346
S. Zacek et al.
∂ n t gp
∂δ ci
=
∂ n t
∂δ ci
1
n δ
[β
2 n
δ + (1 − β
2 )(
n
δ · n)n].
(20)
From Equation (3), the partial derivatives of the scalar effective traction are
∂t
∂η i
=
⎧
⎨
⎩
−2σ c
1 −
δ e
δ c1
δ e
δ 2
c1
0 ≤ δ e < δ c1
0
e l s e
for η i = δ c1
6σ c
δ e −δ c2
δ c3 −δ c2
− 1
δ e −δ c2
δ c3 −δ c2
δ e −δ c3
(δ c3 −δ c2 ) 2
δ c2 ≤ δ e < δ c3
0
e l s e
for η i = δ c2
−6σ c
δ e −δ c2
δ c3 −δ c2
− 1
δ e −δ c2
δ c3 −δ c2
δ e −δ c2
(δ c3 −δ c2 ) 2
δ c2 ≤ δ e < δ c3
0
e l s e .
for η i = δ c3
(21)
Acknowledgements This work has been supported through a grant No. FA9550-12-1-0445 to the
Center of Excellence on Integrated Materials Modeling (CEIMM) at Johns Hopkins University
(partners JHU, UIUC, UCSB), awarded by the AFOSR/RSL (Computational Mathematics Program, Manager Dr. A. Sayir) and AFRL/RX (Monitors Dr. C. Woodward and C. Przybyla).
References
1. I. Daniel, O. Ishai, Engineering Mechanics of Composite Materials (Oxford University Press,
New York, 2005)
2. J. Nairn, Matrix microcracking in composites, Polymer Matrix Composites, 2, 403–432 (2000)
3. K. Garrett, J. Bailey, Multiple transverse fracture in 90 ◦ cross-ply laminates of a glass fibrereinforced polyester. J. Mater. Sci. 12, 157–168 (1977)
4. J. Mayugo, P. Camanho, P. Maimi, C. Davila, Analytical modelling of transverse matrix
cracking of [±θ/90 n ] s composite laminates under multiaxial loading. Mech. Adv. Mater.
Struct. 17, 237–245 (2010)
5. L. Canal, J. Segurado, J. Llorca, Failure surface of epoxy-modified fiber-reinforced composites
under transverse tension and out-of-plane shear. Int. J. Solids Struct. 46, 2265–2274 (2009)
6. D. O’Dwyer, N. O’Dowd, C. McCarthy, Numerical micromechanical investigation of interfacial strength parameters in a carbon fibre composite material. J. Compos. Mater. 48(6),
749–760 (2014)
7. A. Louhghalam, S. Arwade, Prediction of incipient damage sites in composites using classifiers. Int. J. Damage Mech. 19, 233–260 (2010)
8. T. Okabe, M. Nishikawa, H. Toyoshima, A periodic unit-cell simulation of fiber arrangement
dependence on the transverse tensile failure in unidirectional carbon fiber reinforced composites. Int. J. Solids Struct. 48, 2948–2959 (2011)
9. T. Okabe, H. Sekine, K. Ishii, M. Nishikawa, N. Takeda, Numerical method for failure
simulation of unidirectional fiber-reinforced composites with spring element model. Compos.
Sci. Technol. 65, 921–933 (2005)
10. S. Torquato, Random Heterogeneous Materials, Microstructure and Macroscopic Properties
(Springer, New York, 2002)
11. T. Vaughan, C. McCarthy, Micromechanical modelling of the transverse damage behaviour in
fibre reinforced composites. Compos. Sci. Technol. 71, 388–396 (2011)
S. Zacek et al.
∂ n t gp
∂δ ci
=
∂ n t
∂δ ci
1
n δ
[β
2 n
δ + (1 − β
2 )(
n
δ · n)n].
(20)
From Equation (3), the partial derivatives of the scalar effective traction are
∂t
∂η i
=
⎧
⎨
⎩
−2σ c
1 −
δ e
δ c1
δ e
δ 2
c1
0 ≤ δ e < δ c1
0
e l s e
for η i = δ c1
6σ c
δ e −δ c2
δ c3 −δ c2
− 1
δ e −δ c2
δ c3 −δ c2
δ e −δ c3
(δ c3 −δ c2 ) 2
δ c2 ≤ δ e < δ c3
0
e l s e
for η i = δ c2
−6σ c
δ e −δ c2
δ c3 −δ c2
− 1
δ e −δ c2
δ c3 −δ c2
δ e −δ c2
(δ c3 −δ c2 ) 2
δ c2 ≤ δ e < δ c3
0
e l s e .
for η i = δ c3
(21)
Acknowledgements This work has been supported through a grant No. FA9550-12-1-0445 to the
Center of Excellence on Integrated Materials Modeling (CEIMM) at Johns Hopkins University
(partners JHU, UIUC, UCSB), awarded by the AFOSR/RSL (Computational Mathematics Program, Manager Dr. A. Sayir) and AFRL/RX (Monitors Dr. C. Woodward and C. Przybyla).
References
1. I. Daniel, O. Ishai, Engineering Mechanics of Composite Materials (Oxford University Press,
New York, 2005)
2. J. Nairn, Matrix microcracking in composites, Polymer Matrix Composites, 2, 403–432 (2000)
3. K. Garrett, J. Bailey, Multiple transverse fracture in 90 ◦ cross-ply laminates of a glass fibrereinforced polyester. J. Mater. Sci. 12, 157–168 (1977)
4. J. Mayugo, P. Camanho, P. Maimi, C. Davila, Analytical modelling of transverse matrix
cracking of [±θ/90 n ] s composite laminates under multiaxial loading. Mech. Adv. Mater.
Struct. 17, 237–245 (2010)
5. L. Canal, J. Segurado, J. Llorca, Failure surface of epoxy-modified fiber-reinforced composites
under transverse tension and out-of-plane shear. Int. J. Solids Struct. 46, 2265–2274 (2009)
6. D. O’Dwyer, N. O’Dowd, C. McCarthy, Numerical micromechanical investigation of interfacial strength parameters in a carbon fibre composite material. J. Compos. Mater. 48(6),
749–760 (2014)
7. A. Louhghalam, S. Arwade, Prediction of incipient damage sites in composites using classifiers. Int. J. Damage Mech. 19, 233–260 (2010)
8. T. Okabe, M. Nishikawa, H. Toyoshima, A periodic unit-cell simulation of fiber arrangement
dependence on the transverse tensile failure in unidirectional carbon fiber reinforced composites. Int. J. Solids Struct. 48, 2948–2959 (2011)
9. T. Okabe, H. Sekine, K. Ishii, M. Nishikawa, N. Takeda, Numerical method for failure
simulation of unidirectional fiber-reinforced composites with spring element model. Compos.
Sci. Technol. 65, 921–933 (2005)
10. S. Torquato, Random Heterogeneous Materials, Microstructure and Macroscopic Properties
(Springer, New York, 2002)
11. T. Vaughan, C. McCarthy, Micromechanical modelling of the transverse damage behaviour in
fibre reinforced composites. Compos. Sci. Technol. 71, 388–396 (2011)