# Linearization of Composite Material Damage Model Results and Its Impact on the Subsequent Stress–Strain Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Workflow Overview

#### 2.2. Basic Calculus

#### 2.3. Graphic Interpretation of Hashin’s Violation

#### 2.4. Consolidation Function

#### 2.5. Material Card Generation

#### 2.6. Material Properties

#### 2.7. Damage Variation Matrix and Response of Consolidation Functions

#### 2.8. Testing Examples

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$\left\{\tilde{\mathsf{\sigma}}\right\}$ | (MPa) | Undamaged material stress response tensor |

$\left\{\hat{\mathsf{\sigma}}\right\}$ | (MPa) | Effective stress tensor |

$\left\{\mathsf{\sigma}\right\}$ | (MPa) | General stress tensor |

$\left\{\mathsf{\epsilon}\right\}$ | (1) | General strain tensor |

${\mathrm{E}}_{1}^{0}$ | (MPa) | Elastic modulus of non-damaged material in the direction 1 |

${\mathrm{E}}_{2}^{0}$ | (MPa) | Elastic modulus of non-damaged material in the direction 2 |

${\mathrm{G}}_{12}^{0}$ | (MPa) | Shear modulus of non-damaged material in plane 12 |

${\mathrm{G}}_{13}^{0}$ | (MPa) | Transverse shear modulus of non-damaged material in plane 13 |

${\mathrm{G}}_{23}^{0}$ | (MPa) | Transverse shear modulus of non-damaged material in plane 23 |

${\mathsf{\nu}}_{12}^{0}$ | (1) | Poisson’s ratio of undamaged material in plane 12 |

${\mathrm{E}}_{1}^{\mathrm{d}}$ | (MPa) | Consolidated elastic modulus of damaged material in the direction 1 |

${\mathrm{E}}_{2}^{\mathrm{d}}$ | (MPa) | Consolidated elastic of damaged material in the direction 2 |

${\mathrm{G}}_{12}^{d}$ | (MPa) | Consolidated shear modulus of damaged material in plane 12 |

${\mathrm{G}}_{13}^{d}$ | (MPa) | Consolidated transverse shear modulus of damaged material in plane 13 |

${\mathrm{G}}_{23}^{d}$ | (MPa) | Consolidated transverse shear modulus of damaged material in plane 23 |

${\mathsf{\nu}}_{12}^{0}$ | (1) | Poisson’s ratio of undamaged material in plane 12 |

${\mathrm{E}}_{1}^{\mathrm{d}+}$ | (MPa) | Tensile elastic modulus of damaged material in the direction 1 |

${\mathrm{E}}_{1}^{\mathrm{d}-}$ | (MPa) | Compressive elastic of damaged material in the direction 1 |

${\mathrm{E}}_{2}^{\mathrm{d}+}$ | (MPa) | Tensile elastic modulus of damaged material in the direction 2 |

${\mathrm{E}}_{2}^{\mathrm{d}-}$ | (MPa) | Compressive elastic of damaged material in the direction 2 |

${\mathrm{d}}_{1}$ | (1) | Consolidated damage factor in the direction 1 |

${\mathrm{d}}_{2}$ | (1) | Consolidated damage factor in the direction 2 |

${\mathrm{d}}_{12}$ | (1) | Consolidated damage factor in plane 12 |

${\mathrm{d}}_{\mathrm{s}}$ | (1) | Consolidated shear damage factor |

${\mathrm{d}}_{1}^{+}$ | (1) | Tensile damage factor in the direction 1 |

${\mathrm{d}}_{1}^{-}$ | (1) | Compressive damage factor in the direction 1 |

${\mathrm{d}}_{2}^{+}$ | (1) | Tensile damage factor in the direction 2 |

${\mathrm{d}}_{2}^{-}$ | (1) | Compressive damage factor in the direction 2 |

$\left[{\mathrm{C}}^{0}\right]$ | (MPa) | Stiffness matrix of undamaged material |

$\left[{\mathrm{S}}^{0}\right]$ | (MPa^{−1}) | Compliance matrix of undamaged material |

$\left[{\mathrm{C}}^{\mathrm{d}}\right]$ | (MPa) | Stiffness matrix of damaged material |

$\left[{\mathrm{S}}^{\mathrm{d}}\right]$ | (MPa^{−1}) | Compliance matrix of damaged material |

$\left[\mathrm{D}\right]$ | (1) | Damage operator matrix |

${\mathrm{F}}_{1,\mathrm{u}}^{+}$ | (MPa) | Ultimate tensile stress in direction 1 |

${\mathrm{F}}_{1,\mathrm{u}}^{-}$ | (MPa) | Ultimate compressive stress in direction 1 |

${\mathrm{F}}_{2,\mathrm{u}}^{+}$ | (MPa) | Ultimate tensile stress in direction 2 |

${\mathrm{F}}_{2,\mathrm{u}}^{-}$ | (MPa) | Ultimate compressive stress in direction 2 |

${\mathrm{F}}_{12,\mathrm{u}}$ | (MPa) | Ultimate shear stress in plane 12 |

${\mathrm{F}}_{13,\mathrm{u}}$ | (MPa) | Ultimate shear stress in plane 13 |

${\mathrm{F}}_{23,\mathrm{u}}$ | (MPa) | Ultimate shear stress in plane 23 |

${\mathsf{\epsilon}}_{1,\mathrm{u}}^{+}$ | (1) | Ultimate tensile strain in direction 1 |

${\mathsf{\epsilon}}_{1,\mathrm{u}}^{-}$ | (1) | Ultimate compressive strain in direction 1 |

${\mathsf{\epsilon}}_{2,\mathrm{u}}^{+}$ | (1) | Ultimate tensile strain in direction 2 |

${\mathsf{\epsilon}}_{2,\mathrm{u}}^{-}$ | (1) | Ultimate compressive strain in direction 2 |

${\mathrm{U}}_{1}^{+}$ | (mJ/mm^{2}) | Tensile fracture energy in direction 1 |

${\mathrm{U}}_{1}^{-}$ | (mJ/mm^{2}) | Compressive fracture energy in direction 1 |

${\mathrm{U}}_{2}^{+}$ | (mJ/mm^{2}) | Tensile fracture energy in direction 2 |

${\mathrm{U}}_{2}^{-}$ | (mJ/mm^{2}) | Compressive fracture energy in direction 2 |

$F1$ | Consolidation damage function in direction 1 | |

$F2$ | Consolidation damage function in direction 2 | |

$F12$ | Consolidation damage function in plane 12 |

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**Figure 13.**Damage of model Layup 4; Ply oriented at 0°; Selection (${\mathrm{d}}_{1}\to 0,{\mathrm{d}}_{2}\to 0$): (

**a**) Arithmetic mean; (

**b**) weighted mean; (

**c**) maximum; (

**d**) product function.

**Figure 14.**Damage of model Layup 4; Ply oriented at +45°; Selection (${\mathrm{d}}_{1}\to 0,{\mathrm{d}}_{2}\to 0$): (

**a**) Arithmetic mean; (

**b**) weighted mean; (

**c**) maximum; (

**d**) product function.

**Figure 15.**Damage of model Layup 4; Ply oriented at −45°; Selection (${\mathrm{d}}_{1}\to 0,{\mathrm{d}}_{2}\to 0$): (

**a**) Arithmetic mean; (

**b**) weighted mean; (

**c**) maximum; (

**d**) product function.

**Figure 16.**Damage of model Layup 4; Ply oriented at 90°; Selection (${\mathrm{d}}_{1}\to 0,{\mathrm{d}}_{2}\to 0$): (

**a**) Arithmetic mean; (

**b**) weighted mean; (

**c**) maximum; (

**d**) product function.

Function | Consolidate Parameter | Consolidate Variable |
---|---|---|

$F1$ | $\left\{{\mathrm{d}}_{1}^{+},{\mathrm{d}}_{1}^{-}\right\}\to {\mathrm{d}}_{1}$ | $F1:\left\{{\mathrm{E}}_{1}^{+}({\mathrm{d}}_{1}^{+}),{\mathrm{E}}_{1}^{-}({\mathrm{d}}_{1}^{-})\right\}\to {\mathrm{E}}_{1}^{\mathrm{d}}\left({\mathrm{E}}_{1}^{0},{\mathrm{d}}_{1}\right)$ |

$F2$ | $\left\{{\mathrm{d}}_{2}^{+},{\mathrm{d}}_{2}^{-}\right\}\to {\mathrm{d}}_{2}$ | $F2:\left\{{\mathrm{E}}_{2}^{+}({\mathrm{d}}_{2}^{+}),{\mathrm{E}}_{2}^{-}({\mathrm{d}}_{2}^{-})\right\}\to {\mathrm{E}}_{2}^{\mathrm{d}}\left({\mathrm{E}}_{2}^{0},{\mathrm{d}}_{2}\right)$ |

$F12$ | $\left\{{\mathrm{d}}_{1}^{+},{\mathrm{d}}_{1}^{-},{\mathrm{d}}_{2}^{+},{\mathrm{d}}_{2}^{-}\right\}\to {\mathrm{d}}_{12}$ | $F12:\left\{{\mathrm{G}}_{12}\left({\mathrm{d}}_{1}^{+},{\mathrm{d}}_{1}^{-},{\mathrm{d}}_{2}^{+},{\mathrm{d}}_{2}^{-}\right)\right\}\to {\mathrm{G}}_{12}^{\mathrm{d}}\left({\mathrm{G}}_{12}^{0},{\mathrm{d}}_{12}\right)$ |

Method | Consolidate Condition | |
---|---|---|

Arithmetic mean | ${\mathrm{d}}_{\mathrm{i}}={\mathrm{d}}_{\mathrm{i},\mathrm{a}}$ | $\mathrm{i}=\left\{1,2\right\}$ |

Weighted mean | ${\mathrm{d}}_{\mathrm{i}}=\left\{\begin{array}{cc}0,& {\mathrm{if}\Vert \mathrm{d}\Vert}_{\mathrm{i}}=0\\ {\mathrm{d}}_{\mathrm{i},\mathrm{w}},& {\mathrm{if}\Vert \mathrm{d}\Vert}_{\mathrm{i}}1\end{array}\right.$ | |

Maximum | ${\mathrm{d}}_{\mathrm{i}}=\mathrm{Max}\left\{{\mathrm{d}}_{\mathrm{i}}^{+},{\mathrm{d}}_{\mathrm{i}}^{-}\right\}$ | |

Product function | ${\mathrm{d}}_{\mathrm{i}}=1-\left(1-{\mathrm{d}}_{\mathrm{i}}^{+}\right)\left(1-{\mathrm{d}}_{\mathrm{i}}^{-}\right)$ | |

Shear function | ${\mathrm{d}}_{\mathrm{ij}}=1-\left(1-{\mathrm{d}}_{\mathrm{i}}\right)\left(1-{\mathrm{d}}_{\mathrm{j}}\right)$ | $\mathrm{i},\mathrm{j}=\left\{1,2\right\};\mathrm{i}\ne \mathrm{j}$ $\mathrm{i},\mathrm{j}=\left\{1,3\right\};\mathrm{i}\ne \mathrm{j}$ $\mathrm{i},\mathrm{j}=\left\{2,3\right\};\mathrm{i}\ne \mathrm{j}$ |

Plane Stress | Shear-Bending Coupling |
---|---|

${\mathrm{E}}_{1}^{\mathrm{d}},{\mathrm{E}}_{2}^{\mathrm{d}},{\mathrm{G}}_{12}^{\mathrm{d}}0$ | ${\mathrm{E}}_{1}^{\mathrm{d}},{\mathrm{E}}_{2}^{\mathrm{d}},{\mathrm{G}}_{12}^{\mathrm{d}},{\mathrm{G}}_{13}^{\mathrm{d}},{\mathrm{G}}_{23}^{\mathrm{d}}0$ |

$\left|{\mathsf{\nu}}_{12}^{\mathrm{d}}\right|<{\left(\frac{{\mathrm{E}}_{1}^{\mathrm{d}}}{{\mathrm{E}}_{2}^{\mathrm{d}}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ |

${\mathbf{E}}_{1}^{\mathbf{d}}$ | ${\mathbf{E}}_{2}^{\mathbf{d}}$ | ${\mathsf{\nu}}_{12}^{\mathbf{d}}$ | ${\mathsf{\nu}}_{21}^{\mathbf{d}}$ | ${\mathbf{G}}_{12}^{\mathbf{d}}$ |
---|---|---|---|---|

(MPa) | (MPa) | (1) | (1) | (MPa) |

${\mathrm{E}}_{1}^{0}\left(1-{\mathrm{d}}_{1}\right)$ | ${\mathrm{E}}_{2}^{0}\left(1-{\mathrm{d}}_{2}\right)$ | ${\mathsf{\nu}}_{12}^{0}\left(1-{\mathrm{d}}_{1}\right)$ | ${\mathsf{\nu}}_{21}^{0}\left(1-{\mathrm{d}}_{2}\right)$ | ${\mathrm{G}}_{12}^{0}\left(1-{\mathrm{d}}_{1}\right)\left(1-{\mathrm{d}}_{2}\right)$ |

${\mathbf{E}}_{1}^{0}$ | ${\mathbf{E}}_{2}^{0}$ | ${\mathsf{\nu}}_{12}^{0}$ | ${\mathbf{G}}_{12}^{0}$ | ${\mathbf{G}}_{13}^{0}$ | ${\mathbf{G}}_{23}^{0}$ |
---|---|---|---|---|---|

(MPa) | (MPa) | (1) | (MPa) | (MPa) | (MPa) |

129,840 | 13,340 | 0.26 | 4890 | 4890 | 4630 |

${\mathbf{F}}_{1,\mathbf{u}}^{+}$ | ${\mathbf{F}}_{1,\mathbf{u}}^{-}$ | ${\mathbf{F}}_{2,\mathbf{u}}^{+}$ | ${\mathbf{F}}_{2,\mathbf{u}}^{-}$ | ${\mathbf{F}}_{12,\mathbf{u}}$ | ${\mathbf{F}}_{13,\mathbf{u}}$$({\mathbf{F}}_{23,\mathbf{u}})$ |
---|---|---|---|---|---|

(MPa) | (MPa) | (MPa) | (MPa) | (MPa) | (MPa) |

2965.41 | 2911.81 | 100.88 | 109.42 | 100.76 | 98.41 |

${\mathbf{U}}_{1}^{+}$ | ${\mathbf{U}}_{1}^{-}$ | ${\mathbf{U}}_{2}^{+}$ | ${\mathbf{U}}_{2}^{-}$ |
---|---|---|---|

(mJ/mm^{2}) | (mJ/mm^{2}) | (mJ/mm^{2}) | (mJ/mm^{2}) |

35.56 | 34.28 | 0.92 | 1.08 |

Layup Id. | Layup Design | Layup Id. | Layup Design |
---|---|---|---|

1 | [0°/0°/0°/0°] | 5 | [45°/−45°/−45°/45°] |

2 | [0°/+45°/−45°/0°] | 6 | [45°/90°/90°/45°] |

3 | [0°/90°/90°/0°] | 7 | [90°/90°/90°/90°] |

4 | [0°/+45°/−45°/90°] |

Consolidate Function | Layup 1 | Layup 2 | Layup 3 | Layup 4 | Layup 5 | Layup 6 | Layup 7 |
---|---|---|---|---|---|---|---|

[−] | [%] | [%] | [%] | [%] | [%] | [%] | [%] |

Arithmetic mean | 0.38 | 0.03 | 3.75 | 0.81 | 12.94 | 65.19 | 93.05 |

Weighted mean | 0.38 | 2.17 | 0.58 | 4.09 | 12.03 | 57.44 | 85.23 |

Maximum | 0.38 | 3.24 | 4.51 | 6.98 | 30.23 | 95.41 | 90.50 |

Product function | 0.38 | 1.05 | 0.66 | 1.06 | 0.02 | 0.50 | 0.08 |

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**MDPI and ACS Style**

Vlach, J.; Doubrava, R.; Růžek, R.; Raška, J.; Horňas, J.; Kadlec, M.
Linearization of Composite Material Damage Model Results and Its Impact on the Subsequent Stress–Strain Analysis. *Polymers* **2022**, *14*, 1123.
https://doi.org/10.3390/polym14061123

**AMA Style**

Vlach J, Doubrava R, Růžek R, Raška J, Horňas J, Kadlec M.
Linearization of Composite Material Damage Model Results and Its Impact on the Subsequent Stress–Strain Analysis. *Polymers*. 2022; 14(6):1123.
https://doi.org/10.3390/polym14061123

**Chicago/Turabian Style**

Vlach, Jarmil, Radek Doubrava, Roman Růžek, Jan Raška, Jan Horňas, and Martin Kadlec.
2022. "Linearization of Composite Material Damage Model Results and Its Impact on the Subsequent Stress–Strain Analysis" *Polymers* 14, no. 6: 1123.
https://doi.org/10.3390/polym14061123