The statistical second-order two-scale analysis for dynamic thermo-mechanical performances of the composite structure with consistent random distribution of particles

doi:10.1016/j.commatsci.2012.12.011

Computational Materials Science

Volume 69, March 2013, Pages 359-373

https://doi.org/10.1016/j.commatsci.2012.12.011 Get rights and content

Abstract

This study presents a statistical second-order two-scale (SSOTS) method to predict dynamic coupled thermo-mechanical performances of 3-D composite materials with consistent random distribution of particles. In the dynamic thermo-mechanical problem considered, a mutual interaction exists between the displacement and temperature fields. The statistical second-order two-scale asymptotic formulations for calculating the effective thermal and mechanical parameters, temperatures, displacements, heat flux densities and stresses of the problem are developed firstly. And then, the procedure of numerical computation based on the SSOTS method is discussed in detail. The numerical results obtained by using the SSOTS algorithm are compared with those by FEM in a very fine mesh. It shows that the SSOTS method is not only feasible, but also accurate and efficient to predict dynamic thermo-mechanical performances of composite materials with consistent random distribution of particles. Numerical results also demonstrate that the SSOTS method can capture the microscopic characteristics effectively.

Highlights

► The statistical second-order two-scale (SSOTS) asymptotic expressions are developed. ► The SSOTS numerical algorithms are presented. ► The SSOTS method is feasible and efficient to predict dynamic thermo-mechanical problem. ► The SSOTS analysis method can capture the microscopic characteristics effectively.

Introduction

With the rapid development of material science and technology, random composites have been widely used in a variety of engineering and industrial products owing to their advantageous properties. According to the arrangement characteristics of particles, the random composites can be classified into the materials with consistent random distribution and those with inconsistent random distribution. The consistent random distribution denotes that the volume fraction and the probability distribution model of particles are the same everywhere in composites (Fig. 1). With the appearance of various complex and extreme environments, the composite structures usually work under multi-scale and multi-physical fields coupled circumstances. And as a significant physical property, the thermo-mechanical performances of composite materials have attracted a lot of attention from scientists and engineers. There have been some works about the thermo-mechanical problem of composites and some valuable results have been made. However, most of these studies were devoted to one-way thermo-mechanical coupling problems [1], [2], [3], [4], namely, the thermal effects affect the mechanical filed but not vice versa, or periodic composites [5], [6], [7], [8]. Though Khan et al. [9] studied the one-way thermo-mechanical coupling problem of the composites with random distribution of spheres, the Random Sequential Adsorption Algorithm was used and the algorithm cannot generate a random particle sample with high volume fraction or better stochastic behavior. Moreover, the effective material parameters were calculated by the volume averaging method and the cost of calculation was enormous to get the statistical displacement and temperature fields by calculating lots of samples. In many actual situations, the dynamic thermo-mechanical problem which incorporates a mutual interaction between displacement and temperature fields should be considered. And the fully coupled analysis will lead to more accurate results. Besides, the random composites have better physical properties and agree with the real working environment of materials more. So it is significant and meaningful to study the dynamic thermo-mechanical problems of random composites.

The random parameters of the particles, including the shape, orientation, spatial location and the volume fraction, should be taken into account together to simulate the random composite materials. Though lots of works have been done, Yu et al. [10] gave a computer generation algorithm with high computing efficiency to generate random particle samples not only with high particle volume fraction but also better stochastic behavior, which will be used in this paper. It is well known that the traditional way to obtain the displacement and temperature fields of the thermo-mechanical coupling problem for the random composites is based on statistical average, namely, the displacement and temperature fields of each sample are solved by using the Monte Carlo method and the expected displacement and temperature fields are taken as the effective displacement and temperature fields. This method leads to double increase of memory and computing amount due to the complex microstructure of random composites and large numbers of samples. In recent years, based on the homogenization methods [11], [12], [13], [14], Feng et al. [15] developed a multi-scale method for static thermo-mechanical coupling problem of periodic composites; Wan et al. [16] studied the dynamic thermo-mechanical performances of the periodic composites. After that, Cui et al. established a statistical second-order two-scale method to predict thermal or mechanical properties of the composites with random distribution of particles [17], [18], [19]. In this paper, we focus on the calculation of temperatures, displacements, heat flux densities and stresses for the dynamic thermo-mechanical problem of the composite materials with consistent random distribution of particles. And a statistical second-order two-scale algorithm is presented to understand the effects of micro-structures.

The reminder of this paper is organized as follows. In Section 2, the main computational formulas on the statistical second-order two-scale method are developed in detail. Then the discrete models based on FEM and FDM and the algorithm procedure of SSOTS method are given in Section 3. In Section 4, some numerical results are shown. Finally the conclusions are given.

For convenience, we use the Einstein summation convention on repeated indices in this paper.

Section snippets

Statistical second order two-scale formulation

Suppose that the investigated composite material in this paper is made from matrix and particles with one-scale. Each particle is considered as an ellipsoid with nine random parameters, i.e. the coordinates of the central point (x₁, x₂, x₃), the sizes of the long, middle and short axis (a, b, c) and three orientation parameters of the long and middle axis $(θ_{{ax}_{1} x_{2}}, θ_{{ax}_{1}}, θ_{{bx}_{1} x_{2}})$ . If there are I ellipsoids in a cell εY^s, then a sample can be defined as follows [20]: $ω^{s} = (x_{11}, x_{21}, x_{31}, a_{1}, b_{1}, c_{1}, θ_{{ax}_{1} x_{2} 1}, θ_{{ax}_{1} 1}$

Computer simulation of random composites and generation of FE meshes

The computer simulation algorithm of generating a sample with consistent random distribution of particles is developed by Yu et al. [10]. The algorithm consists of two parts: particle generating part, filtering and location part. It can efficiently generate a sample, which has higher volume fraction and preserves stochastic property. Moreover, a new efficient algorithm [22] is used to generate the FE meshes of the composite structures. And Fig. 2 shows the FE meshes of some sample.

Finite element computations of cell functions

It can be

Numerical simulations and discussion

In this section, the dynamic and static thermo-mechanical examples are shown to verify the validity and feasibility of SSOTS analysis. And then, a plate structure with random distribution of particles is investigated to demonstrate the capability of the SSOTS method for predicting dynamic thermo-mechanical performances.

Conclusions

The SSOTS analysis method and related numerical algorithms for predicting the dynamic thermo-mechanical behaviors of composite structures with consistent random distribution of particles are presented. A newly SSOTS method is stated, including the second-order two-scale asymptotic expressions on temperature, heat flux density, displacement, stress and strain fields, and the formulations of the expected homogenized parameters. The SSOTS algorithm is validated by two examples, and the results

Acknowledgements

This work is supported by the National Basic Research Program of China (973 Program 2012CB025904), the National Natural Science Foundation of China (90916027), and also supported by the State Key Laboratory of Science and Engineering Computing and the Center for high performance computing of Northwestern Polytechnical University.

References (29)

C. Jeng-Shian et al.
Computers & Structures
(1992)
V. Levin
Journal of Applied Mathematics and Mechanics
(1982)
K.A. Khan et al.
Mechanics of Materials
(2011)
Y. Yu et al.
Composites Science and Technology
(2008)
Y. Yu et al.
Computational Materials Science
(2009)
F. Han et al.
Computational Materials Science
(2009)
Y.Y. Li et al.
Applied Mathematical Modelling
(2009)
Y.Y. Li et al.
Composites Science and Technology
(2005)
H.G. Matthies et al.
Computer Methods in Applied Mechanics and Engineering
(2006)
Z. Hashin et al.
A variational approach to the theory of the elastic behavior of multiphase materials
Journal of the Mechanics and Physics of Solids
(1963)

J. Aboudi

Applied Mechanics Reviews

(1996)

A.K. Noor et al.

Applied Mechanics Reviews

(1992)

R.B. Hetnarski et al.

Thermal Stresses: Advanced Theory and Applications

(2008)

J. Aboudi

Journal of Engineering Mathematics

(2008)

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The statistical second-order two-scale analysis for dynamic thermo-mechanical performances of the composite structure with consistent random distribution of particles

Abstract

Highlights

Introduction

Section snippets

Statistical second order two-scale formulation

Computer simulation of random composites and generation of FE meshes

Finite element computations of cell functions

Numerical simulations and discussion

Conclusions

Acknowledgements

Computers & Structures

Journal of Applied Mathematics and Mechanics

Mechanics of Materials

Composites Science and Technology

Computational Materials Science

Computational Materials Science

Applied Mathematical Modelling

Composites Science and Technology

Computer Methods in Applied Mechanics and Engineering

Journal of the Mechanics and Physics of Solids

Applied Mechanics Reviews

Applied Mechanics Reviews

Thermal Stresses: Advanced Theory and Applications

Journal of Engineering Mathematics