# Inverse Identification of Elastic Properties of Constituents of Discontinuously Reinforced Composites

## Abstract

**:**

## 1. Introduction

## 2. Identification Procedure

#### 2.1. Optimization Problem

_{1}, x

_{2}… x

_{k}are the elastic constants of the material phases (variables), y

_{i}are the elastic constants of the composite predicted by micromechanical model depending on variables, Y

_{i}—given elastic constants of the composite (input data), n denotes the number of known elastic constants of the composite. The aim of identification is presented graphically in Figure 1.

#### 2.2. Micromechanical Modeling

_{m}and C

_{i}are isotropic stiffness tensors of matrix and inclusion respectively, I is an identity tensor, f

_{i}is volume fraction of the inclusion and I is strain concentration tensor that depends on the Eshelby’s tensor S in the following way [34]:

_{ijkl}can be determined in terms of stiffness tensor of unidirectional composite C

_{pqrs}as follows:

_{ij}is coordinate system transformation matrix [39]. An effectiveness of the orientation-averaging procedure has been presented in numerous works [39,40,41]. The drawback of the M-T method is that it is limited to analysis of spheroidal shape of inclusions only; moreover, error of homogenization increases with increasing volume fraction of reinforcement; thus, only composites with low volume fractions of the reinforcement (approximately up to 0.25) can be successfully analyzed. However, the numerical solution of the equivalent inclusion problem, instead of using the Eshelby’s tensor, allowing the M-T method to be extended so as to involve the arbitrary shapes of the inclusions. The equivalent inclusion problem relates to analysis of single inclusion embedded in a large matrix [42]. The medium is typically approximated by a rectangular prism whose finite dimensions are large enough in comparison with the size of the inclusion [42,43]. The strain concentration tensor A defines the relation between the average strain in the single inclusion embedded in infinite matrix ε

^{i}and the far field strain (macro strain) ε:

^{i}as follows:

## 3. Results and Discussion

#### 3.1. Composite Reinforced with Short Fibers

_{m}= 10

^{4}MPa, ν

_{m}= 0.3, E

_{i}= 105 MPa, ν

_{i}= 0.2.

^{NUM_CYLINDER}(hybrid solution) and A

^{ELLIPSOID}(analytical solution) have been compared (Figure 5). A minor difference between the results can be noticed and, therefore, usage of the pure analytical Mori–Tanaka method (accounting for ellipsoidal shape of fiber) should provide reasonable accuracy of identification in this case.

_{1}and s

_{2}as indicated in Table 4. For each simulation, the input data was selected randomly with probability given by the gaussian distribution (1500 simulations for each standard deviation have been carried out). Monte Carlo simulations exposed a difference in outcome of identification which may occur when uncertain input data is applied. Figure 6 and Figure 7 present the results obtained for the standard deviation s

_{1}and Figure 8 and Figure 9 present the results obtained for the standard deviation s

_{2}. Histograms that represents a distribution of identified quantities have been determined on the basis of Monte Carlo simulations (Figure 10).

#### 3.2. Composite Reinforced with Cubic Particles

_{m}= 10

^{4}MPa, ν

_{m}= 0.3, E

_{i}= 10

^{5}MPa, ν

_{i}= 0.2.

## 4. Concluding Remarks

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Representative volume elements representing composites reinforced with following volume fractions of cylindrical fibers: (

**a**) 0.05; (

**b**) 0.10; (

**c**) 0.15.

**Figure 4.**Model of equivalent inclusion problem for cylindrical fiber: (

**a**) geometry; (

**b**) finite element mesh.

**Figure 5.**Normalized Young moduli in terms of volume fraction of the reinforcement determined by using pure analytical and hybrid method.

**Figure 6.**Normalized Young modulus of matrix and fiber identified during 1500 independent simulations involving standard deviation s

_{1}.

**Figure 7.**Normalized Poisson ratio of matrix and fiber identified during 1500 independent simulations involving standard deviation s

_{1}.

**Figure 8.**Normalized Young modulus of matrix and fiber identified during 1500 independent simulations involving standard deviation s

_{2}.

**Figure 9.**Normalized Poisson ratio of matrix and fiber identified during 1500 independent simulations involving standard deviation s

_{2}.

**Figure 10.**Histograms representing the distributions of identified quantities based on Monte Carlo simulations conducted for two error levels (error level 1 corresponds to the standard deviation s

_{1}, error level 2 corresponds to the standard deviation s

_{2}): (

**a**) Young modulus of matrix; (

**b**) Poisson ratio of matrix; (

**c**) Young modulus of fiber; (

**d**) Poisson ratio of fiber.

**Figure 11.**Normalized Young moduli in terms of volume fraction of the reinforcement determined by using pure analytical and hybrid method.

**Figure 12.**Geometrical models of: (

**a**) equivalent inclusion problem for cubic particle (view on one eight of model) (

**b**) representative volume element of composite reinforced with 15% of cubic particles.

Phase | Young Modulus (MPa) | Poisson Ratio |
---|---|---|

Matrix | 70,000 | 0.33 |

Fiber | 300,000 | 0.20 |

Volume Fraction of Fibers | Young Modulus (MPa) | Poisson Ratio |
---|---|---|

0.05 | E_{c}^{5%} = 74,955.83 | ν_{c}^{5%} = 0.32446 |

0.10 | E_{c}^{10%} = 80,486.48 | ν_{c}^{10%} = 0.31897 |

0.15 | E_{c}^{15%} = 85,820.04 | ν_{c}^{15%} = 0.31383 |

**Table 3.**Elastic constants of composite constituents identified by using different input data and corresponding errors.

Input Data | Analysis Number | Identified Elastic Constants | |||
---|---|---|---|---|---|

E_{m} | E_{r} | v_{m} | v_{r} | ||

E_{c}^{5%}, ν_{c}^{5%} | 1 | 68,383.3, 2.3% | 512,943.6, 71.0% | 0.33013, 0.0% | 0.16642, 16.8% |

2 | 69,287.1, 1.0% | 393,597.7, 31.2% | 0.32500, 1.5% | 0.38210, 91.1% | |

3 | 72,330.0, 3.3% | 152,002.5, 49.3% | 0.32536, 1.4% | 0.31336, 56.7% | |

E_{c}^{10%}, ν_{c}^{10%} | 1 | 65,642.5, 6.2% | 663,675.3,121.2% | 0.32863, 0.4% | 0.21423, 7.1% |

2 | 79,190.7, 13.1% | 92,946.0, 69.0% | 0.32298, 2.1% | 0.27911, 39.6% | |

3 | 71,523.3, 2.2% | 250,707.2, 16.4% | 0.32895, 0.3% | 0.19039, 4.8% | |

E_{c}^{15%}, ν_{c}^{15%} | 1 | 77,654.0, 10.9% | 155,795.0, 48.1% | 0.31281, 5.2% | 0.32879, 64.4% |

2 | 62,787.5, 10.3% | 709,421.2, 136.5% | 0.32567, 1.3% | 0.25722, 28.6% | |

3 | 60,758.4, 13.2% | 929,659.3, 209.9% | 0.32381, 1.9% | 0.32073, 60.4% | |

E_{c}^{5%}, ν_{c}^{5%}, E_{c}^{10%}, ν_{c}^{10%} | 1 | 69,708.4, 0.4% | 331,100.2, 10.4% | 0.33001, 0.0% | 0.16767, 16.2% |

2 | 69,708.4, 0.4% | 331,097.8, 10.4% | 0.33001, 0.0% | 0.16766, 16.2% | |

3 | 69,708.4, 0.4% | 331,098.0, 10.4% | 0.33001, 0.0% | 0.16766, 16.2% | |

E_{c}^{10%}, ν_{c}^{10%}, E_{c}^{15%}, ν_{c}^{15%} | 1 | 70,608.3, 0.9% | 288,238.8, 3.9% | 0.32927, 0.2% | 0.18351, 8.2% |

2 | 70,608.3, 0.9% | 288,237.8, 3.9% | 0.32927, 0.2% | 0.18351, 8.2% | |

3 | 70,608.4, 0.9% | 288,237.4, 3.9% | 0.32927, 0.2% | 0.18353, 8.2% | |

E_{c}^{5%}, ν_{c}^{5%}, E_{c}^{15%}, ν_{c}^{15%} | 1 | 69,934.1, 0.1% | 308,332.5, 2.8% | 0.32982, 0.1% | 0.17573, 12.1% |

2 | 69,934.1, 0.1% | 308,332.6, 2.8% | 0.32982, 0.1% | 0.17573, 12.1% | |

3 | 69,934.04, 0.1% | 308,333.81, 2.8% | 0.32982, 0.1% | 0.17573, 12.1% | |

E_{c}^{5%}, ν_{c}^{5%}, E_{c}^{15%}, ν_{c}^{15%}, E_{c}^{15%}, ν_{c}^{15%} | 1 | 70,001.2, 0.0% | 308,842.0, 2.9% | 0.32976, 0.1% | 0.17571, 12.1% |

2 | 70,001.2, 0.0% | 308,841.9, 2.9% | 0.32976, 0.1% | 0.17571, 12.1% | |

3 | 70,001.2, 0.0% | 308,841.8, 2.9% | 0.32976, 0.1% | 0.17571, 12.1% |

**Table 4.**Statistical properties of composites reinforced with three different volume fractions of reinforcement which serves as an input data to Monte Carlo simulations.

Volume Fraction of Fibers | Young Modulus (Mpa) | Poisson Ratio |
---|---|---|

0.05 | Μ = 74,955.83 | μ = 0.32446 |

s_{1} = 500.00 | s_{1} = 0.0015 | |

s_{2} = 1000.00 | s_{2} = 0.0030 | |

0.10 | μ = 80,486.48 | μ = 0.31897 |

s_{1} = 500.00 | s_{1} = 0.0015 | |

s_{2} = 1000.00 | s_{2} = 0.0030 | |

0.15 | μ = 85,820.04 | μ = 0.31383 |

s_{1} = 500.00 | s_{1} = 0.0015 | |

s_{2} = 1000.00 | s_{2} = 0.0030 |

Phase | Young Modulus (MPa) | Poisson Ratio |
---|---|---|

Matrix | 70,000 | 0.30 |

Particle | 415,000 | 0.16 |

Volume Fraction of Fibers | Young Modulus(MPa) | Poisson Ratio |
---|---|---|

0.10 | 82,012.65 | 0.28828 |

0.15 | 88,707.98 | 0.28269 |

Micromechanical Model | Identified Elastic Constants | |||
---|---|---|---|---|

E_{m} | E_{i} | v_{m} | v_{i} | |

Mori–Tanaka (M-T) | 70,180.5, 0.3% | 518,546.8, 22.2% | 0.29849, 0.5% | 0.12000, 18.2% |

Hybrid M-T/FE | 70,042.9, 0.1% | 432,482.4, 4.1% | 0.29924, 0.3% | 0.15423, 3.7% |

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**MDPI and ACS Style**

Ogierman, W.
Inverse Identification of Elastic Properties of Constituents of Discontinuously Reinforced Composites. *Materials* **2018**, *11*, 2332.
https://doi.org/10.3390/ma11112332

**AMA Style**

Ogierman W.
Inverse Identification of Elastic Properties of Constituents of Discontinuously Reinforced Composites. *Materials*. 2018; 11(11):2332.
https://doi.org/10.3390/ma11112332

**Chicago/Turabian Style**

Ogierman, Witold.
2018. "Inverse Identification of Elastic Properties of Constituents of Discontinuously Reinforced Composites" *Materials* 11, no. 11: 2332.
https://doi.org/10.3390/ma11112332