Weibull scale parameters

In association with each invariant, the a values correspond to the Weibull shape parameters and the /3 values correspond to Weibull scale parameters. A two-parameter Weibull probability density function has the following form... 

Weibull scale parameter monomeric friction coefficient craze fibril extension ratio (=l/vj) average extension ratio in stretch zone... 

Creep measurements of PRD 49 (PpBA) fibers at 65 °C and 150 °C exhibit logarithmic creep, as is also observed for PpPTA and PpBAT fibers at room temperature, and the creep rate increases with temperature [190]. Wu et al. determined Weibull parameters of the strength and lifetime distributions of PpPTA fibers at room and elevated temperatures [207]. They found that the mean filament strength (and Weibull scale parameter) varies inversely with temperature, while the strength variability (and Weibull modulus) remains practically constant over... 

Weibull scale parameter incorporating the effect of ks = characteristic strength (value of Of where 63.21% of specimens fail)... 

For the CARES/Lz/e reliability analysis the Weibull parameters obtained from the three-point flexure bars were used to predict the strength response of the disks. Utilizing Eq. (2) with maximum likelihood analysis and assuming volume flaws a Weibull modulus my = 11.96, a characteristic strength uej/= 612.7 MPa, and a Weibull scale parameter uok = 453.8 MPa mm / was obtained for the flexure... 

The slope of the curve is the Weibull scaling parameter m, which indicates the shape of the distribution. Low m-values indicate a large range of surface flaw sizes, while high m-values indicate high surface quality and a close approach to the tensile strength of a flaw-free flbre under normal condition." m-Values for pristine fibres are normally of the order of 80-150. 

Gn L) is often difficult to determine for a given load distribution, but when is large, an approximation is given by the Maximum Extreme Value Type I distribution of the maximum extremes with a scale parameter, 0, and location parameter, v. When the initial loading stress distribution,/(L), is modelled by a Normal, Lognormal, 2-par-ameter Weibull or 3-parameter Weibull distribution, the extremal model parameters can be determined by the equations in Table 4.11. These equations include terms for the number of load applications, n. The extremal model for the loading stress can then be used in the SSI analysis to determine the reliability. 

Three cases have been constructed by Weibull functions according to Eqs. (la) and (lb), as these best reflect systematic differences in the sequence. In all cases, a reference profile is defined by extent Eo l.O, scale parameter / = 2.0, and shape parameter a = 1.5. In each case, one parameter is altered to illustrate its influence. 

One has simply to assume a particular probability distribution for A with the survival function available in a closed form, namely the exponential, Erlang, Rayleigh, and Weibull. Table 9.1 summarizes the probability density functions, survival functions, and hazard rates for the above-mentioned distributions. In these expressions, A is the scale parameter and p and v are shape parameters with k, A, p > 0 and v = 1, 2,.... ... 

P = shape parameter or the Weibull slope, p > 0 6 = scale parameter or the characteristic life 8 = location parameter or the minimum life... 

The Weibull function has two parameters. The first is 8 or a shape parameter and the second is j a scale parameter. The scale parameter determines when, in time, a given portion of the population will fail (say 75%) at a given time /(t). The shape parameter enables the Weibull distribution to be applied to any portion of the bathtub cmve as follows ... 

Components with increasing failure rate have Weibull distributed lifetimes with scale parameter equal to 0.903/MTTF and shape parameter equal to 1.5. This distribution has a mean which approximately equals the MTTF. 

Weibull rfP, m ), where t]P (i = 0,1,..., k) is the scale parameter of the dth failure mode at stress 5), and is its shape parameter, which is changeless at any stress level. Then the observed lifetime... 

Following approach is based onto Weibull regress model where the scale parameter is modelled using both two-parametric function and covariate (see e.g. Finn 1998). 

Prepare Prediction of an unobserved random variable is a fundamental problem in statistics. The aim of this paper is to construct lower (upper) prediction limits under parametric uncertainty that are exceeded with probability 1—a (a) by future observations or functions of observations. The prediction limits depend on early-failure data of the same sample from the two-parameter Weibull distribution, the shape and scale parameters of which are not known. 

In this equation, ctu is the stress below which Pr = 0 (a is often taken as CTu = 0). ct. is a scale parameter and m is the Weibull modulus that characterizes the width of the distribution. The right hand side of equation (7a) should be dimensionless which means that does not have the dimension of a stress. One way to take this feature into account is to normalize the volume (e.g., to replace V by V/Vo where V. is a reference volume). Additionally, if one considers a lot of cylindrical fibers of length L with a constant diameter, V can be replaced by L (or in a similar manner by L/U where L is a reference length). Assuming Ou = 0, equation (7a) can be rewritten in a linear form and used to derive the value of m from tensile test data. 

The tensile strength of Si-C-0 fibers decreases after exposure to elevated temperatures. When Nicalon NL 200 fibers are exposed for 1 hour to 1300 C in argon (P = 100 kPa), their mean tensile strength and scale parameter, Co, decrease by 45% while their Weibull modulus remains unchanged [80-83]. Fibers exposed to more severe conditions (e.g., for 5 hours in a vacuum at 1500°C) are so weak that they cannot be tested. Finally, the fact that oxygen-free fibers maintain their tensile strength under similar conditions relates to the absence of silicon oxycarbide and its decomposition process. 

Lq = standard gage length m — Weibull modulus cr = tensile strength (To = a scaling parameter (T = an arbitrary parameter normally set to 0 F= cumulative probability of failure... 

Material Weibull modulus - m Volume scale parameter - sov (MPa.mm3/m)... 

The total Pf analysis to follow is based on the reduced thermal load gradient of 91 °C mentioned in the previous section which induces a conditional Pf of 0.186. Table V summarizes the 16 random variables, the statistical distribution functions assigned to them, and the corresponding distribution parameters reflecting their scatter. Some of the random parameters from the previous PDS analysis were removed since they did not strongly influence the maximum stresses in the TE and substrate materials. The four Weibull parameters (two Weibull moduli and two scale parameters) for the TE and substrate materials were assumed to be RIV with Gaussian distributions (see table V). The statistical distribution types used to describe these RIV are not based on data, but still realistically describe their uncertainty. 

Figure 13Sensitivity plot for the probability of failure of the TE device using 94 Monte Carlo simulations. The significant parameters influencing the Pf in order of importance are Weibull modulus of the TE material, scale parameter of the TE material, and elastic modulus of the TE material. 

Table 1. Typical Value Ranges for the Weibull Scale and Shape Parameters of Fibers ...

Weibull life distribution model is selected which has previously been used successfully for the same or a similar failure mechanism. The Weibull distribution is used to find the reliability of the life data and it helps in selecting the particular data that is to be used in life prediction model. The Weibull distribution uses two parameters, namely, b and to estimate the reliability of the life data, b is referred to as shape parameter and is referred to as scale parameter. 

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