A phase-field damage model for orthotropic materials is proposed and used to simulate delamination of orthotropic laminated composites. Using the deviatoric and hydrostatic tensile components of the stress tensor for elastic orthotropic materials, a degraded elastic free energy that can accommodate damage is derived. The governing equations follow from the principle of virtual power and the resulting damage model, by its construction, conforms with the physical relevant condition of no matter interpenetration along the crack faces. The model also dispenses with the traction separation law, an extraneous hypothesis conventionally brought in to model the interlaminar zones. The model is assessed through numerical simulations on delaminations in mode I, mode II, and another such problem with multiple initial notches. The present method is able to reproduce nearly all the features of the experimental load displacement curves, allowing only for small deviations in the softening regime. Numerical results also show forth a superior performance of the proposed method over existing approaches based on a cohesive law.

References

References
1.
Whitney
,
J. M.
, and
Nuismer
,
R. J.
,
1974
, “
Stress Fracture Criteria for Laminated Composites Containing Stress Concentrations
,”
J. Compos. Mater.
,
8
(
3
), pp.
253
265
.
2.
Irwin
,
G. R.
,
1957
, “
Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate
,”
ASME J. Appl. Mech.
,
24
(3), pp. 361–364.
3.
Rice
,
J. R.
,
1968
, “
A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks
,”
ASME J. Appl. Mech.
,
35
(
2
), pp.
379
386
.
4.
Hellen
,
T.
,
1975
, “
On the Method of Virtual Crack Extensions
,”
Int. J. Numer. Methods Eng.
,
9
(
1
), pp.
187
207
.
5.
Parks
,
D. M.
,
1974
, “
A Stiffness Derivative Finite Element Technique for Determination of Crack Tip Stress Intensity Factors
,”
Int. J. Fract.
,
10
(
4
), pp.
487
502
.
6.
Verhoosel
,
C. V.
, and
Borst
,
R.
,
2013
, “
A Phase-Field Model for Cohesive Fracture
,”
Int. J. Numer. Methods Eng.
,
96
(
1
), pp.
43
62
.
7.
May
,
S.
,
Vignollet
,
J.
, and
de Borst
,
R.
,
2015
, “
A Numerical Assessment of Phase-Field Models for Brittle and Cohesive Fracture: Γ-Convergence and Stress Oscillations
,”
Eur. J. Mech. -A/Solids
,
52
, pp.
72
84
.
8.
Simo
,
J. C.
,
Oliver
,
J.
, and
Armero
,
F.
,
1993
, “
An Analysis of Strong Discontinuities Induced by Strain-Softening in Rate-Independent Inelastic Solids
,”
Comput. Mech.
,
12
(
5
), pp.
277
296
.
9.
Belytschko
,
T.
, and
Gracie
,
R.
,
2007
, “
On Xfem Applications to Dislocations and Interfaces
,”
Int. J. Plasticity
,
23
(
10
), pp.
1721
1738
.
10.
Oliver
,
J.
,
Huespe
,
A.
,
Pulido
,
M.
, and
Chaves
,
E.
,
2002
, “
From Continuum Mechanics to Fracture Mechanics: The Strong Discontinuity Approach
,”
Eng. Fract. Mech.
,
69
(
2
), pp.
113
136
.
11.
Huespe
,
A. E.
,
Needleman
,
A.
,
Oliver
,
J.
, and
Sánchez
,
P. J.
,
2009
, “
A Finite Thickness Band Method for Ductile Fracture Analysis
,”
Int. J. Plasticity
,
25
(
12
), pp.
2349
2365
.
12.
Huespe
,
A.
,
Needleman
,
A.
,
Oliver
,
J.
, and
Sánchez
,
P.
,
2012
, “
A Finite Strain, Finite Band Method for Modeling Ductile Fracture
,”
Int. J. Plasticity
,
28
(
1
), pp.
53
69
.
13.
Rahaman
,
M. M.
,
Deepu
,
S.
,
Roy
,
D.
, and
Reddy
,
J.
,
2015
, “
A Micropolar Cohesive Damage Model for Delamination of Composites
,”
Compos. Struct.
,
131
, pp.
425
432
.
14.
Bazant
,
Z. P.
, and
Pijaudier-Cabot
,
G.
,
1988
, “
Nonlocal Continuum Damage, Localization Instability and Convergence
,”
ASME J. Appl. Mech.
,
55
(
2
), pp.
287
293
.
15.
Frémond
,
M.
, and
Nedjar
,
B.
,
1996
, “
Damage, Gradient of Damage and Principle of Virtual Power
,”
Int. J. Solids Struct.
,
33
(
8
), pp.
1083
1103
.
16.
Alfano
,
G.
, and
Crisfield
,
M.
,
2001
, “
Finite Element Interface Models for the Delamination Analysis of Laminated Composites: Mechanical and Computational Issues
,”
Int. J. Numer. Methods Eng.
,
50
(
7
), pp.
1701
1736
.
17.
Škec
,
L.
,
Jelenić
,
G.
, and
Lustig
,
N.
,
2015
, “
Mixed-Mode Delamination in 2D Layered Beam Finite Elements
,”
Int. J. Numer. Methods Eng.
,
104
(
8
), pp.
767
788
.
18.
Aranson
,
I.
,
Kalatsky
,
V.
, and
Vinokur
,
V.
,
2000
, “
Continuum Field Description of Crack Propagation
,”
Phys. Rev. Lett.
,
85
(
1
), pp.
118
121
.
19.
Lemaitre
,
J.
,
1986
, “
Local Approach of Fracture
,”
Eng. Fract. Mech.
,
25
(
5–6
), pp.
523
537
.
20.
Spatschek
,
R.
,
Brener
,
E.
, and
Karma
,
A.
,
2011
, “
Phase Field Modeling of Crack Propagation
,”
Philos. Mag.
,
91
(
1
), pp.
75
95
.
21.
Hakim
,
V.
, and
Karma
,
A.
,
2009
, “
Laws of Crack Motion and Phase-Field Models of Fracture
,”
J. Mech. Phys. Solids
,
57
(
2
), pp.
342
368
.
22.
Da Silva
,
M. N.
,
Duda
,
F. P.
, and
Fried
,
E.
,
2013
, “
Sharp-Crack Limit of a Phase-Field Model for Brittle Fracture
,”
J. Mech. Phys. Solids
,
61
(
11
), pp.
2178
2195
.
23.
Fried
,
E.
, and
Gurtin
,
M. E.
,
2003
, “
The Role of the Configurational Force Balance in the Nonequilibrium Epitaxy of Films
,”
J. Mech. Phys. Solids
,
51
(
3
), pp.
487
517
.
24.
Karma
,
A.
, and
Lobkovsky
,
A. E.
,
2004
, “
Unsteady Crack Motion and Branching in a Phase-Field Model of Brittle Fracture
,”
Phys. Rev. Lett.
,
92
(
24
), p.
245510
.
25.
Henry
,
H.
, and
Levine
,
H.
,
2004
, “
Dynamic Instabilities of Fracture Under Biaxial Strain Using a Phase Field Model
,”
Phys. Rev. Lett.
,
93
(
10
), p.
105504
.
26.
Bourdin
,
B.
,
Francfort
,
G. A.
, and
Marigo
,
J.-J.
,
2000
, “
Numerical Experiments in Revisited Brittle Fracture
,”
J. Mech. Phys. Solids
,
48
(
4
), pp.
797
826
.
27.
Miehe
,
C.
,
Welschinger
,
F.
, and
Hofacker
,
M.
,
2010
, “
Thermodynamically Consistent Phase-Field Models of Fracture: Variational Principles and Multi-Field Fe Implementations
,”
Int. J. Numer. Methods Eng.
,
83
(
10
), pp.
1273
1311
.
28.
Francfort
,
G. A.
, and
Marigo
,
J.-J.
,
1998
, “
Revisiting Brittle Fracture as an Energy Minimization Problem
,”
J. Mech. Phys. Solids
,
46
(
8
), pp.
1319
1342
.
29.
Griffith
,
A. A.
,
1921
, “
The Phenomena of Rupture and Flow in Solids
,”
Philos. Trans. R. Soc. London. Ser. A
,
221
(582–593), pp.
163
198
.
30.
Gurtin
,
M. E.
,
1996
, “
Generalized Ginzburg-Landau and Cahn-Hilliard Equations Based on a Microforce Balance
,”
Phys. D: Nonlinear Phenom.
,
92
(
3–4
), pp.
178
192
.
31.
Duda
,
F. P.
,
Ciarbonetti
,
A.
,
Sánchez
,
P. J.
, and
Huespe
,
A. E.
,
2015
, “
A Phase-Field/Gradient Damage Model for Brittle Fracture in Elastic–Plastic Solids
,”
Int. J. Plasticity
,
65
, pp.
269
296
.
32.
Coleman
,
B. D.
, and
Noll
,
W.
,
1963
, “
The Thermodynamics of Elastic Materials With Heat Conduction and Viscosity
,”
Archive Rational Mech. Anal.
,
13
(
1
), pp.
167
178
.
33.
Borden
,
M. J.
,
Hughes
,
T. J.
,
Landis
,
C. M.
, and
Verhoosel
,
C. V.
,
2014
, “
A Higher-Order Phase-Field Model for Brittle Fracture: Formulation and Analysis Within the Isogeometric Analysis Framework
,”
Comput. Methods Appl. Mech. Eng.
,
273
, pp.
100
118
.
34.
Miehe
,
C.
,
Hofacker
,
M.
, and
Welschinger
,
F.
,
2010
, “
A Phase Field Model for Rate-Independent Crack Propagation: Robust Algorithmic Implementation Based on Operator Splits
,”
Comput. Methods Appl. Mech. Eng.
,
199
(
45
), pp.
2765
2778
.
35.
Farrell
,
P.
, and
Maurini
,
C.
,
2017
, “
Linear and Nonlinear Solvers for Variational Phase-Field Models of Brittle Fracture
,”
Int. J. Numer. Methods Eng.
,
109
(
5
), pp.
648
667
.
36.
De Morais
,
A.
,
De Moura
,
M.
,
Marques
,
A.
, and
De Castro
,
P.
,
2002
, “
Mode-I Interlaminar Fracture of Carbon/Epoxy Cross-Ply Composites
,”
Compos. Sci. Technol.
,
62
(
5
), pp.
679
686
.
37.
Turon
,
A.
,
Camanho
,
P. P.
,
Costa
,
J.
, and
Dávila
,
C.
,
2006
, “
A Damage Model for the Simulation of Delamination in Advanced Composites Under Variable-Mode Loading
,”
Mech. Mater.
,
38
(
11
), pp.
1072
1089
.
38.
Robinson
,
P.
,
Besant
,
T.
, and
Hitchings
,
D.
,
2000
, “
Delamination Growth Prediction Using a Finite Element Approach
,”
Eur. Struct. Integrity Soc.
,
27
, pp.
135
147
.
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