SOME ADDITIONAL THEOREMS FOR A NON-STATIONARY STOCHASTIC PROCESS WITH A CONTINUOUS, NON-RANDOM, TIME-DEPENDENT COMPONENT

1963 ◽  
Author(s):  
R. C. McCarty ◽  
G. W. Evans
Keyword(s):  
Stochastic Process ◽  
Time Dependent ◽  
Random Time ◽  
Stationary Stochastic Process ◽  
Dependent Component

scholarly journals Coagulation Processes with Gibbsian Time Evolution

2012 ◽  
Vol 49 (03) ◽  
pp. 612-626
Author(s):  
Boris L. Granovsky ◽  
Alexander V. Kryvoshaev
Keyword(s):  
Stochastic Process ◽  
Recurrence Relation ◽  
Time Evolution ◽  
Nonnegative Function ◽  
Gibbs Distribution ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Gibbs Distributions ◽  
Giant Cluster ◽  
Positive Integers

We prove that a stochastic process of pure coagulation has at any timet≥ 0 a time-dependent Gibbs distribution if and only if the rates ψ(i,j) of single coagulations are of the form ψ(i;j) =if(j) +jf(i), wherefis an arbitrary nonnegative function on the set of positive integers. We also obtain a recurrence relation for weights of these Gibbs distributions that allow us to derive the general form of the solution and the explicit solutions in three particular cases of the functionf. For the three corresponding models, we study the probability of coagulation into one giant cluster by timet> 0.

Contributions of Time Dependent and Cyclic Crack Growth to the Crack Growth Behavior of Non Strain-Crystallizing Elastomers

2002 ◽  
Vol 75 (4) ◽  
pp. 643-656 ◽  
Author(s):  
J. J. C. Busfield ◽  
K. Tsunoda ◽  
C. K. L. Davies ◽  
A. G. Thomas
Keyword(s):  
Crack Growth ◽  
Growth Behavior ◽  
Time Dependent ◽  
Crack Growth Behavior ◽  
Wide Range ◽  
Dependent Component ◽  
Tearing Energy ◽  
Quasi Crystals ◽  
Crack Growth Rates ◽  
Time Dependent Crack Growth

Abstract Engineering components are observed to fail more rapidly under cyclic loading than under static loading. This reflects features of the underlying crack growth behavior. This behavior is characterized by the relation between the tearing energy, T, and the crack growth per cycle, dc/dn. The increment of crack growth during each cycle is shown here to result from the sum of time dependent and cyclic crack growth components. The time dependent component represents the crack growth behavior that would be present in a conventional constant T crack growth test. Under repeated stressing additional crack growth, termed the cyclic crack growth component, occurs. For a non-crystallizing elastomer, significant effects of frequency have been found on the cyclic crack growth behavior, reflecting the presence of this cyclic element of crack growth. The cyclic crack growth behavior over a wide range of frequencies was investigated for unfilled and swollen SBR materials. The time dependent crack growth component was calculated from constant T crack growth tests and the cyclic contribution derived from comparison with the observed cyclic growth. It is shown that decreasing the frequency or increasing the maximum tearing energy during a cycle results in the cyclic crack growth behavior being dominated by time dependent crack growth. Conversely at high frequency and at low tearing energy, cyclic crack growth is dominated by the cyclic crack growth component. A large effect of frequency on cyclic crack growth behavior was observed for highly swollen SBR. The cyclic crack growth behavior was dominated by the time dependent crack growth component over the entire range of tearing energy and/or crack growth rate. The origin of the cyclic component may be the formation/melting of quasi crystals at the crack tip, which is absent at fast crack growth rates in the unswollen SBR and is absent at all rates in the swollen SBR.

scholarly journals Crack-Considered Elastic Net Monitoring Model of Concrete Dam Displacement

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Jiingmei Zhang ◽  
Chongshi Gu
Keyword(s):  
Water Pressure ◽  
Elastic Net ◽  
Arch Dam ◽  
Time Dependent ◽  
Concrete Dams ◽  
Displacement Monitoring ◽  
Concrete Dam ◽  
Monitoring Model ◽  
Proposed Model ◽  
Dependent Component

Displacement monitoring data modeling is important for evaluating the performance and health conditions of concrete dams. Conventional displacement monitoring models of concrete dams decompose the total displacement into the water pressure component, temperature component, and time-dependent component. And the crack-induced displacement is generally incorporated into the time-dependent component, thus weakening the interpretability of the model. In the practical engineering modeling, some significant explaining variables are selected while the others are eliminated by applying commonly used regression methods which occasionally show instability. This paper proposes a crack-considered elastic net monitoring model of concrete dam displacement to improve the interpretability and stability. In this model, the mathematical expression of the crack-induced displacement component is derived through the analysis of large surface crack’s effect on the concrete dam displacement to improve the interpretability of the model. Moreover, the elastic net method with better stability is used to solve the crack-considered displacement monitoring model. Sequentially, the proposed model is applied to analyze the radial displacement of a gravity arch dam. The results demonstrate that the proposed model contributes to more reasonable explaining variables’ selection and better coefficients’ estimation and also indicate better interpretability and higher predictive precision.

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