m « ~ ~ ~ ~ ~ ~ », «B66 / 2 B26 / 2 B86 / 2 D16 / 2 D66 / 4 D67 / 4 » « ~ ~ ~ ~ ~ ~ » B62 B22 B82 D12 D26 / 2 D27 / 2 « ~ ~ ~ ~ ~ ~ » ¬«B68 / 2 B28 / 2 B88 / 2 D18 / 2 D68 / 4 D78 / 4 ¼»
ª ~ ~ ~ ~ ~ ~ º A66 A26 A68 − B − B / 2 − B / 2 « 61 66 67 » ~ ~ ~ − ~ − ~ − ~ (C.36e) « A26 A22 A28 B21 B26 / 2 B27 / 2 » « ~ ~ ~ » ~ ~ ~ − B − B / 2 − B / 2 = « A68 A28 A88 81 86 87 » . Tm ~ ~ ~ ~ ~ ~ « − B − B − B − D − D / 2 − D / 2» « 61 21 81 11 16 17 » − ~ − ~ − ~ − ~ − ~ − ~ « B66 / 2 B26 / 2 B86 / 2 D16 / 2 D66 / 4 D67 / 4» «− ~ − ~ − ~ − ~ − ~ − ~ » ¬ B67 / 2 B27 / 2 B87 / 2 D17 / 2 D67 / 4 D77 / 4¼ Generalized eigen relation Elasticity relation (Section 13.2): a θ ξ = μ θ ξ, θ = θ ξ = ½ N( ) ( ) N( ) It Nm ( )It , ® ¾, (C.37a) ¯b¿ ªN (θ ) N (θ ) º (N (θ )) (N (θ )) θ = 1 2 θ = ª m 1 m 2 º N( ) « », Nm ( ) , (C.37b) θ T θ «(N (θ )) (N (θ ))T » ¬N3 ( ) N1 ( )¼ ¬ m 3 m 1 ¼ θ = − −1 θ T θ θ = −1 θ (Nm ( ))1 Tm ( )R m ( ), (N m ( ))2 Tm ( ), ς = θ −1 θ T θ − θ (C.37c) (Nm ( ))3 R m ( )Tm ( )R m ( ) Qm ( ), θ = 2 θ + + T θ θ + 2 θ (C.37d) Qm ( ) Qm cos (R m R m )sin cos Tm sin , θ = 2 θ + − θ θ − T 2 θ R m ( ) R m cos (Tm Qm )sin cos R m sin , (C.37e) θ = 2 θ − + T θ θ + 2 θ Tm ( ) Tm cos (R m R m )sin cos Qm sin . (C.37f) Appendix C 623
Thermal relation: θ η = τ θ η +τ −1 θ γ θ θ η = τ θ η +τ −1 θ γ θ N( ) t t ( ) t ˆt ( ) t ( ), N( ) h h ( ) h ˆh ( ) h ( ), (C.38a) η η τ θ τ θ τ θ τ θ γ θ γ θ t , h , t ( ), h ( ), ˆt ( ), ˆh ( ), t ( ) , h ( ) : extended symbols 5. (C.38b) ~t* ~ h* ⎡ S H ⎤⎧ct ⎫ ⎧ct ⎫ ⎪⎧γ1 ⎪⎫ ⎡ S H ⎤⎧ch ⎫ ⎧ch ⎫ ⎪⎧γ1 ⎪⎫ ⎢ ⎥⎨ ⎬ = i⎨ ⎬ + ⎨ ⎬, ⎢ ⎥⎨ ⎬ = i⎨ ⎬ + ⎨ ⎬, − T d d ~t* − T d d ~ h* ⎣ L S ⎦⎩ t ⎭ ⎩ t ⎭ ⎩⎪γ 2 ⎭⎪ ⎣ L S ⎦⎩ h ⎭ ⎩ h ⎭ ⎩⎪γ 2 ⎭⎪ (C.38c) ~t* ~γ h* = γi , i , i 1,2 : extended symbols 5.
Stress resultants (Section 13.3)
Stresses in xi -coordinate system: 1 1 N = −φ , N = φ , M = −ψ − λ ψ , M =ψ − λ ψ , (C.39a) i1 i,2 i2 i,1 i1 i,2 2 i1 k,k i2 i,1 2 i2 k,k
1 1 Q = − ψ , Q = ψ , V = −ψ , V =ψ , 1 2 k,k 2 2 2 k,k1 1 2,22 2 1,11 (C.39b) λ = λ = λ = −λ = 11 22 0, 12 21 1. Stresses in s-n coordinate system (s: tangential direction, n: normal direction): = φ = −φ = ψ −η = −ψ +η t n ,s , t s ,n , m n ,s s, m s ,n n , (C.40a) N = nT φ , N = −sT φ , N = sT φ = −nT φ = N , (C.40b) n ,s s ,n ns ,s ,n sn = T ψ = − T ψ = T ψ −η = − T ψ +η = M n n ,s , M s s ,n , M ns s ,s n ,n M sn , (C.40c) = η = −η = T ψ = − T ψ η = T ψ + T ψ Qn ,s , Q s ,n , V n (s ,s ),s , V s (n ,n ),n , (s ,s n ,n ) / 2. (C.40d) Stresses in polar coordinate system:
TTT11 T (C.41a) NNθθ=nsnsφφφ,,, = =− θ N =− φ θ ,,,rr rrr r , (C.41b) 11 MM==−=−+=−nsnTTψψ,,,ηη T ψ M s T ψ θθ,,rr r , θ r , θ rr 111TT (C.41c) QQθθ==−=+ηηη, , (snψψ θ ), ,,rrrr2 ,, r
TT11 T 1 TT VVθθθθθθ=()ssψψ = , =− ( n ψ ) =− ( ns ψψ − ), ,,rr , rr r rr ,, r2 , , (C.41d)
Resultant forces and bending moments: ~ = φ ]B ~ = φ ]B ~ =η]B t1 1 A , t2 2 A , t3 A , (C.42a) ~ = − ψ − η ]B ~ = ψ − η ]B (C.42b) m1 ( 2 x2 ) A , m2 ( 1 x1 ) A , B m~ = x dφ − x dφ = (x φ − x φ − Φ)]B , 3 ∫A 1 2 2 1 1 2 2 1 A φ = −Φ φ = Φ (C.42c) 1 ,2 , 2 ,1 .
624 Appendix C
C.3 Dimensions of Matrices Used in Stroh Formalism
Two-dimensional problems In-plane anti-plane Electro elastic In-plane Anti-plane Coupling coupling
aα ,bα 2×1 scalar 3×1 4×1 A,B 2×2 scalar 3×3 4×4 c,d (thermal) 2×1 scalar 3×1 4×1 f (z) 2×1 function 3×1 4×1
g(zt ) (thermal) Scalar function – Scalar function Scalar function
N i 2×2 scalar 3×3 4×4 N 4×4 2×2 6×6 8×8
Q,R,T 2×2 C55 ,C45 ,C44 3×3 4×4 u(z),φ(z) 2×1 scalar 3×1 4×1
μα 2 pairs 1 pairs 3 pairs 4 pairs τ (thermal) 1 pairs – 1 pairs 1 pairs Coupled stretching–bending problems Stretching bending Electro elastic Stretching Bending coupling coupling
Aij , Bij , Dij 3×3 ( Aij ) 3×3 ( Dij ) 3×3 5×5
aα ,bα 2×1 2×1 4×1 6×1 A,B 2×2 2×2 4×4 6×6
ct ,dt (thermal) 2×1 2×1 4×1 6×1
ch ,dh (hygro) 2×1 2×1 4×1 6×1 f (z) 2×1 2×1 4×1 6×1
gt (zt ) (thermal) Scalar function – Scalar function Scalar function
gh (zh ) (hygro) Scalar function – Scalar function Scalar function
N i 2×2 2×2 4×4 6×6 N 4×4 4×4 8×8 12×12
Qm ,Rm ,Tm 2×2 2×2 4×4 6×6 φ φ ud (z), d (z) 2×1)(u, 2×1)(β,ψ 4×1 6×1
μα 2 pairs 2 pairs 4 pairs 6 pairs τ t (thermal) 1 pairs – 1 pairs 1 pairs τ h (hygro) 1 pairs – 1 pairs 1 pairs Note: aα ,bα : complex vectors, material properties g(zt ) : complex function, problem dependent
A,B : complex matrices, material properties gt (zt ), gh (zh ) : complex functions, problem dependent
Aij ,Bij , Dij : real tensors, material properties N, Ni ,Q,R,T : real matrices, material properties c, d : complex vectors, material properties u(z),φ(z) : real vectors, physical quantities φ ct ,dt : complex vectors, material properties ud (z), d (z) : real vectors, physical quantities μ τ ch ,dh : complex vectors, material properties α , : complex numbers, material properties τ τ f (z) : complex function vector, problem dependent t , h : complex numbers, material properties Appendix D Collection of the Problem Solutions
Note 1: Two-dimensional problems: u = 2 Re{Af (z)}, φ = 2 Re{Bf (z)} . = φ = Note 2: Stretching–bending coupling problems: ud 2 Re{Af (z)}, d 2 Re{Bf (z)} . Note 3: Mechanical properties are given in all the following problems.
σ ∞ 4.1.1 Infinite space: uniform loading (given ij )
σ ∞ 22 ∞ T ∞ T ∞
∞ σ ∞
•
σ 23 • • •
• σ 13 • =< > + σ ∞ 12 σ ∞ f (z) z (A t B ε ) . 11 ⊗ 12 α σ ∞ 2 1 12 σ ∞
13 ∞ ⊗ • σ 11 ∞ ∞ ∞ ∞ x = + φ = −
2 Real-form solution: u x ε x ε , x t x t . ⊗ • 1 1 2 2 1 2 2 1
⊗ ∞ ∞ ∞ ∞ • x 1 ε1 ,ε2 ,t1 , t 2 : (4.2b).
⊗ •
⊗ Special cases: (4.6)–(4.9). •
⊗ • ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ σ ∞ σ ∞ 23 12 σ ∞ 22 4.1.2 Infinite space: pure in-plane bending (given M ,α, I )
x2 M −M −M u = (sinαu + cosαu ), φ = (x sinα − x cosα)2 s(α), 2I 1 2 2I 1 2
α for γ = 0. x1 γ u1,u2 , : (4.11b,c). I: moment of inertia of the plate cross section. M 4.1.3 Infinite space: point load (given pˆ, zˆα )
x 2 1 f (z) = < ln(z − zˆ ) > AT pˆ. π α α = 2 i pˆ ( pˆ1, pˆ 2 , pˆ 3 ) Real-form solution: = xˆ (x1, x2 ) x 1 1 ~ ~ u = − {(ln r)H + π[N (θ )H + N (θ )ST ]}pˆ, 2π 1 2
1 ~ ~ φ = − {(ln r)ST + π[N (θ )H + NT (θ )ST ]}pˆ. 2π 3 1 Stresses: (4.20); isotropic medium: (4.21). 4.1.4 Infinite space: point moment (given mˆ )
x 2 mˆ u = {Hn(θ ) −[NT (θ )H + N (θ )ST ]s(θ )}, 4πr 1 2
ˆ φ = m T θ − T θ T + θ θ mˆ {S n( ) [N1 ( )S N3 ( )H]s( )}. x 1 4πr Stresses: (4.33a,b) or (4.34); isotropic medium: (4.36).
625 626 Appendix D
4.1.5 Infinite space: dislocation (given bˆ, zˆα ) T x2 ˆ f (z) =< ln(zα − zˆα ) > B b / 2πi. Real-form solution:
bˆ = (bˆ ,bˆ ,bˆ ) 1 ~ ~ 1 2 3 = − + π θ − θ ˆ u {(ln r)S [N1( )S N2 ( )L]}b, xˆ = (x , x ) π 1 2 x 2 1 1 ~ ~ φ = {(ln r)L + π[NT (θ )L − N (θ)S]}bˆ. π 1 3 2 Stresses: (4.39b).
4.2.1 Half space: point load (given pˆ, zˆα ) x 2 3 1 ⎧ − T ⎫ f (z) = < ln(z − zˆ ) > AT + < ln(z − zˆ ) > B 1BI A pˆ. π ⎨ α α ∑ α j j ⎬ 2 i ⎩ j=1 ⎭ x 1 = pˆ Along the half-space surface x2 0 : (4.57) and (4.59). ()xˆ , xˆ 1 2 Isotropic medium: (4.60).
4.2.2 Half space: surface point force (given pˆ, xˆ 1)
x2 1 − f (z) = < ln(z − xˆ ) > B 1pˆ . π α 1 pˆ 2 i = − + π~ θ −1 π φ = −~ θ −1 x1 ˆ ˆ () Real-form solution: u [ln rI N1 ( )]L p/ , N3( )L p. xˆ1,0 Stresses: (4.63); isotropic medium: (4.64).
4.2.3 Half space: distributed load (given pˆ(xˆ1) )
x2 = 1 xb < − > −1 f (z) ln(zα xˆ1) B pˆ(xˆ1)dxˆ1 . pˆ (x ) ∫xa 1 2πi
x1 xa xb
ˆ 4.2.5 Half space: dislocation (given b, zˆα ) x 2 3 1 ⎧ − T ⎫ f (z) = < ln(z − zˆ ) > BT + < ln(z − zˆ ) >B 1BI B ] bˆ. π ⎨ α α ∑ α j j ⎬ 2 i ⎩ j =1 ⎭
x 1 Along the half-space surface x = 0 : (4.67). b 2
(xˆ1, xˆ2 ) Appendix D 627
ˆ (1) 4.3.1 Bimaterials: point load and dislocation (given pˆ ,b, zˆα ) = (1) + (1) φ = (1) + (1) pˆ u1 2 Re{A1[f0 (z ) f1(z )]}, 1 2 Re{B1[f0 (z ) f1(z )]}, bˆ = (2) φ = (2) u2 2 Re{A2f2 (z )}, 2 2 Re{B2f2 (z )}. (xˆ1, xˆ2 ) S2
S 1 1 (1) (1) (1) T T ˆ f (z ) = < ln(zα − zˆα ) > (A pˆ + B b), 0 2πi 1 1 1 3 f(z(1) )=
pˆ ˆ ~ ~ ~ ~ b φ = + π (k ) θ − (k ) α + (k )T θ − (k )T α k ln rg {[N3 ( ) N3 ( 0 )]h [N1 ( ) N1 ( 0 )]g},
α 0 k = 1,2. (x1, x2 ) ~ ˆ ~ ~ ˆ ~T Material 2 h = −(Sb + Hpˆ) / 2π , g = (Lb − S pˆ) / 2π.
5.1 Wedge: uniform traction (given t + ,t − ,θ + ,θ − )
x2 x* 2 Wedge angle Loading Solution 2α ≠ π , 2π tˆ = 0 (4.2) or (5.4)
x* 1 ˆ = ˆ πΔ − (5.11) with h t / , t α ≠ π π α 0 − ˆ ≠ θ x 2 , 2 , 2 t 0 1 c − θ + = − t + h0 : (5.24) with k=0, g0 t
+ − θ −θ = 2α α = π ˆ = ˆ πΔ = = − − 2 (5.30) (5.11) with h0 t / , h0 0, g0 t α = π ˆ − T − + T + − 2 2 (5.32) (5.11) with h = tˆ/πΔ, h = 0, g = −t tˆ = n (θ )t + n (θ )t , 0 0 0 − ~ + − + Δ = nT (θ )N (θ ,θ )n(θ ), α = π π 3 2 or 2 any (5.38) with h0 : (5.37) + − 2α = θ −θ α = α 2 2 c tˆ = 0 (4.2) or (5.4) − Δ(α ) (5.12) with g = 0, g = −t , c 0 1 − ~ + − + = nT (θ )N (θ ,θ )n(θ ) = 0. 2α = 2α c 3 c c c c any h0 : (5.34) and (5.20), ˆ where h0 : (5.35) and (5.16), α = θ + −θ − h : (5.35) and (5.20) 2 c c c 1 628 Appendix D
5.2.1 Wedge: point force (given pˆ,θ + ,θ − )
x2 1 T ~ + ~ − −1 f (z) = < lnzα > B [N (θ ) − N (θ )] pˆ. π 3 3 real-form solution: pˆ = + π~ θ ~ θ + − ~ θ − −1 π θ − u [(ln r)I N1( )][N3 ( ) N3 ( )] pˆ / , x1 θ + φ = ~ θ ~ θ + − ~ θ − −1 N3 ( )[N3 ( ) N3 ( )] pˆ. θ + −θ − = 2α
5.2.2 Wedge: point moment (given mˆ ,θ + ,θ − )
x2 −mˆ − + 2α ≠ π , 2π , 2α : u = Nˆ (θ ,θ )n(θ ), c πrΔ 1 −mˆ − + φ = Nˆ (θ ,θ )n(θ ) . π Δ 3 mˆ r θ − x1 θ + α = π α = π = ˆ θ − θ φ = ˆ θ − θ 2 or 2 2 : ru N1 ( , )h0 , r N3 ( , )h0 , θ + −θ − = 2α h0 : arbitrary.
α = α 2 2 c : (5.49), (5.54) and (5.55). ˆ θ θ 5.2.3 Multi-material wedge space: point force and dislocation (given pˆ,b, 0 ,..., n ) x 2 = + π ~ θ θ + π ~ θ θ u [(ln r)I N1( , 0 )]h N2 ( , 0 )g, ~ T ~ bˆ φ = + π θ θ + π θ θ 2 [(ln r)I N1 ( , 0 )]g N3 ( , 0 )h, θ 2 θ pˆ 1 ~ ~ ~ ~T 1 = − ˆ + ˆ π = ˆ − ˆ π θ = θ h (Sb Hp) / 2 , g (Lb S p) / 2 . θ 0 n k k x1 θ n−1
n−1
θ θ 5.2.4 Multi-material wedge: point force (given pˆ, 0 ,..., n ) x 2 ~ ~ u = [(ln r)I + πN (θ,θ )]h, φ = πN (θ,θ )h, 1 0 3 0 ~ −1 pˆ h = N (θ ,θ )pˆ /π. 2 3 n 0 θ 2 θ 1 1 θ k θ 0 k x1 θ n n
θ θ 5.3 Multi-material wedge and wedge space: near-tip solutions (given 0 ,..., n ) x 2 ⎧u (r,θ )⎫ ⎧u ⎫ k = r1−δ Nˆ 1−δ (θ,θ )(K ) 0 , k = 1,2,3,....,n, ⎨φ θ ⎬ k k−1 e k−1 ⎨φ ⎬ 2 ⎩ k (r, )⎭ ⎩ 0 ⎭ θ 2 θ 1 1 T θ = θ − = = φ θ 0 n Bonded : (Ke I)w0 0, w 0 (u0 0 ) , k k x1 θ (3) n−1 Free–free : K u = 0, φ = 0, n−1 e 0 0 (2) x φ = = 2 Fixed –fixed: Ke 0 0, u0 0, (1) = φ = Free –fixed : K e u0 0, 0 0, 2 (4) θ φ = = 2 θ 1 Fixed–free: K 0, u 0. 1 e 0 0 θ k θ 0 k x1 θ K ,(K ) − : (5.82b), (5.83b), and (5.94); n e e k 1 n (1) (2) (3) (4) K e ,K e ,K e ,K e : (5.88). Appendix D 629
σ ∞ 6.1.1 Elliptical hole: uniform loading (given ij , a,b )
σ ∞ 22 ∞ ∞ − − ∞ ∞ ∞ 1 1
∞ σ ∞
• σ • 23 • • • σ = + − < ζ > −
σ ∞ 13 • 12 σ ∞ u x ε x ε Re{A α B (at ibt )}, 11 ⊗ 12 1 1 2 2 2 1 σ ∞ 12 σ ∞
13 ∞ ⊗ • σ ∞ ∞ − − ∞ ∞ x 11 1 1 2 φ = − − < ζ > −
x1t 2 x2t1 Re{B α B (at 2 ibt1 )}, ⊗ •
⊗ Deformation of the hole boundary: u: (6.11). b • n x1 s ⊗ θ σ
• Hoop stress ss : (6.18a,b); ⊗ a • unidirectional tension: (6.19a,b); isotropic materials: (6.20)
⊗ • ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ σ ∞ 23 σ ∞ 12 σ ∞ 22 = ψ = ψ x1 a cos x2 b cos α 6.1.2 Elliptical hole: in-plane bending (given M , I, , a,b ) x2 M 2 = ∞ + Ma α + α < ζ −2 > −1 α α u u Re{[c ( ) is ( )]A α B }s( ), 2I 2 2 2 φ = φ∞ + Ma α + α < ζ −2 > −1 α b Re{[c2 ( ) is2 ( )]B α B }s( ),
x1 2I ∞ ,φ∞ α α u : (6.21b,c); c2 ( ),s2 ( ) : (6.23b) M Hoop stress: (6.25a,b); isotropic materials: (6.27) a ˆ 6.1.3 Elliptical hole: arbitrary loading (given t n , a,b ) ∞ x 2 i −− =<ζζ >Tk −{} < >1 + fAcBcd()zi lnαα0 ∑ (kk i ), k=1 2k ππ =1ρˆψ =1ρˆψψ b c0 tnd, ck tncos kd, ˆ −π−π tn ∫∫ x ππ 1 2 1π d=ρtˆsin kψdψ k∫−πn a π Hoop stress: (6.38) ρ 2 = 2 2 ψ + 2 2 ψ a sin b cos Point force on the hole surface: (6.49) and (6.50) ˆ 6.1.4 Elliptical hole: point force (given pˆ,ζ α )
x 2 3 1 ⎧ − − T ⎫ pˆ = < ζ −ζˆ > T + < ζ 1 −ζˆ > 1 ˆ f (z) ⎨[ ln( α α ) A ∑ ln( α k ) B BI k A ]⎬p, (x1, x2 ) π 2 i ⎩ k=1 ⎭
2 T iψ iψ −1 T b σ = θ θ < − ζˆ > Hoop stress: ss s ( )G3 ( )Re{B ie (e α ) A }pˆ. x 1 πρ
a
ˆ ˆ 6.1.5 Elliptical hole: dislocation (given b,ζ α ) x 2 3 1 ⎧ − − T ⎫ f (z) = [< ln(ζ −ζˆ ) > BT + < ln(ζ 1 −ζˆ ) > B 1 BI B ] bˆ bˆ ⎨ α α ∑ α k k ⎬ (x , x ) π = 1 2 2 i ⎩ k 1 ⎭
b
x1
a 630 Appendix D
σ ∞ ε 6.2.2 Polygon-like hole: uniform loading (given ij , a,c, ) σ ∞ = ∞ + ∞ + < ζ −1 > + < ζ −k > u x1ε1 x2ε2 2Re{A[ α q1 α qk ]}, α φ = ∞ − ∞ + < ζ −1 > + < ζ −k > x1t2 x2t1 2Re{B[ α q1 α qk ]}, * x2 * x2 x1 α x 1 −1 ∞ ∞ 1 −1 ∞ ∞ 1 = − − = − ε + n s q1 aB (t 2 ict1 ) , qk a B (t 2 ict1 ). θ 2 2 Hoop stress: (6.79); σ ∞ α unidirectional tension: (6.81); isotropic materials: (6.82); end points: (6.83) = ψ + ε ψ x1 a(cos cosk ), = ψ − ε ψ x2 a(csin sink ), 6.2.3 Polygon-like hole: in-plane bending (given M , I,α, a,c,ε )
M ∞ − ∞ − = + < ζ l > φ = φ + < ζ l > α u u 2Re {A α q }, 2 Re {B α q }, ∑ l ∑ l = − = − l 2,k 1 l 2,k 1 + + * x2 * k 1,2k k 1,2k x2 x1 α x1 2 n s Ma −1 θ q = (c + is )B s(α), l 4I l l α c = [1− c2 − (1+ c2 )cos2α]/ 4, s = −csin 2α/2, M 2 2 c − = ε[(1− c) − (1+ c)cos2α]/ 2, s − = ε (1+ c)sin 2α / 2 , k 1 k 1 = ε + − − α = ε − α ck+1 [(1 c) (1 c)cos2 ]/ 2, sk+1 (1 c)sin 2 / 2 , = −ε 2 α = ε 2 α c2k cos2 /2, s 2 k sin 2 / 2 . Hoop stress: (6.87a,b); isotropic materials: (6.88) 7.1.1 Crack: near-tip solution and fracture parameters −1 ⎧u⎫ 2 1/ 2 1/ 2 ⎧L k⎫ ⎨ ⎬ = − r Nˆ (θ,−π )⎨ ⎬ , φ π 0 θ ＝π ⎩ ⎭ ⎩ ⎭ θ ＝- π 1 − Energy release rate: G = k T L 1k , 2 k: vector of stress intensity factors Appendix D 631
7.1.2 and 7.4.5 Interface crack: near-tip solution and fracture parameters
ΙΙ 1−δ −1 ⎧u ⎫ − r −δ ⎧L h⎫ 1 = N1 (θ,−π ) 1 , ⎨φ ⎬ − δ πδ 1 ⎨ ⎬ ⎩ 1 ⎭ (1 )sin ⎩ 0 ⎭ θ ＝π θ ＝- π 1−δ −1 ⎧u ⎫ − −δ −δ ⎧L h⎫ 2 = r 1 θ 1 −π 1 ⎨ ⎬ N2 ( ,0)N1 (0, )⎨ ⎬, φ (1− δ )sinπδ Ι ⎩ 2 ⎭ ⎩ 0 ⎭ 1 δ = + iε : (7.13a,b) and (7.11c); h: arbitrary 2 Along the interface (θ = 0) :
1 iεα −1 φ'= Λ < (r / l) > Λ k, 2πr
iεα 2r − (r / ) − Δu = Λ T < l > Λ 1k π ()1+ 2iεα cos h(πεα )
Λ : (7.67b) and (7.68); εα : (7.69b) and (7.13b); l : reference length 1 − Energy release rate: G = k T Ek, E = D + WD 1W 4 Orthotropic bimaterials: (7.123) and (7.116); Isotropic bimaterials: (7.117a,b)
σ ∞ 7.2.1 Crack: uniform loading (given ij , a )
σ ∞ 22 ∞ ∞ − − ∞ ∞ 1 1
∞ σ ∞
• σ 23 • • •
• σ • = + − < ζ > σ ∞ 13 12 σ ∞ u x ε x ε a Re{A α B t }, 11 ⊗ 12 1 1 2 2 2 σ ∞ 12
σ ∞
13 ∞ ⊗ • σ 11 ∞ ∞ −1 −1 ∞ x φ = − − < ζ >
2 x1t 2 x2t1 a Re{B α B t 2 }, ⊗ •
⊗ ∞ − ∞ • 2 2 1 x1 = π Δ = − Fracture parameters: k at2 , u 2 a x1 L t 2 , ⊗ a a • π ⊗ a ∞T −1 ∞ • G = t L t ⊗ 2 2
• 2 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ σ ∞ σ ∞ 23 12 σ ∞ Isotropic materials: (7.21) 22 7.2.2 Crack: in-plane bending (given M , I,α, a ) 2 ∞ α − − M M (a sin ) 2 1 u = u + Re{A < ζ α > B }s(α), x 2 4I α 2 x ∞ M (a sin ) −2 −1 1 φ = φ + < ζ > α a a Re{B α B }s( ), 4I − 2 M Ma πa sin α k = s(α), 2I
2 − M sin α − Δu = x a2 − x2 L 1s(α) I 1 1 632 Appendix D
ˆ 7.2.3 Crack: arbitrary loading on the crack surface (given t n , a ) ∞ − − x T 1 k 1 2 = {}< ζ > − {}< ζ > ()+ u 2Im A ln α A c0 ∑ Im A α B ck id k , k =1 k ∞ φ = {}< ζ > T − 1 {}< ζ −k > −1()+ x1 2Im B ln α A c Im B α B c id , ⊗ ⊗ ⊗ ⊗ 0 ∑ k k k =1 k ∞ a a π = ⎛ T − ⎞ ck ,dk : (6.29); k ⎜S c0 ∑d k ⎟ a ⎝ k =1 ⎠
7.2.4 Crack: point force (given pˆ, xˆ1, xˆ2 ) 3 = 1 ⎧ < ζ −ζˆ > T + < ζ −1 −ζˆ > −1 T ⎫ x f (z) [ ln( α α ) A ln( α ) B BI A ] pˆ . 2 π ⎨ ∑ k k ⎬ pˆ 2 i ⎩ k=1 ⎭
(xˆ , xˆ ) 1 2 2 − x ˆ 1 T 1 k = Im{B < (1− ζ α ) > A }pˆ . a a πa 1 ⎪⎧ a + c ⎪⎫ When (xˆ , xˆ ) = (c,0) : k = − ST + I pˆ . 1 2 ⎨ − ⎬ 2 πa ⎩⎪ a c ⎭⎪ ˆ 7.2.5 Crack: dislocation (given b, xˆ1, xˆ2 )
x 2 3 T ˆ 1 ⎧ − − ⎫ b = < ζ −ζˆ > T + < ζ 1 −ζˆ > 1 ˆ f (z) ⎨[ ln( α α ) B ln( α k ) B BI k B ]⎬b (xˆ1, xˆ2 ) ∑ x1 π 2 i ⎩ k=1 ⎭ a a
7.3.1 Collinear cracks: general solutions (given a1,b1,...., an ,bn ) n x k 2 = < > + < > f (z) ∑ ∫ zα / X (zα )dzα ck zα c k =0 = Π n − 1/ 2 − 1/ 2 X (zα ) k=1(zα ak ) (zα bk ) x a 1 a1 b1 a2 b2 ak bk n bn
σ ∞ 7.3.2 Two collinear cracks: uniform loading (given a1,b1, a2 ,b2 , ij ) 2 λ − ∞ σ ∞ σ ∞ k 1 22 22 = < > f (z) fk (zα ) B t ,
∞ ∞ ∞ ∞ ∑ • σ • σ σ σ 23 12 23 12 k=0 λ
∞
∞
∞ σ 13⊗ σ ∞ σ x2 12 11
∞ ∞ t : (7.33b); f (zα ), f (zα ), f (zα ) : (7.34b,c); ∞ σ • σ 0 1 2 σ 13 11 12
∞
σ ∞ ∞ 13⊗ σ σ x1 12 11 λ λ λ λ a2 b a 2 1 b1 , , , : (7.41b) and (7.36b)
∞ ∞ 0 1 2 ∞ σ • σ σ 13 11 12
= ∞ Δ = Δ −1 ∞
∞ ∞ ∞ ∞ ⊗ σ ⊗ σ σ σ Fracture parameters: k kt , u f (x )L t , 12 23 12 23 1
σ ∞ σ ∞ 22 22 1 ∞ − ∞ G = k 2t T L 1t 2 Δ k: (7.48c); f (x1) : (7.49b) Orthotropic materials: (7.50) and (7.54); Iso t ropic materials: (7.51a,b) λ λ λ λ Δ Two equal cracks: , 0 , 1, 2 : (7.55a,b); k , u,G : (7.56a,b,c) Appendix D 633
σ ∞ 7.3.3 Collinear periodic cracks: uniform loading (given l,W , ij ) σ ∞ σ ∞
22 22 π ∞ ∞ ∞ ∞
• σ • σ σ σ 1 sin( z /W ) − ∞ 23 23 α 12 12 = − < − > 1 f (z) zα ∫ dzα B t ∞ 2 2
∞
∞ σ 13⊗ σ 2 σ 12 π − π 11 x sin ( z /W ) sin ( /W )
2 α l ∞ ∞ ∞ σ • σ σ 13 11 12
∞
σ ∞ 13⊗ ∞ − ∞ 1 ∞ − ∞ ∞ σ σ w 12 1 2 T 1 11 x = Δ = Δ =
1 Fracture parameters: k kt , u f (x )L t , G k t L t
∞ ∞ ∞ σ • σ 1 σ 2l 2l 2l 13 11 12 2
∞ ∞ ∞ ∞ ⊗ σ σ σ ⊗ σ 12 23 12 23 Δ k: (7.58); f (x1) : (7.59b) σ ∞ σ ∞ 22 22 ˆ 7.4.1 Collinear interface cracks: general solution (given a1,b1,...., an ,bn ,t(s) )
x2 u = A f (z (1) ) + A f (z (1) ), φ = B f (z (1) ) + B f (z (1) ), 1 1 1 1 1 1 1 1 1 1 = (2) + (2) φ = (2) + (2) u2 A2f2 (z ) A2 f2 (z ), 2 B2f2 (z ) B2 f2 (z ), S 1 − − − f (z) = B 1ψ(z), z ∈ S ; f ( z) = B 1M* 1M*ψ(z) , z ∈ S x1 1 1 1 2 2 2 a b a b a b a b 1 1 2 2 k k n n − S 1 1 + − 2 ψ'(z) = X (z) [X (s)] 1tˆ(s)ds + X (z)p (z) 2πi 0 ∫L s − z 0 0 n n − − = < − δα − δα 1 > X0 (z) Λ ∏(z a j ) (z b j ) ; j=1 1 Λ : (7.67b) and (7.68); δα = + iεα : (7.69a,b) and (7.13b) 2 ˆ 7.4.2 A semi-infinite interface crack: point force on the crack surface (given a,t 0 ) x 2 −1/ 2+iε 3 (1) k 1 πε − ε z − − (1) = < k 1/ 2 i k α (1) > 1 1ˆ tˆ f1(z ) e a dzα B1 Ik Λ t0, 2 π ∑ ∫ (1) + 2 k=1 zα a (2)−1/ 2+iε
tˆ1 3 k ˆ • − t3 1 πε − ε z − 1 −
(2) = < k 1/ 2 i k α (2) > 1 * * 1ˆ ⊗ f2 (z ) ∑ e a (2) dzα B2 M M Ik Λ t0, tˆ tˆ x1 ∫ 1 3 2π k=1 zα + a a
tˆ 2 − ε − 2 i α 1 k = Λ < (a / ) cos hπεα > Λ tˆ ; πa l 0 Orthotropic bimaterials: (7.126) ˆ 7.4.3a A finite interface crack: point force on the crack surface (given a,c,t0 )
x2 ε πε iε i k 3 k k 2 2 e ⎛ a + c ⎞ 1 a − c ⎛ zα − a ⎞ tˆ = − < 2 f (z) ⎜ ⎟ ⎜ ⎟ dzα 1 ∑ π − ∫ − 2 − 2 ⎜ + ⎟ k=1 2 ⎝ a c ⎠ c zα zα a ⎝ zα a ⎠
tˆ1 tˆ • 3 − −
x 1 1 ⊗ 1 > B ΛI Λ tˆ , tˆ1 tˆ3 1 k 0 − −1 (2) −1 * 1 * f (z): from f (z) w ith zα and B replaced by zα and B M M . tˆ2 2 1 1 2 ε c + + i α 1 a c ⎡ (a c) ⎤ −1 a a = < l πε > Λ ˆ k Λ ⎢ ⎥ cosh α t0. πa a − c ⎣2a(a − c)⎦
orthotropic bimaterials: (7.128) 634 Appendix D
7.4.3b A finite interface crack: uniform loading on the crack surface (given a,tˆ ) x 2 iε 3 k ⎛ z − a ⎞ − − −1 − f (z) = − < z − z 2 − a2 ⎜ α ⎟ >B 1ΛI Λ 1(I + M* M* ) 1tˆ. 1 ∑ α α ⎜ ⎟ 1 k k=1 zα + a
tˆ ⎝ ⎠
• • • • • −
−1 (2) −1 * 1 *
x
⊗ ⊗ ⊗ ⊗ ⊗ 1 f2 (z): from f1(z) with zα a nd B1 replaced by zα and B2 M M . − ε − = π < + ε i α > 1ˆ k aΛ (1 2i α )(2a / l) Λ t. orthotropic bimaterials: (7.130) a a σ ∞ 7.4.4 Two collinear interface cracks: uniform loading at infinity (given a1,b1, a2 ,b2 , ij ) σ ∞ σ ∞ 22 22 3
∞ 1
∞ x2 ∞ ∞ 2 • σ • σ σ σ 23 12 23 12 f (z) = < (λ z + λ z + λ )X (z ) 1 ∑ ∫ 1k α 2k α 3k k α
σ ∞ ∞ λ
∞ σ = σ 13⊗ 12 k 1 11 1k −1
∞ ∞ σ • σ * σ ∞ 13 11 −1 −1 * −1 ∞ 12 > B ΛI Λ (I + M M ) t x1 1 k ∞
σ a1 b1 a2 b2 σ ∞ σ ∞ 13⊗ 12 11 ∞
∞ ∞ σ • ∞ σ λ λ λ σ 13 11 , , : (7.92b) and (7.91b); t : (7.89b) 12 1k 2k 3k
∞ ∞ ∞ ∞ ⊗ σ σ σ ⊗ σ 12 23 12 23 − −1 (2) −1 * 1 * σ ∞ σ ∞ f (z): from f (z) wit h z and B replaced b y z and B M M . 22 22 2 1 α 1 α 2 −1 ∞ k = 2π Λ < kα > Λ t , kα : (7.131b) 8.1 Elliptical elastic inclusion: general solution (given a,b)
x 2 uAf=+++[()ζζ f ()] Af [() ζζ f ()]⎫ 1101 101⎪ ζ ∈ ζ ⎬,S1, α : (8.2c) φ =+++ζζ ζζ 1101Bf[() f ()] Bf 101 [() f ()]⎭⎪
b * * x = ζ + ζ ⎫ 1 u A f ( ) A f ( ) 2 2 2 2 2 ⎪ ζ * ∈ ζ * ⎬, S2 , α : (8.2c) a φ = ζ * + ζ * 2 B2f2 ( ) B2 f2 ( ) ⎭⎪ ζ ζ f0 ( ) : (8.7) or (8.15); f1( ) : (8.12c,d) or (8.16a,b) ζ f2 ( ) : (8.4a,b) in which ck : (8.11a,b) Interfacial stresses: (8.13a,b) and (8.14) Superscript •* : quantity related to the inclusion. σ ∞ 8.1.1 Elliptical elastic inclusion: uniform loading at infinity (given a,b, ij ) ζ =< > T ∞ + T ∞ f0 ( ) zα (A1 t 2 B1 ε1 ) , σ ∞ 22 ∞ *
∞ σ ∞
• σ 23 • • • • ∞ 13 • σ ∞ − σ ⊗ 12 σ 1 2zα 11 σ ∞ 12 ζ =< ζ > ζ =< > 12 f ( ) α g , f ( ) c , σ ∞ 1 1 2 * 1
x2 13 ∞ ⊗ • σ
11 a − ibμα ⊗ • = − −1 + < γ > − − < γ * > γ ⊗ g B {B1e1 B α e B2 c1 B α c }, α : ( 8.4b)
b • 1 1 1 1 2 1 x1 ⊗ − −
• 1 −1 −T 1 −T a = − − +
⊗ c1 i{G 0 G1G 0 G1} {A1 e1 G1G 0 A1 e1}, G 0 , G 1:(8.11b) • ⊗
• 1 T ∞ T ∞ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ σ ∞ = < − μ > + 23 e a ib α (A t B ε ). σ ∞ 1 1 2 1 1 12 σ ∞ 2 22
Appendix D 635
ˆ 8.1.2 Elliptical elastic inclusion: point force at the matrix (given a,b,pˆ,ζ α )
T ζ =< ζ − ζˆ > π f0 ( ) ln( α α ) A1 pˆ / 2 i 3 ∞ 1 − − − ζ = < ζ 1 −ζˆ > 1 T ˆ + < ζ k f1 ( ) ∑ ln( α k ) B1 B1I k A1 p ∑ α 2πi k=1 k=1 > −1{}+ < γ *k > B1 B 2 ck B 2 α ck
∞ ζ * = {}< ζ *k + γ *kζ *−k > γ * f2 ( ) ∑ α α α c k , α : (8.4b) k=1
ck : (8.11a,b) in which ek : (8.23) 8.2.1a Elliptical rigid inclusion: general loading condition (given a,b) =+++ζζ ζζ⎫ uAf[()01 f ()][() Af 01 f ()]⎪ ζ ∈ ⎬,,S1 φ =+++ζζ ζζ Bf[()()][()()]01 f Bf 01 f ⎭⎪ ω −1 −1 f (ζ ) = −A Af 0 (1/ζ ) + A k , 1 2ζ k: (8.30b); ω : (8.34a,b) and (8.35) ∞ ζ = ζ k =< γ k > When f0 ( ) ∑ek , e-k α ek , k=−∞ ζ 1 f0 ( ) e = dζ , γ α : (8.4b) k 2π i ∫C ζ k+1 ∞ ω ζ = − −1 {}+ < γ k > ζ −k + −1 f1( ) A ∑ Aek A α ek A k, k=1 2ζ −T T − 2Im{k A e1} ω = . kT Mk Interfacial stresses: (8.36a,b) σ ∞ 8.2.1b Elliptical rigid inclusion: uniform loading at infinity (given a,b, ij )
ζ =< ζ + γ ζ −1 > f0 ( ) α α α e1,
1 T ∞ T ∞ e = < a − ibμα > (A t + B ε ) , γ α : (8.4b) 1 2 2 1
1 −1 −1 ∞ ∞ f (ζ ) = − < ζ α > A (aε + ibε − ω k), k: (8.30b) 1 2 1 2 a2 (H−1ε∞ ) − ab[(H−1Sε∞ ) + (H−1Sε∞ ) ]− b2 (H−1ε∞ ) ω = 1 2 1 1 2 2 2 1 2 −1 + −1 + 2 −1 a (H )22 2ab(H S)21 b (H )11 636 Appendix D
ˆ 8.2.1c Elliptical rigid inclusion: point load at the matrix (given a,b,pˆ,ζ α ) ζ =< ζ − ζˆ > T π f0 ( ) ln( α α ) A pˆ / 2 i
3 1 − − ω − − ζ = < ζ 1 − ζˆ > 1 T ˆ + < ζ 1 > 1 f1( ) ∑ ln( α k ) A AIk A p α A k 2πi k=1 2 T Re{k A−T < ζˆ −1 > AT }pˆ ω = α , k: (8.30b) πkT Mk If the load is applied on the interface boundary, ˆ iψˆ i.e., ζ α = e , ω : (8.47)
σ ∞ 8.2.2a Rigid line inclusion: uniform loading at infinity (given a, ij )
σ ∞ ∞ ∞ − − ∞ 22 1 1 ∞ = + − < ζ > −ω σ 23 σ ∞ u x1ε1 x2ε2 a Re{A α A }(ε1 i2 ), 12 ∞ ∞ σ σ 13 ∞ 11 σ 12 ∞ ∞ − − ∞ ∞ 1 1 σ ∞ σ σ ∞ 12 13 11 φ = − − < ζ > −ω x x t x t a Re{B A }(ε i ), 2 1 2 2 1 α 1 2 a Re{kT Mε∞} x 1 1 ω = , k = ai a a kT Mk 2
∞ ∞ σ σ 23 12
σ ∞ 22
ˆ 8.2.2b Rigid line inclusion: point load at the matrix (given a,pˆ,ζ α )
3 1 ⎧ − − ⎫ = < ζ −ζˆ > T + < ζ 1 −ζˆ > 1 T ˆ pˆ f (z) ⎨ ln( α α ) A ∑ ln( α k ) A AI k A ⎬p x π 2 2 i ⎩ k=1 ⎭ ()xˆ , xˆ 1 2 −1 −1 +{ω < ζ α > A k}/ 2, x T 1 −T < ζˆ −1 > T ˆ a a ω = Re{k A α A }p = . k ai2 π kT Mk xˆ pˆ If the load is applied on the rigid line, i.e., xˆ = 0 , ω = 1 2 2 T πk Mk ε σ ∞ 8.2.3 Polygon-like rigid inclusion: uniform loading at infinity (given a,c, , ij ) = ∞ + ∞ + < ζ −1 > + < ζ −k > u x1ε1 x2ε2 2Re{A[ α q1 α qk ]}, φ = ∞ − ∞ + < ζ −1 > + < ζ −k > x1t 2 x2t1 2Re{B[ α q1 α qk ]}, q = AT g + BT h , = 1,k l l l l = ω − ∞ + ∞ = − −1 ω − ∞ − ω + ∞ h1 a( i2 ε1 ) x2ε2 , g 1 aH {S( i2 ε1 ) c( i1 ε2 )} = ε ω − ∞ = − ε −1 ω − ∞ + ω + ∞ hk a ( i2 ε1 ) , g k a H {S( i2 ε1 ) ( i1 ε2 )} ω : (8.55) Appendix D 637
ˆ ˆ 8.3.1 Elliptical elastic inclusion: dislocations outside the inclusions (given a,b,b,ζ α )
1 ˆ T ˆ f (ζ ) = < ln(ζ α − ζ α ) > B b, 0 2πi 1 3 ∞ ζ = 1 < ζ −1 − ζˆ > −1 T ˆ + 1 < ζ −k > ˆ f1( ) ∑ ln( α k ) B1 B1I k B1 b ∑ α Ekb, 2πi k=1 2πi k=1 ∞ ζ * = 1 < ζ *k + γ *kζ *−k > ˆ f2 ( ) ∑ α α α Ckb, 2πi k=1 E ,C : (8.56d); γ * : (8.4b) k k α Rigid inclusion: 1 ˆ T ˆ f (ζ ) = < ln(ζ α − ζ α ) > B b, 0 2πi 1 3 ω ζ = 1 < ζ −1 −ζˆ > −1 T ˆ + < ζ −1 > −1 f1( ) ∑ ln( α k ) A1 A1Ik B1 b α A1 k, 2πi k =1 2 ω : (8.57c); k: (8.30b) ˆ * 8.3.2 Elliptical elastic inclusion: dislocations inside the inclusions (given a,b,b, zˆα ) uAf=+++[()ζζ f ()] Af [() ζζ f ()]⎫ 1101 101⎪ ζ ∈ ⎬,,S1 φ =+++ζζ ζζ 1101Bf[() f ()] Bf 101 [() f ()]⎭⎪ uAf=+++[(**ζζ ) f ( * )] Af [( ** ζζ ) f ( * )]⎫ 2202 202 ⎪ ζ * ∈ ⎬,,S2 φ =+++**ζζ * ** ζζ * 220Bf[()()][()()] f 2 Bf 20 f 2 ⎭⎪ 1 T ˆ f (ζ ) = < lnζ α > B b,, 0 2πi 1 * 1 * * T ˆ f (ζ ) = < ln(zα − zˆα ) > B b, 0 2πi 2 − ∞ ∞ ζ = 1 < ζ −k > −1 < * > T ˆ + 1 < ζ −k > ˆ f1( ) ∑ α B1 B2 ekα B2 b ∑ α Ek b, 2πi k=1 2πi k=1 ∞ ζ * = 1 < ζ *k + γ *kζ *−k > ˆ f2 ( ) ∑ α α α Ck b. 2πi k=1 * γ * ekα : (8.61b); Ek ,Ck : (8.56d) in which Tk : (8.71b); α : (8.4b) 638 Appendix D
ˆ * 8.3.3 Elliptical elastic inclusions: dislocation on the interface (given a,b,b, zˆα ) φ φ u11, and u22, : same expressions as Section 8.3.2. 1 ˆ T ˆ 1 −1 ˆ ˆ f (ζ ) = < ln(ζ α −ζ α ) > B b + < ln(ζ α −ζ α ) > Q b, 0 2πi 1 2πi 1 * 1 * * ˆ f (ζ ) = < ln(zα − zˆα ) > Q b, 0 2πi 2 − 3 ∞ ζ = 1 < ζ −k > −1 < * > T ˆ + 1 < ζ −k > ˆ f1( ) ¦ α B1 B2 " kα Q2 b ¦ α Ekb, 2πi k =1 2πi k =1 ∞ ζ * = 1 < ζ *k + γ *kζ *−k > ˆ f2 ( ) ¦ α α α Ckb, 2πi k =1 * γ * " kα : (8.79b); α : (8.4b)
Q1,Q2 ,Ek ,Ck : (8.82b) and (8.56d) in which Tk : (8.82c)
9.1 Rigid punches on a half-plane: general solution (given uˆ,qˆ k )
− 1 1 + − f '(z) = B 1ș'(z) , ș'(z) = X (z) [X (t)] 1 Muˆ'(t)dt 2ʌ 0 ³L t − z 0 + X0 (z)pn (z), X (z) = ȁī(z), ȁ = [Ȝ Ȝ Ȝ ], 0 1 2 3 n =< − −(1+δα ) − δα > ī(z) ∏(z a j ) (z bj ) . j =1 = + + + n−1 pn (z) d0 d1z dn−1z .
Ȝα ,δα : (9.11) and (9.12a,b); = − dk , k 0,1,2,..., n 1 : (9.15) and (9.20) = − −T ′ − − ′ ∈ Contact pressure: t(x1) A f (x1 ) iMuˆ (x1), x1 L. ′ = T ′ − ∉ Surface deformation: u (x1) B f (x1 ), x1 L. 9.1.2 Indentation by a flat-ended punch (given a,qˆ ) −iε 3 k 1 1 § z + a · − − = < ¨ α ¸ > 1 1 ˆ f '(z) ¦ B ȁIk ȁ q. = 2 2 ¨ − ¸ 2ʌi k 1 zα − a © zα a ¹ 1 − c c = 1 + R T 2 + I T ≤ t(x1 ) [I (S ) S ]qˆ, | x1 | a, π 2 − 2 β 2 β a x1 * * 1 − ª 1 − c c º ′ = 1 + R T 2 + I T ˆ > u (x1 ) L «I (S ) S »q, | x1 | a, # 2 2 2 β π x − a ¬ β ¼ 1 * * ε β cR ,cI ,cR ,cI : (9.27b); k , : (9.12b)
Orthotropic half-plane: (9.29); Isotropic half-plane: (9.31) and (9.30) Appendix D 639
9.1.3 A flat-ended punch tilted by a couple (given a,ω or mˆ ) ω 3 ′ = i ⎧ −1 − < + ε > −1 −1 ⎫ f (z) ⎨B ∑ Γk (zα )(zα 2ia k ) B ΛIk Λ ⎬Li 2 , 2 ⎩ k=1 ⎭
ω ⎡ − 2 ⎤ = x1 + 1 cR T + cI T ≤ t(x1) ⎢I 2 S S ⎥Li 2 , | x1 | a, 2 − 2 β β a x1 ⎣ ⎦
⎧ x ⎡ 1− c* c* ⎤ ⎫ ′ = ω⎪ 1 + R 2 − I ⎪ > < − u (x1) ⎨I m ⎢I 2 S S⎥i2 ⎬, x1 a and x1 a, 2 − 2 β β ⎩⎪ x1 a ⎣ ⎦ ⎭⎪ π ε 2 2 ⎪⎧⎡ 4 T 2 ⎤ ⎪⎫ mˆ = a ω⎨⎢I − (S ) ⎥L⎬ 2 ⎪ β 2 ⎪ ⎩⎣ ⎦ ⎭22 * * ε ε β Γ k (zα ), cR ,cI , cR ,cI : (9.36b) ; , k , : (9.12b) 9.1.4 Indentation by a parabolic punch (given a, R,qˆ ) 3 ′ = − i < 2 + ε − + ε 2 2 > −1 −1 f (z) ∑ Γk (zα )[zα 2ia k zα (1 4 k )a ] B Λ I k Λ Li 2 4R k=1 3 i − 1 − − + < > 1 + < > 1 1 ˆ zα B Li 2 ∑ Γk (zα ) B Λ I k Λ q, 2R 2πi k=1 2 2 ~ ~ − ⎡ − 2 ⎤ = 2x1 a + 1 (cc )R T + (cc )I T t(x1) ⎢I 2 S S ⎥Li 2 2 − 2 β β 2R a x1 ⎣ ⎦ 1 ⎡ 1− c c ⎤ + + R T 2 + I T ˆ ≤ ⎢I 2 (S ) S ⎥q, | x1 | a, π 2 − 2 β β a x1 ⎣ ⎦
2 − 2 ⎡ − *~ *~ ⎤ ′ = x1 2x1 a + 1 (c c )R 2 − (c c )I u (x1) i 2 m ⎢I 2 S S⎥i 2 R 2 − 2 β β 2R x1 a ⎣ ⎦ * * 1 − ⎡ 1− c c ⎤ 1 + R T 2 + I T ˆ > < − m L ⎢I 2 (S ) S ⎥q, x1 a and x1 a, π 2 − 2 β β x1 a ⎣ ⎦ ε β Γ k (zα ) : (9.36b)1; k , : (9.12b) * * ~ ~ *~ *~ cR ,cI , cR ,cI ,(cc ) R ,(cc ) I ,(c c ) R ,(c c ) I : (9.38b,c) 9.2 Rigid stamp indentation on a curvilinear hole boundary: general solution (given ε a,c, ,uˆ,qˆ k ) f′(ζ ) = B−1θ′(ζ )/ζ
1 ρ + − θ'(ζ ) = − X (ζ ) [X (s)] 1Muˆ'(s)ds + X (ζ )p (ζ ), 2πi c ∫L s − ζ c c c ζ = ζ = ζ Xc ( ) ΛcΓc ( ) ΛΓ( ), n = =< − −(1+δα ) − δα > Λ [λ1 λ 2 λ3 ], Γ(z) ∏(z a j ) (z bj ) . j =1
ρ : (9.42b) and (6.64); λα ,δα : (9.11) and (9.12a,b);
pc : (9.47), (9.48b) and (9.49a,b) 640 Appendix D
9.2.2 Rigid stamp indentation on an elliptical hole boundary (given a,b,φ,qˆ ) φε 3 2 k 1 ⎡ − − e f ′(ζ ) = < Γ (ζ ) > B 1Λ I Λ 1BAT + < Γ (ζ ) ∑ ⎢ k α c k c k α ζ 2πi k =1 ⎣ α ⎤ > −1 −1 T B Λ c I k Λ c BA ⎥qˆ ⎦ − φ − − ε φ + ε Γ ζ =< ζ − i 1/ 2 i k ζ − i 1/ 2 i k > ε k ( α ) ( α e ) ( α e ) ; k : (9.12b) Isotropic medium: (9.57) 9.2.3 Rigid stamp indentation on a polygon-like hole boundary (given a,c,ε,φ,qˆ ) φε 3 2 k 1 ⎡ − − e f ′(ζ ) = < Γ (ζ ) > B 1Λ I Λ 1BAT + < Γ (ζ ) ∑ ⎢ k α c k c k α ζ 2πi k =1 ⎣ α ⎤ > −1 −1 T B Λ c I k Λ c BA ⎥qˆ ⎦ − φ − − ε φ + ε Γ ζ =< ζ − i 1/ 2 i k ζ − i 1/ 2 i k > ε k ( α ) ( α e ) ( α e ) ; k : (9.12b) Note: the solution form of this problem is the same as that of
9.2.2 except ζ α . 9.3.1 Rigid punch on a boundary perturbed by a straight line: general solution (given ε(ϕ (x1),uˆ,uˆ 0 ,uˆ 1 ) = + ε( + ε(2 + f (z) f0 (z) f1(z) f2 (z) L.
= + ε( + ε(2 + uˆ(x1) uˆ 0 (x1) uˆ 1(x1) uˆ 2 (x1) L f ′(z) = B−1θ′ (z) 0 0 , 1 1 + − θ′ (z) = X (z) [X (t)] 1 Muˆ′ (t)dt + X (z)p (z), 0 2π 0 ∫L t − z 0 0 0 n = −1 − μϕ ′ f1(zˆ) B {θ1(zˆ) (zˆ)Bf0 (zˆ)},
1 1 + − θ (zˆ) = X (zˆ) [X (t)] 1 Muˆ (t)dt + X (zˆ)p(1) (zˆ). 1 2π 0 ∫L t − zˆ 0 1 0 n Appendix D 641
ε( 9.3.2 Rigid stamp on an elliptical perturbed hole: general solution (given a,ck ,dk , , ϕ ζ ( ),uˆ,uˆ 0 ,uˆ 1 )
= ζ + ε( ζ + ϕ ζ ′ ζ + ε( 2 ζ + ϕ ζ ′ ζ f (z) f0 ( ) [f1 ( ) ( )f0 ( )] [f2 ( ) ( )f1 ( ) 1 + ϕ 2 (ζ )f ′′(ζ )] + 2 0 L σ = σ + ε( σ + ε( 2 σ + uˆ( ) uˆ 0 ( ) uˆ 1 ( ) uˆ 2 ( ) L ′ ζ = −1 ′ ζ ζ f0 ( ) B θ0 ( )/
1 ρ + − θ′ (ζ ) = − X (ζ ) [X (s)] 1Muˆ′ (s)ds + X (ζ )p (ζ ), 0 2πi c ∫L s − ζ c 0 c c N ζ = −1 ⎧ ζ − a + − μ + − μ ζ 2 ζ −k ′ ζ ⎫ f1 ( ) B ⎨θ1 ( ) ∑(ck id k )[1 i (1 i ) ] Bf0 ( )⎬ ⎩ 2 k=1 ⎭
1 1 + − θ (ζ ) = X (ζ ) [X (s)] 1Muˆ * (s)ds + X (ζ )p(1) (ζ ), 1 2πi c ∫L s − ζ c 1 c c N * = − ρ − + μ k−1 ′ uˆ 1 (s) i{uˆ 1 (s) ∑Re[ia (ck idk )(1 i )s ]uˆ 0 (s)}. k=1 ρ : (9.42b) and (9.80) = ε(ϕ 9.3.3a A rigid flat-ended punch on a cosine wavy-shaped boundary (given x2 (x1 ), ϕ = ˆ (x1 ) cos x1 , q )
= + ε( + μϕ ′ f (z) f0 (zˆ) [f1(zˆ) (x1)f0 (zˆ)] − ε ˆ + i α ′ = 1 −1 −1 =< 1 ⎛ z a ⎞ > f0 (zˆ) B ΛΓ(zˆ)Λ qˆ, Γ(zˆ) ⎜ ⎟ . 2πi zˆ2 − a2 ⎝ zˆ − a ⎠
icoszˆ − − f (zˆ) = B 1(πLi + μΛΓ(zˆ)Λ 1qˆ). 1 2π 2
9.3.3b A rigid stamp on a triangular hole boundary (given, N=2, c1=d1=d2=0, c2=1, ε( =0.25,φ , qˆ ) = ζ + ε( ζ +ϕ ζ ′ ζ ϕ ζ f (z) f0 ( ) [f1( ) ( )f0 ( )] , ( ) : (9.83b)
2φεα 1 − ⎡ − e − ⎤ f′(ζ ) = B 1Λ Γ(ζ ) Λ 1BAT + < > Λ 1BAT qˆ 0 c ⎢ c ζ c ⎥ 2πi ⎣ ⎦ ζ = − − μ ζ −2 + − μ ′ ζ f1 ( ) a[(1 i ) (1 i )]f0 ( )/ 2 642 Appendix D
9.4.1 Sliding punches with or without friction: general solution (given ak ,bk , Fk , Nk , gk (t) ) ±ηθ′ − ⎧ 2 (x1 )⎫ = −1 ′ − = ⎪ θ′ − ⎪ η = f '(z) B θ'(z) , θ (x1 ) ⎨ 2 (x1 ) ⎬ , Fk / Nk ⎪ 0 ⎪ ⎩ ⎭ χ n ′ θ′ = 0 (z) gk (t) + χ 2 (z) ∑ ∫ + dt 0 (z) pn (z), 2πτ = χ (t)(t − z) k 1 Lk 0 n χ = − −δ − δ −1 δ = 1 −τ τ ≤ δ < 0 (z) ∏ (z ak ) (z bk ) , arg( / ), 0 1; k=1 2π τ : (9.107b)
= + + + n−1 pn (z) d0 d1z LL dn−1z , dk : (9.109b,c) σ = τ +τ θ′ − − ′ τ ∈ Contact pressure: 22 (x1) {( ) 2 (x1 ) igk (x1)}/ , x1 Lk . ′ = − τ +τ θ′ − Surface deformation: u2 (x1) i( ) 2 (x1 ), ′ = − −1 ′ − ∉ u (x1) 2iL θ (x1 ), x1 L. η = τ = Frictionless surface: 0 , m22 , −1 m22 : (22) components of M 9.4.2 A sliding wedge-shaped punch (given a, F, N,ε * ; a: to be determined for incomplete indentation) iε * iN θ ′(z) = {}1− []z + ()2δ −1 a χ (z) + χ (z), 2 τ +τ 0 2π 0 χ = + −δ − δ −1 0 (z) (z a) (z a) Contact pressure:
− sinπδ 2πε * σ (x ) = {N − [x + (2δ −1)a]}. 22 1 π + δ − 1−δ τ +τ 1 (a x1) (a x1) ( ) 4πε *δ Complete indentation: N ≥ a. τ +τ τ +τ Incomplete indentation: a = N, 4πε *δ δ 2ε * sinπδ ⎛ a − x ⎞ σ (x ) = − `⎜ 1 ⎟ . 22 1 τ +τ ⎜ + ⎟ ⎝ a x1 ⎠ Appendix D 643
9.4.3 A sliding parabolic punch (given ", a,b, F, N, R ; a,b: to be determined for incomplete indentation ) τ +τ θ′ = i − χ ª − ( )RN º½ 2 (z) ®z 0 (z)« j2 (z) »¾ , (τ +τ )R ¯ ¬ 2π ¼¿ χ = + −δ − δ −1 0 (z) (z a) (z b) , 1 j (z) = z 2 + [δ (a + b) − b]z − δ (1− δ )(a + b)2 . 2 2 Contact pressure and surface deformation: − χ − τ +τ σ = i 0 (x1 ) − ( )RN ½ − < < 22 (x1) ® j2 (x1) ¾, a x1 b. τR ¯ 2π ¿ τ +τ ′ = 1 − χ − ª − ( )RN º½ < − > u2 (x1) ®x1 0 (x1 )« j2 (x1) »¾, x1 a or x1 b. R ¯ ¬ 2π ¼¿ χ − 0 (x1 ) : (9.126) 2 1−δ δ ½ π" Complete indentation: N ≥ max® , ¾ . ¯ δ 1−δ ¿ (τ +τ )R δ (τ +τ )RN (1− δ )(τ +τ )RN Incomplete Indentation: a2 = , b2 = , π (1− δ ) πδ
2sinπδ −δ + δ σ (x ) = − (x + a) 1(b − x ) , − a < x < b, 22 1 (τ +τ )R 1 1 1
x 1 −δ + δ u′ (x ) = 1 | x + a | 1| x − b | , x < −a or x > b . 2 1 R # R 1 1 1 1 Frictionless surface: (9.133), (9.134) and (9.135a,b)
9.4.4 Two sliding flat-ended punches (given a1,b1,a2 ,b2 ,F1,F2 , N1, N 2 ) θ′ = χ + 2 (z) 0 (z)(d0 d1z), d0 , d1 : (9.138a,b) and (9.141a,b) χ = − −δ − δ −1 − −δ − δ −1 0 (z) (z a1) (z b1) (z a2 ) (z b2 ) Contact pressure and surface deformation: τ σ = § + ·χ − + < < < < 22 (x1) ¨1 ¸ 0 (x1 )(d0 d1x1), a1 x1 b1, a2 x1 b2. © τ ¹ ′ = − τ +τ χ − + < < < > u2 (x1) i( ) 0 (x1 )(d0 d1x1), x1 a1, b1 x1 a2 , x1 b2. χ − 0 (x1 ) : (9.141a,b) Frictionless surface: (9.142a,b); special case: (9.143) 9.5 Contact between two elastic bodies: general expressions
u = A f (z) + A1f (z),½ 1 1 1 1 ° ∈ ¾, z S1, φ = + 1 B1f1(z) B1f1(z), ¿°
u = A f (z) + A2 f (z),½ 2 2 2 2 ° ∈ ¾, z S2 , φ = + 2 B2f2 (z) B2 f2 (z), ¿° − B f (z),ᇫz ∈ S , ș(z) = 1 1 1 ® ∈ ¯B2f2 (z),ᇫz S2. 644 Appendix D
η η (1) (2) 9.5.1a Contact in the presence of friction: general solution (given 1, 3 , N, g (x1), g (x1) )
θ′ =η θ′ θ′ =η θ′ 1(z) 1 2 (z), 3 (z) 3 2 (z), χ ⎧ n ′ ⎫ θ′ = 0 (z) ⎪ g (t) + ⎪ 2 (z) ⎨∑ ∫ + dt iN ⎬ 2π k=1 τχ (t)(t − z) ⎩⎪ Lk 0 ⎭⎪
χ = − −δ − δ −1 δ = 1 −τ τ ≤ δ < 0 (z) (z a) (z b) , arg( / ), 0 1, 2π = (1) − (2) g(x1) g (x1 ) g (x1 ). τ : (9.161b)
contact pressure and surface deformation: σ (x ) = {(τ +τ )θ′(x− ) − ig′(x )}/τ , ⎫ 22 1 2 1 1 ⎪ ∈ ⎬, x 1 L * = τ (2) (1) +τ (1) (2) + τ (1)τ (2) −τ (1)τ (2) θ − τ g (x1) { g (x1) g (x1) i( ) 2 (x1 )}/ ⎭⎪ g (1)* (x ) = g (1) (x ) + i(τ (1) +τ (1) )θ (x − ) ⎫ 1 1 2 1 ⎪ ∉ ⎬, x1 L, (2)* = (2) − τ (2) +τ (2) θ − g (x1 ) g (x1 ) i( ) 2 (x1 )⎭⎪ τ (1) ,τ (2) : (9.163b)
9.5.1b Contact of two parabolic elastic bodies with friction (given F, N, R1, R2 ) i(R + R ) iN θ′(z) = 1 2 {z − χ (z) j (z)}+ χ (z). 2 τ +τ 0 2 π 0 ( )R1R2 2 1 j (z) = z 2 −[δ (a − b) + b]z − δ (1− δ )(a − b)2 ; 2 2 δ : (9.162b) and (9.161b) τ χ : (9.161b); 0 (z) : (9.162b) and (9.161b) x2 1 1 δ τ +τ −δ τ +τ = 1 + 2 ( )R1R2 N 2 (1 )( )R1R2 N g(x1) ( ). a = , b = . 2 R1 R2 π −δ + πδ + (1 )(R1 R2 ) (R1 R2 ) Contact pressure: 2(R + R )sinπδ −δ δ σ (x ) = − 1 2 (x − a)1 (b − x ) , a < x < b 22 1 τ +τ 1 1 1 ( )R1R2
9.5.2 Contact of two parabolic elastic bodies without friction (given N, R1, R2 ) + ⎧ 2 − 2 ⎫ θ′ = i(R1 R2 ) ⎪ − z (b / 2)⎪ + iN 2 (z) ⎨z ⎬ , * 2 2 2 2 2m22R1R2 ⎩⎪ z − b ⎭⎪ 2π z − b Contact pressure and surface deformation: R + R 2m* R R N σ (x ) = − 1 2 b2 − x2 , | x |< b, b = 22 1 2 . 22 1 * 1 1 π + m22R1R2 (R1 R2 ) x2 1 1 (2) − (1) = 1 + * R2m22 R1m22 2 g(x1) ( ). = − < < g (x1) * x1 , b x1 b, 2 R1 R2 2m R R 22 1 2 2 2 (1)* = x1 + (1)θ − (2)* = − x1 − (2)θ − g (x1) 2im22 2 (x1 ), g (x1) 2im22 2 (x1 ), 2R1 2R2 θ − (1) (2) * −1 −1 * 2 (x1 ) : (9.171b); m 22 , m22 , m22: (22) components of M1 ,M 2 ,M Appendix D 645
9.5.3 Contact of two elastic bodies in complete adhesion (given a,b, g(x1),qˆ )
1 g′(t) + − − θ′(z) = X (z) [X (t)] 1 dtM* 1i + X (z)d 2π 0 ∫L t − z 0 2 0 0 1 z − a ε − = < i α > = 1 1 X0 (z) Λ ( ) ; d0 Λ qˆ (z − a)(z − b) z − b 2πi ε = ε ε = −ε ε = ε Λ : (B.9); 1 , 2 , 3 0 in which : (B.10b)
10.2 Elliptical holes: uniform heat flow at infinity (given a,b, h0 )
h0 2 x 2 = < > + u 2Re{∑A fi (zα ) qi cg(zt )}, i=1 2 φ = < > + 2Re{∑B fi (zα ) qi dg(zt )} i=1
a x1 b = e1 ⎧1 2 − 1 2 − 2 +τ 2 2 g(zt ) ⎨ zt zt zt (a b ) = ψ + τ x1 a cos a i b 2 2 = ψ ⎩ x2 bsin 2 2 2 ⎫ a +τ b + 2 − 2 +τ 2 2 h0 + ln(z z (a b )) t t ⎬ 2 ⎭ 1 ⎧1 1 ⎫ f (z ) = z 2 − z z 2 − (a 2 + μ 2b2 ) , 1 α + μ ⎨ α α α α ⎬ a i α b ⎩2 2 ⎭ − μ a i α b 2 2 2 2 f (zα ) = ln(zα + zα − (a + μα b )), 2 2
e1 : (10.27) and (10.19b); q1,q2 : (10.24b) and (10.29a) Hoop stress: σ = − T φ ss s ,n ,
ah − − ψ φ = 0 {}2ρ sinψγ (θ ) − N (θ )L 1 Re[e 2i (a + ibτ )~γ * ] ,n ~ρ 2 3 2 ~ 2k k : (10.19b); θ - ψ relation: (10.20a); ρ : (10.20b); θ ~* γ 2 ( ) : (10.16b), (10.12c), and (10.8b); γ 2 : (10.16b) and (10.13) α 2 + σ = Eh0 a (a b) ψ Isotropic materials: ss ~ sin , 2kρ 2 (1−ν ) 646 Appendix D
10.2.2 Cracks: uniform heat flow at infinity (given a, h0 ) = + φ = + u 2 Re{Af(z) cg(zt )}, 2Re{Bf (z) dg(zt )}
h0 = − ih0 2 − 2 − 2 + 2 + 2 − 2 g(z) ~ {zt zt zt a a ln(zt zt a )} , 4k x2 = ih0 < 2 − 2 − 2 > −1 f (z) ~ zα zα zα a B d 4k 2 x ih a 2 −1 * 1 + 0 < + − > − ~ a a ~ ln(zα zα a ) B [d iRe(γ2 )], 4k ~ ~* k : (10.19b); γ2 : (10.16b) and (10.13)
h π h 0 = 0 3/ 2 ~* Fracture parameters: k ~ a Re{γ2}, 2k π 2 3 = h0 a ~* T −1 ~* G ~ Re{γ2} L Re{γ2}, 8k 2 Δ = 2 − 2 −1 ~* ~ u h0 x1 a x1 L Re{ γ2}/ k , Isotropic materials: (10.42) ϕ 10.3.1 Insulated elliptical rigid inclusions: uniform heat flow at infinity (given a,b, h0 , )
h0 2 x 2 2 2 = < > + < > + + u 2Re{A zα q0 ∑ A fi (zα ) qi c[e0 zt e1g0 (zt )]}, = i 1 2 φ = < 2 > + < > + 2 + 2Re{B zα q0 ∑B fi (zα ) qi d[e0 zt e1g0 (zt )]}, i=1 x1 b τ τ ~ a e0 ,e1 : (10.47a) in which I : imaginary part of ; k : (10.19b)
q0 ,q1,q2 : (10.24b) and (10.47a,b,c,d) h 0 Interfacial stresses: (10.48a,b) ϕ 10.3.2 Insulated rigid line inclusions: uniform heat flow at infinity (given a, h0 , ) 2 2 2 u = 2Re{A < zα > q + A < f (zα ) > q + c[e z + e g (z )]}, h 0 ∑ i i 0 t 1 0 t 0 = i 1 2 x 2 φ = < 2 > + < > + 2 + 2Re{B zα q0 ∑B fi (zα ) qi d[e0 zt e1g0 (zt )]}, i=1 τ τ ~ e0 ,e1 : (10.49) in which I : imaginary part of ; k : (10.19b) x1 a a q0 ,q1,q2 : (10.24b), (10.49), and (10.47d) Strength of thermal stress singularity: (10.51) and (10.52)
h 0 Appendix D 647
ˆ ˆ 10.4.1 Collinear interface cracks: general thermal loading (given a1,b1,...., an ,bn , h(s),t(s) )
hˆ (i) (i) (i) (i) x = ′ = − + τ ′′ 2 Ti 2Re{gi (zt )}, hi 2Re{(k1 ik 2 )gi (zt )}, = (i) + (i) ui 2Re{Aifi (z ) ci gi (zt )},
S 1 φ = (i) + (i) = i 2Re{Bifi (z ) di g(zt )}, i 1,2. x1 a b a b a b a b 1 1 2 2 k k n n h ,k (i) ,k (i) : h,k ,k (defined in (10.6c)) of ith material S2 i 1 2 1 2
g1(z), g2 (z),f1(z),f2 (z) : (10.61b,c,d) in which k + k hˆ(s)ds θ′′(z) = 1 2 χ (z) + χ (z) p (z), π 0 ∫L χ + − 0 n 2 k1k2 0 (s)(s z)
1 1 + − + − ψ′(z) = X (z) [X (s)] 1[−tˆ(s) +θ′(s )e +θ′(s )e ]ds + 2πi 0 ∫L s − z 0 1 2
X0 (z)pn (z), ~ ki: k (defined in (10.19b)) of ith material with the tilde dropped.
e1,e2 : (10.61c,d) χ 0 (z),X0 (z) : (10.63b,c), (7.68), (7.69a,b), and (7.13b) σ Stresses i2 along the interface: (10.65) Crack opening displacement: (10.66)
10.4.2 Interface crack: uniform heat flow (given a, h0 ,t0 ) + θ ′ = − * − 2 − 2 * = h0 (k1 k2 ) (z) ih0 (z z a ), where h0 2k1k2 x2 ψ′(z) = −Λ{J (z)t* + ih* (J (z)e* + J (z)e* )}, h0 0 0 0 1 1 2 2 Λ : (10.63c), (7.68) t0 J (z), J (z), J (z) : (10.78b), (10.75b), (7.69b), and (7.13b) x1 0 1 2 t* , e* , e* : (10.75b), (10.63c), and (7.68) h0 0 1 2 a a
648 Appendix D
θ θ 10.5 Multi-material wedges under thermal loading: near- tip solution (given 0 ,..., n )
x2 ⎧v (r,θ ) ⎫ ⎡ Γ* (θ ) 0 ⎤⎧v (θ ) ⎫ 1 = 1 1 0 ⎨ ⎬ ⎢ 2 * * ⎥⎨ ⎬ 2 θ θ ⎩w1(r, )⎭ ⎢r F (θ ) E (θ )⎥ w1( 0 ) θ ⎣ 1 1 ⎦⎩ ⎭ 2 θ 1 1 θ θ 0 * k k x θ θ θ 1 ⎧v (r, ) ⎫ ⎡ Γ ( ) 0 ⎤⎡ (KT )k−1 0 ⎤⎧v ( ) ⎫ θ k k 1 0 n ⎨ ⎬ = ⎢ ⎥⎢ ⎥⎨ ⎬, n θ 2 * * 2 θ w (r, ) r F (θ ) E (θ ) r (K ) − (K ) − w ( ) ⎩ k ⎭ ⎣⎢ k k ⎦⎥⎣ c k 1 e k 1 ⎦⎩ 1 0 ⎭ k = 2,3,....,n ⎧T (r,θ )⎫ u (r,θ ) θ = ⎪ ,r ⎪ θ = ⎧ k ⎫ vk (r, ) ⎨ ⎬ , w k (r, ) ⎨ ⎬ ⎪ * θ ⎪ φ (r,θ ) ⎩h (r, )⎭k ⎩ k ⎭ * θ * θ * θ Γk ( ),Ek ( ),Fk ( ) : (10.116b) in which Λ k , Uk : Λ, U (defined in (10.90)) of kth material
(KT )k−1,(K c )k−1,(K e )k−1 : (10.116c) in which
Γk ,Ek ,Fk : (10.95b), (10.96b) θ θ δ v1( 0 ), w1( 0 ) and singular order : (10.103)–(10.115) Special cases (including single wedge, bi-wedge): (10.119)–(10.122) θ θ 11.4.2 Piezoelectric multi-material wedges: near- tip solutions (given 0 ,..., n )
x2 1 −δ − ε − θ = 1 R θ < − δ + ε 1 i α > 1 u(r, ) r V( ) (1 R i α ) (r / l) Λ k, 2π 2 θ 2 θ 1 1 −δ − ε − θ 1 1 R 1 i α 1 θ 0 φ θ = θ < − δ + ε > k k x (r, ) r Λ( ) (1 i ) (r / ) Λ k, 1 R α l θ n 2π n 1 −δ ε − φ θ = R θ < i α > 1 ,r (r, ) r Λ( ) (r / l) Λ k. 2π V(θ ), Λ(θ ) : (11.70b) and (11.69a,b,c); Λ = Λ(0) δ = δ + ε α R i α : (11.69c), (5.83b), (5.82b), and (5.88) 11.5.1 Cracks in piezoelectric materials: near-tip solution θ = − π 1/ 2 θ −1 φ θ = − π 1/ 2 θ −1 u(r, ) 2/ r V( )L k, (r, ) 2/ r Λ( )L k, 1/ 2 T 1/ 2 T θ ＝π V(θ ) = A < μˆα (θ,−π ) > B + A < μˆα (θ,−π ) > B θ ＝- π Λ(θ ) = B < μˆ 1/ 2 (θ,−π ) > BT + B < μˆ 1/ 2 (θ,−π ) > BT α α 2 − 1 − Δu = 2 r1/ 2L 1k , G = kT L 1k π 2 Piezoelectric ceramics poling in x -axis: (11.85a,b,c) and (11.7d,e) 3 Piezoelectric ceramics poling in x2-axis: (11.86a,b,c), (11.62b,c), (11.56b) and (11.53) Appendix D 649
11.5.2 Interface cracks between two dissimilar piezoelectric materials: near-tip solution 2 2r − ε − u (r,θ ) = V (θ ) < (1+ 2iε ) 1(r / )i α > Λ 1k, k = 1,2, k π k α l
2r − ε − φ (r,θ ) = Λ (θ ) < (1+ 2iε ) 1(r / )i α > Λ 1k, k = 1,2, k π k α l
φ = φ = 1 < iεα > −1 1,r (r,0) 2,r (r,0) Λ (r / l) Λ k, 1 2πr
iεα 2r − (r / ) − Δu(r) = Λ T < l > Λ 1k, π (1+ 2iεα )coshπεα
1 − G = k T (D + WD 1W)k. 4 θ θ Vk ( ), Λk ( ) : (11.96b,c,d), (11.87b,c), (11.95a,b), and ( 11.94b) in which λ : (11.89) Λ = Λ(0) ; ε α : (11.95a,b) and (11.94b); l : reference length
Piezoelectric ceramics poling in x3-axis: (11.98)
Piezoelectric ceramics poling in x2-axis: (11.99) σ ∞ ∞ 11.6.1 Collinear cracks: uniform load/induction at infinity (given a1,b1,...., an ,bn , ij , D2 )
σ ∞ σ ∞ 22 22
∞ ∞ ∞ ∞
• σ • σ σ σ 23 12 23 12 = ∞ Δ = Δ −1 ∞ = 1 2 ∞ T −1 ∞ ∞ x k kt , u f (x )L t , G k (t ) L t , t : (11.100b) 2 2 1 2 2 2 2 ∞
∞
∞ σ 13⊗ σ σ 12 11 2
∞ ∞ ∞ σ • σ σ 13 11 12 2 2 x = π Δ = −
∞ a a a a 1 A single crack: k a, f (x ) 2 a x . σ 1 b1 2 b2 k bk n bn ∞ ∞ 13⊗ σ 1 1 σ 12 11
∞ ∞ ∞ σ • σ σ 13 11 12 Δ Two collinear cracks: k : (7.48c), f (x1) : (7.49b) in which
∞ ∞ ∞ ∞ ⊗ σ σ σ ⊗ σ 12 23 12 23 λ : (7.41b) and (7.36b) σ ∞ σ ∞ k 22 22 Δ Evenly spaced collinear periodic cracks: k:(7.58), f (x1):(7.59b) . 11.6.2a A semi-infinite interface crack: point force/charge on crack surfaces (given a,pˆ )
x2 − ε 2 i α −1 k = Λ < (a / ) cos hπεα > Λ pˆ Λ,εα , : §11.5.2 pˆ l , l
2 πa • Piezoelectric ceramics poling in x -axis: (11.110a,b)
pˆ 3 pˆ1 3 pˆ ⊗ pˆ x1 1 3 Piezoelectric ceramics poling in x2 -axis: (11.111a,b) a pˆ 2 11.6.2b A finite interface crack: point force/charge on the crack surfaces (given a,c,pˆ ) x 2 ε + + i α pˆ 1 a c ⎡ (a c) ⎤ −1 2 = < l πε > k Λ ⎢ ⎥ cosh α Λ pˆ ,
πa a − c ⎣2a(a − c)⎦ pˆ3 • pˆ1
x ⊗ 1 ε pˆ1 pˆ Λ, α , : §11.5.2 3 l
pˆ 2 Piezoelectric ceramics poling in x3 -axis: (11.113a,b,c) c Piezoelectric ceramics poling in x2 -axis: (11.114a,b) a a 650 Appendix D
11.6.2c A finite interface crack: uniform load/induction on the crack surfaces (given a,tˆ ) x 2 − ε − = π < + ε i α > 1ˆ ε k aΛ (1 2i α )(2a / l) Λ t, Λ , α ,l : §11.5.2
ˆ
t Piezoelectric ceramics poling in x3 -axis: (11.116a,b,c)
• • • • •
x ⊗ ⊗ ⊗ ⊗ ⊗ 1 Piezoelectric ceramics poling in x2 -axis: (11.117a,b)
a a 11.6.2d Two collinear interface cracks: uniform load/induction at infinity σ ∞ (given a1,b1, a2 ,b2 , ij ) σ ∞ σ ∞ 22 22 −1 ∞
∞ ∞ ∞ = π < >
∞ x2 • σ • σ σ σ k 2 Λ kα Λ t , kα : (7.131b), 23 12 23 12 2
σ ∞ ∞
∞ σ σ 13⊗ 12 ε 11 Λ, α ,l : Section 11.5.2
∞ ∞ σ • σ σ ∞ 13 11 12
x1 ∞
σ a1 b1 a2 b2 σ ∞ σ ∞ 13⊗ 12 11
∞ ∞ σ • σ σ ∞ 13 11 12
∞ ∞ ∞ ∞ ⊗ σ σ σ ⊗ σ 12 23 12 23
σ ∞ σ ∞ 22 22 ˆˆˆ 12.4.1 Elliptical hole: uniform bending at infinity (given abM,,x , Myxy , M )
Mˆ y ββ=∞−−∞∞−Re{ABmm<ζ 11> (aib− )}, α 21 =−∞ <ζ −−∞11 > − ∞ ˆ 2a ˆ ψψ Re{BBmmα (aib )}, M x M x 21 2b βψ, : (12.39b); A , B : (12.39b), (12.40), and (12.14);ζ α : (12.61) ∞∞, − * −1 βψ: (12.57) (12.60) in which D ij: components of D , ˆ M y D: bending stiffness Moments around the hole boundary: (12.65) = ˆ = = Circular holes with M x M , M y M xy 0 : =∞−ˆ<ζ−1>−1 ββ aM Im{AαB}i1, =∞−ˆ<ζ−1>−1 ψψ aM Im{BBiα} 1. Orthotropic plate: (12.68a,b) ˆˆˆ 12.4.2 Elliptical rigid inclusion: uniform bending at infinity (given abM,,x , Myxy , M )
Mˆ y ββ=−∞−−∞∞Re{AA <ζ 11 > (aibββ + )}, α 12 =−∞−−∞∞ <ζ 11 > + ˆ 2a ˆ ψψ Re{BAα (aibββ )}, M x M x 12 2b , ζ ∞∞ βψ,A,B, k: Section12.4.1; β 1, β2: (12.58) and (12.59) in which * ˆ Dij : Section 12.4.1 M y Moments around the inclusion boundary: (12.72) = ˆ = = Circular inclusions with M x M , M y M xy 0 : (12.73) Orthotropic plate: (12.75) Appendix D 651
ˆˆˆ 12.4.3 Crack: uniform bending at infinity (given aM,,,x Myxy M )
Mˆ y =∞− <ζ−1>−1∞=∞− <ζ−1>−1∞ ββ aRe{AαB}m2, ψ ψ aRe{BαB}m2, , ∞∞, ζ ˆ 2a ˆ βψ,A,B: Section 12.4.1; βψ , α : (12.57)–(12.60), (12.76b) x M x x 63 ∞ K=πaMˆ, K=πa{Mˆ−[Gm ] } Ih2yIIh2xy12(2) Mˆ − ∞ y =−T 1 GNNSL11 3, m2 : (12.57b) z y The subscript (2): the second component of the vector. 14.1 Holes in laminates: uniform stretching and bending moments at infinity ∞ ∞ (given a,b, Nij , M ij )
N ∞ 22 ∞ −1 −1 ∞ ∞ ∞ = − < ζ > − ∞ M u u Re{A B (am ibm )}, 12 α N12 d d 2 1 ∞ M N ∞ 22 ∞ −1 −1 ∞ ∞ 12 φ = φ − < ζ > − ∞ 2a ∞ Re{B B (am ibm )}, ∞ α M11 M11 ∞ d d 2 1 N11 N11 2b ∞ ∞ ∞ M ∞ ∞ M 12 N 12 12 φ ζ ud , d : (14.3a,b), (14.6), (14.5a,b,c,d,e) and ( 14.4c); α : ( 14.7); M ∞ ∞ 22 N12 ∞ ∞ M ∞ 12 m ,m : (14.3b) ∞ 1 2 N 22 Stress resultants and bending moments along the hole boundary: (14.12a–f)
14.2.1 Holes in laminates: uniform heat flow and moisture transfer in x1x2 plane (given a,b, qˆi , mˆ i ) = ′ = ′ T 2Re{gt (zt )}, H 2Re{gh (zh )}, θˆ qˆ,mˆ q( = −2Re{(K t +τ K t )g′′(z )}, m( = −2Re{(K h +τ K h )g′′(z )}, i i1 t i2 t t i i1 h i2 h h
θ = + + n s ud 2Re{Af (z) ct gt (zt ) ch gh (zh )}, x1 φ = + + d 2Re{Bf (z) dt gt (zt ) dh gh (zh )},
qˆ,mˆ f (z), g (z ), g (z ) : (14.17a,b,c), (14.18a,b,c,d), (14.19a,b) x2 x t t h h 3 Stress resultants and bending moments along the hole boundary: (14.20), (13.79)–(13.81), and (13.70b)
14.2.2 Holes in laminates: uniform heat flow and moisture transfer in x3 direction
(given a,b,h,Tl ,Tu ,H l ,H u)
= 0 + * = 0 + * ( = − t * ( = − h * T T x3T , H H x3H , qi Ki3T , mi Ki3H = < ζ −1 > −1 ϑ − ϑ θ u Re{A B (a ib )}, n s d α 2 1 x 1 φ = < ζ −1 > −1 ϑ − ϑ − ϑ + ϑ d Re{B α B (a 2 ib 1)} x1 2 x2 1, ϑ ϑ ζ 0 0 * * x , : (13.125b,c); : (14.7); T , H ,T , H : (14.21) 2 x 3 1 2 α qˆ,mˆ Stress resultants and bending moments along the hole boundary: (14.27), (13.79)–(13.81), and (13.70b) 652 Appendix D
∞ ∞ 14.3 Holes in electroelastic laminates (given a,b, N ij , M ij )
N ∞ 22 ∞ −1 −1 ∞ ∞ ∞ = − < ζ > − ∞ M u u Re{A B (am ibm )}, 12 α N12 d d 2 1 ∞ M N ∞ 22 ∞ −1 −1 ∞ ∞ 12 φ = φ − < ζ > − ∞ 2a ∞ Re{B B (am ibm )}, ∞ α M11 M 11 ∞ d d 2 1 N11 N11 2b ∞ ∞ ∞ M ∞ ∞ M 12 N 12 12 φ ζ ud , d : (14.28a,b), (14.29a,b) and (13.145); α : (14.7); M ∞ ∞ 22 N12 ∞ ∞ ∞ M 12 ∞ m1 ,m 2 : (14.28b) N 22 Generalized stress resultants and bending moments along the hole boundary: (14.31a,b,c) ˆ ˆ 14.4.1 Concentrated inplane forces and out-of-plane moments (given zˆα , f1, f2 , mˆ1,mˆ 2 ) T u = Im{A < ln(zα − zˆα ) > A }pˆ /π , d pˆ = ( f ˆ fˆ mˆ − mˆ )T φ = < − > T π 1 2 2 1 d Im{B ln(zα zˆα ) A }pˆ / , Real form solution: (14.39)
ˆ 14.4.2 Concentrated transverse force (given zˆα , f3 ) u = Im{A < (z − zˆ )ln(z − zˆ ) − (z − zˆ ) > AT } fˆ i /π , d α α α α α α 3 3 φ = < − − − − > T ˆ π d Im{B (zα zˆα ) ln(zα zˆα ) (zα zˆα ) A } f3i3 / . Real-form solution: (14.49a,b)
14.4.3 Concentrated in-plane torsion (given zˆα , mˆ 3 ) u = −Im{A < (z − zˆ )−1 > AT }mˆ i /π , d k k 3 2 φ = − < − −1 > T π d Im{B (zk zˆk ) A }mˆ 3i2 / . Real-form solution: (14.57)
ˆ ˆ ˆ 14.5 Holes in laminates: point forces and moments (given a,b, zˆα , f1, f2 , f3 , mˆ 1, mˆ 2 , mˆ 3 ) =+ζζφ =+ ζζ uAffdpdp2Re{ [00 ( ) ( )]}, 2Re{ Bff [ ( ) ( )]} ˆ ˆ ζ + ζ Case 1 f1, f2 , mˆ1,mˆ 2 : f0 ( ) f p ( ) : (14.79) ˆ ζ + ζ Case 2 f3: f0 ( ) f p ( ): (14.85), (14.74b), (14.65b) and (14.82b) ζ + ζ Case 3 mˆ 3: f 0 ( ) f p ( ): (14.89), (14.87b), (14.65b) and (14.74b) Stress resultants and moments along the hole boundary:
(14.91a–c) and (14.74b) in which ρ : (14.16b); ζ α : (14.7) Appendix D 653
ˆ ˆ ˆ 14.5.4 Cracks in laminates: point forces and moments (given a, zˆα , f1, f2 , f3 , mˆ 1, mˆ 2 , mˆ 3 ) = ζ φ = ζ ζ ud 2 Re{Af( )}, d 2 Re{Af ( )} , f ( ) : (14.94a–e) Stress intensity factors: (14.95), (14.96) and (14.97a,b,c),
in which ζ α : (14.94d) When the force is applied on the crack surface: (14.98)
ˆ ˆ ˆ 14.6.1 Point forces/moments outside the Inclusions (given a,b, zˆα , f1, f2 , f3 , mˆ 1, mˆ 2 , mˆ 3 )
(1) = − ζ + ( ζ + − ζ + ( ζ ud A1[f0 ( ) f1( )] A1[f0 ( ) f1( )], − ( − ( φ(1) = B [f (ζ ) + f (ζ )] + B [f (ζ ) + f (ζ )], d 1 0 1 1 0 1 (2) = + ζ * + − ζ * + + ζ * + − ζ * ud A2[f2 ( ) f2 ( )] A2[f2 ( ) f2 ( )], φ(2) = + ζ * + − ζ * + + ζ * + − ζ * d B2[f2 ( ) f2 ( )] B2[f2 ( ) f2 ( )]. f − (ζ ) : (14.106a–d) and (14.74b); 0 ( ζ f1 ( ) : (14.114a,b), (14.111b,c,d) and (14.113a,b,c); + ζ * + − ζ * = ζ * f2 ( ) f2 ( ) f2 ( ) : (14.112a–c) and (14.113a–c)
in which cε ,γ α : (14.74b), ζ α : (14.7) and the superscript * denotes the quantities of the inclusion Stress resultants and moments along the interface: (14.115) and (14.117) ˆ ˆ ˆ 14.6.2 Point forces/moment inside the Inclusion (given a,b, zˆα , f1, f2 , f3 , mˆ 1, mˆ 2 , mˆ 3 ) (1) = ζ + + ζ + − ζ + ζ + ζ + + ζ + − ζ ud A1[fs ( ) f0 ( ) f0 ( ) f1 ( )] A1[fs ( ) f0 ( ) f0 ( ) + ζ f1 ( )], φ(1) = B [f (ζ ) + f + (ζ ) + f − (ζ ) + f (ζ )] + B [f (ζ ) + f + (ζ ) + f − (ζ ) d 1 s 0 0 1 1 s 0 0 + ζ f1( )], (2) = * ζ * + *+ ζ * + *− ζ * + + ζ * + − ζ * ud A 2[fs ( ) f0 ( ) f0 ( ) f2 ( ) f2 ( )] + * ζ * + *+ ζ * + *− ζ * + + ζ * + − ζ * A2[fs ( ) f0 ( ) f0 ( ) f2 ( ) f2 ( )], φ(2) = * ζ * + *+ ζ * + *− ζ * + + ζ * + − ζ * d B2[fs ( ) f0 ( ) f0 ( ) f2 ( ) f2 ( )] + * ζ * + *+ ζ * + *− ζ * + + ζ * + − ζ * B2[fs ( ) f0 ( ) f0 ( ) f2 ( ) f2 ( )]. ζ + ζ − ζ fs ( ),f0 ( ),f0 ( ) : (14.120a–c), (14.123a–c), (14.124) and (14.125a–e) ζ f1( ) : (14.126b,c,d,e) and (14.127b,c,d); * ζ * + *+ ζ * + *− ζ * = * ζ * fs ( ) f0 ( ) f0 ( ) f0 ( ): (14.118a–c) and (14.119a–c) + ζ * + − ζ * = ζ * f2 ( ) f2 ( ) f2 ( ) : (14.126a,d,e) and (14.127a–d)
in which cε ,γ α : (14.74b), ζ α : (14.7) and the superscript * denotes the quantities of the inclusion Stress resultants and moments along the interface: (14.128a–c) References
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A Erdogan, F., 256 Anderson, T. L., 131 Eshelby, J. D., 53, 55, 239, 246, 266 Atkinson, C., 337 F Fan, C. W., 277, 297 B Florence, A. L., 337, 343 Barber,J.R.,348 Barnett, D. M., 333, 389 G Becker, W., 436 Galin, L. A., 277 Berger,J.R.,132 Gangadharan, A. C., 266 Bhargave, R. D., 239 Gao, H., 140, 216 Bogy, D. B., 132 Gerasoulis, A., 271 Brebbia, C. A., 545, 567, 581 Gladwell, G. M. L., 277, 289 Brock, L. M., 289 Goodier, J. N., 165, 184, 337, 343 Broek, D., 144, 205 Griffith, A. A., 187
C H Carlsson, L. A., 228 Hartmann, F., 515, 585 Chadwick, P., 67, 93, 333 Hartmann, H. H., 132 Chai, H., 232 Hein, V. L., 132 Chen,D.H.,132 Herrmann, G., 239 Chen,H.P.,132 Hirth, J. P., 256 Chen,W.T.,239 Honein, T., 239 Cheng, Z. Q., 435–436, 444–445 Hong, C. S., 225 Choi, N. Y., 147 Hsieh, M. C., 411, 434, 436, 453, 469, 498, Chou, Y. T., 239 503, 505 Clements, D. L., 337 Hwu, C., 64, 69, 75, 85–86, 103, 120, 132, Comninou, M., 348 139–140, 144–145, 149, 156–158, 159, 167, 174, 180, 182, 185–186, 195–196, 206–207, 225–226, 230–231, 233, 235, D 237, 245–246, 257, 268, 271, 273, 275, Desmorat, R., 132 277, 284, 290, 295, 297, 314, 321, Ding, S., 132 328, 333, 336–337, 346–348, 357, 367, Dongye, C., 69 369, 377, 379, 411, 434–436, 440, 453, Dundurs, J., 132, 256, 262–263, 266 460, 469, 498, 503–505, 519, 522, 536, Dunn, M. L., 140 542–543, 545, 560–561, 565 Dwight, H. B., 319 I E Ikeda, T., 369 Earmme, Y. Y., 147 Im, S., 147 England, A. H., 277, 286, 295, 308, 317–318 Irwin, G. R., 193, 218, 344
663 664 Author Index
J R Jaswon, M. A., 239 Reddy, J. N., 435–436, 444–445 Jih, C. J., 226 Reedy, J. E. D., 132 Johnson, K. L., 131, 277 Rekach,V.G.,330 Jones, R. M., 23, 131 Rice, J. R., 140, 143, 147, 216, 219, 223, 231, 397 K Rogacheva, N. N., 370 Keer, L. M., 239, 256 Russell, A. J., 225 Kim, K. S., 147 Kirchner, H. O. K., 76, 93 S Kuo, C. M., 369, 401, 402 Santare, M. H., 239, 256 Kuo, T. L., 140, 145, 149, Savin, G. N., 159 156–158 Sendeckyi, G. P., 239 Sih, G. C., 195, 202, 223, 337, 344 L Sinclair, G. B., 140, 147 Labossiere, P. E. W., 140 Smith, G. D., 67, 93, 333 Leckie,F.A.,132 Sneddon, I. N., 289, 337 Lee, W. J., 132, 333, 367 Soh, A. K., 369, 383 Lekhnitskii, S. G., 16, 29–30, 34, 42, 50, Sokolnikoff, I. S., 2, 5, 8, 12, 15, 147, 377 87, 186, 302, 374, 411, 415–417, 420, Soni, M. L., 132 431, 433 Sosa, H., 369 Liang, Y. C., 369, 545, 561 Stagni, L., 239, 256 Liao, C. Y., 545, 560 Stern, M., 132, 140, 515, 576, 585–586 Lin, C. C., 333, 346–348 Street, K. N., 225 Lin, K. Y., 132 Stroh, A. N., 29, 53, 256, 333, 374 Lizzio, R., 256 Sturla, F. A., 337, 344 Lothe, J., 67, 73, 93, 256, 369, 389 Sumi, N., 348 Lowengrub, M., 289 Sun, C. T., 225–226, 369, 400 Lu, P., 435–436 Suo, Z., 140, 216, 218, 220, 369, 377, 390, 402 M Mahrenholtz, O., 435–436 T Malen, K., 67 Tan, C. Z., 536, 542–543 Manoharan, M. G., 226 Theocaris, P. S., 132 Martin-Moran, C. J., 348 Timoshenko, S. P., 165, 184, 288 Mura, T., 256 Ting, T. C. T., 6, 9, 11–12, 17, 53, 58, Murakami, Y., 204 60–62, 64, 66–67, 74–76, 95, 98–99, 101, Muskhelishvili, N. I., 86, 101, 165, 196, 208, 108, 117, 120, 128, 130, 137, 174, 190, 277, 284–285, 295, 308 209, 245–246, 253, 334, 360, 365, 377, 379–380, 402 N Tsai, Y. M., 337 Nishitani, H., 132 Nowacki, W., 333, 474 U Ueda, S., 348 O Olesiak, Z., 337 W Wang, A. S. D., 235 P Wang, W. Y., 180, 182, 186, 239, 255–256 Pagano, N. J., 225 Wang, Z. Y., 239 Pak, Y. E., 369 Warren, W. E., 256 Park, S. B., 369, 400 Wilkins, D. J., 225 Patton, E. M., 256 Williams, M. L., 131, 206 Pipes, R. B., 225, 228 Willis, J. R., 289 Author Index 665
Wu, C. H., 333 Yin, W. L., 460 Wu, K. C., 140, 195, 216–217, 220 Yoon, S. H., 225
Y Yan, G., 253, 365 Z Yang, H. C., 239 Zakharov, D. D., 436 Yen, W. J., 174, 195, 239, 246, 248, 256–257, Zhou, S. G., 225 266, 377, 460, 469, 519 Zotemantel, R., 515 Subject Index
A Boundary integral equation, 545–546, 552, Airy stress function, 33, 44–45, 55, 444 561, 565–571, 573–580, 586–588 Analogy technique, 249, 288–290 Boundary value problem Analytical continuation, 101–103, 109, 174, first (traction-prescribed), 6 246, 264, 279, 325, 350, 517–519, second (displacement-prescribed), 6 529–532, 537 third (mixed), 6 Analytic function, 101, 174 Branch cut, 112, 129, 297–298, 315, 319 Angular bracket, 64, 504 Burgers vector, 99, 107–108, 111–112, 128, Anisotropic 176, 195, 249, 257, 266 elasticity, 3, 5, 29, 87, 197, 295, 374–375, 386, 420, 440, 446 C elastic solid, 5–6, 29, 67, 333, 557 Cauchy principal value, 575–576 matrix, 239, 266, 268 Cauchy’s formula, 2, 58, 152, 377, 417, 464 plates, 182, 186, 337, 411–415, 424, 454 Chain rule, 54, 85, 164, 169, 244, 291, Antiplane deformation, 18 522, 533 Antiplane shear, 18, 189 Characteristic equation, 34, 61, 358, 383, 386, Arbitrary loading, 87, 159, 168–173, 194, 239, 388, 422, 424, 427, 468 245, 250, 494 Chentsov coefficient, 13–14 Axial force, 29, 39, 43, 46 Classical lamination theory, 23–27, 412, 445, 482 B Coefficient Barnett–Lothe tensor, 69–71, 73, 77, 86, 98, matrix, 383, 386, 485, 491, 554 188–189, 203, 206, 377, 389–392, of moisture expansion, 474, 476, 478 540, 582 of mutual influence, 13 Bending of thermal expansion, 342–343 moment, 25, 39, 46–48, 50, 412, 416, Commutative properties, 76–77, 127 428–431, 434, 437–438, 440–443, 477, Compatibility equation, 5, 29, 88, 374, 381, 493–498, 505, 585 420–421, 448 rigidity, 51 Complementary solution, 140, 148–151, theory, 411–415, 436–439 154–155 Biaxial loading, 90, 162, 197 Complete adhesion, 278, 322, 325, 331–332 Bimaterial Complex matrix, 145, 190, 208, 217, 221, 325–326 conjugate, 34–35, 56, 133, 143, 151, 157, stress intensity factors, 144, 216–220, 223, 357, 388, 415, 444, 485 226, 275 method, 229, 234 Body force, 3, 6, 29–30, 40, 47–48, 147, 413, parameter, 45, 415, 420 483, 566 reduced elastic, 17–18, 32, 47 Boundary element, 92, 149, 207, 247, 494, variable formulation, 6, 29, 86, 417, 545–588 440–458
667 668 Subject Index
Compliance tip, 144, 147, 187–189, 191–194, 199, method, 229, 234 202–205, 207, 216, 237, 289, 344, reduced elastic, 17–18, 32, 47, 53, 591 403, 563 Concentrated force, 87, 92, 95–96, 99, 111, Cracked lap shear (CLS), 225–229, 233–235 131, 173–176, 194–195, 246–248, 263, Critical point, 161, 179–180, 240–241 505–507, 526, 529–539, 570 Cross section, 14–15, 20, 24–25, 39–41, Concentrated moment, 95–99, 107, 510, 512 48–51, 237 Conformal mapping function, 277, 299 Curvilinear boundaries, 277, 290, 435 Constitutive law, 1, 5, 7, 277, 369–375, Curvilinear interface, 273 436–437, 451, 482–484, 503 Contact D pressure, 282–284, 286, 311, 313–317, Degenerate material, 61, 73, 84–86, 196, 284, 327–329 342–343, 386, 389–390, 420, 560 surface, 278, 329 Degrees of freedom, 580, 586, 588 Continuity condition, 111, 113, 129, 134–135, Delaminated composite, 228, 230, 237 208, 259–260, 359–360, 527, 530–531, Delamination, 187, 206, 225–237 536 Delamination fracture criterion, 187, 225, 234 Contracted notation, 8–9, 11, 21, 30, 60, 88, Delta function, 172, 210, 569 370–371, 441, 484 Diagonalization, 64, 73, 76, 133 Coordinate Dielectric permittivities, 370, 560 bipolar, 283, 315 Direction cosine, 2, 10 cartesian, 3, 567, 575 Dislocation, 87, 99–100, 107–108, 111–112, global, 558–560 176, 239–240, 256–275, 288–289 local, 202, 558–560 Dislocation density, 269, 271, 273–275 polar, 59, 78, 83, 94, 104, 111, 283, Displacement formalism, 436, 440–445, 297–298, 315, 319, 378, 463, 575 447–451, 454–459, 465, 478, 487 s–n, 59, 347, 462–463, 522 Displacement gradient, 93, 287, 292, 312, 316 Double cantilever beam (DCB), 225–230, 232, tangent–normal, 567, 575, 587 234–235 transformation, 225, 338, 358, 499, 560 Dual coordinate systems, 66, 83 Correspondence relation, 86, 284, 295 Coupled stretching–bending analysis, 411, 435–491, 565–588 E Couple moment, 95–99, 107 Edge notch, 156 Coupling effect, 454, 474, 498 Effective transverse shear force, 414, 416–417, Crack 419, 423, 429, 442–443, 458–459, 463 collinear, 187, 195–207, 225, 349, 405 Eigenfunction, 115, 124, 132, 136–137, 143, 157, 395 collinear interfacial, 187 Eigen-relation collinear periodic, 200–202, 204–206, 406 generalized, 73–74, 76, 85–86, 337, 425, curvilinear, 268, 273–275 461, 469, 479 edge, 156 material, 60–68, 424–425, 459–461, 560 elliptical central, 156 sextic, 61–67 insulated, 353, 366 Eigenvalue interface, 140, 144–145, 190, 206–226, elasticity, 335, 337, 342–343 229, 278, 288–290, 333, 348–356, 370, material, 55–56, 60–61, 72–74, 84, 86, 221, 396–397, 408–410 262, 303, 307–308, 376–377, 384, 386, isothermal, 366 422, 427, 459–460, 468, 534 opening displacement, 187, 193, 196, repeated, 64, 84, 337, 386 201–206, 210, 216, 344, 354, 405 thermal, 334, 337, 342–343 penetrating, 272–273 Eigenvector penny-shaped interface, 156 elasticity, 342 semi-infinite, 132, 138, 144, 188, 366, 399 left, 62 Subject Index 669
material, 55, 61, 86, 145, 248, 284, Fracture parameter, 187, 196, 201–207, 289–290, 380–389, 436, 451, 467–469, 216–225, 236, 356 491, 541, 583 Fracture toughness, 146, 226–232 right, 62–63, 377 Free term coefficient, 580–588 thermal, 342, 351 Friction Elastically homogeneous, 7 coefficient of, 314 Elastic constant, 1, 5, 7–8, 10–12, 18, 21–22, force, 278, 308–309, 312 54–55, 60–61, 73, 297, 326, 362, 446, surface, 309, 312, 319–320, 329 474, 478, 541 Fundamental elasticity matrix, 62, 68–69, Elastic material 71–73, 84, 132, 133, 436, 446, anisotropic (or triclinic), 1–27 469–472, 501, 513 isotropic, 239 Fundamental solution, 92, 494, 539, 545–551, monoclinic (or aelotropic), 10 556–558, 560–561, 569–576, 588 orthotropic (or rhombic), 11 transversely isotropic, 11, 20, 60 G Elastic stiffness, 17–18, 68–69, 370, 375, Gauss elimination, 560 389, 466 Gaussian quadrature rule, 553, 560 Electric Gauss point, 553 displacement, 370, 376–378, 394, 397, Generalized Hooke’s law, 5, 7–9, 12 403, 481–482, 505, 560–563 Glide force, 268 field, 369–370, 372, 376, 394, 481, Green’s function 560–562 interfacial, 111–113 intensity factor, 396–400, 403–410 surface, 105–106, 112, 176, 330 potential, 372, 378, 393–394, 398–400, 403, 481 H singularity, 394 Half-plane, 277–292, 300, 306, 308–309, Electro-elastic analysis, 560–565 311–315, 317–318, 326, 330, 332 Electromechanical analysis, 560 Half-space, 87–113, 174, 545–547, 550, 557 Electronic package, 156 Heat Elliptic integral, 204, 320 conduction, 333–334, 338, 358, 362, End-notched flexure (ENF), 225–230, 232, 474, 476 234–235 flux, 333–334, 337–339, 343, 345–346, Energy equation, 333 349–354, 356–365, 474–476, 498–499, Energy release rate, 193, 196, 201, 203–206, 502 218, 220–222, 226, 228–230, 400, 405 Hilbert problem, 208–209, 280, 292, 302, 306, Engineering constant, 1, 12–15, 18–19, 21, 23, 310, 326, 331, 351 70–71, 221 H-integral, 141, 147–158, 557 Equilibrium equation, 3, 5, 29–30, 33, 52, 293, Holder-continuous, 270–271, 275 413, 417, 420–421, 436–438, 447–448, Hole 474–475, 566 circular, 165, 172, 178, 411, 431, 562 curvilinear, 290–298, 302–303 F elliptical, 159–176, 178, 182, 289–290, Failure initiation, 132, 140 293–295, 338–343, 428, 516, 543 Fiber bridging, 227 oval, 177, 184, 186 Fiber orientation, 23, 225, 227–232, 234 pin-loaded, 172–173, 290 Fiber-reinforced composite, 19–20, 481 polygon-like, 159, 177–186, 290, 295–296, Field solution, 115, 137, 141–144, 183, 194, 549 200, 207, 209, 283, 352, 405, 496–497, square, 186 516–522, 533 triangular, 181, 299, 308 Finite element, 149, 207, 228–230, 234, 539, Holomorphic function, 242, 260, 291, 303, 557, 570 305, 307, 331, 376, 415, 477, 517, 529 Fracture mechanics, 141, 187–188, 225, Homogeneous material, 52, 111, 130, 144, 236–237, 268, 399 155, 188, 191, 266, 406, 415 670 Subject Index
Homogeneous solution, 115, 120, 122–124, Isotropic plate, 87, 89, 162, 165–166, 168, 335, 484 173–174, 176–179, 182–184, 186, Hoop stress, 105, 163–165, 167, 170–172, 194–195, 338, 344–345, 412–415, 424, 176, 183–186, 342–343, 542–543, 427–428, 469, 473, 581, 584–586 562–563 Hygrothermal stress, 474–481, 498 J J-integral, 147, 557 I Identities, 61, 68–84, 104–105, 112, 127, K 165–167, 183, 200, 336–337, 341–343, Kernel function, 99, 256, 270, 275, 494 457, 511, 581, 583 Key matrix, 132–134, 356, 358 Image force, 101 Kinematic relation, 277, 447–448, 475 Image singularities, 174, 519 Kirchhoff force, 414, 439 Kirchhoff’s assumption, 436, 445, 475, Impedance matrix, 81, 110, 243, 280, 312, 325, 481, 565 540, 548 Kronecker delta, 3, 54, 63, 441, 546, 581 Inclusion circular, 256, 263, 432 L elastic, 239–248, 250–252, 256–258, Lame constant, 12, 15, 60 526–543, 546–547, 574 Lamina elliptical, 239–240, 256–257, 272, 274, generally orthotropic, 20–23 526, 542–543 specially orthotropic, 20–21 hard, 248, 268 Laminate polygon-like rigid, 249–250, 255–256 balanced, 439 rigid, 239, 248–256, 268, 288–289, composite, 206, 227, 234–237, 474–475, 344–348, 431–433, 546–548, 550 481–491, 493–495, 498, 526 rigid line, 248–249, 254–255, 288–289, cross-ply, 427 347–348 degenerate, 583–584 soft, 268, 543 electro-elastic, 484, 503–505 Indentation symmetric, 411, 435, 454–457, 465, complete, 312–317 468–469, 472–474, 503 depth of, 278, 306 unsymmetric, 435, 472, 503 incomplete, 312–317 Laurent’s expansion, 241, 244, 251 normal, 278, 311–312, 314 Lekhnitskii bending formalism, 411, 415–420, punch, 278, 311 422 rigid stamp, 290–298, 302 Lekhnitskii formalism, 29–52, 87, 380, 411, rotary, 278 420, 424, 440, 451 Indenter, 278, 289 L’Hospital’s rule, 96 Induction, 370, 393, 405–406, 409–410 Linear algebraic equation, 215, 469, 509, Infinite space, 87–113, 505, 545–547, 550, 557 557, 560 Integral formalism, 73, 184, 389 Line force, 92 Interaction coefficient matrices, 560 Line integral, 212–213, 286–287, 327, Interaction effect, 320–321 353–355, 566 Interaction energy, 266–268 Liouville’s theorem, 102, 110, 175, 242, 261, Interface corner, 115–158, 396 310, 326, 350, 517 Interfacial stress, 191, 244, 251, 256, 347 Logarithmic function, 93, 112, 129, 169, Interlaminar stress, 225 248, 588 Internal Logarithmic singularity, 137, 142, 365, 395 point, 92, 99, 107, 545, 554–555, 565–570, Longitudinal direction, 19–20 581, 586 strain, 554 M stress, 505, 555, 560 Macromechanics, 3, 6, 493 Interpolation function, 552, 556 Mapping function, 180, 277, 299, 302 Isothermal condition, 337, 339 Material axes, 20–21, 24 Subject Index 671
Material symmetry, 1, 9–12 Path-independent property, 148–149 Matrices of influence coefficients, 552, 554 Perturbation Matrix elliptical perturbed boundary, 304 diagonal, 64, 154, 203, 206, 217, 221, 282, higher order, 299, 302, 306–308 359, 504 straight perturbed boundary, 301, 305 identity, 62, 72, 445 surface, 277, 299–308 nonsemisimple, 64, 84 technique, 277, 299, 303 rotation, 65, 75 zero-order, 301, 305, 307–308 semisimple, 64, 73, 78 Piezoeffect equation, 370 simple, 21–23, 557 Piezoelectric Matrix differential equation, 67–68 ceramics, 385, 387, 390–393, 400, 404, Mechanical energy release rate, 400 406–410 Membrane analogy, 288 material, 369–410, 453, 481, 561 Micromechanics, 3, 6, 493 tensor, 560 Mid-plane strain, 437–438, 446, 475, 482, Plane 504–505, 565 generalized plane strain, 16, 18, 43–44, Mixed formalism, 436, 440, 446–460, 465, 69–70, 372, 376, 385, 505 467, 487–489 generalized plane stress, 15–16, 18, 20–21, Mixed-mode fracture, 225, 229, 232–234 47, 69–71, 88–89, 203, 221, 285, Mode I, II, III, 189, 207, 225–233 372–373 Modified end-notched flexural (MENF), strain (plane deformation), 14, 43–44 225–229, 233–235 stress, 15–16, 18, 20–21, 44, 47, 69–71, Moisture 87–89, 156, 162, 193, 203, 221, 295, content, 474–476, 479–480, 501 372, 466, 585 diffusion, 474, 476 Plate bending analysis, 411–434, 454, 457 transfer, 474–475, 493, 498–503 Plate curvature, 24–25, 475 Moment–curvature relation, 413 Plemelj function, 209–211, 216, 280, 292, 296, Moment of inertia, 91 331, 352 Moment intensity factor, 433–434 Point charge, 378, 561 Multi-valued function, 92–93, 506, 508, 510 Point force, 92, 100–101, 105, 107, 131, 173, 176, 195, 210, 288–289, 408–409, N 549–550, 570, 575 Near-tip solution, 140–142, 146–147, 149, Poisson’s ratio, 1, 12–13, 285, 295, 314, 342, 154–155, 189, 206, 216, 356, 364–365, 427, 431, 466 394–397, 399, 401, 403 Poling direction, 385, 387 Nodal displacement, 552, 557 Positive definite, 12, 54–56, 60, 62, 64, 119, point, 552–553, 559 188, 190, 221, 251, 312, 474 traction, 552 Positive semi-definite, 62 Non-permittivities, 370, 560 Potential energy, 266 Normalization factor, 57, 69, 389–390, 423 Principal axes of inertia, 40, 48 Principal material direction, 11, 14, 18, 20–22 O Principal moments of inertia, 40 Open circuit, 372, 484 Pseudo-inverse, 120 Opening mode (or tensile mode), 189 Punch Orthogonality relation, 57, 63–64, 85–86, 94, flat-ended, 283–287, 289, 299, 306, 314, 105, 377, 384, 387, 389, 507, 541, 560 318–322 Oscillatory characteristics, 140, 207, 226, 289 parabolic, 278, 287–288, 314–319 Oscillatory index, 191, 221–223, 295, 300, 403 rigid, 277–290, 299–309, 318, 332 sliding, 277–278, 299, 308–321, 326 P wedge-shaped, 312–317 Parabolic elastic bodies, 327, 329–330 Pure in-plane bending, 87, 91–92, 159, Particular solution, 34, 49, 115, 124, 335, 184–185 415–416, 484, 503 Pure shear, 12–13, 90, 162, 510–511 672 Subject Index
R engineering shear, 4, 22 Reciprocal theorem of Betti and Rayleigh, 140, Eulerian, 4 147, 152 Lagrangian, 4, 7 Reference length, 145, 157 normal, 4, 18, 22 Residue theorem, 253 shear, 4, 14, 22–23, 372, 441, 447, 565 Rigid body Strain energy, 8, 12, 34, 55–56, 64, 132, 141, motion, 4, 102, 120, 123, 162, 175, 258, 193, 201, 218, 356, 377, 475 260–261, 350, 517, 532, 586 Strength of stress singularity, 189, 320 rotation, 118, 120, 248, 256 Stress translation, 32, 48, 89, 175, 312 concentration, 159, 165, 184, 206, 337, Root 344, 493 double, 141, 143, 155, 395 contour, 297 non-repeated, 141 expanded, 369, 374–379, 394 triple, 60–61, 84, 141, 143, 155, 395 expanded Stroh-like formalism, 484–491, Rotated coordinate system, 59, 83, 336 493 extended Stroh formalism, 333–337, S 348, 375 Scaling factor, 383–384, 387, 389, 407, 468 extended Stroh-like formalism, 476–481, Sectionally holomorphic, 279, 292, 301, 305, 498 307, 310, 331 function, 29, 33, 43–45, 55–59, 65–67, Semi-inverse method, 174 141, 172, 189, 270, 374, 457, 501, 530 Sensors and actuators, 369, 481 intensity factor, 115, 139–147, 152–158, Shearing mode (or sliding mode), 189 196, 201–207, 226, 273, 345, 396–398, Shear modulus, 60, 71, 221, 229, 314, 431 555–557, 562–564 Short circuit, 372, 376, 385, 505 normal, 2, 13, 18, 22, 311 Sign convention, 414 principal, 2, 396 Sign function, 271 resultant, 58, 92, 436–438, 446, 462–464, Single-valued displacement, 4, 33, 92, 100, 485, 497–498, 522–523, 570 129, 169, 270, 275, 507, 510 shear, 2, 10, 21–22, 205, 372, 483, 510 Singular singularity, 115, 124, 135, 155, 189, 207, characteristics, 187–192, 370, 320, 348, 362, 365, 525 399–404 –strain relation, 1, 8–10, 16–18, 20–21, 47, field, 141, 148–149, 153 88–89, 162 integral, 270–273, 275, 553 Stroh formalism integral equation, 270–273 Stroh-like bending formalism, 411–412, order, 133–134, 137–143, 145–146, 420–429, 435, 454, 457, 504 154–155, 187–188, 190–191, 356–357, Stroh-like formalism, 411, 436, 451, 366–367, 394–395, 402–403 458–474, 476–481, 484–491, 507, 581 point, 101, 212, 240, 242, 246, 250, 259, St. Venant’s principle, 46 263, 528, 534, 536 Sub-inverse, 119, 346, 501 Smooth boundary, 545, 575, 581, 586 Subregion technique, 557–559 Source point, 575, 577, 579, 588 Superposition method, 95, 101 Specimen fabrication, 228–229 Surface Stacking sequence, 23, 236 deformation, 104, 282–284, 286, 288, Stiffness 311–312, 315, 317, 319–321, bending, 25, 413, 421, 427, 469–471, 329–330 489, 541 integral, 153, 565–566, 570 coupling, 412, 421, 439–440, 452, 454, moment, 418, 462 460, 465–466, 469, 495 traction, 6, 29, 40, 58, 94, 100–101, 121, extensional, 25, 437, 452, 473, 495, 541 123, 172–173, 208, 239, 291, 377–378, reduced, 17, 19 462, 545 Strain System of equation, 5, 382, 509, 553–554, Cauchy’s infinitesimal, 4 558–560 Subject Index 673
T U Taylor’s series expansion, 117 Unidirectional tension, 90, 162, 165, 183 Tearing mode (or antiplane mode), 189, 207, Uniform 226, 229 heat flow, 337–339, 342–348, 493, 498–503 Technical constant, 12 loading, 87–91, 161–165, 170, 177, Temperature field, 333, 350, 353, 357, 362, 365 182–186, 245–247, 255, 268, 338, Thermal 354, 493 eigenvalue, 334, 337, 342–343 stress solution, 89, 116–117, 122, 124 eigenvector, 342, 351 traction, 115–124, 213 expansion coefficient, 342 Unperturbed elastic field, 101, 240, 244–245, moduli, 334, 352, 362 247, 251–253, 517, 528–529 properties, 362, 365, 367 stress, 337, 344–345, 348 V Thermoelasticity, 333, 335, 349, 362, 375, Virtual crack closure method, 193, 218, 344 474–475, 477–478 Tilted angle, 287 Torsion W generalized, 50–51 Wavy-shaped surface, 299, 306 generalized torsional rigidity, 50 Wedge pure torsion deformation, 50 angle, 115, 117–125, 127, 132, 188, 191, Transfer matrix, 132, 135, 360 356, 365–367, 399 Transformation apex, 115, 121, 125–134, 137–138, 145, function, 159–160, 178–182, 240, 246, 356–357, 394, 396, 398 257, 295–296, 302–303 bi-, 132, 138–139, 144, 190, 366–367, 401 law, 4, 58, 414, 417–418, 420, 463, critical wedge angle, 117–123, 127–128 505, 580 insulated, 363, 366 matrix, 271 isothermal, 363, 366–367 Translating technique, 102–103, 111, 175, 209, multi-material, 115, 128–132, 134–137, 211, 244, 253, 265, 283, 295–296, 352, 356–367, 393–398 518, 520, 522, 540–541 multi-material wedge space, 115, 128–132, Transverse direction, 19–20, 474 134–137, 361–362, 365–366 Transverse loading, 411, 493 non-critical wedge angle, 117–122 Transverse shear, 23, 71, 87, 372, 413–414, tri-, 132, 139, 367 416–417, 419, 442–443, 462–463, 569 Twisting moment, 39–40, 47–51, 428, 431 Y Two-dimensional deformation, 29, 53–55, Young’s modulus, 71, 221, 229, 326, 342, 67, 435 427, 466