Recursions and limit theorems for the strength and lifetime distributions of a fibrous composite

1987 ◽  
Vol 24 (1) ◽  
pp. 137-159 ◽  
Author(s):  
Chia-Chyuan Kuo ◽  
S. Leigh Phoenix
Keyword(s):  
Distribution Function ◽  
Limit Theorem ◽  
Fibrous Composite ◽  
Probability Model ◽  
Failure Process ◽  
Static Strength ◽  
Fatigue Lifetime ◽  
Lifetime Distributions ◽  
The Matrix ◽  
A New Technique

A composite material is a parallel arrangement of stiff brittle fibers in a flexible matrix. Under load fibers fail, and the loads of failed fibers are locally redistributed onto nearby survivors through the matrix. In this paper we develop a new technique for computing the probability of failure under a previously studied model of the failure process. A recursion and limit theorem are obtained which apply separately to static strength and fatigue lifetime depending on the composite loading and the probability model for the failure of individual fibers under their own loads. The limit theorem yields an approximation for the distribution function for composite lifetime which is of the form 1 – [1 – W(t)]mn where W(t) is a characteristic distribution function and mn is the composite volume, reflecting a size effect. A similar result holds also for static strength. In both cases such a result was conjectured several years ago. This limit theorem is obtained from the recursion upon applying a key theorem in the theory of the renewal equation. In the proofs three technical conditions arise which must be verified in specific applications. In the case of static strength these conditions are quite easy to verify, but in the case of fatigue lifetime the verification is generally difficult, and entails considerable numerical computation.

Recursions and limit theorems for the strength and lifetime distributions of a fibrous composite

1987 ◽  
Vol 24 (01) ◽  
pp. 137-159 ◽  
Author(s):  
Chia-Chyuan Kuo ◽  
S. Leigh Phoenix
Keyword(s):  
Distribution Function ◽  
Limit Theorem ◽  
Fibrous Composite ◽  
Probability Model ◽  
Failure Process ◽  
Static Strength ◽  
Fatigue Lifetime ◽  
Lifetime Distributions ◽  
The Matrix ◽  
A New Technique

A composite material is a parallel arrangement of stiff brittle fibers in a flexible matrix. Under load fibers fail, and the loads of failed fibers are locally redistributed onto nearby survivors through the matrix. In this paper we develop a new technique for computing the probability of failure under a previously studied model of the failure process. A recursion and limit theorem are obtained which apply separately to static strength and fatigue lifetime depending on the composite loading and the probability model for the failure of individual fibers under their own loads. The limit theorem yields an approximation for the distribution function for composite lifetime which is of the form 1 – [1 – W(t)] mn where W(t) is a characteristic distribution function and mn is the composite volume, reflecting a size effect. A similar result holds also for static strength. In both cases such a result was conjectured several years ago. This limit theorem is obtained from the recursion upon applying a key theorem in the theory of the renewal equation. In the proofs three technical conditions arise which must be verified in specific applications. In the case of static strength these conditions are quite easy to verify, but in the case of fatigue lifetime the verification is generally difficult, and entails considerable numerical computation.

scholarly journals A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Fa-mei Zheng
Keyword(s):  
Distribution Function ◽  
Limit Theorem ◽  
Wiener Process ◽  
Continuous Distribution ◽  
Random Variables ◽  
Standard Wiener Process ◽  
Trimmed Sums ◽  
Fixed Constant ◽  
Random Products ◽  
Trimmed Sum

Let be a sequence of independent and identically distributed positive random variables with a continuous distribution function , and has a medium tail. Denote and , where , , and is a fixed constant. Under some suitable conditions, we show that , as , where is the trimmed sum and is a standard Wiener process.

Reliability Evaluation of Structural Ceramics Under Multiaxial Stress State

1998 ◽  
Author(s):  
Takashi Ono ◽  
Masaki Kaji ◽  
Michiaki Nishimura
Keyword(s):  
Experimental Data ◽  
Distribution Function ◽  
Stress State ◽  
Fracture Probability ◽  
Test Method ◽  
Structural Ceramics ◽  
Multiaxial Stress ◽  
Fatigue Lifetime ◽  
Empirical Parameter ◽  
Multiaxial Stress State

Strength and fatigue lifetime of structural ceramics under multiaxial stress state have been estimated and compared with experimental data. Biaxial strength tests were done by an anticlastic bending test method at room temperature. Biaxial fatigue tests were done by anticlastic bending and also ring-on-ring test method at 1200°C in air. Fracture probability and lifetime were predicted on the basis of a Weibull multiaxial distribution function and subcritical crack growth, using the results of stress analyses by the finite element method. Modified maximum hoop stress theory including an empirical parameter, T, was applied to the equivalent normal stress in the multiaxial distribution function. The empirical parameter T represents a shear stress sensitivity to mixed-mode fracture due to a grain interlocking effect. It has been confirmed that the predicted fracture probability and the fatigue lifetime agrees well with the experimental data if grain interlocking effects are taking into account.

scholarly journals On a class of norms generated by nonnegative integrable distributions

2019 ◽  
Vol 7 (1) ◽  
pp. 259-278 ◽  
Author(s):  
Michael Falk ◽  
Gilles Stupfler
Keyword(s):  
Distribution Function ◽  
Limit Theorem ◽  
Pointwise Convergence ◽  
Distribution Functions ◽  
Compact Sets ◽  
Wasserstein Metric ◽  
Normal Distributions ◽  
The Central Limit Theorem ◽  
Integrability Conditions ◽  
Characteristic Sets

AbstractWe show that any distribution function on ℝd with nonnegative, nonzero and integrable marginal distributions can be characterized by a norm on ℝd+1, called F-norm. We characterize the set of F-norms and prove that pointwise convergence of a sequence of F-norms to an F-norm is equivalent to convergence of the pertaining distribution functions in the Wasserstein metric. On the statistical side, an F-norm can easily be estimated by an empirical F-norm, whose consistency and weak convergence we establish.The concept of F-norms can be extended to arbitrary random vectors under suitable integrability conditions fulfilled by, for instance, normal distributions. The set of F-norms is endowed with a semigroup operation which, in this context, corresponds to ordinary convolution of the underlying distributions. Limiting results such as the central limit theorem can then be formulated in terms of pointwise convergence of products of F-norms.We conclude by showing how, using the geometry of F-norms, we may characterize nonnegative integrable distributions in ℝd by simple compact sets in ℝd+1. We then relate convergence of those distributions in the Wasserstein metric to convergence of these characteristic sets with respect to Hausdorff distances.

Theoretical Relations Between Static Strength and Lifetime Distributions of Composites: An Evaluation

1984 ◽  
Vol 6 (4) ◽  
pp. 164 ◽  
Author(s):  
WW Feng ◽  
KL Reifsnider ◽  
GP Sendeckyj ◽  
TT Chiao ◽  
W Steve Johnson ◽  
...  
Keyword(s):  
Static Strength ◽  
Lifetime Distributions

Unimodality of Wave Amplitude in the Northern Hemisphere

2008 ◽  
Vol 65 (3) ◽  
pp. 1077-1086 ◽  
Author(s):  
Maarten H. P. Ambaum
Keyword(s):  
Distribution Function ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Wave Amplitude ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Normal Distributions ◽  
The Central Limit Theorem ◽  
Local Wave

Abstract A novel statistic for local wave amplitude of the 500-hPa geopotential height field is introduced. The statistic uses a Hilbert transform to define a longitudinal wave envelope and dynamical latitude weighting to define the latitudes of interest. Here it is used to detect the existence, or otherwise, of multimodality in its distribution function. The empirical distribution function for the 1960–2000 period is close to a Weibull distribution with shape parameters between 2 and 3. There is substantial interdecadal variability but no apparent local multimodality or bimodality. The zonally averaged wave amplitude, akin to the more usual wave amplitude index, is close to being normally distributed. This is consistent with the central limit theorem, which applies to the construction of the wave amplitude index. For the period 1960–70 it is found that there is apparent bimodality in this index. However, the different amplitudes are realized at different longitudes, so there is no bimodality at any single longitude. As a corollary, it is found that many commonly used statistics to detect multimodality in atmospheric fields potentially satisfy the assumptions underlying the central limit theorem and therefore can only show approximately normal distributions. The author concludes that these techniques may therefore be suboptimal to detect any multimodality.

scholarly journals Effective elastic shear stiffness of a periodic fibrous composite with non-uniform imperfect contact between the matrix and the fibers

2014 ◽  
Vol 51 (6) ◽  
pp. 1253-1262 ◽  
Author(s):  
Juan C. López-Realpozo ◽  
Reinaldo Rodríguez-Ramos ◽  
Raúl Guinovart-Díaz ◽  
Julián Bravo-Castillero ◽  
J.A. Otero ◽  
...  
Keyword(s):  
Fibrous Composite ◽  
Shear Stiffness ◽  
Imperfect Contact ◽  
The Matrix ◽  
Elastic Shear

scholarly journals The Size of the Largest Part of Random Weighted Partitions of Large Integers

2013 ◽  
Vol 22 (3) ◽  
pp. 433-454 ◽  
Author(s):  
LJUBEN MUTAFCHIEV
Keyword(s):  
Distribution Function ◽  
Limit Theorem ◽  
Gumbel Distribution ◽  
Random Variable ◽  
Ideal Gas ◽  
Negative Numbers ◽  
Generating Series ◽  
Einstein Model ◽  
Bose Einstein ◽  
Weighted Partition

We consider partitions of the positive integernwhose parts satisfy the following condition. For a given sequence of non-negative numbers {bk}k≥1, a part of sizekappears in exactlybkpossible types. Assuming that a weighted partition is selected uniformly at random from the set of all such partitions, we study the asymptotic behaviour of the largest partXn. LetD(s)=∑k=1∞bkk−s,s=σ+iy, be the Dirichlet generating series of the weightsbk. Under certain fairly general assumptions, Meinardus (1954) obtained the asymptotic of the total number of such partitions asn→∞. Using the Meinardus scheme of conditions, we prove thatXn, appropriately normalized, converges weakly to a random variable having Gumbel distribution (i.e., its distribution function equalse−e−t, −∞<t<∞). This limit theorem extends some known results on particular types of partitions and on the Bose–Einstein model of ideal gas.

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