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Based on the iterated statistically multiscale analysis (SMSA), we present the convergence of the equivalent mechanical parameters (effective moduli), obtain the error result, and prove the symmetric, positive and definite property of the equivalent mechanical parameters tensor computed by the finite element method. The numerical results show the proved results and illustrate that the SMSA-FE algorithm is a rational method for predicting the equivalent mechanical parameters of the composite material with multiscale random grains. In conclusion, we discuss the future work for the inhomogeneous composite material with multiscale random grains.

Predicting the mechanical parameters of a composite material with the multiscale random grains is a very difficult problem because there are too many random grains in the composite material and the range of the scale of the grains is very large in the material field shown in Figure

(a)

In the recent decades, for the problems with the stationary random distribution, Jikov et al. [

For the porous medias with the random distribution, Hou and Wu [

Duschlbauer et al. developed the homogenization method with the Mori-Tanaka scheme averaging microfields extracted for individual fibers and the finite element analysis to estimate the linear thermoelastic and thermophysical behavior of a short fiber reinforced composite material with planar random fiber arrangement [^{−1} m to 10^{−6} m [

In these previous papers [

The next section reviews a representation of a composite material with multiscale random grains, some results, and the SMSA-FE procedure [

In the previous papers [

For the brief, all of the grains are assumed as the ellipsoids. Set a domain

Obtain the statistical data of the composites and specify the distributions

Set

Set the domain

Set

For example, the asphalt concrete [

In the previous section, we introduced the equivalent composites

For the domain

In the paper [

If a composite material with random grains is subjected to the probability distribution

Therefore, the equivalent mechanical parameters can be determined by the following SMSA-FE algorithm.

Specify the scale number

Model

If

If

Compute the FE approximation

Set

If

If

From (

If

From the definition of the long axis

If

Because

If

Set

That is,

Based on the SMSA-FE algorithm, if the equivalent mechanical parameter tensor is computed, three kinds of errors are considered: the homogenization error, the random error based on Monte Carlo simulation method, and the finite element computation error. For the homogenization error, the composite materials with multiscale random grains are the special cases of the random coefficient problems whose convergence was proved in [

Firstly, we give the finite element error estimation of the statistical two-scale analysis. Then we give the error estimation of the SMSA-FE algorithm.

If

Since

Let

In fact, based on the idea and the method in [

Let

From the above algorithm, the following equation is held.

Since

Since there exists one constant

Based on the SMSA-FE algorithm, the equivalent mechanical parameter tensor of the equivalent material with the

Let

From Theorem

From [

Therefore, if the parameters of the ellipsoids are subjected to the uniform probability

Let

Taking into consideration the fact that

From the integral identity for solution of problem (

Based on the relations

Hence, we see that

In the sequel, we shall prove (

By the iterated multiscale analysis and Lemma

Let

To test the validity of the error result, the convergence, and the symmetric, positive definite property of the mechanical parameter tensor computed by the SMSA-FE algorithm, two numerical examples are given as follows.

The first example models a composite material. The grains are divided into two classes according to the sizes of their long axis shown in Table

Probability distributions of grains in composite material.

Small grains | Large grains | ||

Mechanical parameters of the matrix and grains.

Matrix | Grains | ||||

0 | 0 | ||||

0 | 0 | ||||

0 | 0 | 0 | 0 |

Equivalent mechanical parameters

0.04 | |||||

0.02 | |||||

0.01 | |||||

0.005 |

From Table

The second example is a concrete named as C30 with three-scale random rock grains whose sizes are from 0.3 mm to 19 mm. Its matrix is made up of the cement and the sand. Their sizes of the three-scale rock grains in the concrete are shown in Table

Size of rock random grains and the number in a statistical window.

Class | Grain size range | Average size | Number of rock grains | Sizes of statistical windows |
---|---|---|---|---|

Large grains | 10–19 mm | 14.5 mm | 8 | |

Middle grains | 1–10 mm | 5 mm | 20 | |

Small Grains | 0.3–1 mm | 0.6 mm | 28 |

Elasticity mechanical parameters of the matrix, grains, and joint interface materials.

Class | Young’s modulus (GPa) | Poisson's ratio |
---|---|---|

Rock grains | 74.5 | 0.15 |

Matrix | 13.5 | 0.25 |

Joint interface | 50.5 | 0.20 |

Numerical results for the mechanical parameter tensor of a composite with only small, middle, and large rock grains are listed in Tables

Equivalent mechanical parameters of a composites with small rock random grains by the SMSA-FE algorithm (kPa).

20341770.6 | 6054537.6 | 6054645.8 | −119951.6 | 164516.8 | −12764.7 |

6054537.6 | 20128556.8 | 6060485.8 | −106579.5 | 19649.2 | −141765.4 |

6054645.8 | 6060485.8 | 20061719.4 | −18765.7 | 140779.4 | −116731.9 |

0.000000 | 0.000000 | 0.000000 | 6923128.0 | 0.000000 | 0.000000 |

164516.8 | 19649.25 | 140779.48 | 0.000000 | 6902337.6 | −81061.0 |

−12764.7 | −141765.4 | 0.000000 | 98172.3 | −81061.0 | 6859146.1 |

Equivalent mechanical parameter tensor for concrete with middle rock random grains by the SMSA-FE algorithm (kPa).

25464900.2 | 6882577.0 | 6881924.2 | −135341.116667 | 111669.7 | −2238.3 |

6882577.0 | 25410941.8 | 6916568.5 | −123921.9 | 7075.0 | −187899.58 |

6881924.2 | 6916568.5 | 25291123.2 | −17145.9 | 89634.5 | −192916.5 |

0.000000 | 0.000000 | 0.000000 | 9071093.3 | 0.000000 | 0.000000 |

111669.7 | 7075.0 | 89634.5 | 0.000000 | 9042380.8 | −82547.1 |

−2238.3 | -187899.5 | 0.000000 | 69773.7 | −82547.1 | 9060481.0 |

Equivalent mechanical parameter tensor for concrete with large rock random grains by the SMSA-FE algorithm (kPa).

34510679.4 | 8073028.8 | 8076036.2 | −75074.2 | 111127.6 | −4769.6 |

8073025.0 | 34121732.2 | 8092588.8 | −36814.0 | 5862.0 | −110837.9 |

8076036.8 | 8092589.3 | 34065284.2 | −531.7 | 123664.7 | −79363.6 |

0.000000 | 0.000000 | 0.000000 | 12860237.8 | 0.000000 | 0.000000 |

111127.6 | 5862.0 | 123664.7 | 0.000000 | 12862877.8 | −30079.5 |

−4769.6 | −110837.9 | 0.000000 | 78302.2 | −30079.5 | 12783465.6 |

Equivalent mechanical parameter tensor for concrete with large rock random grains by the SMSA-FE algorithm (GPa).

Iterative number | 5 | 10 | 15 | 20 | 25 | 30 |
---|---|---|---|---|---|---|

17.4985392 | 16.9160384 | 16.9151317 | 17.0004448 | 17.0245196 | 16.9342101 | |

0.231375 | 0.234653 | 0.234271 | 0.233951 | 0.233942 | 0.233926 | |

22.5917136 | 22.2098256 | 22.2298752 | 22.1302736 | 22.1057408 | 22.0321386 | |

0.214631 | 0.215408 | 0.215342 | 0.215443 | 0.215552 | 0.215537 | |

30.644806.4 | 30.3689408 | 30.5923968 | 30.6815936 | 30.6936934 | 30.6862218 | |

0.192951 | 0.193419 | 0.193065 | 0.193116 | 0.193066 | 0.193082 |

Young's modulus and Poisson's ratio for 5, 10, 15,…, 30 samples with different scale random grains by the SMSA-FE procedure.

A comparison of the numerical results for Young's modulus and Poisson's ratio by the SMSA-FE procedure, by the mixed volume method, and by the experiment method in the lab is also shown in Table

Equivalent mechanical parameters for a concrete by the different methods (GPa).

In this paper, we proved that the equivalent mechanical parameter tensor for the composite materials with multiscale random grains is convergent and obtained the error result by the finite element analysis. At the same time, we also prove that the equivalent parameter tensor matrix should satisfy the symmetric, positive, and definite property.

Various test examples were solved by the SMSA-FE procedure. The numerical results show that the SMSA-FE procedure is feasible and valid and that these data satisfy the properties of the equivalent mechanical parameter tensor proved in the pervious sections.

The procedure can also be extended to other composite materials with random short fibers, random foams, random cavities, and so forth. Although we have given the specific steps, some theory results and numerical examples to carry out the SMSA-FE method and to illustrate the validity, the influence of shape, size, component, orientation, spatial distribution, and volume fraction of inclusions on inhomogeneous macromechanical properties, analyze the calculated results, and capture the information of microbehaviors to the macromechanical properties are also our important future work for the composite materials with random grains.

This work is supported by National Natural Science Foundation of China (NSFC) Grants (11072041), by State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology (GZ1005), by Hunan province Natural Science Foundation Grants of China (10JJ6065), by Scientific Research Starting Foundation for Returned Overseas Chinese Scholars, Ministry of Education, China (20091001), and by China Postdoctoral Science Foundation (20100480944).