# An Energy-Based Concept for Yielding of Multidirectional FRP Composite Structures Using a Mesoscale Lamina Damage Model

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## Abstract

**:**

## 1. Introduction

## 2. Damage Model of FRP Composite Lamina

#### 2.1. Damage Initiation

^{T}, Y

^{T}, X

^{C}, Y

^{C}, S

^{L}, and S

^{T}are the strength properties. In Equations (1)–(4), ${d}_{f}^{t},$ ${d}_{f}^{c}$ and ${d}_{m}^{t},$ ${d}_{m}^{c}$ are the internal damage variables in the fiber and matrix phases of the lamina, under tension or compression loadings. Since no plastic deformation is observed in the FRP composite [2,10,55], the permanent deformation of the lamina is considered in the damage evolution processes.

#### 2.2. Damage Propagation

_{p}, defined as

#### 2.3. Damage Dissipation Energy

## 3. Materials and Experimental Procedures

## 4. Finite Element Simulation

## 5. Results and Discussion

#### 5.1. Structural Response and Damage Evolution of GFRP Composite Beam under Three-Point Bending

#### 5.2. Structural Response and Damage Evolution of CFRP Composite Beam under Three-Point Bending Load

#### 5.3. Structural Response and Damage Evolution of CFRP Composite Beam under Four-Point Bending

#### 5.4. Comparison of the Estimated Yield Limits Based on UD Hashin Criteria and Energy-Based Criterion

## 6. Conclusions

- The yield point of the FRP composite laminate structures could be identified by a 5% increase in the initial slope of the DDE evolution curve with respect to the applied load parameters.
- At the yield point, the extent of damage by the various modes depended on material, lay-ups, load, and test configurations.
- The yield points of the MD GFRP and CFRP composite laminates (cases 1, 2, and 3) were identified to occur upon flexural loading when the rate of the DDE reached 0.914, 2.1, and 11.1 $\mathrm{N}/(\mathrm{mm}\xb7\mathrm{s}),$ respectively. The corresponding deflections were 13.5, 9, and 3 mm, respectively.
- The initial flexural stiffness of the MD GFRP and CFRP composite structures (cases 1, 2, and 3) were measured at 28, 17.6, and 108.26 N/mm, reduced to 27.2, 17.44, and 107.1 N/mm at the yield point, indicating 3%, 0.91%, and 1.1% reductions in the stiffness of the beams, respectively. Therefore, an average 2% reduction in flexural stiffness could be suggested as a mean for the determination of the yield point in MD FRP composite structures under three- and four-point bending loads.
- In general, the UD criteria resulted in the assumption of structural yielding at 10%–20% maximum capacity of the structure (displacement or load), whereas, using the energy-based criterion, the yield limit could be safely increased to 30%–50% of the maximum capacity of the structure.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\left[\widehat{\sigma}\right]$ | Effective stress of lamina (MPa) |

Y^{T} | Transverse tensile strength (MPa) |

S^{L} | Longitudinal shear strength (MPa) |

S^{T} | Transverse shear strength (MPa) |

Y^{C} | Transverse compressive strength (MPa) |

X^{T} | Longitudinal tensile strength (MPa) |

X^{C} | Longitudinal compressive strength (MPa) |

${d}_{m}^{t}$ | Matrix damage initiation variable in lamina under tensile load |

${d}_{m}^{c}$ | Matrix damage initiation variable in lamina under compressive load |

${d}_{f}^{t}$ | Fiber damage initiation variable in lamina under tensile load |

${d}_{f}^{c}$ | Fiber damage initiation variable in lamina under compressive load |

D | Damage operator in post-damage initiation process |

${\mathrm{d}}_{\mathrm{f}},{\text{}\mathrm{d}}_{\mathrm{m}}$, and ${\mathrm{d}}_{\mathrm{s}}$ | Internal damage variable corresponding to lamina fiber, matrix, and shear damage modes |

G_{C} | Fracture energy (N/mm) |

G^{XT} | Longitudinal tensile fracture energy (N/mm) |

G^{XC} | Longitudinal compressive fracture energy (N/mm) |

G^{YT} | Transverse tensile fracture energy (N/mm) |

G^{YC} | Transverse compressive fracture energy (N/mm) |

G_{DDE} | Damage dissipation energy (N/mm) |

d_{p} | Damage evolution variable |

${k}_{eq}^{0}$ | Equivalent elastic stiffness at the onset of damage (MPa) |

${\delta}_{eq}^{0}$ | Equivalent displacement at the onset of damage (mm) |

${\delta}_{eq}^{f}$ | Equivalent displacement at the separation of the material point (mm) |

${\mathsf{\sigma}}_{ij}^{o},$${\mathsf{\tau}}_{ij}^{o},$ and ${\mathsf{\epsilon}}_{ij}^{o}$ | Effective stresses (MPa) and strains at the onset of damage |

${\mathsf{\delta}}_{\mathrm{eq}}^{0}$ | Equivalent displacement at the state of damage initiation (mm) |

${\mathsf{\delta}}_{\mathrm{eq}}^{\mathrm{f}}$ | Equivalent displacement at failure (mm) |

${\mathrm{L}}^{\mathrm{c}}$ | Characteristic length of an element (mm) |

${E}_{U}$ | Internal energy (N/mm) |

${\mathsf{\sigma}}^{\mathrm{c}}$ | Stress (MPa) |

${\dot{\mathit{\epsilon}}}^{el},{\dot{\mathit{\epsilon}}}^{pl},{\dot{\mathit{\epsilon}}}^{cr}$ | Time rate of elastic, plastic, and creep strains |

${E}_{S}$ | Elastic strain energy (N/mm) |

${\mathit{\sigma}}^{u}$ | Un-damaged stress (MPa) |

${f}^{u}$ | Undamaged elastic energy function |

${f}^{c}$ | Damage strain energy function |

E_{E} and E_{D} | Recoverable and irrecoverable energy (N/mm) |

$h$ and $s$ | Interaction between the nodes-in-contact as separation and sliding motions (mm) |

${\mathrm{N}}_{\mathrm{i}}$ | Interpolation function of the interface segments |

${\mathsf{\rho}}_{\mathrm{n}}$ | Interface segment curvature |

${D}_{ij}$ | Stiffness matrix of interface with linear coupled elastic behavior (MPa) |

${F}_{i}$ | Force of interface node ${i}^{th}$ (N) |

${u}_{i}$ | Motion of interface node ${i}^{th}$ (mm) |

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**Figure 1.**(

**a**) Local (1–2) and global (x–y) axes of an angle lamina; (

**b**) bilinear stress–strain behavior of fiber-reinforced polymer (FRP) lamina in orthogonal axes for various failure modes. (

**c**) Each colored curve corresponds to the loading as shown by the same colored arrows in the inset figure.

**Figure 2.**(

**a**) Damage dissipation energy in the stress–displacement curve; (

**b**) evolution of the damage initiation (solid brown line) and propagation (dotted brown line) variables at the material point.

**Figure 3.**(

**a**) Longitudinal cross-section of (left) glass fiber-reinforced polymer (GFRP) and (middle and right) carbon fiber-reinforced polymer (CFRP) composite specimens, (

**b**) A schematic view of the composite beam on the test set-up.

**Figure 4.**Configuration of (

**a**) single-layer and (

**b**) multi-layer finite element (FE) model-based constructions of FRP composite laminates for vacuum-assisted infusion molding (VAIM) and prepreg/autoclave manufacturing processes, respectively.

**Figure 5.**FE model of the CFRP composite specimen under the three-point bending (3PB) test set-up, showing the discretized geometry.

**Figure 6.**(

**a**) Load–deflection response, structural stiffness curve, (

**b**) strain and damage dissipation energies, and (

**c**) final failure of GFRP composite beam under 3PB load.

**Figure 7.**(

**a**) Load–deflection response, structure stiffness curve, (

**b**) strain and damage dissipation energies, and (

**c**) time to matrix damage initiation in the laminas of CFRP composite beam under 3PB load.

**Figure 8.**(

**a**) Load–deflection response, structure stiffness curve, (

**b**) strain and damage dissipation energies, and (

**c**) time to matrix damage initiation in the laminas of CFRP composite beam under 4PB load.

**Table 1.**Configuration of multidirectional (MD) fiber-reinforced polymer (FRP) composite specimens and loading type. ID—identifier; CF—carbon fiber; GF—glass fiber; PB—point bending.

Composite Panel (Case ID) | Laminate Sequences | Dimensions of the Beam Specimen (mm) | Loading Rate(mm/min) | Loading Type | ||||
---|---|---|---|---|---|---|---|---|

Length L | Width W | Laminate Thickness t | Lamina Thickness | Support Span Length L _{SS} | ||||

GFRP (Case 1) | [0/90/45/0/−45/90/45/0] | 210 | 25 | 4 | 0.5 | 170 | 2 | 3PB |

CFRP (Case 2) | [45/−45/45/0/−45/0 /0/45/0/−45/45/−45] | 140 | 20 | 2.4 | 0.2 | 112 | 2 | 3PB |

CFRP (Case 3) | [−45/45/−45/90/45/90/ 90/−45/90/45/−45/45] | 70 | 20 | 2.4 | 0.2 | 60 | 1 | 4PB |

Lamina Constants | Constitutive Damage Model Parameters of Lamina | |||||
---|---|---|---|---|---|---|

GFRP | CFRP | GFRP | CFRP | |||

E_{11}, GPa | 36.9 | 105.5 | Longitudinal tensile strength, MPa | X_{T} | 820 | 1340 |

E_{22}, GPa | 10 | 7.2 | Longitudinal compressive strength, MPa | X_{C} | 500 | 1192 |

E_{33}, GPa | 10 | 7.2 | Transverse tensile strength, MPa | Y_{T} | 80.6 | 19.6 |

G_{12}, GPa | 3.3 | 3.4 | Transverse compressive strength, MPa | Y_{C} | 322 | 92.3 |

G_{13}, GPa | 3.3 | 3.4 | Longitudinal shear strength, MPa | S_{L} | 54.5 | 51 |

G_{23}, GPa | 3.6 | 2.52 | Transverse shear strength, MPa | S_{T} | 161.2 | 23 |

ν_{12} | 0.32 | 0.34 | Longitudinal tensile fracture energy, N/mm | G_{XT} | 32 | 48.4 |

ν_{13} | 0.32 | 0.34 | Longitudinal compressive fracture energy, N/mm | G_{XC} | 20 | 60.3 |

ν_{23} | 0.44 | 0.378 | Transverse tensile fracture energy | G_{YT} | 4.5 | 4.5 |

Transverse compressive fracture energy, N/mm | G_{YC} | 4.5 | 8.5 |

**Table 3.**Results of the yield values (UD and energy-based criteria) of the MD composite structure under flexural loading condition.

Composite Panel (Case ID) | Yield Parameter | Maximum Capacity (MC) | Yield Point | Damage Type | |||
---|---|---|---|---|---|---|---|

UD Hashin Criteria | Energy-based Criteria | ||||||

Value | Percentage to MC | Value | Percentage to MC | ||||

GFRP (Case 1) | Deflection, mm | 23.4 | 13 | 55.5% | 13.5 | 57.7% | Single mode Fiber failure in first lamina (0°) |

Load, N | 554.7 | 337 | 60.7% | 350 | 63.1% | ||

CFRP (Case 2) | Deflection, mm | 28 | 4.7 | 16.8% | 9 | 32.1% | Mixed-matrix cracking and crushing events in different laminas |

Load, N | 301 | 80.5 | 26.7% | 150 | 49.8% | ||

CFRP (Case 3) | Deflection, mm | 8 | 1.5 | 18.7% | 3 | 37.5% | |

Load, N | 600 | 161 | 26.8% | 313 | 52.2% |

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## Share and Cite

**MDPI and ACS Style**

Rahimian Koloor, S.S.; Karimzadeh, A.; Yidris, N.; Petrů, M.; Ayatollahi, M.R.; Tamin, M.N.
An Energy-Based Concept for Yielding of Multidirectional FRP Composite Structures Using a Mesoscale Lamina Damage Model. *Polymers* **2020**, *12*, 157.
https://doi.org/10.3390/polym12010157

**AMA Style**

Rahimian Koloor SS, Karimzadeh A, Yidris N, Petrů M, Ayatollahi MR, Tamin MN.
An Energy-Based Concept for Yielding of Multidirectional FRP Composite Structures Using a Mesoscale Lamina Damage Model. *Polymers*. 2020; 12(1):157.
https://doi.org/10.3390/polym12010157

**Chicago/Turabian Style**

Rahimian Koloor, Seyed Saeid, Atefeh Karimzadeh, Noorfaizal Yidris, Michal Petrů, Majid Reza Ayatollahi, and Mohd Nasir Tamin.
2020. "An Energy-Based Concept for Yielding of Multidirectional FRP Composite Structures Using a Mesoscale Lamina Damage Model" *Polymers* 12, no. 1: 157.
https://doi.org/10.3390/polym12010157