# A Three-Phase Model Characterizing the Low-Velocity Impact Response of SMA-Reinforced Composites under a Vibrating Boundary Condition

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Three-Phase Model

#### 2.1. Material Property of the Glass Fiber/Epoxy Composite Laminate

#### 2.2. Material Property of the SMA

#### 2.3. Material Property of the Interphase

#### 2.4. Boundary Condition

## 3. The Effect of the Fixed Boundary Condition on Impact Resistance

#### 3.1. Composite Laminates

_{8}; 0° and 90° are the glass fiber’s layer angles in the X-direction. The sample (${L}_{x}\times {L}_{y}\times {L}_{\mathrm{z}}$ = 100 mm × 100 mm × 3.2 mm) was subjected to an impact from a rigid half-ball cylinder at the center of the top surface, as shown in Figure 1. The half ball’s diameter is 14 mm, and its mass is 8 kg (Steel). Two impact energies were considered in the tests (32 J and 64 J), and the corresponding impact velocities are 2.83 m/s and 4 m/s, respectively.

#### 3.2. Simulation Result: Damage State During the Impact Process

**Group A1**: The simulation results on the composite laminate under an impact at different times (0.0015 s, 0.0030 s, 0.0045 s, 0.0060 s, 0.0075 s, and 0.0090 s) are shown in Figure 3 and Figure 4. Under the lower impact energy (32 J), a generally elastic behavior of the composite laminate can be expressed as: the deformation is increased with time t from the initial state (t = 0) to the maximum deformation (t ≈ 0.0045 s), then the deformation is decreased.

**Group A2**: Different from group A1, the composite is destroyed completely under the higher impact energy (64 J), as shown in Figure 5. During the simulation process, the impactor was found to move along the top layer to the bottom layer of the composite without being bounced back (t > 0.0030 s). This can also be confirmed by the experimental result, as shown in Figure 5g. The velocity of the impactor was reduced in the breakdown process and then remained as a constant.

**Groups A3 and A4**: Embedding SMA alloys is an effective way to improve the impact resistance of composite laminates. As shown in Figure 7a, the SMA was stretched to a larger strain in the case of 32 J. In Figure 7b, a broken or invalid state of the SMA is obtained due to the larger strain, which is beyond the critical value. More specifically, five SMAs in the center region were chosen to demonstrate the working mechanism, as shown in Figure 8c,d.

#### 3.3. Simulation Result: Absorbed Energy and Contact Force

## 4. The Effect of the Vibrating Boundary Condition on Impact Resistance

#### 4.1. The Effect of Amplitude

_{tot}= 0.01 s). Several amplitudes A were chosen for the study, as shown in Table 3.

#### 4.2. The Effect of Frequency

#### 4.3. Statistical Analysis of the Damage State

_{max}, the energy at time 0.01 s, E

_{t = 0.01}, the maximum force, F

_{max}, and the average force, F

_{avg}, of different groups are shown. An E

_{max}= 32 J denotes a rebound behavior of the impactor. The average force is defined as:

_{i}within a time of 0.01 s. More importantly, the average value is still in accordance with A and f.

#### 4.4. Mathematical Expression: Effect of Amplitude and Frequency

_{a}and m

_{i}are parameters related to amplitude; and k

_{f}and n

_{i}are parameters related to frequency. Inserting Equation (16) into the energy equation, the relationship between absorbed energy and amplitude (or frequency) is obtained as follows:

## 5. Conclusions

- (1)
- Under a smaller amplitude (A < 0.0032 m) and a lower frequency (f < 500 cycles/s), the absorbed energy and contact force of composite laminates are similar to that under a fixed boundary condition. In contrast, both a high frequency and a high amplitude can weaken the impact resistance of composite laminates, where extensive damage can be observed rather than a hole-shaped damage region.
- (2)
- The absolute value of amplitude has a greater influence on the impact resistance than the movement direction of the laminates at the initial time. The absorbed energy and contact force in the positive direction are about 20% larger than that in the negative direction.
- (3)
- Embedding an SMA can improve the impact resistance of composite laminates due to the superelasticity. In this study, embedding an SMA was found to increase the absorbed energy and contact force by about 15–30%. Also, embedding an SMA can change the damage morphology with respect to shape and proportion.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$A$ | amplitude |

${A}_{s}$, ${A}_{f}$ | start and finish temperature of austenite phase |

${a}_{m}$, ${a}_{a}$ | material parameters of SMA |

${C}_{m}$, ${C}_{a}$ | thermal coefficient of martensite phase and austenite phase |

${c}_{ij}$ | stiffness matrix |

${d}_{mt}$, ${d}_{mc}$, ${d}_{ft}$, ${d}_{fc}$ | damage variables |

$E$, ${E}_{\mathrm{max}}$, ${E}_{\mathrm{t}=0.01\mathrm{s}}$ | absorbed energy, maximum value of energy, and absorbed energy at 0.01 s |

${E}_{i}$, ${E}_{ij}$ | elastic modulus of laminate in the i-direction, shear modulus in the ij-direction |

${E}_{s}$, ${E}_{a}$, ${E}_{m}$ | elastic modulus of SMA, elastic modulus for martensite phase and austenite phase |

${F}_{i}$, ${F}_{\mathrm{max}}$, ${F}_{\mathrm{avg}}$ | contact force, maximum value of force, average value of force |

$f$ | frequency |

${g}_{0}$, ${g}_{i}$ | parameters of the relaxation modulus |

${k}_{a}$, ${k}_{f}$ | parameter related to amplitude, parameter related to frequency |

${L}_{x}$, ${L}_{y}$, ${L}_{z}$ | sample dimension in the x, y, and z direction |

${M}_{s}$, ${M}_{f}$ | start and finish temperature of martensite phase |

$m$, ${m}_{i}$ | mass, parameters related to amplitude |

${n}_{i}$ | parameters related to frequency |

${S}_{23}$, ${S}_{13}$, ${S}_{12}$ | ultimate shear strength in the 23, 13, and 12 direction |

${s}_{mt}$, ${s}_{mc}$ | reduction factors |

$T$ | temperature |

$t$, ${t}^{\prime}$, ${t}_{i}$, ${t}_{tot}$ | time, new time variable, time parameters of the relaxation modulus, total time |

${V}_{0}$, $\Delta V$ | initial velocity, velocity increment |

${X}_{T}$, ${X}_{C}$ | tensile strength and compressive strength in the longitudinal direction |

${Y}_{T}$, ${Y}_{C}$ | tensile strength and compressive strength in the transverse direction |

${\epsilon}_{ij}$ | strain of laminate in the ij-direction |

${\epsilon}_{s}$, ${\epsilon}_{L}$ | strain of SMA, maximum residual strain of SMA |

${\epsilon}_{in}$, ${\dot{\epsilon}}_{in}$ | strain of interphase, strain rate of interphase |

$\mathsf{\Theta}$ | thermal coefficient |

${\sigma}_{ij}$ | stress in the ij-direction |

${\sigma}_{s}$, ${\sigma}_{in}$ | stress of SMA, stress of interphase |

${\upsilon}_{ij}$ | Poisson’s ratio of laminate in the ij-direction |

$\mathsf{\Omega}$ | transformation coefficient |

$\xi $, ${\xi}_{S}$, ${\xi}_{T}$ | martensite fraction, martensite fraction of stress, and temperature effect |

${\xi}_{0}$, ${\xi}_{{S}_{0}}$, ${\xi}_{{T}_{0}}$ | martensite fraction at the initial state, initial stress, and initial temperature |

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**Figure 1.**A schematic of the shape memory alloy (SMA)-reinforced composite samples: (

**a**) Fixed boundary; (

**b**) Vibrating boundary.

**Figure 3.**The simulation results on the composite laminates at different times (group A1), and the bottom view of the sample after impact: (

**a**) t = 0.0015 s; (

**b**) t = 0.0030 s; (

**c**) t = 0.0045 s; (

**d**) t = 0.0060 s; (

**e**) t = 0.0075 s; (

**f**) t = 0.0090 s; (

**g**) bottom view of sample.

**Figure 4.**The middle section of different layers during the impact, group A1: (

**a**) layer 16; (

**b**) layer 8; (

**c**) layer 1.

**Figure 5.**The simulation results on the composite laminates at different times, group A2: (

**a**) t = 0.0015 s; (

**b**) t = 0.0030 s; (

**c**) t = 0.0045 s; (

**d**) t = 0.0060 s; (

**e**) t = 0.0075 s; (

**f**) t = 0.0090 s; (

**g**) bottom view of sample.

**Figure 6.**The middle section of different layers during the impact, group A2: (

**a**) layer 16; (

**b**) layer 8; (

**c**) layer 1.

**Figure 7.**The simulation results on the SMA-reinforced composite laminates at different times: (

**a**) group A3, t = 0.0050 s; (

**b**) group A4, t = 0.0045 s.

**Figure 8.**The middle section of different layers during the impact: (

**a**) layers in group A3; (

**b**) layers in group A4; (

**c**) the SMA in group A3; (

**d**) the SMA in group A4.

**Figure 9.**A comparison between the simulation results and the experimental results. (

**a**) absorbed energy–time history; (

**b**) contact force–time history.

**Figure 10.**The fracture morphology of the top layer of the composite laminate under different amplitudes: (

**a**) Group B1; (

**b**) Group B2; (

**c**) Group B3; (

**d**) Group B4; (

**e**) Group B5; (

**f**) Group B6; (

**g**) Group B7; (

**h**) Group B8.

**Figure 11.**The fracture morphology of the top layer of the SMA-reinforced composite laminate under different amplitudes: (

**a**) Group C1; (

**b**) Group C2; (

**c**) Group C3; (

**d**) Group C4.

**Figure 12.**The middle section of the half model during the impact: (

**a**) group B1; (

**b**) group B3; (

**c**) group B5; (

**d**) group B7; (

**e**) group B4.

**Figure 13.**The middle section of the half model during the impact: (

**a**) group C2; (

**b**) group C3; (

**c**) group C4.

**Figure 14.**The analysis of the impact resistance of the composite laminate in group B and C: (

**a**) absorbed energy; (

**b**) contact force.

**Figure 15.**The fracture morphology of the top layer with different frequencies: (

**a**) Group D1; (

**b**) Group D2; (

**c**) Group D3; (

**d**) Group D4; (

**e**) Group D5; (

**f**) Group E1; (

**g**) Group E2; (

**h**) Group E3; (

**i**) Group E4; (

**j**) Group E5.

**Figure 17.**The analysis of the impact resistance of the composite laminate in group D and group E: (

**a**) absorbed energy; (

**b**) contact force.

**Figure 18.**The statistics of the damage area in different groups: (

**a**) layer 16; (

**b**) layer 8; (

**c**) layer 1.

**Figure 20.**A comparative study of the statistical results and the simulation results: (

**a**) E

_{t = 0.01}; (

**b**) F

_{avg}.

Mechanical Constants | Values |
---|---|

Young’s modulus/GPa (E_{1}, E_{2}, E_{3}) | 55.2, 18.4, 18.4 |

Poisson’s ratio (${\upsilon}_{12}$, ${\upsilon}_{13}$, ${\upsilon}_{23}$) | 0.27, 0.27, 0.43 |

Shear modulus/GPa (E_{12}, E_{13}, E_{23}) | 13.8, 13.8, 13.8 |

Ultimate tensile stress/MPa (X_{T}, Y_{T}, Z_{T}) | 1656, 73.8, 73.8 |

Ultimate compressive stress/MPa (X_{C}, Y_{C}, Z_{C}) | 1656, 91.8, 91.8 |

Ultimate shear stress/MPa (S_{12}, S_{13}, S_{23}) | 117.6, 117.6, 117.6 |

Stacking Sequence | Impact Energy/J | |
---|---|---|

Group A1 | [0°,90°]_{8} | 32 |

Group A2 | [0°,90°]_{8} | 64 |

Group A3 | [(0°,90°)_{4},SMA, (0°,90°)_{4}] | 32 |

Group A4 | [(0°,90°)_{4},SMA, (0°,90°)_{4}] | 64 |

Group B1 | Group B2 | Group B3 | Group B4 | Group B5 | Group B6 | Group B7 | Group B8 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A (m) | 0.0016 | −0.0016 | 0.0032 | −0.0032 | 0.008 | −0.008 | 0.016 | −0.016 | ||||||||

f (c/s) | 1000 | |||||||||||||||

SMA Stacking | NO [0°,90°] _{8} | |||||||||||||||

Group C1 | Group C2 | Group C3 | Group C4 | |||||||||||||

A (m) | 0.0016 | 0.0032 | 0.008 | 0.016 | ||||||||||||

f (c/s) | 1000 | |||||||||||||||

SMA Stacking | YES [(0°,90°) _{4}, SMA, (0°,90°)_{4}] | |||||||||||||||

Group D1 | Group D2 | Group D3 | Group D4 | Group D5 | ||||||||||||

A (m) | 0.0032 | |||||||||||||||

f (c/s) | 100 | 200 | 500 | 2000 | 10,000 | |||||||||||

SMA Stacking | NO [0°,90°] _{8} | |||||||||||||||

Group E1 | Group E2 | Group E3 | Group E4 | Group E5 | ||||||||||||

A (m) | 0.0032 | |||||||||||||||

f (c/s) | 100 | 200 | 500 | 2000 | 10,000 | |||||||||||

SMA Stacking | YES [(0°,90°) _{4}, SMA, (0°,90°)_{4}] |

A1 | B1 | B2 | B3 | B4 | B5 | B6 | A7 | B8 | |
---|---|---|---|---|---|---|---|---|---|

E_{max} (J) | 32 | 32 | 32 | 21.67 | 18.27 | 4.30 | 3.72 | 0.30 | 0.80 |

E_{t}_{= 0.01s} (J) | 28.70 | 30.04 | 30.79 | 9.24 | 5.90 | 4.24 | 3.56 | 0.30 | 0.80 |

F_{max} (N) | 7.04 | 7.74 | 7.69 | 7.92 | 7.35 | 8.39 | 6.86 | 0.86 | 1.13 |

F_{avg} (N) | 1.93 | 1.70 | 1.81 | 0.83 | 0.65 | 0.16 | 0.13 | 0.01 | 0.02 |

A3 | C1 | C2 | C3 | C4 | |||||

E_{max} (J) | 32 | 32 | 32 | 13.20 | 3.74 | ||||

E_{t}_{= 0.01s} (J) | 25.14 | 28.60 | 31.97 | 13.17 | 3.74 | ||||

F_{max} (N) | 6.89 | 10.77 | 10.56 | 10.52 | 8.71 | ||||

F_{avg} (N) | 3.32 | 2.98 | 2.31 | 0.52 | 0.12 | ||||

D1 | D2 | D3 | D4 | D5 | |||||

E_{max} (J) | 32 | 32 | 25.38 | 6.21 | 1.75 | ||||

E_{t}_{= 0.01s} (J) | 31.98 | 31.92 | 23.68 | 4.69 | 1.74 | ||||

F_{max} (N) | 7.40 | 7.23 | 7.45 | 7.29 | 1.44 | ||||

F_{avg} (N) | 2.31 | 2.36 | 1.11 | 0.17 | 0.06 | ||||

E1 | E2 | E3 | E4 | E5 | |||||

E_{max} (J) | 32 | 32 | 32 | 11.86 | 1.79 | ||||

E_{t}_{= 0.01s} (J) | 28.77 | 29.94 | 30.70 | 11.86 | 1.79 | ||||

F_{max} (N) | 10.15 | 9.79 | 11.92 | 9.02 | 2.36 | ||||

F_{avg} (N) | 2.95 | 2.81 | 2.69 | 0.46 | 0.05 |

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## Share and Cite

**MDPI and ACS Style**

Chang, M.; Kong, F.; Sun, M.; He, J. A Three-Phase Model Characterizing the Low-Velocity Impact Response of SMA-Reinforced Composites under a Vibrating Boundary Condition. *Materials* **2019**, *12*, 7.
https://doi.org/10.3390/ma12010007

**AMA Style**

Chang M, Kong F, Sun M, He J. A Three-Phase Model Characterizing the Low-Velocity Impact Response of SMA-Reinforced Composites under a Vibrating Boundary Condition. *Materials*. 2019; 12(1):7.
https://doi.org/10.3390/ma12010007

**Chicago/Turabian Style**

Chang, Mengzhou, Fangyun Kong, Min Sun, and Jian He. 2019. "A Three-Phase Model Characterizing the Low-Velocity Impact Response of SMA-Reinforced Composites under a Vibrating Boundary Condition" *Materials* 12, no. 1: 7.
https://doi.org/10.3390/ma12010007