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Abstract

This paper proposes a new thermodynamically consistent anisotropic continuum damage mechanics model that the degradation of material is described by a set of damage variables. We adopt a set of novel equivalent strains which are functions of both the strain and the damage state variables as well. A viscous stabilization method is considered to improve the convergence during the material softening. The evolution of damage state variables is updated through a Newton–Raphson iterative process. The expression of the consistent tangent stiffness matrix is also derived. A smoothed exponential damage evolution shape function with rescaling regularization method is proposed to describe the material softening process and the mesh dependency phenomenon is relieved. Several sets of numerical examples including three-point-bending of a notched beam, uniaxial tension of unnotched and notched plate of composite material with multiple layups are presented to verify the anisotropic damage models.

References

1.
Kachanov
,
L. M.
,
1986
,
Introduction to Continuum Damage Mechanics
,
Springer
,
Dordrecht
.
2.
Krajcinovic
,
D.
, and
Lemaitre
,
J.
,
1987
,
Continuum Damage Mechanics: Theory and Applications
,
Springer Verlag
,
Wien
.
3.
Lemaitre
,
J.
,
1992
,
A Course on Damage Mechanics
, 2nd ed.,
Springer Berlin
,
Heidelberg
.
4.
Chaboche
,
J.-L.
,
1988
, Continuum Damage Mechanics: Part I–General Concepts.
5.
Voyiadjis
,
G. Z.
, and
Kattan
,
P. I.
,
2005
,
Damage Mechanics
,
CRC Press
,
Boca Raton, FL
.
6.
Kachanov
,
L. M.
,
1999
, “
Rupture Time Under Creep Conditions
,”
Int. J. Fract.
,
97
, pp.
11
18
.
7.
Murakami
,
S.
, and
Kamiya
,
K.
,
1997
, “
Constitutive and Damage Evolution Equations of Elastic-Brittle Materials Based on Irreversible Thermodynamics
,”
Int. J. Mech. Sci.
,
39
(
4
), pp.
473
486
.
8.
Kuna-Ciskał
,
H.
, and
Skrzypek
,
J. J.
,
2004
, “
CDM Based Modelling of Damage and Fracture Mechanisms in Concrete Under Tension and Compression
,”
Eng. Fract. Mech.
,
71
(
4
), pp.
681
698
.
9.
Chow
,
C.
, and
Wang
,
J.
,
1988
, “
A Finite Element Analysis of Continuum Damage Mechanics for Ductile Fracture
,”
Int. J. Fract.
,
38
(
2
), pp.
83
102
.
10.
Bonora
,
N.
,
1997
, “
A Nonlinear CDM Model for Ductile Failure
,”
Eng. Fract. Mech.
,
58
(
1
), pp.
11
28
11.
Bonora
,
N.
, and
Newaz
,
G.
,
1998
, “
Low Cycle Fatigue Life Estimation for Ductile Metals Using a Nonlinear Continuum Damage Mechanics Model
,”
Int. J. Solids Struct.
,
35
(
16
), pp.
1881
1894
12.
Pandey
,
V.
,
Singh
,
I.
,
Mishra
,
B.
,
Ahmad
,
S.
,
Venugopal Rao
,
A.
, and
Kumar
,
V.
,
2019
, “
A New Framework Based on Continuum Damage Mechanics and XFEM for High Cycle Fatigue Crack Growth Simulations
,”
Eng. Fract. Mech.
,
206
, pp.
172
200
13.
Brünig
,
M.
,
2003
, “
An Anisotropic Ductile Damage Model Based on Irreversible Thermodynamics
,”
Int. J. Plast.
,
19
(
10
), pp.
1679
1713
14.
Basaran
,
C.
, and
Nie
,
S.
,
2004
, “
An Irreversible Thermodynamics Theory for Damage Mechanics of Solids
,”
Int. J. Damage Mech.
,
13
(
3
), pp.
205
223
15.
Voyiadjis
,
G. Z.
, and
Deliktas
,
B.
,
2000
, “
A Coupled Anisotropic Damage Model for the Inelastic Response of Composite Materials
,”
Comput. Methods Appl. Mech. Eng.
,
183
(
3
), pp.
159
199
16.
Alabdullah
,
M.
, and
Ghoniem
,
N. M.
,
2020
, “
A Thermodynamics-Based Damage Model for the Non-Linear Mechanical Behavior of Sic/sic Ceramic Matrix Composites in Irradiation and Thermal Environments
,”
Int. J. Damage Mech.
,
29
(
10
), pp.
1569
1599
17.
Ju
,
J.
,
1990
, “
Isotropic and Anisotropic Damage Variables in Continuum Damage Mechanics
,”
J. Eng. Mech.
,
116
(
12
), pp.
2764
2770
18.
Voyiadjis
,
G. Z.
, and
Kattan
,
P. I.
,
2009
, “
A Comparative Study of Damage Variables in Continuum Damage Mechanics
,”
Int. J. Damage Mech.
,
18
(
4
), pp.
315
340
19.
Richard
,
B.
,
Ragueneau
,
F.
,
Cremona
,
C.
, and
Adelaide
,
L.
,
2010
, “
Isotropic Continuum Damage Mechanics for Concrete Under Cyclic Loading: Stiffness Recovery, Inelastic Strains and Frictional Sliding
,”
Eng. Fract. Mech.
,
77
(
8
), pp.
1203
1223
20.
Hottin
,
A.
,
Naït Abdelaziz
,
M.
,
Talha
,
A.
, and
Charrier
,
P.
,
2023
, “
Continuum Damage Mechanics to Predict Rubber Fatigue Life Under Multiaxial Loadings
,”
Int. J. Fatigue
,
170
, p.
107559
21.
Fassin
,
M.
,
Eggersmann
,
R.
,
Wulfinghoff
,
S.
, and
Reese
,
S.
,
2019
, “
Gradient-Extended Anisotropic Brittle Damage Modeling Using a Second Order Damage Tensor – Theory, Implementation and Numerical Examples
,”
Int. J. Solids Struct.
,
167
, pp.
93
126
22.
Holthusen
,
H.
,
Brepols
,
T.
,
Reese
,
S.
, and
Simon
,
J.-W.
,
2022
, “
A Two-Surface Gradient-Extended Anisotropic Damage Model Using a Second Order Damage Tensor Coupled to Additive Plasticity in the Logarithmic Strain Space
,”
J. Mech. Phys. Solids
,
163
, p.
104833
23.
Chaboche
,
J.-L.
,
1992
, “
Damage Induced Anisotropy: On the Difficulties Associated With the Active/Passive Unilateral Condition
,”
Int. J. Damage Mech.
,
1
(
2
), pp.
148
171
24.
Lubarda
,
V.
, and
Krajcinovic
,
D.
,
1993
, “
Damage Tensors and the Crack Density Distribution
,”
Int. J. Solids Struct.
,
30
(
20
), pp.
2859
2877
25.
Ahmed
,
B.
,
Voyiadjis
,
G. Z.
, and
Park
,
T.
,
2020
, “
Damaged Plasticity Model for Concrete Using Scalar Damage Variables With a Novel Stress Decomposition
,”
Int. J. Solids Struct.
,
191–192
, pp.
56
75
.
26.
Pantidis
,
P.
, and
Mobasher
,
M. E.
,
2023
, “
Integrated Finite Element Neural Network (I-FENN) for Non-Local Continuum Damage Mechanics
,”
Comput. Methods Appl. Mech. Eng.
,
404
, p.
115766
27.
Chaboche
,
J.
,
1984
, “
Anisotropic Creep Damage in the Framework of Continuum Damage Mechanics
,”
Nucl. Eng. Des.
,
79
(
3
), pp.
309
319
28.
Chaboche
,
J.-L.
,
1993
, “
Development of Continuum Damage Mechanics for Elastic Solids Sustaining Anisotropic and Unilateral Damage
,”
Int. J. Damage Mech.
,
2
(
4
), pp.
311
329
29.
Chow
,
C.
, and
Wang
,
J.
,
1987
, “
An Anisotropic Theory of Elasticity for Continuum Damage Mechanics
,”
Int. J. Fract.
,
33
, pp.
3
16
30.
Chow
,
C.
, and
Wang
,
J.
,
1987
, “
An Anisotropic Theory of Continuum Damage Mechanics for Ductile Fracture
,”
Eng. Fract. Mech.
,
27
(
5
), pp.
547
558
31.
Murakami
,
S.
,
1987
,
Anisotropic Aspects of Material Damage and Application of Continuum Damage Mechanics
,
Springer Vienna
,
Vienna, Austria
, pp.
91
133
.
32.
Desmorat
,
R.
,
Gatuingt
,
F.
, and
Ragueneau
,
F.
,
2007
, “
Nonlocal Anisotropic Damage Model and Related Computational Aspects for Quasi-Brittle Materials
,”
Eng. Fract. Mech.
,
74
(
10
), pp.
1539
1560
33.
Reinoso
,
J.
,
Catalanotti
,
G.
,
Blázquez
,
A.
,
Areias
,
P.
,
Camanho
,
P.
, and
París
,
F.
,
2017
, “
A Consistent Anisotropic Damage Model for Laminated Fiber-Reinforced Composites Using the 3D-Version of the Puck Failure Criterion
,”
Int. J. Solids Struct.
,
126–127
, pp.
37
53
34.
Yuan
,
Z.
, and
Fish
,
J.
,
2016
, “
Are the Cohesive Zone Models Necessary for Delamination Analysis?
,”
Comput. Methods Appl. Mech. Eng.
,
310
, pp.
567
604
35.
Zapara
,
M.
,
Tutyshkin
,
N.
,
Müller
,
W.
, and
Wille
,
R.
,
2012
, “
Constitutive Equations of a Tensorial Model for Ductile Damage of Metals
,”
Contin. Mech. Thermodyn.
,
24
(
4–6
), pp.
697
717
36.
Vilppo
,
J.
,
Kouhia
,
R.
,
Hartikainen
,
J.
,
Kolari
,
K.
,
Fedoroff
,
A.
, and
Calonius
,
K.
,
2021
, “
Anisotropic Damage Model for Concrete and Other Quasi-Brittle Materials
,”
Int. J. Solids Struct.
,
225
, p.
111048
37.
Kassar
,
S.
,
Ayoub
,
G.
, and
Kridli
,
G.
,
2019
, “
Anisotropic Time Dependent and Continuum Damage Coupled Plasticity Model: An Application for Mg AZ31B
,”
Int. J. Solids Struct.
,
178–179
, pp.
199
211
38.
Negi
,
A.
,
Soni
,
A.
, and
Kumar
,
S.
,
2022
, “
An Anisotropic Localizing Gradient Damage Approach for Failure Analysis of Fiber Reinforced Composites
,”
Compos. Struct.
,
294
, p.
115677
39.
Liu
,
P.
, and
Zheng
,
J.
,
2008
, “
Progressive Failure Analysis of Carbon Fiber/epoxy Composite Laminates Using Continuum Damage Mechanics
,”
Mater. Sci. Eng. A
,
485
(
1
), pp.
711
717
40.
Liu
,
Y.
,
Hou
,
Y.
,
Sapanathan
,
T.
,
Meng
,
L.
, and
Xu
,
Y.
,
2023
, “
Multiscale Modeling of the Mechanical Behavior of 3D Braided Cfrp Composites Under Uniaxial Tension
,”
Compos. Struct.
,
306
, p.
116601
41.
Kim
,
E.-H.
,
Rim
,
M.-S.
,
Lee
,
I.
, and
Hwang
,
T.-K.
,
2013
, “
Composite Damage Model Based on Continuum Damage Mechanics and Low Velocity Impact Analysis of Composite Plates
,”
Compos. Struct.
,
95
, pp.
123
134
42.
Huang
,
S.
,
Yue
,
J.
,
Liu
,
X.
,
Ren
,
P.
,
Zu
,
L.
, and
Yuan
,
Z.
,
2023
, “
A Framework of Defining Constitutive Model for Fibrous Composite Material Through Reduced-Order-homogenization Method With Analytical Influence Functions
,”
Compos. Struct.
,
314
, p.
116968
43.
Shah
,
S.
,
Lee
,
J.
,
Megat-Yusoff
,
P.
,
Hussain
,
S. Z.
,
Sharif
,
T.
, and
Choudhry
,
R.
,
2023
, “
Multiscale Damage Modelling of Notched and Un-Notched 3D Woven Composites With Randomly Distributed Manufacturing Defects
,”
Compos. Struct.
,
318
, p.
117109
44.
Xiao
,
Y.
,
Zhang
,
X.
, and
Ghosh
,
S.
,
2022
, “
Parametrically-Upscaled Continuum Damage Mechanics (PUCDM) Model for Plain Weave Woven Composites: Part I Model Development
,”
Compos. Struct.
,
296
, p.
115825
45.
Xiao
,
Y.
,
Zhang
,
X.
, and
Ghosh
,
S.
,
2022
, “
Parametrically-Upscaled Continuum Damage Mechanics (PUCDM) Model for Plain Weave Woven Composites: Part Ii Model Validation and Parametric Studies
,”
Compos. Struct.
,
296
, p.
115826
46.
Mazars
,
J.
, and
Pijaudier-Cabot
,
J.
,
1989
, “
Continuum Damage Theory–Application to Concrete
,”
J. Eng. Mech.
,
115
(
2
), pp.
345
365
.
47.
Zolochevsky
,
A.
,
Itoh
,
T.
,
Obataya
,
Y.
, and
Betten
,
J.
,
2000
, “
A Continuum Damage Mechanics Model With the Strain-Based Approach to Biaxial Low Cycle Fatigue Failure
,”
Forsch. Ingenieurw.
,
66
(
2
), pp.
67
73
48.
de Vree
,
J.
,
Brekelmans
,
W.
, and
van Gils
,
M.
,
1995
, “
Comparison of Nonlocal Approaches in Continuum Damage Mechanics
,”
Comput. Struct.
,
55
(
4
), pp.
581
588
49.
Liu
,
N.
, and
Yuan
,
Z.
,
2023
, “
Evaluation of Dissipation Energy of Isotropic Continuum Damage Mechanics Model With Adaptive Time-Step Control Approach
,”
Int. J. Multiscale Comput. Eng.
,
21
(
6
), pp.
49
62
50.
Yuan
,
Z.
,
Crouch
,
R.
,
Wollschlager
,
J.
,
Shojaei
,
A.
, and
Fish
,
J.
,
2017
, “
Assessment of Altair Multiscale Designer for Damage Tolerant Design Principles (DTDP) of Advanced Composite Aircraft Structures
,”
J. Compos. Mater.
,
51
(
10
), pp.
1379
1391
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