Equation (4.7.34):
DELTA eta = DELTA eta sub 0 = ( Z over{S sub 1}) sup m
Equation (4.7.35):
DELTA eta mark = 1 over{S sub 1 sup m} "{" psi sup m Z sup m + (1 - psi ) sup m Z sup m (e sup{- alpha T/2} + e sup{- alpha T}) sup m [ 1 + e sup{- alpha Tm} + e sup {-2 alpha T m} + cdot cdot cdot ] "}" lineup = ( Z over{S sub 1} ) sup m "{" psi sup m + (1 - psi ) sup m {(1 + e sup{- pi lambda}) sup m}over{2 sinh pi lambda m} "}"
Equation (4.7.36):
DELTA eta = ( Z over{S sub 1} ) sup m "{" psi sup 3 + 15 (1 - psi ) sup 3 "}"
Section 4.7.4 - Natural Dispersion.
Equation (4.7.37):
DELTA eta sub 1 , DELTA eta sub 2 , DELTA eta sub 3 , cdot cdot cdot DELTA eta sub j cdot cdot cdot
Equation (4.7.38):
eta (t) = eta sub n = DELTA eta sub 1 + DELTA eta sub 2 + DELTA eta sub 3 + cdot cdot cdot + DELTA eta sub n
Equation (4.7.39):
xi = 1 over{N(S)} = ( S over{S sub 1}) sup m = r S sup m roman with r = S sub 1 sup -m
Equation (4.7.40):
f( xi ) = g(d, k over m , rD sup m ; xi )
Equation (4.7.41):
xi bar = M sub 1 ( xi ) = int from 0 to inf xi f( xi ) d xi = r D sup m {GAMMA (d + m over k )}over{GAMMA (d)} = TU
Equation (4.7.42):
M sub 2 ( xi ) = int from 0 to inf xi sup 2 f( xi ) d xi = (r D sup m ) sup 2 {GAMMA (d + 2m over k}over{GAMMA (d)} = TV
Equation (4.7.43):
M sub 3 ( xi ) = int from 0 to inf xi sup 3 f( xi ) d xi = (r D sup m ) sup 3 {GAMMA (d + 3m over k}over{GAMMA (d)} = TW
Equation (4.7.44):
U = {xi bar}over T = {M sub 1 ( xi )}over T V = {M sub 2 ( xi )}over T W = {M sub 3 ( xi )}over T
Equation (4.7.45):
mu sub 2 ( xi ) = sigma sub xi sup 2 = M sub 2 ( xi ) - M sub 1 sup 2 ( xi ) = nu sup 2 xi bar sup 2 roman where
nu sup 2 = ( {sigma sub xi}over{xi bar} ) sup 2 = {GAMMA (d + 2m over k ) GAMMA (d) - GAMMA (d + m over k ) sup 2}over {GAMMA (d + m over k ) sup 2}
Equation (4.7.46):
mu sub 3 ( xi ) = M sub 3 ( xi ) - 3M sub 2 ( xi ) M sub 1 ( xi ) + 2M sub 1 ( xi ) sup 3 = lambda sigma sub xi sup 3 = lambda nu sup 3 xi bar sup 3 roman where lambda = {GAMMA (d + 3m over k ) GAMMA (d) sup 2 -
3 GAMMA (d + 2m over k ) GAMMA (d) GAMMA (d + m over k ) + 2 GAMMA (d + m over k ) sup 3}over
{[ GAMMA (d + 2m over k ) GAMMA (d) - GAMMA (d + m over k ) sup 2 ] sup 3/2}
Equation (4.7.47):
phi (s) = int from 0 to inf e sup{s xi} f( xi ) d xi Re "{" s "}" < 0
Equation (4.7.48):
phi (s) = int from 0 to inf [ 1 + s xi + 1 over 2 s sup 2 xi sup 2 + 1 over 6 s sup 3 xi sup 3 + cdot cdot ] f( xi ) d xi
Equation (4.7.49):
phi (s) = 1 + M sub 1 ( xi ) s mark + 1 over 2 M sub 2 ( xi ) s sup 2 + 1 over 6 M sub 3 ( xi ) s sup 3 + cdot cdot
lineup = 1 + T U s + 1 over 2 T V s sup 2 + 1 over 6 T W s sup 3 + cdot cdot
Equation (4.7.50):
PHI (s, t) = int from 0 to inf e sup{s eta } rho ( eta , t) d eta Re "{" s "}" < 0
Equation (4.7.51):
eta (t + T) = eta sub n+1 = eta sub n + xi
Equation (4.7.52):
PHI (s, t+ T ) = PHI (s, t) phi (s)
Equation (4.7.53):
{partial PHI (s, t)}over{partial t} = 1 over T [ PHI (s, t + T ) - PHI (s, t) ]
Equation (4.7.54):
int from 0 to inf e sup{s eta} {partial rho ( eta , t)}over{partial t} d eta mark = 1 over T PHI (s, t) [ phi (s) - 1 ]
lineup = U s PHI (s, t) + 1 over 2 V s sup 2 PHI (s, t) + 1 over 6 W s sup 3 PHI (s, t)
Equation (4.7.55):
int from 0 to inf e sup{s eta} [ {partial rho}over{partial t} + U{partial rho}over{partial eta} - 1 over 2 V{partial sup 2 rho}over{partial eta sup 2} + 1 over 6 W{partial sup 3 rho}over{partial eta sup 3} ] d eta - [ e sup{s eta} "{"
rho (U + 1 over 2 sV + 1 over 6 s sup 2 W) - {partial rho}over{partial eta}( 1 over 2 V + 1 over 6 sW) + {partial sup 2 rho}over{partial eta sup 2}1 over 6 W "}" ] from{eta = 0} to {eta = inf} = 0
Equation (4.7.56):
{partial rho}over{partial t} + U{partial rho}over{partial eta} - 1 over 2 V {partial sup 2 rho}over{partial eta sup 2} + 1 over 6 W {partial sup 3 rho}over{partial eta sup 3} = 0
Equation (4.7.57):
{eta sub n}bar mark = sum{DELTA eta}bar = n xi bar
Equation (4.7.58):
mu sub 2 ( eta sub n ) lineup = sum mu sub 2 ( DELTA eta sub i ) = n cdot sigma sub xi sup 2 = n nu sup 2 xi bar sup 2
Equation (4.7.59):
mu sub 3 ( eta sub n ) lineup = sum mu sub 3 ( DELTA eta sub i ) = n lambda sub 3 sigma sub xi sup 3 = n lambda nu sup 3 xi bar sup 3
Equation (4.7.60):
{sigma sub {eta sub n}}over{{eta sub n}bar} = {sqrt{mu sub 2 ( eta sub n )}}over{{eta sub n}bar} = nu over sqrt n
Equation (4.7.61):
lambda sub 3 = {mu sub 3 ( eta )}over{mu sub 2 ( eta ) sup 3/2} = lambda over sqrt n
Equation (4.7.62):
rho ( eta , t) = |h| over{GAMMA (a)} e sup{ah( eta - u )} e sup{-e sup{h( eta - u )}}
Equation (4.7.63):
{| psi '' (a) |}over{psi ' (a) sup 3/2} = {lambda sub 3}over sqrt n
Equation (4.7.64):
h = \(+- {sqrt{psi '(a)}}over {sqrt n sigma sub xi} + for lambda sub 3 < 0 and - for lambda sub 3 > 0
Equation (4.7.65):
u = n{DELTA eta}bar - 1 over h psi (a) = n xi bar + sqrt n sigma sub xi {psi (a)}over{sqrt{psi ' (a)}}
Equation (4.7.66):
a mark approx n over{lambda sup 2}
Equation (4.7.67):
h lineup approx - n lambda over{sigma sub xi}
Equation (4.7.68):
u lineup approx n "{" xi bar - {sigma sub xi}over lambda ln [ n over{lambda sup 2a} ] "}"
Equation (4.7.69): (xxx)
rho ( eta , t) = 1 over sqrt{2 pi n} 1 over{sigma sub xi} e sup{- {( eta - n xi bar ) sup 2}over{2 n sigma sub xi sup 2}} t = n T
Equation (4.7.70):
j = eta over L roman or eta = j L
Equation (4.7.71):
Pr ( eta = j L ) = Pr (j; n) = ( cpile{n above j} ) p sup j (1 - p) sup n-j n \(>= j
Equation (4.7.72):
p = (1 - p) = 1 over 2
Equation (4.7.73):
Pr(j; n) = ( cpile{n above j} ) 1 over{2 sup n}
Equation (4.7.74):
{eta sub n}bar = L n p and sigma sub eta sup 2 = L sup 2 n p (1 - p)
Equation (4.7.75):
{sigma sub eta}over{eta bar} = 1 over sqrt n sqrt{{1 - p}over p}
Equation (4.7.76):
L = xi bar (1 + nu sup 2 ) and p = 1 over{1 + nu sup 2}
Equation (4.7.77):
L = {M sub 2 ( xi )}over{M sub 1 ( xi )} and p = {M sub 1 ( xi ) sup 2}over{M sub 2 ( xi )}
Section 4.7.5 - Fracture Mechanics Approach.
Equation (4.7.78):
sigma sub ij = R(r) THETA sub ij ( theta )
Equation (4.7.79):
R(r) = r sup {n over 2 - 1}
Equation (4.7.80):
sigma sub ij = K over sqrt{2 pi r} THETA sub ij ( theta )
Equation (4.7.81):
sigma sub ij = sqrt{x over 2r} sigma sub inf THETA sub ij ( theta ) roman {so that} K = sqrt{pi x} sigma sub inf
Equation (4.7.82):
DELTA K = K sub max - K sub min
Equation (4.7.83):
DELTA x = left { lpile{ C( DELTA K ) sup m above above 0} for lpile{ DELTA K > DELTA K sub 0 above above DELTA K < DELTA K sub 0}
Equation (4.7.84):
DELTA K = sqrt{pi x} g'(x) S = g(x) S g(x) = g'(x) sqrt{pi x}
Equation (4.7.85):
DELTA x = left { lpile{ C g(x) sup m S sup m above above 0} for lpile{ S > S sub 0 (x) = {DELTA K sub }over{g(x)} above above S < S sub 0 (x)}
Equation (4.7.86):
DELTA x sub 1 , DELTA x sub 2 , DELTA x sub 3 , cdot cdot cdot DELTA x sub j cdot cdot cdot
Equation (4.7.87):
eta = {x - x sub 0}over{x sub f - x sub 0} and DELTA eta = {DELTA x}over{x sub f - x sub 0}
Equation (4.7.88):
{DELTA x}bar = C g(x) sup m int from{S sub 0} to inf S sup m f(S) dS = C g(x) sup m D sup m { GAMMA (d + m over k ; ({DELTA K sub 0}over{g(x) D}) sup k )} over{GAMMA (d)}
Equation (4.7.89): (xxx)
U = dx over dt = 1 over T dx over dN = {{DELTA x}bar}over T = 1 over T C D sup m {GAMMA (d + m over k }over{GAMMA (d)} g(x)
Equation (4.7.90):
Pr( roman{crack depth} \(<= x roman{at time} t) = F(x, t)
Equation (4.7.91):
Q(x, t) = 1 - F(x, t)
Equation (4.7.92):
rho (x, t sub 1 ) = {partial F(x, t sub 1 )}over{partial x} = - {partial Q(x, t sub 1 )}over{partial x}
Equation (4.7.93):
{partial Q}over{partial t} dt = -{partial Q}over{partial x} dx = -{partial Q}over{partial x} U(x) dt
Equation (4.7.94):
{D F(x, t)}over{D t} \(== ({partial F}over{partial t} + U {partial F}over{partial x}) = - ({partial Q}over{partial t} + U {partial Q}over{partial x}) = 0
Equation (4.7.95):
{partial rho}over{partial t} + {partial rho U}over{partial x} = {partial rho}over{partial t} + U {partial rho}over{partial x} + rho {partial U}over{partial x} = 0
Equation (4.7.96):
int from 0 to inf rho (x, t) dx = 1
Equation (4.7.97):
chi (x, t) = {partial Q(x, t)}over{partial t} = - {partial F(x, t)}over{partial t}
Equation (4.7.98):
chi (x, t) = U rho (x, t)
Equation (4.7.99):
{partial chi}over{partial t} + U {partial chi}over{partial x} = 0
Equation (4.7.100):
P sub f (t) = Q(x sub f , t) = 1 - F(x sub f , t)
Section 4.7.6 - Life-time Probability.
Equation (4.7.101):
Q(x, 0) = e sup{- ( x over{x sub 0} ) sup gamma} t = 0
Equation (4.7.102):
E[x] = x sub 0 GAMMA (1 + 1 over gamma ) t = 0
Equation (4.7.103):
sigma sub x = x sub 0 [ GAMMA (1 + 2 over gamma ) - GAMMA (1 + 1 over gamma ) sup 2 ] sup{1 over 2} t = 0
Equation (4.7.104):
Q(x, t) = Q( xi ) xi = xi (x, t) xi (x, 0) = x
Equation (4.7.105):
{partial Q}over{partial t} + U{partial Q}over{partial x} = ( {partial xi}over{partial t} + U{partial xi}over{partial x} ) {partial Q}over{partial xi} = 0
Equation (4.7.106):
U = 1 over T da over dN = dx over dt = - {partial xi / partial t} over{partial xi / partial x}
Equation (4.7.107):
xi = x - Ut U = 1 over T da over dN = roman constant
Equation (4.7.108):
P sub f (t) = Q (x sub f , t) = e sup{-({x sub f - Ut}over{x sub 0}) sup gamma} = e sup{-({x sub f /U - t}over{x sub 0 /U}) sup gamma} t < {x sub f}over U
Equation (4.7.109):
E[t] = 1 over U [ x sub f - x sub 0 GAMMA (1 + 1 over gamma ) ]
Equation (4.7.110):
sigma sub t = {sigma sub x}over U = {x sub 0}over U [ GAMMA (1 + 2 over gamma ) - GAMMA (1 + 1 over gamma ) sup 2 ] sup{1 over 2}
Equation (4.7.111):
da over dN = C x roman and U(x) = C over T x = cx
Equation (4.7.112):
xi = x e sup -ct
Equation (4.7.113):
P sub f (t) = Q(x sub f , t) = e sup{-({x sub f}over{x sub 0 e sup ct}) sup gamma} = e sup{-e sup{- gamma c ( t - 1 over c ln {x sub f}over{x sub 0})}}
Equation (4.7.114):
t sub c = 1 over c ln {x sub f}over{x sub 0}
Equation (4.7.115):
E[t] = 1 over c [ ln {x sub f}over{x sub 0} + 0.5772 over gamma ]
Equation (4.7.116):
sigma sub t = pi over sqrt 6 1 over{gamma c}
Equation (4.7.117):
{sigma sub t}over{t sub c} = pi over{sqrt 6 gamma ln {x sub f}over{x sub 0}}
Equation (4.7.118):(xxx)
da over dN = C x sup s roman or U = C over T x sup s = cx sup s s \(!= 1
Equation (4.7.119):
xi = [ x sup 1-s - (1 - s) ct ] sup{1 over 1-s}
Equation (4.7.120):
P sub f (t) = Q (x sub f , t) = e sup{-({x sub f sup 1-s - (1-s)ct}over {x sub 0 sup 1-s} ) sup{gamma over{(1-s)}}}
Equation (4.7.121):
t sub c = {x sub f sup 1-s - x sub 0 sup 1-s}over{(1 - s) c}
Equation (4.7.122):
{sigma sub t}over{t sub c} = { [ GAMMA (1 + 2(1-s) over gamma ) - GAMMA (1 + 1-s over gamma ) sup 2 ] sup 1/2} over{( x sub f / x sub 0 ) sup 1-s - 1} gamma > 2(s - 1)