Fracture Mechanics Solutions for Interfacial Cracks between Compressible Thin Layers and Substrates
Abstract
:1. Introduction
2. Materials and Methods
2.1. Analytical Model: Materials with
2.2. Computational Model
2.3. Semi-Analytical Model—General Material Mismatch
3. Results
3.1. Finite Element Results for Materials with
3.2. Tabulated Coefficients for the Semi-Analytic Solutions in Equation (13)—Materials with
3.3. Explicit Solutions for Soft/Stiff Material Substrates and Nearly Homogeneous Systems—
- Phase angle associated with the axial force, (in radians) [28]:
- Phase angle associated with shear, (in radians):
- The dimensionless function which defines the energy release rate associated with shear, :
4. Discussion
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
Appendix B. Finite Element Model
References
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Dundurs’ Parameters | ||
---|---|---|
Glass-epoxy/polymer foam | 0.99 | 0.28 |
Glass/epoxy | 0.94 | 0.19 |
Al/PMMA. | 0.91 | 0.23 |
ITO/PET. | 0.97 | 0.2 |
Ni/Polycarbonate | 0.975 | 0.2 |
Al/Alfoam. | 0.82 | 0.23 |
Y2O3-ZrO2/β-(Ni,Pt)Al. | 0.63 | 0.31 |
Au/PDMS | ≈1 | ≈0 |
SiO2/Epoxy | 0.89 | 0.21 |
SiC/SiO2 | 0.72 | 0.38 |
−1 | −0.999 | −0.99 | −0.9 | −0.8 | −0.6 | −0.4 | −0.2 | 0 | |
0.794183 | 0.794268 | 0.795042 | 0.802955 | 0.812157 | 0.832039 | 0.854295 | 0.879592 | 0.908934 | |
0.416984 | 0.417119 | 0.418335 | 0.430885 | 0.445719 | 0.478693 | 0.517208 | 0.563241 | 0.619981 | |
0.2 | 0.4 | 0.6 | 0.8 | 0.9 | 0.99 | 0.998 | 0.999 | 1 | |
0.943943 | 0.987518 | 1.045755 | 1.136137 | 1.214548 | 1.395760 | 1.466932 | 1.48809 | π/2 | |
0.693045 | 0.793630 | 0.949054 | 1.258612 | 1.637033 | 3.676521 | 6.33202 | 7.98856 | – |
β = −0.4 | β = −0.3 | β = −0.2 | β = −0.1 | β = 0 | β = 0.1 | β = 0.2 | β = 0.3 | β = 0.4 | |
---|---|---|---|---|---|---|---|---|---|
α = −0.99 | 64.0° | 59.0° | 54.3° | 49.8° | 45.2° (45.553°) | ||||
α = −0.9 | 64.7° | 59.8° | 55.2° | 50.6° | 46° (46.006°) | ||||
α = −0.8 | 64.4° | 60.2° | 55.5° | 51.0° | 46.7° (46.533°) | ||||
α = 0.8 | 65.7° (65.096°) | 62.0° | 57.0° | 54.9° | 50.3° | ||||
α = 0.9 | 69.6° (69.588°) | 66.1° | 62.6° | 58.9° | 55.1° | ||||
α = 0.99 | 80° (79.971°) | 77.7° | 75.4° | 73.1° | 70.7° | ||||
α = 0.998 | 84.1° (84.049°) | 82.7° | 81.4° | 80.1° | 78.8° | ||||
α = 0.999 | 85.3° (85.261°) | 84.3° | 83.4° | 82.6° | 81.8° |
β = −0.4 | β = −0.3 | β = −0.2 | β = −0.1 | β = 0 | β = 0.1 | β = 0.2 | β = 0.3 | β = 0.4 | |
---|---|---|---|---|---|---|---|---|---|
α = −0.99 | 18.4° | 15.0° | 11.7° | 8.6° | 5.4° (5.634°) | ||||
α = −0.9 | 17.9° | 14.6° | 11.4° | 8.3° | 5.2° (5.252°) | ||||
α = −0.8 | 17.2° | 14.0° | 10.9° | 7.8° | 4.8° (4.818°) | ||||
α = −0.6 | 12.7° | 9.7° | 6.8° | 3.9° (3.920°) | |||||
α = −0.4 | 11.3° | 8.5° | 5.7° | 3.0° (2.978°) | 0.2° | ||||
α = −0.2 | 7.2° | 4.6° | 2.0° (1.993°) | −0.7° | |||||
α = 0 | 5.8° | 3.4° | 1.0° (0.963°) | −1.5° | −4.1° | ||||
α = 0.2 | 2.1° | −0.1° (−0.107°) | −2.4° | −4.8° | |||||
α = 0.4 | 0.7° | −1.2° (−1.208°) | −3.2° | −5.4° | −7.6° | ||||
α = 0.6 | −2.3° (−2.301°) | −4.0° | −5.8° | −7.7° | |||||
α = 0.8 | −3.2° (−3.238°) | −4.4° | −5.7° | −7.0° | −8.5° | ||||
α = 0.9 | −3.4° (−3.427°) | −4.1° | −4.8° | −5.7° | −6.6° | ||||
α = 0.99 | −2.3° (−2.284°) | −1.5° | −0.7° | 0.2° | 1.0° | ||||
α = 0.998 | −1.3° (−1.436°) | 0.4° | 2.2° | 4.1° | 6.0° |
β = −0.4 | β = −0.3 | β = −0.2 | β = −0.1 | β = 0 | β = 0.1 | β = 0.2 | β = 0.3 | β = 0.4 | |
---|---|---|---|---|---|---|---|---|---|
α = −0.99 | 1.444 | 1.489 | 1.528 | 1.563 | 1.593 (1.597) | ||||
α = −0.9 | 1.465 | 1.510 | 1.550 | 1.585 | 1.616 (1.617) | ||||
α = −0.8 | 1.487 | 1.533 | 1.573 | 1.609 | 1.640 (1.641) | ||||
α = −0.6 | 1.586 | 1.627 | 1.663 | 1.695 (1.696) | |||||
α = −0.4 | 1.650 | 1.692 | 1.729 | 1.762 (1.762) | 1.791 | ||||
α = −0.2 | 1.773 | 1.811 | 1.844 (1.845) | 1.874 | |||||
α = 0 | 1.878 | 1.916 | 1.951 (1.951) | 1.981 | 2.007 | ||||
α = 0.2 | 2.058 | 2.093 (2.093) | 2.123 | 2.150 | |||||
α = 0.4 | 2.261 | 2.297 (2.298) | 2.328 | 2.355 | 2.378 | ||||
α = 0.6 | 2.627 (2.628) | 2.658 | 2.685 | 2.708 | |||||
α = 0.8 | 3.316 (3.317) | 3.347 | 3.374 | 3.395 | 3.411 | ||||
α = 0.9 | 4.190 (4.193) | 4.222 | 4.247 | 4.266 | 4.279 | ||||
α = 0.99 | 9.1 (9.089) | 9.1 | 9.1 | 9.1 | 9.1 | ||||
α = 0.998 | 15.3 (15.559) | 15.6 | 15.6 | 15.5 | 15.5 |
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Massabò, R.; Ustinov, K.; Barbieri, L.; Berggreen, C. Fracture Mechanics Solutions for Interfacial Cracks between Compressible Thin Layers and Substrates. Coatings 2019, 9, 152. https://doi.org/10.3390/coatings9030152
Massabò R, Ustinov K, Barbieri L, Berggreen C. Fracture Mechanics Solutions for Interfacial Cracks between Compressible Thin Layers and Substrates. Coatings. 2019; 9(3):152. https://doi.org/10.3390/coatings9030152
Chicago/Turabian StyleMassabò, Roberta, Konstantin Ustinov, Luca Barbieri, and Christian Berggreen. 2019. "Fracture Mechanics Solutions for Interfacial Cracks between Compressible Thin Layers and Substrates" Coatings 9, no. 3: 152. https://doi.org/10.3390/coatings9030152
APA StyleMassabò, R., Ustinov, K., Barbieri, L., & Berggreen, C. (2019). Fracture Mechanics Solutions for Interfacial Cracks between Compressible Thin Layers and Substrates. Coatings, 9(3), 152. https://doi.org/10.3390/coatings9030152