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A paper on: Mathematical modeling of the overall time-dependent behavior of non-ageing viscoelastic reinforced composites.

El Kouri mohamed's picture

  New mathematical and numerical formulations are developed for effective time-dependent non ageing viscoelastic behavior of linear viscoelastic composites. The modeling is based on the dynamic Green's functions, integral equations, and Volterra product. The time concentration tensor is derived through numerical solution of the integral equation and the Mori-Tanaka micromechanical

model. The developed modeling gives explicit expressions of effective properties through convolution products in the Stieltjes space. A numerical procedure is developed allowing the calculation of the tensoriel convolution product and its inverse. Based on Carson-Laplace transform, the frequency dependent effective properties are also obtained and converted to the time domain.    For viscoelastic Prony laws and inclusions with various shapes, numerical results are presented and compared with the approach based on the Laplace transform and the correspondence principle.

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EL Kouri Mohamed

Congratulation! I have a question to discuss with you. As we know, multiscale properties of ageing rate-dependent composites can not analyzed by asymptotic homogenization method. So, what is your method adopted?  In addition, if your materials include ageing, which method you will use? Self-consistent method ot Mori-Tanaka? Thank you!

Liu Pengfei

Zhejiang University

El Kouri mohamed's picture

thank you for your interest in my paper. i'm sorry my late reply.


In this paper we introduce a new method of homogenization which is based on an integral equation method and was motivated by the methodology and numerical algorithm. This method has a number of advantages.

The Mori-Tanaka micromechanical model is known to give accurate prediction of the effective properties of composite materials. So if my materials include ageing, the time dependent Mori-Tanaka concentration tensor used.


This newly developed integral equation approach is an attractive homogenization technique for number of reasons. Firstly it is a direct approach to the computation of effective dynamic material behavior in time domain. Secondly, unlike the method of asymptotic homogenization, there is no requirement to solve a cell problem. Although this was not an issue in the one dimensional case, in higher dimensions the solution to such a cell problem, although often mathematically elegant is also very difficult to construct and one must often resort to numerical solutions, especially in the three dimensional context.

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