INVESTIGATION OF THE M-INTEGRAL IN CRACK-DAMAGED PIEZOELECTRIC CERAMICS

The physical interpretation of the M-integral is investigated in the analysis of crackdamaged piezoelectric problems. The relation between the M-integral and the change of the total electric enthalpy （CTEE）, i.e., M = 2CTEE, is derived with a theoretical derivation procedure for two^dimensional piezoelectric problems. It is shown that the M-integral may provide a more natural description of electric enthalpy release due to the formation of the pre-existing microcracks associated with the damaged body, rather than the description of the total potential energy release rate as interpreted for conventional brittle solids. For crack-damaged piezoelectric ceramics, numerical calculation of the M-integral is discussed. Based on the pseudo-traction electric displacement method, M = 2CTEE has also been proved by the numerical results.

NEW DESCRIPTION OF MICROCRACK DAMAGE BASED ON CONSERVATION LAWS

This paper presents a new description for brittle solids with micro- cracks under plane strain assumption.The basic idea is to extend the conservation laws such as the J_j-vector and M-integral analysis used in single crack problems to strongly interacting crack problems.The M-integral contains two distinct parts.One of them is a summation from the well-known relation between the M-integral and the stress intensity factors(SIF)at both tips of each crack.The other,called as the additional contribution,is obtained from the two components of the J_j-vector and the coordinates of each microcrack center in a global system.Of great significance is the clarification of the confusion about the dependence of the M-integral on the origin selection of global coordinates,provided that the vector vanishes at infinity and that the closed contour chosen to calculate the integral and the vector encloses all the microcracks completely.The M-integral is equivalent to the decrease of the total potential energy of the microcracking solids with the strong interaction being taken into account.The M-integral analysis,from a physical point of view,does play an important role in evaluating the damage level of brittle solids with strongly interacting microcracks.

Analysis of series RL and RC circuits with time-invariant source using truncated M,Atangana beta and conformable derivatives

Complex processes of the physical world require novel and sophisticated mathematical notions to get deep insights.In this research analysis,two standard mathematical models for the series RL and RC circuits having time-invariant sources taken from the discipline of electrical engineering have been investigated with the help of differential operators known with the name of truncated M-derivative,Atangana beta-derivative,and the conformable derivative operators.The exact solutions for these two models have been found in terms of the transcendental exponential function of time under the truncated M-derivative,Atangana beta-derivative,and the conformable derivative operators.The numerical simulations carried out via MATLAB”9.4.0.813654(R2018a)”have been interpreted to explore new behavior for solutions of the models not possible to obtain through standard classical calculus wherein one is restricted to have integer-order derivatives unlike the differential operators used in the present research study.The models under consideration for the three differential operators have been investigated with varying parameters’values including the differential ordersαandβwhereupon the classical case is resumed forα=β=1 andτ=0.