A c,a f,a m E c.e f,e m E e E s G f,g m L M x, M y, M xy M H N H N x, N y, N xy P c,p f,p m Q S S ijkl T T V V v V crit W h k t c,t f,t m u 0 v c,v f,v m w c,w f,w m cross-sectional area of composite,fiber and matrix material elastic modulus of composite,fiber and matrix material transverse modulus elastic modulus of longitudinal direction elastic modulus of tranverse direction effective modulus secondary modulus in-plane shear modulus shear modulus longitudinal direction moment per unit length hygroscopic moment hygroscopic force force per unit length load carried by the composite,fiber and matrix material stiffness matrix strength reduction factor compliance tensor transverse direction axis perpendicular to the L and T axes volume fraction volume fraction of void A critical fiber volume fraction weight fraction thickness plate curvature thickness of composite,fiber and matrix material displacement in x direction volume of composite,fiber and matrix material weight of composite,fiber and matrix material
α slope of the laminate midplane in the x direction α L, α T coefficient of tharmal expansion in longitudinal and transverse direction α xy apparent coefficient of thermal expansion β L, β T coefficient of moisture in longitudinal and transverse direction β xy apparent coefficient of moisture δ c, δ f, δ m elongation of the composite,fiber and matrix material ɛ c, ɛ f, ɛ m strains experienced by the composite,fiber and matrix material ɛ mb,ɛ cb,ɛ fb, braking strain of composite,fiber and matrix material (ε T ) c, (ε T ) f, (ε T ) m transverse strain of composite,fiber and matrix material ε T thermal strain ε H hygroscopic strain ε M mechanical strain ε 0 midplane strain λ c, λ f, λ m shear strain of composite,fiber and matrix material ν c, ν f, ν m poisson ratio of composite,fiber and matrix material ν LT major Poisson s ratio ν T L minor Poisson s ratio ν L, ν T Poissons ratio on longitudinal and transverse direction ρ c density of the composite material ρ f density of the fiber material ρ m density of the matrix material ρ ct theoretical composite density ρ ce experimentally determined density σ c, σ f, σ m stresss of composite,fiber and material σ cb, σ fb, σ mb breaking stress of composite,fiber and material σ cu longitudinal strength of the composite σ fu ultimate strength of the fiber σ mu ultimate strength of the matrix (σ m ) ε matrix stress at the fiber fracture strain ε f f σ T U composite tranverse strength σ A gross stress σ F fracture stress τ f, τ c, τ m shearing stress of fiber, composite and matrix material dσ dɛ slope of the corresponding stress-strain curve at the given strain c, f, m shear deformation T change in temperature C change in moisture 2
CHAPTER-2 Table 2. Typical Composition of E-glass and s-glass fibers % weight % weight Material E-Glass S-Glass Silicon oxide 54.3 64.20 Aluminum oxide 5.2 24.80 Ferrous oxide - 0.2 Calcium oxide 7.2 0.0 Magnesium oxide 4.7 0.27 Sodium oxide 0.6 0.27 Boron oxide 8.0 0.0 Barium oxide - 0.20 Miscellaneous - 0.03 Table 2.2 Properties of E-glass and S-glass fibers Property,units E-Glass S-Glass Density,g/cm 3 2.54 2.49 Tensile Strength, a Mpa 3448 4585 Elastic modulas, Gpa 72.4 85.5 Range of diameter,µm 3-20 8-3 Coeffecient of thermal expansion, 0 6 / 0 C 5.0 2.9 Table 2.3 Properties of graphite fibers Property,units pitch Rayon PAN Tensile Strength,Mpa 550 2070-2760 2480-300 Tensile modulas, Gpa 380 45-550 200-345 Specific gravity 2.0.7.8 Elongation - 0.6-.2 Coeffecient of thermal expansion Axial (0 6 / 0 C) Axial (0 6 / 0 C) -.6 to -0.9 - -0.7to-0.5 Transverse(0 6 / 0 C) 7.8-7-0 Fiber diameter,µm 0-6.5 7.5 Table 2.4 typical properties of Kevlar fibers
Property,units Kevlar 29 Kevlar 49 diameter,µm 2 2 Density,g/cm 3.44.44 Tensile Strength,Mpa 2760 3620 Tensile modulas, Gpa 62 24 Tesile Elongation,% 3-4 2-8 Coeffecient of thermal expansion (0-00 0 C), m/m/ 0 C In Axial direction -2 0 6-2 0 6 In radial direction 60 0 6 60 0 6 Table 2.5 Properties of Boron fiber (with tungsten,core) Property,units 00 µm 40 µm 200 µm Ultimate Tensile strength,mpa 3450 3450 3450 Modulas,Gpa 400 400 400 Coeffecient of thermal expansion,m/m 0 C 4.9 0 6 4.9 0 6 4.9 0 6 Density,g/Cm 3 2.6 2.47 2.39 Table 2.6 Properties of ceramic fibers Fiber Fiber Fiber Property,units Alumina(fiber FP) SiC(CVD) SiC(pyrolysis) Diameter,µm 20±5 40 0-20 Density,g/Cm 3 3.95 3.3 2.6 Tensile strength,mpa 380 3500 2000 Modulas,Gpa 379 430 80 Table 2.0 Typical properties of cast thermosetting polyesters Density,g/Cm 3.-.4 Tensile strength,mpa 34.5-03.5 Tensile Modulas,Gpa 2-4.4 Thermal expansion,0 6 / 0 C 55-00 Water absorption,% in 24 h 0.5-0.6 Table 2. typical properties of cast epoxy resins(at 23 0 C) Density,g/Cm 3.2-.3 Tensile strength,mpa 55-30 Tensile Modulas,Gpa 2.75-4.0 Thermal expansion,0 6 / 0 C 45-65 Water absorption,% in 24 h 0.08-0.5 Table 2.2 Typical properties of polyimides and phenolics 2
Property,units Phenolics polyimide Density,g/Cm 3.30.46 Tensile strength,mpa 50-55 20 Flexural modulas,gpa - 3.5-4.5 Continuous service temperature, 0 C 50-75 260-425 Coeffecient of thermal expansion,0 6 / 0 C 45-0 90 Water absorption,% in 24 h 0.-0.2 0.3 Table 2.3 Typical properties of thermoelastic resins Property,units PEEK Polyamide-imide Polyetherimide Polysu Density,g/Cm 3.30.38 -.25 Tensile strength,mpa 92 95 05 75 Flexural modulas,gpa 4.0 5.0 3.5 2.8 Continuous service temperature, 0 C 30-70 75-9 Coeffecient of thermal expansion,0 6 / 0 C - 63 56 94-00 Water absorption,% in 24 h 0. 0.3 0.25 0.2 3
swarup α Chapter-3 3..2 Volume and weight fractions v c = v f + v m () V f = v f v c, V m = v m v c (2) and w c = w f + w m (3) w f = w f w c, W m = w m w c (4) ρ c v c = ρ f v f + ρ m v m (5) ρ c = ρ f v f v c + ρ m v m vc ρ c = ρ f V f + ρ m V m (6) ρ c = ( ) ( ) wf ρ f + Wm (7) ρm W f = w f w c = ρ f v f ρ c v c = ρ f ρ c W f = ρ f ρ c V f (8) W m = ρ m ρ c V m V f = ρ c ρ f W f
V m = ρ c ρ m W m (9) n ρ c = ρ i V i (0) i= ρ c = n ( i= ) W i ρ i W i = ρ i ρ c V i () V i = ρc ρ i W i V v = ρ ct ρ ce ρ ct (2) 3.2. Initial Behaviour ɛ f = ɛ m = ɛ c (3) P c = P f + P m (4) P c = σ c A c = σ f A f + σ m A m or σ c = σ f A f A c + σ m A m A c (5) V f = A f A c, V m = A m A c (6) Thus σ c = σ f V f + σ m V m (7) 2
dσ c dɛ = dσ f dɛ V f + dσ m dɛ V m (8) E c = E f V f + E m V m (9) n σ c = σ i V i (20) i= n E c = E i V i (2) i= 3.2.2 behaviour initial Deformation E c = E f V f + ( ) dσm dɛ m 3.2.3 Failure Mechanism and strength ɛ c V m (22) σ cu = σ fu V f + (σ m ) ε f ( V f ) (23) σ cu = σ mu ( V f ) (24) V min = σ mu (σ m ) ε f σ fu + σ mu (σ m ) ε f (25) σ cu = σ fu V f + (σ m ) ε f ( V f ) σ mu (26) V crit = σ mu (σ m ) ε f σ fu (σ m ) ε f (27) 3
3.3. constant-stress Model σ f = σ m = σ c (28) δ c = δ f + δ m (29) δ c = ɛ c t c δ f = ɛ f t f (30) δ m = ɛ m t m Substituting eq.(3.30) in eq(3.29) ɛ c t c = ɛ f t f + ɛ m t m (3) ɛ c = ɛ f t f t c + ɛ m t m t c = ɛ f V f + ɛ m V m (32) σ c E c = σ f E f V f + σ m E m V m (33) E c = V f E f + V m E m (34) E c = n (V i /E i ) i= 3.3.3 Helpin-Tsai Equation for Transverse Modulas (35) E m = + ξηv f ηv f (36) where η = (E f/e m ) (E f /E m ) + ξ (37) 4
In which ξ is a measure of reinforcement and depends o the fiber geomtry ξ = 2 a b (38) a/b is the rectangular cross section aspect ratio 3.3.4.2 prediction of tranverse strength σ T U = σ mu S σ T U =composite tranverse strength (39) σ mu =matrix ultimate strength SCF = stressconcentrationf actor = V f [ (E m /E f )] (4V f /π) 2 [ (Em /E f )] (40) SMF = stressmagnificationfactor = (4V f /π) 2 [ (Em /E f )] (4) S = (U max) /2 σ c (42) ( ) /3 ɛ CB = ɛ mb V f (43) 3.4 prediction of shear modulas τ f = τ m = τ c (44) c = f + m (45) c = γ c t c f = γ f t f (46) 5
m = γ m t m γ c t c = γ f t f + γ m t m (47) γ c = γ f t f t c + γ m t m t c = γ f V f + γ m V m (48) τ c = τ f G f V f + τ m G m V m (49) = V f G f + V m G m (50) = G f G m G m V f + G f V m (5) G m = + ξηv f ηv t (52) W here η = (G f/g m ) (G f /G m ) + ξ (53) where is the in-plane shear modulas of the composite and G f and G m is the shear modulas of fiber and matrix. 3.5 prediction of poisson s ratio (ε T ) f = ν f (ε L ) f (ε T ) m = ν m (ε L ) m (54) (ε T ) c = ν c (ε L ) c δ f = t f (ε T ) f = t f ν f (ε L ) f δ m = t m (ε T ) m = t m ν m (ε L ) m (55) 6
δ c = t c (ε T ) c = t c ν c (ε L ) c t c ν LT (ε L ) c = t f ν f (ε L ) f t m ν m (ε L ) m (56) t c ν LT = t f ν f + t m ν m (57) ν LT = ν f V f + ν m V m (58) ν LT = ν LT (59) Table 3. Typical properties of unidirectional-fiber reinforced epoxy resins Fiber type Fiber type Fiber type Property E-Glass Kevlar 49 Graphite(Thornel 300) Fiber volume fraction 46 60-65 63 specific gravity.80.38.6 Tensile strength,0 0 (Mpa) 04 30 725 Tensile modulas,0 0 (Gpa) 39 83 59 Tensile strength,90 0 (Mpa) 36 39 42 Tensile modulas,90 0 (Gpa) 0 5.6 0.9 Compression strength,0 0 (Mpa) 600 286 366 Compression modulas,0 0 (Gpa) 32 73 38 Compression strength,90 0 (Mpa) 38 38 230 Compression modulas,90 0 (Mpa) 8 5.6 In-plane shear strength (Mpa) - 60 95 In-plane shear modulas (Mpa) - 2. 6.4 Longitudinal poisson ratio(ν LT ) 0.25 0.34 0.38 Interlaminar shear strength (Mpa) 3 69 3 Longitudinal coeff,of th.exp(0 6 / 0 C) 5.4-2.3-4.0 a 0.045 Transverse coeff,of th.exp(0 6 / 0 C) 36 35 b 20.2 a 79 0 C to + 00 0 C b 95 0 C to + 20 0 C Table 3.2 7
Composite property Fiber Matrix Interface Tensile property Longitudinal modulas S W N Longitudinal strength S W N Transverse modulas W S N Transverse strength W S S Compression property Longitudinal modulas S W N Longitudinal strength S S N Transverse modulas W S N Transverse strength W S N Shear properties In-plane shear modulas W S N In-plane shear strength W S S Interlaminar shear strength N S S a S =strong influence; w=weal influence; N=negligible influence 8
Chapter-5 5.2 Hook s Law for orthotropic material σ ij = E ijkl ɛ kl () E ijkl = E ijlk (2) E ijkl = E jikl (3) U = U (ɛ ij ) (4) with the property U ɛ ij = σ ij (5) ɛ kl U ɛ ij = E ijkl ɛ kl (6) ( ) U = E ijkl (7) ɛ ij ɛ ij ( ) U = E klij (8) ɛ kl ɛ ij ( ) U ɛ kl = ( ) U ɛ kl ɛ ij it is clear that (9) E ijkl = E klij (0)
E mnrs = a im a jn a kr a ls E ijkl () where E mnrs is the elasticity tensor in the transformed (x ) axis system, E ijkl is the elasticity tensorin the original (x) axis system x = x ; x 2 = x 2, x 3 = x 3 (2) x x 2 x 3 x a = a 2 = 0 a 3 = 0 x 2 a 2 = 0 a 22 = a 23 = 0 x 3 a 3 = 0 a 32 = 0 a 33 = (3) E = E ijkl a i a j a k a l = E E 2 = E ijkl a i a j a k a l2 = E 2 (4) E 3 = E ijkl a i a j a k a l3 = E 3 E 3, E 2223, E 23, E 223, E 23, E 223, E 333, E 2333, (5) x = x ; x 2 = x 2, x 3 = x 3 (6) x x 2 x 3 x a = a 2 = 0 a 3 = 0 x 2 a 2 = 0 a 22 = a 23 = 0 x 3 a 3 = 0 a 32 = 0 a 33 = (7) E 233, E 323, E 222, E 2 (8) (E ijkl ) = E E 22 E 33 0 0 0 E 22 E 2222 E 2233 0 0 0 E 33 E 2233 E 3333 0 0 0 0 0 0 E 2323 0 0 0 0 0 0 E 33 0 0 0 0 0 0 E 22 (9) 2
σ i = Q ij ɛ j i, j =, 2, 3, 4, 5, 6 (20) σ σ 2 σ 3 τ 23 τ 3 τ 2 = Q Q 2 Q 3 0 0 0 Q 2 Q 22 Q 23 0 0 0 Q 3 Q 23 Q 33 0 0 0 0 0 0 Q 44 0 0 0 0 0 0 Q 55 0 0 0 0 0 0 Q 66 ɛ ɛ 2 ɛ 3 γ 23 γ 3 γ 2 (2) σ σ 2 τ 2 = Q Q 2 0 Q 2 Q 22 0 0 0 Q 33 ɛ ɛ 2 γ 2 (22) ɛ ij = S ijkl σ kl (23) ɛ ɛ 2 γ 2 = S S 2 0 S 2 S 22 0 0 0 S 33 σ σ 2 τ 2 (24) Q = S 22 S S 22 S2 2 Q 22 = S S S 22 S2 2 S 2 Q 2 = S S 22 S2 2 (25) Q 66 = S 66 5.3 Stress strain relation and enginnering constants 5.3. Specially orthotropic lamina ε L = σ L (26) ε T = ν LT ε L = ν LT σ L (27) 3
γ LT = 0 (28) ε T = σ T (29) ε L = ν T L ε T = ν LT σ T (30) γ LT = 0 (3) ε L = 0 (32) ε T = 0 (33) γ LT = γ τ LT (34) ε L = σ L ν T L σ T ε T = σ T ν LT σ L (35) γ LT = τ LT 5.3.2 Relations betwwen Engineering constant and elements of stiffness σ L = Q ɛ L + Q 2 ɛ T σ T = 0 = Q 2 ɛ L + Q 22 ɛ T (36) ɛ L = Q 22 σ Q Q 22 Q 2 L (37) 2 4
Q 2 ɛ T = σ Q Q 22 Q 2 L (38) 2 = σ L ɛ L = Q Q 22 Q 2 2 Q 22 (39) ν LT = ɛ T ɛ T = Q 2 Q 22 (40) = σ T ɛ T = Q Q 22 Q 2 2 Q (4) ν T L = ɛ L ɛ T = Q 2 Q 22 (42) = τ LT γ LT = Q 66 (43) Q = ν LT ν T L Q 22 = ν LT ν T L (44) Q 2 = ν LT ν LT ν T L = ν T L ν LT ν T L Q 66 = ν LT = ν T L or ν LT = ν T L (45) S = S 22 = 5
S 2 = ν LT = ν T L (46) S 66 = 5.3.3 Restriction on Elastic constants G = E 2 ( + ν) (47) = = ν LT = ν LT (48) G T T = 2(+ν T T ),,,,, G T T > 0 (49) ( ν LT ν T L ), ( ν LT ν T L ), ( ν T T ν T T ) > 0 (50) ν LT ν T L ν LT ν T L 2ν T T ν T T ν T L > 0 (5) ν LT < ( ) /2, ν T L < ( ) /2 ν LT < ( EL ) /2 ( ) /2, ν T L < ET (52) ( ) /2 ( ) /2 E ν T T < T, ν E T T < ET T 5.3.4 Stress-strain relation for genarally orthotropic lamina σ L σ T τ LT = [T ] σ x σ y τ xy (53) 6
and ɛ L ɛ T 2 τ LT = [T ] ɛ x ɛ y 2 τ xy (54) where the transformation matrix [T] is given by [T ] = cos 2 θ sin 2 θ 2 cos θ sin θ sin 2 θ cos 2 θ 2 cos θ sin θ cos θ sin θ cos θ sin θ cos 2 θ sin 2 θ (55) σ x σ y τ xy = [T ] σ L σ T τ LT (56) [T ] = cos 2 θ sin 2 θ 2 cos θ sin θ sin 2 θ cos 2 θ 2 cos θ sin θ cos θ sin θ cos θ sin θ cos 2 θ sin 2 θ (57) σ L σ T τ LT = Q Q 2 0 Q 2 Q 22 0 0 0 2Q 66 ɛ L ɛ T 2 γ LT (58) σ x σ y τ xy = [T ] σ x σ y τ xy = Q Q 2 0 Q 2 Q 22 0 0 0 2Q 66 Q Q2 Q6 Q 2 Q22 Q26 Q 6 Q26 Q66 [T ] ɛ x ɛ y γ xy ɛ x ɛ y 2 τ xy (59) (60) Q = Q cos 4 θ + Q 22 sin 4 θ + 2 (Q 2 + 2Q 66 ) sin 2 θ cos 2 θ Q 22 = Q sin 4 θ + Q 22 cos 4 θ + 2 (Q 2 + 2Q 66 ) sin 2 θ cos 2 θ Q 2 = (Q + Q 22 4Q 66 ) sin 2 θ cos 2 θ + Q 2 ( cos 4 θ + sin 4 θ ) Q 66 = (Q + Q 22 2Q 2 2Q 66 ) sin 2 θ cos 2 θ + Q 66 ( sin 4 θ + cos 4 θ ) 7
Q 6 = (Q Q 2 2Q 66 ) cos 3 θ sin θ (Q 22 Q 2 2Q 66 ) cos θ sin 3 θ Q 26 = (Q Q 2 2Q 66 ) cos θ sin 3 θ (Q 22 Q 2 2Q 66 ) cos 3 θ sin θ (6) ɛ x ɛ y γ xy = S S2 S6 S 2 S22 S26 S 6 S26 S66 σ x σ y τ xy (62) S = S cos 4 θ + S 22 sin 4 θ + (2S 2 + S 66 ) sin 2 θ cos 2 θ S 22 = s sin 4 θ + S 22 cos 4 θ + (2S 2 + S 66 ) sin 2 θ cos 2 θ S 2 = (S + S 22 S 66 ) cos 2 θ sin 2 θ + S 2 ( cos 4 θ + sin 4 θ ) S 66 = 2 (2S + 2S 22 4S 2 S 66 ) cos 2 θ sin 2 θ + S 66 ( cos 4 θ + sin 4 θ ) S 6 = (2S 2S 2 S 66 ) cos 3 θ sin θ (2S 22 2S 2 S 66 ) cos θ sin 3 θ S 26 = (2S 2S 2 S 66 ) cos θ sin 3 θ (2S 22 2S 2 S 66 ) cos 3 θ sin θ (63) 5.3.5 Transformation of Engineering constant σ L = σ x cos 2 θ σ T = σ x sin 2 θ (64) τ LT = σ x sin θ cos θ the strain in the L and T directions ( are given by ) Eq. (5.35) ɛ L = σ L σ ν T LT = σ cos 2 θ sin x ν 2 θ T L ɛ L = σ ( T σ L sin 2 θ cos 2 ) θ ν LT = σ x ν LT (65) γ LT = τ LT σx sin θ cos θ = ɛ x = ɛ L cos 2 θ + ɛ T sin 2 θ γ LT sin θ cos θ ɛ y = ɛ L sin 2 θ + ɛ T cos 2 θ + γ LT sin θ cos θ (66) ( γ xy = 2(ɛ L ɛ T ) sin θ cos θ + γ LT cos 2 θ sin 2 θ ) Substituting of Equ.(5.65) in (5.66) gives the strains 8
[ ( ɛ x = σ cos 4 θ x + sin4 θ + 4 2ν LT ) sin 2 2θ ] [ νlt ɛ y = σ x ( + 2ν LT + ) ] sin 2 2θ 4 γ xy = σ x sin 2θ [ ν LT + 2 cos 2 θ ( + 2ν LT + )] (67) since E x = σ x ɛx = cos4 θ + sin4 θ + ( 2ν ) LT sin 2 2θ (68) E x 4 = sin4 θ + cos4 θ + ( 2ν ) LT sin 2 2θ (69) E y 4 ν xy = ɛ y ɛ x ν xy = ν LT ( + 2ν LT + ) sin 2 2θ (70) E x 4 similarly ν xy E y = ν LT ( + 2ν LT + ) sin 2 2θ (7) 4 γ xy = m x σ x (72) [ m x = sin 2θ ν LT + E ( L cos 2 θ + 2ν LT + E )] L 2 (73) γ xy = m y σ y (74) [ m y = sin 2θ ν LT + E ( L sin 2 θ + 2ν LT + E )] L 2 (75) 9
σ L = σ T = 2τ xy sin θ cos θ τ LT = ( cos 2 θ sin 2 θ ) τ xy (76) ɛ L = 2τ xy sin θ cos θ ( ) + 2ν T L ( ɛ L = 2τ xy sin θ cos θ + ν ) LT ν LT = τ ( xy cos 2 θ sin 2 θ ) [ γ xy = τ xy + 2ν LT + ( + 2ν LT + ) ] cos 2 2θ (77) (78) Now the defination of shear modulas G xy will give = + 2ν LT + ( + 2ν LT + ) cos 2 2θ (79) G xy ɛ x = m x τ xy ɛ y = m y τ xy (80) ɛ x = σ x E x ν yx σ y E y m x τ xy ɛ y = σ y E y ν xy σ x E x m y τ xy (8) γ xy = τ xy G xy m x σ x m y σ y > 2 ( + ν LT ) (82) < 2 [ / + ν LT ] (83) 5.4 Strengths of an orthotropic lamina 0
5.4. Maximum stress theory σ L < σ LU σ T < σ T U (84) τ LT < τ LT U σ L < σ LU σ T < σ T U (85) σ L = σ x cos 2 θ σ T = σ x sin 2 θ (86) τ LT = σ x sin θ cos θ the maximum stress theory is applied to a typical glass-epoxy composite with the following normalized material properties σ T U τ σ LU = 0.025, LT U σ LU = 0.05 σ LU σ σ LU =, T U σ LU = 0.25 ν LT = 0.25, ν T L =0.08 5.4.2 Maximum strain theory ɛ L < ɛ LU ɛ T < ɛ T U (87) γ LT < γ LT U ɛ L < ɛ LU ɛ T < ɛ T U (88) ɛ LU = σ LU ɛ T U = σ T U (89) γ LT U = γ LT U
ɛ L = ( cos 2 θ ν LT sin 2 θ ) σ x ɛ T = ( sin 2 θ ν T L cos 2 θ ) σ x (90) γ LT = (sin θ cos θ) σ x 5.4.3 Maximum work theory ( ) σl 2 ( ) ( ) σl σt σ LU σ LU σ LU + ( ) σt 2 ( τlt + σ T U τ LT U ) 2 < (9) cos 4 θ σ 2 LU cos2 θ sin 2 θ σ 2 LU + sin4 θ σ 2 T U + cos2 θ sin 2 θ σ 2 LT U < σ 2 x (92) 2