Markoff Random Processes and the Statistical Mechanics of Time‐Dependent Phenomena. II. Irreversible Processes in Fluids

1954 ◽  
Vol 22 (3) ◽  
pp. 398-413 ◽  
Author(s):  
Melville S. Green
Keyword(s):  
Statistical Mechanics ◽  
Random Processes ◽  
Time Dependent ◽  
Irreversible Processes

Statistical Mechanics of Irreversible Processes in Polymer Solutions

1958 ◽  
Vol 29 (4) ◽  
pp. 909-913 ◽  
Author(s):  
Jerome J. Erpenbeck ◽  
John G. Kirkwood
Keyword(s):  
Statistical Mechanics ◽  
Polymer Solutions ◽  
Irreversible Processes

Reliability Analysis and Random Vibration of Nonlinear Systems Using the Adjoint Method and Projected Differentiation

2020 ◽  
Vol 143 (6) ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Zhen Hu
Keyword(s):  
Reliability Analysis ◽  
Equations Of Motion ◽  
Random Variables ◽  
Random Processes ◽  
Time Dependent ◽  
Vibratory System ◽  
Standard Normal ◽  
Non Gaussian ◽  
Nonlinear Equations Of Motion ◽  
Adjoint Approach

Abstract This paper proposes a new methodology for time-dependent reliability and random vibrations of nonlinear vibratory systems using a combination of a time-dependent adjoint variable (AV) method and a projected differentiation (PD) method. The proposed approach is called AV-PD. The vibratory system is excited by stationary Gaussian or non-Gaussian input random processes. A Karhunen–Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. The nonlinear equations of motion (EOM) are linearized using a Taylor expansion using the first-order derivatives of the output with respect to the input KL random variables. An adjoint approach obtains the output derivatives accurately and efficiently requiring the solution of as many sets of EOM as the number of outputs of interest, independently of the number of KL random variables. The proposed PD method then computes the autocorrelation function of each output process at an additional cost of solving as many sets of EOM as the number of outputs of interest, independently of the time horizon (simulation time). A time-dependent reliability analysis is finally performed using a KL expansion of the output processes and Monte Carlo simulation (MCS). The number of solutions of the EOM scales only with the number of output random processes which is commonly much smaller than the number of input KL random variables. The efficiency and accuracy of the proposed approach is demonstrated using a four degree-of-freedom (DOF) half-car vibratory problem.

Some Issues in Ice Mechanics

2015 ◽  
Author(s):  
Ian Jordaan
Keyword(s):  
Damage Mechanics ◽  
Yield Condition ◽  
Threshold Value ◽  
Time Dependent ◽  
Irreversible Processes ◽  
Microstructural Changes ◽  
Weakest Link ◽  
Random Fashion ◽  
Viscoelastic Theory ◽  
Spatially Varying

Ice mechanical behavior is time-dependent, as has been known for many decades. But in many references, the attempt is made to use time-independent plasticity theory. The relevant analytical approach that accounts for time is viscoelastic theory. The need for this approach is made quite essential by study of microstructural changes that occur in ice under high stresses. In no case does there appear to be a clear yield condition, with flow occurring after a certain threshold value. Furthermore, the microstructural changes occurring under stress result in a highly significant enhancement of the creep rates. This results in a spatially varying viscoelastic response that is a function of prior stress history. The ice response is then a function of position resulting in a microstucturally modified layer in the region where compressive stress is applied. This can be deep or highly localized, depending on the loading rate. The most promising approach is that based on damage mechanics combined with viscoelasticity, using the thermodynamics of irreversible processes. Ice is also prone to fracture, especially at high loading rates and under high stresses. This is basic to the notion of scale effect. Fracture processes are also time-dependent in viscoelastic materials, a phenomenon that needs to be explored further. Furthermore, failure often will take place in a random fashion, depending on the distribution of flaws in ice. This indicates strongly that a theory based on “weakest-link” hypotheses and probability theory is appropriate. Finally, some aspects relevant to practical data analysis are discussed. These include measurement uncertainties of Molikpaq data, and geometric approximations of ice features, e.g. ridges as uniform beams.

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