Kinetic fracture models and structural reliability

Abstract

A kinetic model is postulated which considers the interaction of cumulative fatigue damage and chance overload on a component or structure under typical probabilistic service load histories. This model recognizes that:[I] materials fail from pre-existing flaws; [2] that flaws develop in a characteristic manner which is determined by the material properties, state and magnitude of the stresses at the flaw perimeter, the history of the imposed tractions, the thermal and environmental histories; and [3] the critical load for structural failure is a decreasing function of the crack length.Since the growing crack may not be detected by inspection procedures before reaching critical proportions, the reliability of a component or structure must include the chance of survival with an undetected crack. Fundamental to this operation is the definition of a "critical load". Failure occurs when the resistance F(t) of an element or structure is decreased through flaw development to the level such that the next load at that level, the "critical load", will produce rapid fracture. The fatigue lifetime, an induced stochastic variable, is simply the time required to produce a sufficient strength degradation to permit failure.Let us assume that the damage rate accumulation may be approximated by a power-law growth equationwhere r is a positive exponent characteristic of material behavior and independent of test or service variables, and C is the dimension of the critical damage zone. The premultiplier M is proportional to the far field work inputed into the element by the service environment M = const. W r ~ ADrF 2r max and F is the load or stress, D is the effective compliance, and A is a constant. In addition assume that a "critical load" is correlated withwhere K is an apparent toughness or work parameter. Integrating (i) 465 Int Journ of Fracture Mech 8 (1972) 466 and employing (2) gives a statement for the time dependent residual strength F.(t) 2(r-l) = F.(to)2(r-l) --(r-l)A4F2rmax(t-t o) (3) 1 1 of the ith element in a population of N elements, where A 4 = ADrK 2(r-l) F(to) is the initial static strength of the component possessing a Weibull (extreme value) distributionwhere a and F(t ) are the shape and scale parameters, respectively, for the°Weibull ~istribution, and the distribution of some function G[F(to) ] = F(t) where statement (3) is required. Using (3) it is easily seen that the set of values where F(t) > F is the same as the set of values whereHence, the probability that F(t) > F is just the probability that (3') holds. Equation (4) gives this probability as P[F(t)>F] = P{F(to)>[F(t)2(r-1) P[F(t)>F] = exp I--] 1/2 (r-+ (r_l)A4F2rmax(t_to) i) } (5) F(t) (r-l) + (r-l)A4F2rmax(t-t o) ~f Fit2 (r-l) ] 1where a_ = a /2(r-l). Note that when t = t , (5) reduced to (4); that when t ~ t o ~he result that P[F(t) > 0] < l°is a positive probability that F(t) = 0; and that when F(t) > 0 the random variable F(t) 2(r-I) is distributed as a translated three parameter Weibull distribution as demonstrated in Figure i.In the experiment...

Effects of loading adjustment on the reliability degradation of composites

“…Experimental fatigue data [6] for unnotched composite laminates under constant amplitude cyclic stress show that the residual strength after n cycles decreases monotonically with n . Halpin et al [7] assumed that the rate of degradation in the residual strength is inversely proportional to a power function of residual strength.…”

“…Initially, Broutman and Sahu [1] proposed a simple linear degradation model for determining the residual strength of composites. Later, Halpin et al [6,7] assumed that the residual strength is a monotonically decreasing function of the number of fatigue cycles. Its rate of degradation is the product of a power function of residual strength and a function of maximum cyclic stress.…”

“…(6) to(8) in a deterministic manner for the derivation of the transition period and reliability drop. Replacing the random variables X (0),…”

A generalized Gompertz model of reliability-dependent hazard rate for composites under cyclic stresses

“…Halpin et al [10] fi rst showed that for unnotched composite laminates under constant-amplitude cyclic stress, the residual strength Z ( n ) after n cycles decreases monotonically with n . Halpin et al [11] also assumed that the slope of residual strength can be approximated by dividing a function of the maximum cyclic stress by a power function of Z ( n ).…”

Statistical Fatigue of Graphite/Epoxy Angle-Ply Laminates in Shear1

“…The fatigue failure is assumed to occur when the residual strength R(n) has been reduced to the value of the maximum cyclic stress amax, i.e., when Substitution of Equation 9 into Equation 8 results in an expression for the fatigue life, N, in the form The ultimate strength R(0) is a statistical variable assumed to follow the twoparameter Weibull distribution [References 1, 3-13] , in which FR (0) (x) is the distribution function denoting the probability that the ultimate strength R(0) is smaller than a value x. In Equation 11, a and j3 are the shape parameter and the scale parameter (or characteristic strength), respectively.…”