- Email: [email protected]
The postbuckling behaviour of adhesively bonded stiffened panels subjected to in-plane shear loading
JID:AESCTE AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.1 (1-12)
Aerospace Science and Technology ••• (••••) •••–•••
1
67
Contents lists available at ScienceDirect
2
68
3
69
Aerospace Science and Technology
4
70
5
71
6
72
www.elsevier.com/locate/aescte
7
73
8
74
9
75
10
76
11 12 13
The postbuckling behaviour of adhesively bonded stiffened panels subjected to in-plane shear loading
14 15
77 78 79 80
a
a,∗
a
81
17
Anaisa de Paula Guedes Villani , Mauricio V. Donadon , Mariano A. Arbelo , Paulo Rizzi a , Carlos V. Montestruque a , Flavio Bussamra a , Marcelo R.B. Rodrigues b
18
a
84
16
19
Instituto Tecnológico de Aeronáutica–ITA/IEA, São José dos Campos-SP, Brazil b Empresa Brasileira de Aeronáutica–Embraer, São José dos Campos-SP, Brazil
86
21
87
a r t i c l e
i n f o
a b s t r a c t
88
23 24 25 26 27
89
Article history: Received 13 March 2015 Received in revised form 21 June 2015 Accepted 26 June 2015 Available online xxxx
28 29 30 31 32
83 85
20 22
82
Keywords: Adhesive Post-buckling Finite elements Mechanical characterization
33 34 35 36
This paper presents a detailed investigation on the post-buckling behaviour of adhesively bonded stiffened panels subjected to in-plane shear loading. An experimental programme was carried to determine the buckling load, buckling shape, collapse load and failure modes of two bonded stiffened panels. A nonlinear finite element based modelling approach, accounting for geometrical and material nonlinearities as well as progressive failure in the adhesively bonded interface between the skin and the stiffener is proposed to predict the structural behaviour of the panels up to failure. This approach consists in modelling the bonded interfaces using a newly developed cohesive zone based constitutive damage model. In order to account for damage in the stiffener and the skin a Von Mises based constitutive damage model is also formulated and presented in the paper. Both constitutive models were implemented into ABAQUS/Explicit finite element code as user-defined material models. A very good agreement between experimental results and numerical predictions is obtained using the proposed modelling approach, with deviations smaller than 8% in buckling load and bonded interface failure load onset. © 2015 Published by Elsevier Masson SAS.
90 91 92 93 94 95 96 97 98 99 100 101 102
37
103
38
104
39
105
40 41
1. Introduction
42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
Today’s high-strength and damage-tolerant materials allow for significant increases in load carrying capacity and stress limits in aircraft structures. In stiffened panels, which are the basic building blocks of these structures, joints between skin and stiffeners will need to transfer more load. It is widely known that adhesive joints have several advantages over other joining techniques such as improved fatigue tolerance, potentially better crack bridging in the presence of serious damage and in many cases improved buckling stability (see Chowdhury et al. [1] and Azari et al. [2]). Stringer-stiffened shear panels are extensively used in many aircraft applications. The fuselage of an airplane is a typical example of this design concept. Thus, most studies of buckling and postbuckling in structures loaded in shear are related to fuselage design. In those investigations, it is intended to replace the conservative design scenario by one that exploits the full potential of the materials in terms of weight reduction. It is important then to pursue more accurate methods to analyse the structure response
60 61 62 63 64 65 66
*
Corresponding author. Tel./fax: +55 12 3947 5944. E-mail address: [email protected] (M.V. Donadon).
http://dx.doi.org/10.1016/j.ast.2015.06.028 1270-9638/© 2015 Published by Elsevier Masson SAS.
when submitted to large strains and displacements. This can only be achieved by considering the geometrical and material nonlinearities on the models formulation, as discussed by Panda and Singh [3–6]. A summary of the previous work on stiffened panels loaded in shear, are presented in Kuhn et al. [7], Agarwal [8], Shuart and Hagaman [9], Farley and Baker [10], Murphy et al. [11]. Detailed numerical and experimental investigation on the structural behaviour of composite shear webs in the post-buckling regime are given in Ambur et al. [12] and Arbelo et al. [13,14]. Debonding in composite panels in postbuckling regime was studied by Degenhardt et al. [15], using simple stress-based degradation criteria and by Orifici et al. [16], using VCCT approach. Boni et al. [17] present numerical and experimental investigation on shear loaded FML (Fiber Metal Laminate) panel with a bonded window frame, using VCCT to model the failure on the adhesive interface. The present work is concerned with the stress analysis of a metallic stiffened panel with adhesively bonded skin-to-stiffener joints, loaded in pure shear in the post-buckling regime up to failure. To this end, a fully nonlinear finite element approach including geometrical and material nonlinearities as well as progressive failure was used. The skin and stringers behaviours were mod-
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.2 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
2
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
Fig. 1. Schematic view of the tested stiffened panels.
elled using an elasto-plastic Von-Mises based constitutive damage model. The bonded skin/stiffener interfacial behaviour was modelled using a newly developed cohesive based damage model. The proposed formulation uses a single damage parameter and accounts for mixed-mode debonding without knowing a priori the debonding mode ratio. The proposed constitutive models were implemented into ABAQUS as user-defined material models. An experimental programme was carried out to validate the proposed postbuckling modelling approach. An extensive compilation of similar experimental setups can be found in Singer et al. [18].
40 41
2. Buckling and postbuckling tests
42 43
2.1. Panel description and test setup
44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
80
Fig. 2. Stiffened panel mounted in the servo-hydraulic testing machine.
15
The test specimen is a 480 × 480 mm2 metallic stiffened panel loaded in pure shear. The skin is made using a 1 mm thickness, 2024-T3 aluminium plate. The stiffeners are made of extruded 7050-T651 aluminium, bonded to the skin using an FM-73 adhesive film. The dimensions of the bonded stiffened panels and the shear loading frame are depicted in Fig. 1. The test specimens were fixed along their edges in a hinged shear loading frame by double rows of bolts and connected along the panel diagonal to a 200 tons servo-hydraulic testing machine available in the Aerospace Structures Laboratory at ITA, as shown in Fig. 2. The load was measured using a 250 kN load cell placed between the upper loading device and the testing machine. The crosshead displacement was measured using an LVDT (Linear Variable Differential Transformer) attached between the moving crosshead and the base of the testing machine. Four pairs of uniaxial strain gages, indicated as E1–2; E3–4; E5–6; E7–8 in Fig. 3 (a), were bonded along the panel diagonal, close to each corner, to characterize diagonal stress state. Each pair was symmetrically placed on the top and bottom surfaces (backto-back) of the skin to characterize the membrane and bending strains. The stress state in the stiffeners was characterized using two additional pairs of symmetrically bonded uniaxial strain
81
gages, named E9–10 & E11–12, in the location shown in Fig. 3 (b). A pair of rosettes strain gages named R1 and R2 was symmetrically placed top to bottom in the central area of the panel between the two stiffeners, in order to provide a full characterization of the strain state in this point. A 3D structured light scanner was used to characterize the out-of-plane displacement, buckling mode shape and postbuckling pattern during the tests. A schematic view of this system is depicted in Figs. 4 (a) and (b). Two cameras are used to obtain a 3D image of a structured light sequence reflected over the stiffened panel. A computer with dedicated software acquires and post processes the images in order to calculate the out-of-plane displacement from an initial “non-loaded” state. A calibration pattern was used to assure an accuracy of ten micrometres. During testing, the out of plane displacement on the back surface (the one without stiffeners) was measured every 2 kN of load increment up to the buckling load onset.
82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
3. Finite element modelling approach
101 102
3.1. Constitutive models
103 104
This section presents the formulations for both elastic/plastic material models with progressive failure used to model the skin and cohesive zone based damage model used to predict skin/stiffener debonding and adhesive behaviour. Both models were implemented into ABAQUS/Explicit finite element code within shell elements and solid elements respectively, as user defined material models via VUMAT Fortran subroutine.
105 106 107 108 109 110 111 112
3.1.1. Elastic/plastic Von Mises material model with isotropic hardening and progressive failure The elastic/plastic material behaviour is defined in terms of the four properties listed below:
113 114 115 116 117
a) A Yield function, which gives the yield condition that specifies the state of multiaxial stress corresponding to start of plastic flow b) A flow rule, which relates the plastic strain increments to the current stresses and to the stress increments c) A hardening rule, which specifies how the yield function is modified during the plastic flow d) A critical value of maximum cumulative plastic strain, which specifies the maximum allowable plastic strain of the material prior to catastrophic failure
118 119 120 121 122 123 124 125 126 127 128
The Von Mises Yield function for a material under plane stress assumption is given as follows [19],
f =
2 1/2
σx2 + σ y2 − σx σ y + 3τxy
129 130 131
− σ0 = σe − σ0
(1)
132
JID:AESCTE AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.3 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
3
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17 18
83
Fig. 3. Strain gauge locations: a) panel; b) stiffeners.
84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63
129
64
130
65 66
131
Fig. 4. 3D structured light scanner used to measure the out-of-plane displacement field.
132
JID:AESCTE
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.4 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
where σe is the effective stress and σ0 the yield stress. In conjunction with Eq. (1) the Prandt–Reuss flow rules for associative plasticity are
Substitution of Eq. (11) into Eq. (10) yields
⎫ ⎧ ⎫ ⎧ ⎨ ε˙ p x ⎬ ∂f λ˙ ⎨ 2σx − σ y ⎬ 2σ y − σ x {˙ε p } = λ˙ = λ˙ {a} = ε˙ p x = ⎭ ⎩ ˙ ⎭ 2σe ⎩ ∂{σ } ε p xy 6τxy
Hence premultiplying Eq. (12) by a T and substituting into Eq. (3) gives
˙ is a poswhere a is the vector normal to the Yield surface and λ itive constant usually referred to as the plastic strain-rate multiplier. By decomposing the total strain rates into elastic and plastic strain rates, the stress change are related to strain change via
⎫ ⎧ ⎫⎤ ⎡⎧ ⎨ ε˙ x ⎬ ⎨ ε˙ p x ⎬ {σ˙ } = [C ] {˙ε } − {˙ε p } = [C ] ⎣ ε˙ y − ε˙ p y ⎦ ⎩˙ ⎭ ⎩˙ ⎭ εxy ε p xy ˙ = [C ] {˙ε } − λ{a} where
⎡
20 21 22
C=
23 24 25 26 27 28 29 30 31
E 1−ν
34 35 36
2
1
ν
⎣ν 1
0 0
0
(1 − ν )/2
0
(3)
⎤ ⎦
(4)
A negative plastic parameter would imply plastic unloading from the yield surface. The latter cannot occur and, consequently, any negative plastic parameter should be replaced by zero so that elastic unloading occurs [19]. For plastic flow to occur, the stresses must remain on the yield surface and hence,
˙f =
32 33
(2)
∂f ∂{σ }
T {σ˙ } = 0
(5)
In the present model, hardening is introduced by changing the fixed yield stress σ0 , so that, σ0 becomes a function of the equivalent plastic strain, which leads to the following expression for yield function,
37 38
f = σe − σ0 (ε ps )
(6)
39 40
with,
41 42
ε˙ ps dt
ε ps =
(7)
43 44
where
45 46 47 48 49 50 51 52 53 54
2
ε˙ ps = √
H=
57 58 59
62 63 64
˙ 2p y
ε +ε
1
1/2
+ ε˙ p x ε˙ p y + γ 4
˙p2xy
(8)
∂ σ0 ∂ σx Et E = = ∂ ε ps ∂ ε px E − Et
(9)
where E t is the tangent modulus. Once hardening is introduced, the tangency condition can be written as follows,
60 61
3
˙ 2p x
Under uniaxial tension ε˙ p x = ε˙ p y = − 12 ε˙ p x so that there is plastic volume change and ε˙ p s = ε˙ p x while σe = σ0 = σx . Consequently, the relationship between σ0 and ε ps can be taken from uniaxial stress/plastic strain relationship. In particular, we will require ∂ σ0 /∂ ε ps . Assuming a linear relationship between σ0 and ε ps we obtain the following expression for the hardening modulus,
55 56
˙f =
∂f ∂{σ }
T {σ˙ } +
∂ f ∂ σ0 ε˙ ps = {a}T {σ˙ } − H ε˙ ps = 0 ∂ σ0 ∂ ε ps
(10)
65 66
ε˙ ps = λ˙
˙f = {a} {σ˙ } − H λ˙ = 0
(11)
(12)
68 69 70 71 72
{a}T [C ]{˙ε } λ˙ = {a}T [C ]{a} + H
(13)
which results in the following stress rate–strain rate relationship
{σ˙ } = [C t ]{˙ε } = [C ] [ I ] −
{a}{a} [C ] {˙ε } {a}T [C ]{a} + H T
73 74 75 76 77
(14)
78 79
The criterion adopted to detect failure initiation is based on the maximum cumulative plastic strain. This criterion states that damage starts when the maximum cumulative plastic strain reaches a critical value, that is,
80
ε ps F (ε ps ) = max − 1 ≥ 0 ε ps
84
(15)
81 82 83 85 86
The updated degraded stresses are written in terms of the updated elastic/plastic stress components via
87
⎧ d⎫ ⎪ ⎨ σx ⎪ ⎬
89
⎫ ⎤⎧ d 0 0 ⎨ σx ⎬ σ yd = ⎣ 0 d 0 ⎦ σ y ⎩ ⎪ ⎩ d ⎪ ⎭ 0 0 d τxy ⎭ ⎡
(16)
d→
91 92 93
where d is defined as the stress degradation parameter defined as follows:
88 90
τxy
= 1.0 for F (ε ps ) < 0 = 0.0 for F (ε ps ) ≥ 0
(17)
In order to avoid numerical instabilities the stresses should be degraded within a certain number of time steps for pseudo-static analysis. The degradation procedure proposed here consists of degrading the stresses in one hundred time steps, so that the expression for the degradation parameter at current time step (t + t) is computed according to the central difference time integration scheme as follows:
d(t + t ) = d(t ) − 0.01(1 − δ)
(18)
94 95 96 97 98 99 100 101 102 103 104 105 106 107 108
where
109
δ = 0 for F (ε ps ) ≥ 0 δ = 1 for F (ε ps ) < 0 d(t 0 ) = 1
110 111 112
(19)
Details on the material model implementation into ABAQUS/Explicit FE code are provided in Appendix A.
w } T = h∗ γxz
h∗ γ yz
h∗ εzz
114 115 116 118
The adhesively bonded interfacial material behaviour is defined in terms of tractions and relative displacements between the upper and lower surfaces of the interface. The relative displacement vector is composed of the resultant normal and sliding components defining by the relative movement between upper and lower surfaces of the finite element (Fig. 5). For a single integration point hexahedron solid element the relative displacement vector can be written in terms of through-thickness normal strain and out-ofplane shear strains as follows,
113
117
3.2. Damage model for adhesively bonded interface
{δ}T = { u v
Substitution from Eq. (2) into Eq. (8) gives
67
T
T
(20)
where u = u T − u b , v = v T − v b , and w = w T − w b . h∗ is the element thickness of the updated geometry. Following the standard
119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.5 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
Similarly to mode I, the interfacial behaviour is defined in terms of resultant shear stress-resultant sliding displacement for both mode II and mode III debonding, that is,
1 2 3
σII = K uu 1 − dII (u max ) u σIII = K v v 1 − dIII ( v max ) v
4 5 6 7 9
dII (u ) = 1 −
10 11
dIII ( v ) = 1 −
Fig. 5. Three dimensional interface element used to model the adhesive.
12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
interface element formulation, the uncoupled linear-elastic constitutive law without membrane effects for the interface element can be written as follows
⎧ ⎫ ⎨ σI ⎬
σII ⎩ σIII
⎡
Kww =⎣ 0 ⎭ 0
σI = K w w
41 42
45 46 47 48 49 50 51 52 53
dI (w) = 1 −
56
kI (w) =
59 60 61 62 63 64 65 66
(23)
71
(29)
72
wf
G Ic =
σ I dw =
σ I0 w f 2
(25)
0
From Eq. (25) w f can be written in terms of the strain energy release rate as follows,
wf =
2G Ic
kII (u ) =
1 + k2III ( v ) 2kIII ( v ) − 3
u − u0
G IIc =
u f − u0
σII du =
2
82 83 84 85 86 87 88 90
93 95
(34)
σII0
96 97 98 99
v − v0 v f − v0
v f G IIIc =
(35)
σIII dv =
100 101 102
σIII0 v f
(36)
2
103 104 105
2G IIIc
(37)
σIII0
106 107 108
In order to account for mixed-mode debonding, a quadratic stress based criterion [20] given by Eq. (38) was used to detect damage initiation,
max(0, σ I )
2
+
σ I0
σII σII0
2
+
σIII σIII0
2
109 110 111 112
=1
(38)
113 114
The formulation enables the prediction of damage propagation within an energy based framework. For this purpose two distinct energy based failure criteria available in the open literature have been incorporated into the formulation. The first criterion is an extension of the power law criterion proposed by Wu et al. [21] for mixed-mode I/II. The criterion is written in terms of interactions between the strain energy release rates and interlaminar fracture toughnesses. It also takes into account the contribution of mode III in the mixed-mode debonding process,
115
124
GI
λ
+
G II
λ
G IIc
+
G III G IIIc
λ =1
(39)
The material behaviour in compression (σ I < 0) is assumed to be linear-elastic in order to avoid element interpenetration,
σI = K w w w
G Ic + (G IIc − G Ic )
(27)
81
94
2G IIc
kIII ( v ) =
G Ic (26)
80
92
(33)
The second criterion incorporated into the formulation is the B–K criterion (Fig. 6) proposed by Benzeggagh and Kenane [22], which is given by, η
σ I0
79
91
σII0 u f
and
78
89
0
uf =
76 77
(31)
(32)
u f
(24)
w f − w0
73 75
(30)
where u 0 = σII0 / K uu and σII0 is the interfacial transverse shear strength in the X–Z plane. K uu is the interfacial shear stiffness given in terms of the adhesive shear modulus, that is, K uu = G xz /h0 where h0 is the initial thickness of the element associ0 ated with the undeformed configuration. Similarly, v 0 = σIII /K v v 0 and σIII is the transverse shear strength in the Y – Z plane. K v v is the interfacial shear stiffness given in terms of the shear modulus, that is, K v v = G yz /h0 . Here
vf =
w − w0
v
69
74
1 + k2II (u ) 2kII (u ) − 3
u v0
68 70
0
where w f is the relative displacement in which the interfacial stress in mode I is equal to zero (complete decohesion) and w 0 = σ I0 / K w w . σ I0 is the through-thickness interfacial strength in mode I. K w w is the through-thickness interfacial stiffness given in terms of the adhesive through-thickness Young modulus, that is, K w w = E zz /h0 where h0 is the initial thickness of the element associated with the undeformed configuration. The strain energy release rate associated with mode I debonding is defined by the area underneath the stress-relative displacement defined by the bi-linear constitutive law, that is
57 58
w0 1 + k2I ( w ) 2k I ( w ) − 3 w
with
54 55
(22)
where the damage evolution law d I ( w max ) is defined in terms of the maximum normal relative displacements. For a linearpolynomial constitutive law, the expression for damage evolution law d I ( w ) is written as follows,
43 44
v
1 − d I ( w max ) w
39 40
Kvv
(21)
where σ I , σII and σIII are the interfacial stresses between upper and lower surfaces associated with mode I, mode II and mode III debonding, respectively. A linear-polynomial constitutive law has been used to define the skin/stiffener interfacial material behaviour. The advantage of the linear-polynomial over others is that it is numerically more stable due to its smoothness on both damage initiation and fully failed displacement onsets. The interfacial behaviour for mode-I debonding (σ I > 0) is given by
37 38
0 K uu 0
⎤⎧ ⎫ 0 ⎨w⎬ 0 ⎦ u ⎩ ⎭
u0
67
(28)
with the damage evolutions given by
8
14
5
GS
GI + GS
= GI + GS
(40)
116 117 118 119 120 121 122 123 125 126 127 128 129 130 131 132
JID:AESCTE
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.6 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
6
The final resultant displacement associated with the fully debonded interfacial behaviour based on the B–K (Benzeggagh and Kenane) criterion is given by
1 2 3
4 6 7 9 11 12 13
16
(( K uu cos(α )) + ( K v v sin(α )) ) 2
Fig. 6. Constitutive law for adhesively bonded interface subjected to mixed-mode debonding.
17
2
K w w cos2 (β) + sin (β)(( K uu cos(α ))2 + ( K v v sin(α ))2 )
¯ 2 ¯ ¯ = 1 − δ0 1 + km ¯ −3 dm (δ) (δ) 2km (δ) ¯δ
20 21 22 23
26
×
Fig. 7. Resultant relative displacement vector.
29 30
35 36 37 38
where G Ic and G IIc are the mode I and mode II interlaminar fracture toughnesses, respectively. G I and G S are the strain energy release rates associated with mode I and resultant mode II/III shear debonding, respectively. It is worth mentioning that B–K criterion assumes that G IIc = G IIIc , since there are no test standards currently available in the literature to characterize G IIIc . The components of the relative displacements vector under mixed-mode loading are illustrated in Fig. 7 and can be written as follows.
39 40 41 42 43 44 45 46 47 48
u = δ¯ sin(β) cos(α )
(41)
v = δ¯ sin(β) sin(α )
(42)
w = δ¯ cos(β)
(43)
51
δ¯0 =
K w w cos(β)
σ
52
53
+
54 56 57 58 59 60
63 64 65 66
0 I
+
K uu sin(β) cos(α )
2
0 II
σ 2
− 12
(44)
0 III
σ
Now writing Eqs. (21), (25), (33), (36), (41), (42) and (43) in terms of the relative displacement components and substituting them into Power Law criterion (Eq. (39)) we obtain, for any mode ratio, the following expression for final resultant displacement associated with the fully debonded interfacial behaviour,
61 62
2
K v v sin(β) sin(α )
55
δ¯ f =
2
δ¯0 +
K w w cos2 (β) G Ic
78 79 80 81
(47)
82 83 85
(48)
86 87
⎧ ⎫ ⎨w⎬ u ⎩ ⎭ v
{σ } = K dm (δ¯max ) {δ}
0 K uu (1 − dm (δ¯max )) 0
0 ⎦ 0 K v v (1 − dm (δ¯max ))
89 91 92 93 94
(49)
95 96 97
(50)
The proposed formulation incorporates a consistent single damage variable dm (δ¯max ) for all debonding modes, which enables the prediction of variable mixed mode debonding without knowing a priori the mixity ratio between different debonding modes. More information regarding this modelling approach can be found in Burger et al. [23] and Mendes and Donadon [24]. Details on the material model implementation into ABAQUS/Explicit FE code are provided in Appendix B.
98 99 100 101 102 103 104 105 106 107
where β = acos(max(0, w ))/δ and α = acos(u )/δs . Combining Eqs. (21), (38), (41), (42), (43) the following expression for the mixed-mode debonding damage onset displacement vector is obtained,
49 50
77
(46)
90
28
34
76
⎤
⎡
III
27
33
75
88
K w w (1 − dm (δ¯max )) 0 σII = ⎣ ⎩ 0 σ ⎭
25
73 74
1 2
and the mixed-mode stress-relative displacement relationships are given by,
⎧ ⎫ ⎨ σI ⎬
24
72
84
¯ ¯ ¯ = δ − δ0 km (δ) δ¯ f − δ¯0
19
32
η
1 2
where
18
31
2
The resultant damage evolution is given by
14
69 71
2 × δ¯0 K w w cos2 (β) + sin2 (β) K uu cos(α ) 2 1 −1 + K v v sin(α ) 2
10
15
×
8
68 70
δ¯ f = 2 G Ic + sin2 (β)(G IIc − G Ic )
5
67
λ
+
K v v sin2 (β) sin2 (α ) G IIIc
K uu sin2 (β) cos2 (α )
λ
G IIc
λ − λ1 (45)
3.3. Finite element setup
108 109
In the finite element model, the out-of-plane displacement for nodes on each member of the test fixture was constrained. Pin joints were modelled by two coincident nodes tied with a multipoint constraint at the four corners of the panel. The displacements and rotations of the dependent node are made the same as that of the independent node, except the rotation around the hinge axis. The independent node diagonally opposite to the loading pin is constrained for axial and transverse displacements. An axial displacement was applied at the loading pin, to simulate the loading condition, as shown in Fig. 8. The test specimen and the members of the loading frame were modelled using the four nodes, reduced integration, shear deformable S4R shell elements available in ABAQUS [25]. A convergence study of the critical buckling loads was performed to define the mesh size. The selected mesh results in a model with 19 639 elements and average element size of 5 mm. The simulations were carried out under displacement control using the Dynamic Relaxation method available in the ABAQUS/Explicit FE code accounting for geometrical and material nonlinearities. Progressive failure modelling was also incorporated into the FE analyses by using the constitutive models described in Section 3.2. The mechanical properties of the skin, stiffeners and adhesive film materials are given in Tables 1 and 2 respectively.
110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.7 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
7
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
Fig. 8. Finite element model: overview and boundary conditions.
22 23 24 25
89
27 28 30 31 32
Density (kg/m3 ) Young modulus (GPa) Poisson Stress–plastic strain (MPa)
33 34 35 36 37 38
90
Table 1 Materials and mechanical properties adopted for the skin and stiffeners [26].
26
29
Maximum plastic strain
Skin (AL 2024-T3) (thickness 0.05 )
Stringer (AL 7050-T765) (extruded)
Loading frame (steel)
2770 72.2 0.33 301.2–0 342.3–0.0019 364.1–0.0118 389.4–0.0244 429.0–0.0491 473.0–0.0887 498.7–0.1207 506.0–0.1328 0.132
2770 73.8 0.33 574.4–0 592.0–0.0038 604.7–0.0097 629.9–0.0251 658.3–0.0479 676.1–0.0733
7850 210 0.29
0.073
–
91 92 93 94 95 96 97 98 99 100
–
101 102 103 104
39
105
40 41 42
106
45 46 47 48 49 50 51 52 53 54 55
Fig. 9. Load versus displacement curves obtained for adhesively bonded stiffened panels.
Table 2 Mechanical properties of the adhesive film.
43 44
Adhesive film Density (kg/m3 ) Thickness of the cured film (mm) E z (MPa) G xz = G yz (MPa) Poisson G IC (J/m2 ) G IIC (J/m2 ) G IIIC (J/m2 ) 0 I 0 II 0 III
σ (MPa) σ (MPa) σ (MPa) B–K law parameter
1105 0.15 2200 840 0.33 2500 9000 9000 45 45
η
45 2
56 57 58
4. Results
59 60 61 62 63 64 65 66
88
The global behaviour of a stiffened panel loaded in pure shear can be represented by the variation of the measured load versus the applied displacement of the loading pin on the frame test, as presented in Fig. 9. The experimental results obtained from the specimens named BND01 and BND03 are compared against the numerical predictions obtained using the proposed finite element modelling approach. Notice that the filled circle indicates the on-
set of skin/stiffener interface failure on each case. The “◦” mark indicates the numerically predicted damage onset. The predicted global stiffness obtained from the load-displacement curve in the pre and post-buckling regime agree very well with the experimental results obtained for both tested specimens. This indicates that the mechanical interaction between the skin and stiffener is well predicted by the proposed model for the adhesively bonded joint. A variation on the experimental skin/stiffener damage onset load was observed between the tested specimens. This variation may be attributed to material heterogeneities effects in the bonded interface and/or differences in the skin/stiffener bonding process. Despite these differences, a good correlation between numerical and experimental results was also obtained for the skin/stiffener damage onset and panel ultimate load. Figs. 10–15 show a comparison between predicted and measured strains and stresses in the positions indicated in Fig. 3 for the tested stiffened panels. The experimental and numerical results are given by solid and dashed lines, respectively. The experimental determination of the buckling load was based on the strainreversal method [17,27,28]. A pair of rosettes strain gages placed on the centre of the skin (indicated as R1 and R2 in Fig. 2) was used for this purpose. The vertical dot-dashed black line indicates
107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE
8
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.8 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16 17 18
82
Fig. 10. Comparison between numerical and experimental strains on the stringer webs of the adhesively bonded stiffened panel BND01.
Fig. 13. Comparison between numerical and experimental strains on the stringer webs of the adhesively bonded stiffened panel BND03.
83 84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36 37 38
Fig. 11. Comparison between numerical and experimental stress on the panel diagonal for the adhesively bonded stiffened panel BND01.
Fig. 14. Comparison between numerical and experimental stress on the panel diagonal for the adhesively bonded stiffened panel BND03.
102 103 104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55 56 57
121
Fig. 12. Comparison between numerical and experimental strains on the centre of the skin for the adhesively bonded stiffened panel BND01.
Fig. 15. Comparison between numerical and experimental strains on the centre of the skin for the adhesively bonded stiffened panel BND03.
58 59 60 61 62 63 64 65 66
122 123 124
the load levels where the first strain reversal occurred (“×” mark), as shown in Figs. 12 and 15. Some differences between predicted and experimental strains were observed, mainly in the stringers web (see Figs. 10 and 13). These differences may be attributed to initial geometrical imperfections inherent to the residual stresses induced during the adhesive curing cycle and the assembly process of the stiffened panel
on the testing frame. A symmetry in stresses acting along the skin diagonal was observed for both tested panels as depicted in Figs. 11 and 14. The same behaviour is also confirmed by the numerical simulations, where the magnitudes of the predicted strains correlate very well with the measured strains. These results indicate that up to buckling the panel skin is subjected to a pure shear stress state where the action plane of the principal stresses is 45◦ . For load levels above the buckling load this symmetry in
125 126 127 128 129 130 131 132
JID:AESCTE AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.9 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
9
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41 42
107
Fig. 16. Comparison between experimental and predicted buckling mode.
43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
108 109
the stresses is no longer valid and the skin is subjected to an unsymmetrical partial diagonal-tension stress state. In this case, the stresses acting along the skin diagonals are not symmetric and the local tension stress is much higher than the local compressive stress, which results in a local principal stresses action plane different from 45◦ . The symmetry in the diagonal stresses prior to buckling also gives some insight on how efficient was the testing fixture used to hold the panels in terms of flexibility. In order to provide a symmetric diagonal stress state the testing fixture must be stiff enough to transfer the axial prescribed load along the panel edges as a uniform shear edge load in a such way that the skin is subject to a pure shear stress state prior to buckling. The predicted strains at the rosettes location also agree very well with the experimentally measured strains for both tested panels, as shown in Figs. 12 and 15. These results also indicate the proposed model for the adhesive provides was able to accurately predict the interaction between the skin and stringers. For comparison purposes the predicted onset buckling load was also determined using the strain reversal method. Table 3 summarizes a comparison between experimental and numerical results in terms of buckling and failure loads. An overall very good agreement between experimental and numerical results
110 111 112 113 114 115 116 117 118 119 120 121 122 123
Fig. 17. Plastic strain deformation field comparison: (a) experimental, (b) numerical.
124 125
was found. The predicted buckling mode also correlates very well with the measured out-of-plane displacement pattern as shown in Fig. 16. A comparison between the observed and predicted failure modes is presented in Figs. 17 and 18, showing a very good correlation. In this case, the experimental catastrophic fracture of the stiffened panels occurs at the bolted joints between the skin and the loading frame. On the numerical model, a perfect joint
126 127 128 129 130 131 132
JID:AESCTE
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.10 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
10
nonlinear dynamic equations. The method assumes a linear interpolation for velocities between two subsequent time steps and no stiffness matrix inversions are required during the analysis. The drawback of the explicit method is that it is conditionally stable for nonlinear dynamic problems and the stability for its explicit operator is based on a critical value of the smallest time increment for a dilatational wave to cross any element in the mesh. A detailed description on the model implementation into ABAQUS VUMAT user-defined material model subroutine is given step-bystep as follows.
1 2 3 4 5 6 7 8 9 10 11
Fig. 18. Skin-stiffener interface failure mode.
13 14 16
Table 3 Numerical and experimental buckling onset and failure loads comparison.
17
Panel
Buckling load (kN)
Failure onset load (kN)
18
BND01 BND03 FE model
16.1 18.9 18.8
106.5 120.1 110.5
19 20 21 22 23
between the skin and the loading frame was used and a small deviation of the predicted ultimate load was expected.
24 25
5. Conclusions
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
This paper presented an investigation on the modelling of buckling and post-buckling behaviour of adhesively bonded stiffened panels subjected to in-plane shear loading. An experimental programme was carried-out to determine the buckling load, buckling shape, collapse load and failure modes of the bonded stiffened panels. Constitutive models to represent the adhesive and skin behaviours were formulated and implemented into ABAQUS/Explicit finite element code as user-defined material models. Nonlinear finite element analyses accounting for geometrical and material nonlinearities using the proposed constitutive models were performed. The numerical predictions obtained using the proposed modelling approach were compared with experimental results. Very good correlation between predictions and experimental results was obtained in terms of onset buckling load, postbuckling behaviour, damage onset of skin/stiffener adhesive interface and ultimate collapse load. The numerical procedure presented herein may be used as a reliable tool to design adhesively bonded stiffened panels in the aeronautical industry. The authors are conducting further tests, using different loading configurations and boundary conditions, to increase the data poll and fully verify this approach.
48 49
Conflict of interest statement
1) Based on the current time step compute strain, stress increments and update strain and trial stresses at current time:
None declared.
52 53
Acknowledgements
54 55 56 57
trial
n +1
(A.1)
n
{σ }
= {σ } + [C ]{ ε }
(A.2)
where [C ] and { ε } T = { εxx ε y y εxy } are the material stiffness matrix and strain increment vector, respectively. The superscripts n and n + 1 refer to previous and current time, respectively. 2) Compute the current deviatoric trial stresses trial { S }n+1 based on the decomposition of the total trial stresses trial {σ }n+1 into deviatoric trial { S }n+1 and hydrostatic σ Hn+1 stresses: trial
{ S }n+1 = trial {σ }n+1 − σ Hn+1 δi j
(A.3)
60
63 64 65 66
71 72 73 74 75 76
{ ε } t 2 n+1 2 n+1 2 n+1 n+1 n +1 ε˙ eff =√ ε˙ x + ε˙ y + ε˙ x ε˙ y 3
1
+
4
n +1 2 ˙xy
(A.4)
σ¯
n +1
=
√
3
The authors acknowledge the financial support received for this work from the National Research Council CNPq, Grant 300990/ 2013-8. Appendix A. Numerical implementation of the Von-Mises based damage model
79 81 82 83 84 85 86 87 88 89 90 91 92 94 95 96 97
101 102
1/2
103
γ
1 trial
(A.5)
2
2 S nxx+1
99 100
+
trial
104 105 106
4) Compute current equivalent stress,
78
98
1
{˙ε }n+1 =
107
trial n+1 2 1 2 + S ny+ S xy y
12 (A.6)
109 111 112
5) Compute current Yield stress (σY )n+1 ,
113
Et E
(σY )n+1 = (σY )n + σY = (σY )n +
108 110
E − Et
n +1 t ε˙ eff
114
(A.7)
115 116
6) Check for Yielding at current time n + 1,
117 118
f n+1 (σ¯ ) = σ¯ n+1 − (σY )n+1
(A.8)
119
– If f (σ¯ ) > 0, return the deviatoric stresses to the Yield surface using the Radial Return Mapping Algorithm,
120
n +1
{S}
n +1
= (σY )
/σ¯
n +1
trial
n +1
{S}
(A.9)
and compute the current equivalent plastic strain increment ( ε p )n+1 ,
61 62
70
93
where δi j is the Dirac delta function. 3) Based on the current time step t and current strain increments { ε }, compute the current strain rate vector {˙ε }n+1 and n+1 current effective strain rate ε˙ eff :
n+1
58 59
69
80
{ε }n+1 = {ε }n + { ε }
50 51
68
77
12
15
67
121 122 123 124 125 126 127
The Von Mises based progressive failure model has been implemented into ABAQUS Explicit finite element code within shell elements. The code formulation is based on the updated Lagrangian formulation which is used in conjunction with the central difference time integration scheme for integrating the resultant set of
n +1
( ε ps )
{a}T [C ]{˙ε } t = {a}T [C ]{a} + H
with {a} = ∂ f n+1 (σ¯ , σY )/∂{ S } – Else, ε p = 0
(A.10)
128 129 130 131 132
JID:AESCTE AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.11 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
1
3) Check for interfacial failure at current time n + 1:
7) Update total equivalent plastic strain,
2 3
(ε ps )n+1 = (ε ps )n + ( ε ps )n+1
(A.11)
4 5 7 8 9 10 11
F
14
εnps+1 = max − 1 ≥ 0 ε ps
(A.12)
17 18
27 28 29 30 31
(A.14)
⎩
σII σIII ⎭
⎡
(A.13)
9) Update total stresses:
{σ }n+1 = { S }n+1 + P n+1 δi j
(33)
10) End of one direct integration cycle. 11) Compute new stable time increment t and strain increments { ε } using the central difference method integration scheme and return to step 1).
32 33 34
69 70 71
(B.6)
δ¯0 δ¯n+1
αn+1 , and update de
2 ¯ n +1 n +1 ¯ n +1 −3 1 + km 2km δ δ
Appendix B. Numerical implementation of the damage model for adhesively bonded interface
37 38 39
42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
1) Based on the current time step and strain increments, compute local relative displacement vector increment and update the local relative displacement vector at current time:
⎧ ⎫ ⎫ ⎧ ⎨ εzz ⎬ ⎨ w ⎬ γxz { δ} = u = w n + h0 ⎩ ⎭ ⎭ ⎩ v γ yz ⎫ ⎧ ⎫n+1 ⎧ ⎫n ⎧ ⎨w⎬ ⎨w⎬ ⎨ w ⎬ {δ}n+1 = u = u + u ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ v v v
64 65 66
(B.1)
(B.2)
where the subscripts n and n + 1 refer to the previous and current time step, respectively. h0 is the initial interface thickness associated with the undeformed configuration. 2) Compute total elastic stresses acting on the interface and the resultant relative displacement δ¯n+1 and resultant stress σ¯ n+1 at current time:
⎧ ⎫n+1 ⎨ σI ⎬ ⎩
62 63
80 81
= dnm
82 83 84 85 86
⎤
87 88 89 90 91
(B.8)
5) End of one direct integration cycle. 6) Compute new stable time increment t and strain increments { ε } using the central difference method integration scheme and return to step 1).
92 93 94 95 96 97 98 99
References
100 101
The damage model for adhesively bonded interface has been implemented into ABAQUS/Explicit FE code within single integration solid elements using VUMAT user-defined material model subroutine. Details on the model implementation are given as follows.
40 41
75
79
35 36
74
78
⎧ ⎫n+1 ⎨w⎬ u ⎩ ⎭ v
73
77
K w w (1 − dnm+1 (δ¯n+1 )) 0 0 ⎦ 0 0 K uu (1 − dnm+1 (δ¯n+1 )) =⎣ 0 0 K v v (1 − dnm+1 (δ¯n+1 ))
×
72
76
⎧ ⎫n+1 ⎨ σI ⎬
25 26
68
σII + σII0
– Else, 4) Compute damaged interfacial stress at current time n + 1:
dn+1 = dn
23
n +1 ¯ n +1 =1− dm δ
dnm+1 (δ¯n+1 )
– Else,
22
67
n+1 2
(B.7)
d(t 0 ) = 1.0
19
σ
– If F n+1 (σ I , σII , σIII ) ≥ 0 compute, β n+1 , damage variable:
(A.13)
δ = 0 for F εnps+1 ≥ 0 n +1 δ = 1 for F ε ps < 0
16
0 I n +1 2 III 0 III
2
σ + σ
where
15
24
ε
n +1 ps
dn+1 = dn − 0.01(1 − δ)
13
21
(σ I , σII , σIII ) =
max(0, σ In+1 )
– If F (εnps+1 ) ≥ 0 update the time step based damage parameter,
12
20
F
n +1
8) Compute failure index at current time step,
6
11
σII σIII ⎭
δ¯n+1 =
σ¯
n +1
=
⎡
Kww
=⎣ 0 0
u n +1
2
0 K uu 0
+ v n +1
max 0, σ
⎤ ⎧ ⎫n+1 0 ⎨w⎬ 0 ⎦ u ⎩ ⎭
n +1 I
2
v
Kvv
2
+ w n +1
+ σ
(B.3)
2
n +1 2 II
(B.4)
+ σ
n +1 2 III
(B.5)
[1] N. Chowdhury, W.K. Chiu, J. Wang, P. Chang, Static and fatigue testing thin riveted, bonded and hybrid carbon fiber double lap joints used in aircraft structures, Compos. Struct. 121 (2015) 315–323. [2] S. Azari, M. Papini, J.A. Schroeder, J.K. Spelt, Fatigue threshold behavior of adhesive joints, Int. J. Adhes. Adhes. 30 (3) (2010) 145–159. [3] S.K. Panda, B.N. Singh, Thermal post-buckling behaviour of laminated composite cylindrical/hyperboloidal shallow shell panel using nonlinear finite element method, Compos. Struct. 91 (3) (2009) 366–384. [4] S.K. Panda, B.N. Singh, Post-buckling analysis of laminated composite doubly curved panel embedded with SMA fibres subjected to thermal environment, Mech. Adv. Mat. Struct. 20 (10) (2013) 842–853. [5] S.K. Panda, B.N. Singh, Thermal post-buckling analysis of laminated composite shell panel using NFEM, Mech. Based Des. Struct. Mach. 41 (4) (2013) 468–488. [6] S.K. Panda, B.N. Singh, Thermal post-buckling analysis of laminated composite spherical shell panel embedded with SMA fibres using nonlinear FEM, Proc. Inst. Mech. Eng., Part C, J. Mech. Eng. Sci. 224 (4) (2010) 757–769. [7] P. Kuhn, J.P. Peterson, L.R. Levin, A summary of diagonal tension, pt I, Vol. I: Methods of analysis, NACA Tech. Note 2661, 1952. [8] B.L. Agarwal, Postbuckling behavior of composite shear webs, AIAA J. 19 (7) (1981) 933–939. [9] M.J. Shuart, J.A. Hagaman, Buckling and failure characteristics of graphite – polyimide shear panels, NASA Technical Paper 2153, 1983. [10] G.L. Farley, D.J. Baker, In-plane shear test of thin panels, Exp. Mech. 23 (1983) 81–88. [11] A. Murphy, M. Price, C. Lynch, A. Gibson, The computational post-buckling analysis of fuselage stiffened panels loaded in shear, Thin-Walled Struct. 43 (2005) 1455–1474. [12] D.R. Ambur, N. Jaunky, M.W. Hilburger, Progressive failure studies of stiffened panels subjected to shear loading, Compos. Struct. 65 (2004) 129–142. [13] M.A. Arbelo, S.F.M. de Almeida, M.V. Donadon, An experimental and numerical analysis for the post-buckling behavior of composite shear webs, Compos. Struct. 93 (2011) 465–473. [14] M.A. Arbelo, S.F.M. de Almeida, M.V. Donadon, A numerical model for postbuckling analysis of composite shear webs, Mech. Adv. Mat. Struct. 17 (2010) 1–7. [15] R. Degenhardt, A. Kling, K. Rohwer, A.C. Orifici, R.S. Thomson, Design and analysis of stiffened composite panels including post-buckling and collapse, Comput. Struct. 86 (2008) 919–929.
102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.12 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
[16] A.C. Orifici, R.S. Thomson, R. Degenhardt, C. Bisagni, J. Bayandor, A finite element methodology for analysing degradation and collapse in postbuckling composite aerospace structures, J. Compos. Mater. 43 (26) (2009) 3239–3263. [17] L. Boni, A. Lanciotti, L. Lazzeri, Experimental tests and numerical analyses of fiber metal laminate panels under shear load with bonded window frame, J. Compos. Mater. 43 (16) (2009) 1729–1750. [18] Singer, J. Arbocz, T. Weller, J., Buckling Experiments: Experimental Methods in Buckling of Thin-Walled Structures: Basic Concepts, Columns, Beams and Plates – Volume 1, John Wiley & Sons, Inc., 1998. [19] M.A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures– vol. 1, John Wiley & Sons Ltd, England, UK, 1991, 345 pp. [20] W.C. Cui, M.R. Wisnson, M. Jones, Criteria to predict delamination of unidirectional glass/epoxi specimens waisted through the thickness, Composites 23 (1992) 158–166. [21] E.M. Wu, R.J.R.C., Crack extension in fiberglass reinforced plastics, T. and AM report N◦ 275, University of Illinois, 1965.
[22] M.L. Benzeggagh, M. Kenane, Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxi composites with mixed-mode bending apparatus, Compos. Sci. Technol. 56 (1996) 439–449. [23] D. Bürger, A.R. de Faria, S.F.M. de Almeida, F.C.L. de Melo, M.V. Donadon, Ballistic impact simulation of an armour-piercing projectile on hybrid ceramic/fiber reinforced composite armours, Int. J. Impact Eng. 43 (2012) 63–77. [24] P.A.A.E. Mendes, M.V. Donadon, Numerical prediction of compression after impact behavior of woven composite laminates, Compos. Struct. 113 (2014) 476–491. [25] ABAQUS v6. 11, ABAQUS Theory Manual, 2010. [26] United States, MIL-HDBK-5J: Metallic Materials and Elements for Aerospace Vehicle Structures, Department of Defense, Washington, DC, 2003. [27] E.F. Bruhn, Analysis and Design of Flight Vehicle Structures, Tri-State Offset Company, New York, 1973. [28] G. Gerard, H. Becker, Handbook of Structural Stability, Part I – Buckling of Flat Plates, NACA TN 3781, 1957.
67 68 69 70 71 72 73 74 75 76 77 78 79 80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63
129
64
130
65
131
66
132
[m5G; v1.157; Prn:8/07/2015; 13:39] P.1 (1-12)
Aerospace Science and Technology ••• (••••) •••–•••
1
67
Contents lists available at ScienceDirect
2
68
3
69
Aerospace Science and Technology
4
70
5
71
6
72
www.elsevier.com/locate/aescte
7
73
8
74
9
75
10
76
11 12 13
The postbuckling behaviour of adhesively bonded stiffened panels subjected to in-plane shear loading
14 15
77 78 79 80
a
a,∗
a
81
17
Anaisa de Paula Guedes Villani , Mauricio V. Donadon , Mariano A. Arbelo , Paulo Rizzi a , Carlos V. Montestruque a , Flavio Bussamra a , Marcelo R.B. Rodrigues b
18
a
84
16
19
Instituto Tecnológico de Aeronáutica–ITA/IEA, São José dos Campos-SP, Brazil b Empresa Brasileira de Aeronáutica–Embraer, São José dos Campos-SP, Brazil
86
21
87
a r t i c l e
i n f o
a b s t r a c t
88
23 24 25 26 27
89
Article history: Received 13 March 2015 Received in revised form 21 June 2015 Accepted 26 June 2015 Available online xxxx
28 29 30 31 32
83 85
20 22
82
Keywords: Adhesive Post-buckling Finite elements Mechanical characterization
33 34 35 36
This paper presents a detailed investigation on the post-buckling behaviour of adhesively bonded stiffened panels subjected to in-plane shear loading. An experimental programme was carried to determine the buckling load, buckling shape, collapse load and failure modes of two bonded stiffened panels. A nonlinear finite element based modelling approach, accounting for geometrical and material nonlinearities as well as progressive failure in the adhesively bonded interface between the skin and the stiffener is proposed to predict the structural behaviour of the panels up to failure. This approach consists in modelling the bonded interfaces using a newly developed cohesive zone based constitutive damage model. In order to account for damage in the stiffener and the skin a Von Mises based constitutive damage model is also formulated and presented in the paper. Both constitutive models were implemented into ABAQUS/Explicit finite element code as user-defined material models. A very good agreement between experimental results and numerical predictions is obtained using the proposed modelling approach, with deviations smaller than 8% in buckling load and bonded interface failure load onset. © 2015 Published by Elsevier Masson SAS.
90 91 92 93 94 95 96 97 98 99 100 101 102
37
103
38
104
39
105
40 41
1. Introduction
42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
Today’s high-strength and damage-tolerant materials allow for significant increases in load carrying capacity and stress limits in aircraft structures. In stiffened panels, which are the basic building blocks of these structures, joints between skin and stiffeners will need to transfer more load. It is widely known that adhesive joints have several advantages over other joining techniques such as improved fatigue tolerance, potentially better crack bridging in the presence of serious damage and in many cases improved buckling stability (see Chowdhury et al. [1] and Azari et al. [2]). Stringer-stiffened shear panels are extensively used in many aircraft applications. The fuselage of an airplane is a typical example of this design concept. Thus, most studies of buckling and postbuckling in structures loaded in shear are related to fuselage design. In those investigations, it is intended to replace the conservative design scenario by one that exploits the full potential of the materials in terms of weight reduction. It is important then to pursue more accurate methods to analyse the structure response
60 61 62 63 64 65 66
*
Corresponding author. Tel./fax: +55 12 3947 5944. E-mail address: [email protected] (M.V. Donadon).
http://dx.doi.org/10.1016/j.ast.2015.06.028 1270-9638/© 2015 Published by Elsevier Masson SAS.
when submitted to large strains and displacements. This can only be achieved by considering the geometrical and material nonlinearities on the models formulation, as discussed by Panda and Singh [3–6]. A summary of the previous work on stiffened panels loaded in shear, are presented in Kuhn et al. [7], Agarwal [8], Shuart and Hagaman [9], Farley and Baker [10], Murphy et al. [11]. Detailed numerical and experimental investigation on the structural behaviour of composite shear webs in the post-buckling regime are given in Ambur et al. [12] and Arbelo et al. [13,14]. Debonding in composite panels in postbuckling regime was studied by Degenhardt et al. [15], using simple stress-based degradation criteria and by Orifici et al. [16], using VCCT approach. Boni et al. [17] present numerical and experimental investigation on shear loaded FML (Fiber Metal Laminate) panel with a bonded window frame, using VCCT to model the failure on the adhesive interface. The present work is concerned with the stress analysis of a metallic stiffened panel with adhesively bonded skin-to-stiffener joints, loaded in pure shear in the post-buckling regime up to failure. To this end, a fully nonlinear finite element approach including geometrical and material nonlinearities as well as progressive failure was used. The skin and stringers behaviours were mod-
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.2 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
2
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
Fig. 1. Schematic view of the tested stiffened panels.
elled using an elasto-plastic Von-Mises based constitutive damage model. The bonded skin/stiffener interfacial behaviour was modelled using a newly developed cohesive based damage model. The proposed formulation uses a single damage parameter and accounts for mixed-mode debonding without knowing a priori the debonding mode ratio. The proposed constitutive models were implemented into ABAQUS as user-defined material models. An experimental programme was carried out to validate the proposed postbuckling modelling approach. An extensive compilation of similar experimental setups can be found in Singer et al. [18].
40 41
2. Buckling and postbuckling tests
42 43
2.1. Panel description and test setup
44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
80
Fig. 2. Stiffened panel mounted in the servo-hydraulic testing machine.
15
The test specimen is a 480 × 480 mm2 metallic stiffened panel loaded in pure shear. The skin is made using a 1 mm thickness, 2024-T3 aluminium plate. The stiffeners are made of extruded 7050-T651 aluminium, bonded to the skin using an FM-73 adhesive film. The dimensions of the bonded stiffened panels and the shear loading frame are depicted in Fig. 1. The test specimens were fixed along their edges in a hinged shear loading frame by double rows of bolts and connected along the panel diagonal to a 200 tons servo-hydraulic testing machine available in the Aerospace Structures Laboratory at ITA, as shown in Fig. 2. The load was measured using a 250 kN load cell placed between the upper loading device and the testing machine. The crosshead displacement was measured using an LVDT (Linear Variable Differential Transformer) attached between the moving crosshead and the base of the testing machine. Four pairs of uniaxial strain gages, indicated as E1–2; E3–4; E5–6; E7–8 in Fig. 3 (a), were bonded along the panel diagonal, close to each corner, to characterize diagonal stress state. Each pair was symmetrically placed on the top and bottom surfaces (backto-back) of the skin to characterize the membrane and bending strains. The stress state in the stiffeners was characterized using two additional pairs of symmetrically bonded uniaxial strain
81
gages, named E9–10 & E11–12, in the location shown in Fig. 3 (b). A pair of rosettes strain gages named R1 and R2 was symmetrically placed top to bottom in the central area of the panel between the two stiffeners, in order to provide a full characterization of the strain state in this point. A 3D structured light scanner was used to characterize the out-of-plane displacement, buckling mode shape and postbuckling pattern during the tests. A schematic view of this system is depicted in Figs. 4 (a) and (b). Two cameras are used to obtain a 3D image of a structured light sequence reflected over the stiffened panel. A computer with dedicated software acquires and post processes the images in order to calculate the out-of-plane displacement from an initial “non-loaded” state. A calibration pattern was used to assure an accuracy of ten micrometres. During testing, the out of plane displacement on the back surface (the one without stiffeners) was measured every 2 kN of load increment up to the buckling load onset.
82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
3. Finite element modelling approach
101 102
3.1. Constitutive models
103 104
This section presents the formulations for both elastic/plastic material models with progressive failure used to model the skin and cohesive zone based damage model used to predict skin/stiffener debonding and adhesive behaviour. Both models were implemented into ABAQUS/Explicit finite element code within shell elements and solid elements respectively, as user defined material models via VUMAT Fortran subroutine.
105 106 107 108 109 110 111 112
3.1.1. Elastic/plastic Von Mises material model with isotropic hardening and progressive failure The elastic/plastic material behaviour is defined in terms of the four properties listed below:
113 114 115 116 117
a) A Yield function, which gives the yield condition that specifies the state of multiaxial stress corresponding to start of plastic flow b) A flow rule, which relates the plastic strain increments to the current stresses and to the stress increments c) A hardening rule, which specifies how the yield function is modified during the plastic flow d) A critical value of maximum cumulative plastic strain, which specifies the maximum allowable plastic strain of the material prior to catastrophic failure
118 119 120 121 122 123 124 125 126 127 128
The Von Mises Yield function for a material under plane stress assumption is given as follows [19],
f =
2 1/2
σx2 + σ y2 − σx σ y + 3τxy
129 130 131
− σ0 = σe − σ0
(1)
132
JID:AESCTE AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.3 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
3
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17 18
83
Fig. 3. Strain gauge locations: a) panel; b) stiffeners.
84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63
129
64
130
65 66
131
Fig. 4. 3D structured light scanner used to measure the out-of-plane displacement field.
132
JID:AESCTE
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.4 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
where σe is the effective stress and σ0 the yield stress. In conjunction with Eq. (1) the Prandt–Reuss flow rules for associative plasticity are
Substitution of Eq. (11) into Eq. (10) yields
⎫ ⎧ ⎫ ⎧ ⎨ ε˙ p x ⎬ ∂f λ˙ ⎨ 2σx − σ y ⎬ 2σ y − σ x {˙ε p } = λ˙ = λ˙ {a} = ε˙ p x = ⎭ ⎩ ˙ ⎭ 2σe ⎩ ∂{σ } ε p xy 6τxy
Hence premultiplying Eq. (12) by a T and substituting into Eq. (3) gives
˙ is a poswhere a is the vector normal to the Yield surface and λ itive constant usually referred to as the plastic strain-rate multiplier. By decomposing the total strain rates into elastic and plastic strain rates, the stress change are related to strain change via
⎫ ⎧ ⎫⎤ ⎡⎧ ⎨ ε˙ x ⎬ ⎨ ε˙ p x ⎬ {σ˙ } = [C ] {˙ε } − {˙ε p } = [C ] ⎣ ε˙ y − ε˙ p y ⎦ ⎩˙ ⎭ ⎩˙ ⎭ εxy ε p xy ˙ = [C ] {˙ε } − λ{a} where
⎡
20 21 22
C=
23 24 25 26 27 28 29 30 31
E 1−ν
34 35 36
2
1
ν
⎣ν 1
0 0
0
(1 − ν )/2
0
(3)
⎤ ⎦
(4)
A negative plastic parameter would imply plastic unloading from the yield surface. The latter cannot occur and, consequently, any negative plastic parameter should be replaced by zero so that elastic unloading occurs [19]. For plastic flow to occur, the stresses must remain on the yield surface and hence,
˙f =
32 33
(2)
∂f ∂{σ }
T {σ˙ } = 0
(5)
In the present model, hardening is introduced by changing the fixed yield stress σ0 , so that, σ0 becomes a function of the equivalent plastic strain, which leads to the following expression for yield function,
37 38
f = σe − σ0 (ε ps )
(6)
39 40
with,
41 42
ε˙ ps dt
ε ps =
(7)
43 44
where
45 46 47 48 49 50 51 52 53 54
2
ε˙ ps = √
H=
57 58 59
62 63 64
˙ 2p y
ε +ε
1
1/2
+ ε˙ p x ε˙ p y + γ 4
˙p2xy
(8)
∂ σ0 ∂ σx Et E = = ∂ ε ps ∂ ε px E − Et
(9)
where E t is the tangent modulus. Once hardening is introduced, the tangency condition can be written as follows,
60 61
3
˙ 2p x
Under uniaxial tension ε˙ p x = ε˙ p y = − 12 ε˙ p x so that there is plastic volume change and ε˙ p s = ε˙ p x while σe = σ0 = σx . Consequently, the relationship between σ0 and ε ps can be taken from uniaxial stress/plastic strain relationship. In particular, we will require ∂ σ0 /∂ ε ps . Assuming a linear relationship between σ0 and ε ps we obtain the following expression for the hardening modulus,
55 56
˙f =
∂f ∂{σ }
T {σ˙ } +
∂ f ∂ σ0 ε˙ ps = {a}T {σ˙ } − H ε˙ ps = 0 ∂ σ0 ∂ ε ps
(10)
65 66
ε˙ ps = λ˙
˙f = {a} {σ˙ } − H λ˙ = 0
(11)
(12)
68 69 70 71 72
{a}T [C ]{˙ε } λ˙ = {a}T [C ]{a} + H
(13)
which results in the following stress rate–strain rate relationship
{σ˙ } = [C t ]{˙ε } = [C ] [ I ] −
{a}{a} [C ] {˙ε } {a}T [C ]{a} + H T
73 74 75 76 77
(14)
78 79
The criterion adopted to detect failure initiation is based on the maximum cumulative plastic strain. This criterion states that damage starts when the maximum cumulative plastic strain reaches a critical value, that is,
80
ε ps F (ε ps ) = max − 1 ≥ 0 ε ps
84
(15)
81 82 83 85 86
The updated degraded stresses are written in terms of the updated elastic/plastic stress components via
87
⎧ d⎫ ⎪ ⎨ σx ⎪ ⎬
89
⎫ ⎤⎧ d 0 0 ⎨ σx ⎬ σ yd = ⎣ 0 d 0 ⎦ σ y ⎩ ⎪ ⎩ d ⎪ ⎭ 0 0 d τxy ⎭ ⎡
(16)
d→
91 92 93
where d is defined as the stress degradation parameter defined as follows:
88 90
τxy
= 1.0 for F (ε ps ) < 0 = 0.0 for F (ε ps ) ≥ 0
(17)
In order to avoid numerical instabilities the stresses should be degraded within a certain number of time steps for pseudo-static analysis. The degradation procedure proposed here consists of degrading the stresses in one hundred time steps, so that the expression for the degradation parameter at current time step (t + t) is computed according to the central difference time integration scheme as follows:
d(t + t ) = d(t ) − 0.01(1 − δ)
(18)
94 95 96 97 98 99 100 101 102 103 104 105 106 107 108
where
109
δ = 0 for F (ε ps ) ≥ 0 δ = 1 for F (ε ps ) < 0 d(t 0 ) = 1
110 111 112
(19)
Details on the material model implementation into ABAQUS/Explicit FE code are provided in Appendix A.
w } T = h∗ γxz
h∗ γ yz
h∗ εzz
114 115 116 118
The adhesively bonded interfacial material behaviour is defined in terms of tractions and relative displacements between the upper and lower surfaces of the interface. The relative displacement vector is composed of the resultant normal and sliding components defining by the relative movement between upper and lower surfaces of the finite element (Fig. 5). For a single integration point hexahedron solid element the relative displacement vector can be written in terms of through-thickness normal strain and out-ofplane shear strains as follows,
113
117
3.2. Damage model for adhesively bonded interface
{δ}T = { u v
Substitution from Eq. (2) into Eq. (8) gives
67
T
T
(20)
where u = u T − u b , v = v T − v b , and w = w T − w b . h∗ is the element thickness of the updated geometry. Following the standard
119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.5 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
Similarly to mode I, the interfacial behaviour is defined in terms of resultant shear stress-resultant sliding displacement for both mode II and mode III debonding, that is,
1 2 3
σII = K uu 1 − dII (u max ) u σIII = K v v 1 − dIII ( v max ) v
4 5 6 7 9
dII (u ) = 1 −
10 11
dIII ( v ) = 1 −
Fig. 5. Three dimensional interface element used to model the adhesive.
12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
interface element formulation, the uncoupled linear-elastic constitutive law without membrane effects for the interface element can be written as follows
⎧ ⎫ ⎨ σI ⎬
σII ⎩ σIII
⎡
Kww =⎣ 0 ⎭ 0
σI = K w w
41 42
45 46 47 48 49 50 51 52 53
dI (w) = 1 −
56
kI (w) =
59 60 61 62 63 64 65 66
(23)
71
(29)
72
wf
G Ic =
σ I dw =
σ I0 w f 2
(25)
0
From Eq. (25) w f can be written in terms of the strain energy release rate as follows,
wf =
2G Ic
kII (u ) =
1 + k2III ( v ) 2kIII ( v ) − 3
u − u0
G IIc =
u f − u0
σII du =
2
82 83 84 85 86 87 88 90
93 95
(34)
σII0
96 97 98 99
v − v0 v f − v0
v f G IIIc =
(35)
σIII dv =
100 101 102
σIII0 v f
(36)
2
103 104 105
2G IIIc
(37)
σIII0
106 107 108
In order to account for mixed-mode debonding, a quadratic stress based criterion [20] given by Eq. (38) was used to detect damage initiation,
max(0, σ I )
2
+
σ I0
σII σII0
2
+
σIII σIII0
2
109 110 111 112
=1
(38)
113 114
The formulation enables the prediction of damage propagation within an energy based framework. For this purpose two distinct energy based failure criteria available in the open literature have been incorporated into the formulation. The first criterion is an extension of the power law criterion proposed by Wu et al. [21] for mixed-mode I/II. The criterion is written in terms of interactions between the strain energy release rates and interlaminar fracture toughnesses. It also takes into account the contribution of mode III in the mixed-mode debonding process,
115
124
GI
λ
+
G II
λ
G IIc
+
G III G IIIc
λ =1
(39)
The material behaviour in compression (σ I < 0) is assumed to be linear-elastic in order to avoid element interpenetration,
σI = K w w w
G Ic + (G IIc − G Ic )
(27)
81
94
2G IIc
kIII ( v ) =
G Ic (26)
80
92
(33)
The second criterion incorporated into the formulation is the B–K criterion (Fig. 6) proposed by Benzeggagh and Kenane [22], which is given by, η
σ I0
79
91
σII0 u f
and
78
89
0
uf =
76 77
(31)
(32)
u f
(24)
w f − w0
73 75
(30)
where u 0 = σII0 / K uu and σII0 is the interfacial transverse shear strength in the X–Z plane. K uu is the interfacial shear stiffness given in terms of the adhesive shear modulus, that is, K uu = G xz /h0 where h0 is the initial thickness of the element associ0 ated with the undeformed configuration. Similarly, v 0 = σIII /K v v 0 and σIII is the transverse shear strength in the Y – Z plane. K v v is the interfacial shear stiffness given in terms of the shear modulus, that is, K v v = G yz /h0 . Here
vf =
w − w0
v
69
74
1 + k2II (u ) 2kII (u ) − 3
u v0
68 70
0
where w f is the relative displacement in which the interfacial stress in mode I is equal to zero (complete decohesion) and w 0 = σ I0 / K w w . σ I0 is the through-thickness interfacial strength in mode I. K w w is the through-thickness interfacial stiffness given in terms of the adhesive through-thickness Young modulus, that is, K w w = E zz /h0 where h0 is the initial thickness of the element associated with the undeformed configuration. The strain energy release rate associated with mode I debonding is defined by the area underneath the stress-relative displacement defined by the bi-linear constitutive law, that is
57 58
w0 1 + k2I ( w ) 2k I ( w ) − 3 w
with
54 55
(22)
where the damage evolution law d I ( w max ) is defined in terms of the maximum normal relative displacements. For a linearpolynomial constitutive law, the expression for damage evolution law d I ( w ) is written as follows,
43 44
v
1 − d I ( w max ) w
39 40
Kvv
(21)
where σ I , σII and σIII are the interfacial stresses between upper and lower surfaces associated with mode I, mode II and mode III debonding, respectively. A linear-polynomial constitutive law has been used to define the skin/stiffener interfacial material behaviour. The advantage of the linear-polynomial over others is that it is numerically more stable due to its smoothness on both damage initiation and fully failed displacement onsets. The interfacial behaviour for mode-I debonding (σ I > 0) is given by
37 38
0 K uu 0
⎤⎧ ⎫ 0 ⎨w⎬ 0 ⎦ u ⎩ ⎭
u0
67
(28)
with the damage evolutions given by
8
14
5
GS
GI + GS
= GI + GS
(40)
116 117 118 119 120 121 122 123 125 126 127 128 129 130 131 132
JID:AESCTE
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.6 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
6
The final resultant displacement associated with the fully debonded interfacial behaviour based on the B–K (Benzeggagh and Kenane) criterion is given by
1 2 3
4 6 7 9 11 12 13
16
(( K uu cos(α )) + ( K v v sin(α )) ) 2
Fig. 6. Constitutive law for adhesively bonded interface subjected to mixed-mode debonding.
17
2
K w w cos2 (β) + sin (β)(( K uu cos(α ))2 + ( K v v sin(α ))2 )
¯ 2 ¯ ¯ = 1 − δ0 1 + km ¯ −3 dm (δ) (δ) 2km (δ) ¯δ
20 21 22 23
26
×
Fig. 7. Resultant relative displacement vector.
29 30
35 36 37 38
where G Ic and G IIc are the mode I and mode II interlaminar fracture toughnesses, respectively. G I and G S are the strain energy release rates associated with mode I and resultant mode II/III shear debonding, respectively. It is worth mentioning that B–K criterion assumes that G IIc = G IIIc , since there are no test standards currently available in the literature to characterize G IIIc . The components of the relative displacements vector under mixed-mode loading are illustrated in Fig. 7 and can be written as follows.
39 40 41 42 43 44 45 46 47 48
u = δ¯ sin(β) cos(α )
(41)
v = δ¯ sin(β) sin(α )
(42)
w = δ¯ cos(β)
(43)
51
δ¯0 =
K w w cos(β)
σ
52
53
+
54 56 57 58 59 60
63 64 65 66
0 I
+
K uu sin(β) cos(α )
2
0 II
σ 2
− 12
(44)
0 III
σ
Now writing Eqs. (21), (25), (33), (36), (41), (42) and (43) in terms of the relative displacement components and substituting them into Power Law criterion (Eq. (39)) we obtain, for any mode ratio, the following expression for final resultant displacement associated with the fully debonded interfacial behaviour,
61 62
2
K v v sin(β) sin(α )
55
δ¯ f =
2
δ¯0 +
K w w cos2 (β) G Ic
78 79 80 81
(47)
82 83 85
(48)
86 87
⎧ ⎫ ⎨w⎬ u ⎩ ⎭ v
{σ } = K dm (δ¯max ) {δ}
0 K uu (1 − dm (δ¯max )) 0
0 ⎦ 0 K v v (1 − dm (δ¯max ))
89 91 92 93 94
(49)
95 96 97
(50)
The proposed formulation incorporates a consistent single damage variable dm (δ¯max ) for all debonding modes, which enables the prediction of variable mixed mode debonding without knowing a priori the mixity ratio between different debonding modes. More information regarding this modelling approach can be found in Burger et al. [23] and Mendes and Donadon [24]. Details on the material model implementation into ABAQUS/Explicit FE code are provided in Appendix B.
98 99 100 101 102 103 104 105 106 107
where β = acos(max(0, w ))/δ and α = acos(u )/δs . Combining Eqs. (21), (38), (41), (42), (43) the following expression for the mixed-mode debonding damage onset displacement vector is obtained,
49 50
77
(46)
90
28
34
76
⎤
⎡
III
27
33
75
88
K w w (1 − dm (δ¯max )) 0 σII = ⎣ ⎩ 0 σ ⎭
25
73 74
1 2
and the mixed-mode stress-relative displacement relationships are given by,
⎧ ⎫ ⎨ σI ⎬
24
72
84
¯ ¯ ¯ = δ − δ0 km (δ) δ¯ f − δ¯0
19
32
η
1 2
where
18
31
2
The resultant damage evolution is given by
14
69 71
2 × δ¯0 K w w cos2 (β) + sin2 (β) K uu cos(α ) 2 1 −1 + K v v sin(α ) 2
10
15
×
8
68 70
δ¯ f = 2 G Ic + sin2 (β)(G IIc − G Ic )
5
67
λ
+
K v v sin2 (β) sin2 (α ) G IIIc
K uu sin2 (β) cos2 (α )
λ
G IIc
λ − λ1 (45)
3.3. Finite element setup
108 109
In the finite element model, the out-of-plane displacement for nodes on each member of the test fixture was constrained. Pin joints were modelled by two coincident nodes tied with a multipoint constraint at the four corners of the panel. The displacements and rotations of the dependent node are made the same as that of the independent node, except the rotation around the hinge axis. The independent node diagonally opposite to the loading pin is constrained for axial and transverse displacements. An axial displacement was applied at the loading pin, to simulate the loading condition, as shown in Fig. 8. The test specimen and the members of the loading frame were modelled using the four nodes, reduced integration, shear deformable S4R shell elements available in ABAQUS [25]. A convergence study of the critical buckling loads was performed to define the mesh size. The selected mesh results in a model with 19 639 elements and average element size of 5 mm. The simulations were carried out under displacement control using the Dynamic Relaxation method available in the ABAQUS/Explicit FE code accounting for geometrical and material nonlinearities. Progressive failure modelling was also incorporated into the FE analyses by using the constitutive models described in Section 3.2. The mechanical properties of the skin, stiffeners and adhesive film materials are given in Tables 1 and 2 respectively.
110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.7 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
7
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
Fig. 8. Finite element model: overview and boundary conditions.
22 23 24 25
89
27 28 30 31 32
Density (kg/m3 ) Young modulus (GPa) Poisson Stress–plastic strain (MPa)
33 34 35 36 37 38
90
Table 1 Materials and mechanical properties adopted for the skin and stiffeners [26].
26
29
Maximum plastic strain
Skin (AL 2024-T3) (thickness 0.05 )
Stringer (AL 7050-T765) (extruded)
Loading frame (steel)
2770 72.2 0.33 301.2–0 342.3–0.0019 364.1–0.0118 389.4–0.0244 429.0–0.0491 473.0–0.0887 498.7–0.1207 506.0–0.1328 0.132
2770 73.8 0.33 574.4–0 592.0–0.0038 604.7–0.0097 629.9–0.0251 658.3–0.0479 676.1–0.0733
7850 210 0.29
0.073
–
91 92 93 94 95 96 97 98 99 100
–
101 102 103 104
39
105
40 41 42
106
45 46 47 48 49 50 51 52 53 54 55
Fig. 9. Load versus displacement curves obtained for adhesively bonded stiffened panels.
Table 2 Mechanical properties of the adhesive film.
43 44
Adhesive film Density (kg/m3 ) Thickness of the cured film (mm) E z (MPa) G xz = G yz (MPa) Poisson G IC (J/m2 ) G IIC (J/m2 ) G IIIC (J/m2 ) 0 I 0 II 0 III
σ (MPa) σ (MPa) σ (MPa) B–K law parameter
1105 0.15 2200 840 0.33 2500 9000 9000 45 45
η
45 2
56 57 58
4. Results
59 60 61 62 63 64 65 66
88
The global behaviour of a stiffened panel loaded in pure shear can be represented by the variation of the measured load versus the applied displacement of the loading pin on the frame test, as presented in Fig. 9. The experimental results obtained from the specimens named BND01 and BND03 are compared against the numerical predictions obtained using the proposed finite element modelling approach. Notice that the filled circle indicates the on-
set of skin/stiffener interface failure on each case. The “◦” mark indicates the numerically predicted damage onset. The predicted global stiffness obtained from the load-displacement curve in the pre and post-buckling regime agree very well with the experimental results obtained for both tested specimens. This indicates that the mechanical interaction between the skin and stiffener is well predicted by the proposed model for the adhesively bonded joint. A variation on the experimental skin/stiffener damage onset load was observed between the tested specimens. This variation may be attributed to material heterogeneities effects in the bonded interface and/or differences in the skin/stiffener bonding process. Despite these differences, a good correlation between numerical and experimental results was also obtained for the skin/stiffener damage onset and panel ultimate load. Figs. 10–15 show a comparison between predicted and measured strains and stresses in the positions indicated in Fig. 3 for the tested stiffened panels. The experimental and numerical results are given by solid and dashed lines, respectively. The experimental determination of the buckling load was based on the strainreversal method [17,27,28]. A pair of rosettes strain gages placed on the centre of the skin (indicated as R1 and R2 in Fig. 2) was used for this purpose. The vertical dot-dashed black line indicates
107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE
8
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.8 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16 17 18
82
Fig. 10. Comparison between numerical and experimental strains on the stringer webs of the adhesively bonded stiffened panel BND01.
Fig. 13. Comparison between numerical and experimental strains on the stringer webs of the adhesively bonded stiffened panel BND03.
83 84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36 37 38
Fig. 11. Comparison between numerical and experimental stress on the panel diagonal for the adhesively bonded stiffened panel BND01.
Fig. 14. Comparison between numerical and experimental stress on the panel diagonal for the adhesively bonded stiffened panel BND03.
102 103 104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55 56 57
121
Fig. 12. Comparison between numerical and experimental strains on the centre of the skin for the adhesively bonded stiffened panel BND01.
Fig. 15. Comparison between numerical and experimental strains on the centre of the skin for the adhesively bonded stiffened panel BND03.
58 59 60 61 62 63 64 65 66
122 123 124
the load levels where the first strain reversal occurred (“×” mark), as shown in Figs. 12 and 15. Some differences between predicted and experimental strains were observed, mainly in the stringers web (see Figs. 10 and 13). These differences may be attributed to initial geometrical imperfections inherent to the residual stresses induced during the adhesive curing cycle and the assembly process of the stiffened panel
on the testing frame. A symmetry in stresses acting along the skin diagonal was observed for both tested panels as depicted in Figs. 11 and 14. The same behaviour is also confirmed by the numerical simulations, where the magnitudes of the predicted strains correlate very well with the measured strains. These results indicate that up to buckling the panel skin is subjected to a pure shear stress state where the action plane of the principal stresses is 45◦ . For load levels above the buckling load this symmetry in
125 126 127 128 129 130 131 132
JID:AESCTE AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.9 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
9
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41 42
107
Fig. 16. Comparison between experimental and predicted buckling mode.
43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
108 109
the stresses is no longer valid and the skin is subjected to an unsymmetrical partial diagonal-tension stress state. In this case, the stresses acting along the skin diagonals are not symmetric and the local tension stress is much higher than the local compressive stress, which results in a local principal stresses action plane different from 45◦ . The symmetry in the diagonal stresses prior to buckling also gives some insight on how efficient was the testing fixture used to hold the panels in terms of flexibility. In order to provide a symmetric diagonal stress state the testing fixture must be stiff enough to transfer the axial prescribed load along the panel edges as a uniform shear edge load in a such way that the skin is subject to a pure shear stress state prior to buckling. The predicted strains at the rosettes location also agree very well with the experimentally measured strains for both tested panels, as shown in Figs. 12 and 15. These results also indicate the proposed model for the adhesive provides was able to accurately predict the interaction between the skin and stringers. For comparison purposes the predicted onset buckling load was also determined using the strain reversal method. Table 3 summarizes a comparison between experimental and numerical results in terms of buckling and failure loads. An overall very good agreement between experimental and numerical results
110 111 112 113 114 115 116 117 118 119 120 121 122 123
Fig. 17. Plastic strain deformation field comparison: (a) experimental, (b) numerical.
124 125
was found. The predicted buckling mode also correlates very well with the measured out-of-plane displacement pattern as shown in Fig. 16. A comparison between the observed and predicted failure modes is presented in Figs. 17 and 18, showing a very good correlation. In this case, the experimental catastrophic fracture of the stiffened panels occurs at the bolted joints between the skin and the loading frame. On the numerical model, a perfect joint
126 127 128 129 130 131 132
JID:AESCTE
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.10 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
10
nonlinear dynamic equations. The method assumes a linear interpolation for velocities between two subsequent time steps and no stiffness matrix inversions are required during the analysis. The drawback of the explicit method is that it is conditionally stable for nonlinear dynamic problems and the stability for its explicit operator is based on a critical value of the smallest time increment for a dilatational wave to cross any element in the mesh. A detailed description on the model implementation into ABAQUS VUMAT user-defined material model subroutine is given step-bystep as follows.
1 2 3 4 5 6 7 8 9 10 11
Fig. 18. Skin-stiffener interface failure mode.
13 14 16
Table 3 Numerical and experimental buckling onset and failure loads comparison.
17
Panel
Buckling load (kN)
Failure onset load (kN)
18
BND01 BND03 FE model
16.1 18.9 18.8
106.5 120.1 110.5
19 20 21 22 23
between the skin and the loading frame was used and a small deviation of the predicted ultimate load was expected.
24 25
5. Conclusions
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
This paper presented an investigation on the modelling of buckling and post-buckling behaviour of adhesively bonded stiffened panels subjected to in-plane shear loading. An experimental programme was carried-out to determine the buckling load, buckling shape, collapse load and failure modes of the bonded stiffened panels. Constitutive models to represent the adhesive and skin behaviours were formulated and implemented into ABAQUS/Explicit finite element code as user-defined material models. Nonlinear finite element analyses accounting for geometrical and material nonlinearities using the proposed constitutive models were performed. The numerical predictions obtained using the proposed modelling approach were compared with experimental results. Very good correlation between predictions and experimental results was obtained in terms of onset buckling load, postbuckling behaviour, damage onset of skin/stiffener adhesive interface and ultimate collapse load. The numerical procedure presented herein may be used as a reliable tool to design adhesively bonded stiffened panels in the aeronautical industry. The authors are conducting further tests, using different loading configurations and boundary conditions, to increase the data poll and fully verify this approach.
48 49
Conflict of interest statement
1) Based on the current time step compute strain, stress increments and update strain and trial stresses at current time:
None declared.
52 53
Acknowledgements
54 55 56 57
trial
n +1
(A.1)
n
{σ }
= {σ } + [C ]{ ε }
(A.2)
where [C ] and { ε } T = { εxx ε y y εxy } are the material stiffness matrix and strain increment vector, respectively. The superscripts n and n + 1 refer to previous and current time, respectively. 2) Compute the current deviatoric trial stresses trial { S }n+1 based on the decomposition of the total trial stresses trial {σ }n+1 into deviatoric trial { S }n+1 and hydrostatic σ Hn+1 stresses: trial
{ S }n+1 = trial {σ }n+1 − σ Hn+1 δi j
(A.3)
60
63 64 65 66
71 72 73 74 75 76
{ ε } t 2 n+1 2 n+1 2 n+1 n+1 n +1 ε˙ eff =√ ε˙ x + ε˙ y + ε˙ x ε˙ y 3
1
+
4
n +1 2 ˙xy
(A.4)
σ¯
n +1
=
√
3
The authors acknowledge the financial support received for this work from the National Research Council CNPq, Grant 300990/ 2013-8. Appendix A. Numerical implementation of the Von-Mises based damage model
79 81 82 83 84 85 86 87 88 89 90 91 92 94 95 96 97
101 102
1/2
103
γ
1 trial
(A.5)
2
2 S nxx+1
99 100
+
trial
104 105 106
4) Compute current equivalent stress,
78
98
1
{˙ε }n+1 =
107
trial n+1 2 1 2 + S ny+ S xy y
12 (A.6)
109 111 112
5) Compute current Yield stress (σY )n+1 ,
113
Et E
(σY )n+1 = (σY )n + σY = (σY )n +
108 110
E − Et
n +1 t ε˙ eff
114
(A.7)
115 116
6) Check for Yielding at current time n + 1,
117 118
f n+1 (σ¯ ) = σ¯ n+1 − (σY )n+1
(A.8)
119
– If f (σ¯ ) > 0, return the deviatoric stresses to the Yield surface using the Radial Return Mapping Algorithm,
120
n +1
{S}
n +1
= (σY )
/σ¯
n +1
trial
n +1
{S}
(A.9)
and compute the current equivalent plastic strain increment ( ε p )n+1 ,
61 62
70
93
where δi j is the Dirac delta function. 3) Based on the current time step t and current strain increments { ε }, compute the current strain rate vector {˙ε }n+1 and n+1 current effective strain rate ε˙ eff :
n+1
58 59
69
80
{ε }n+1 = {ε }n + { ε }
50 51
68
77
12
15
67
121 122 123 124 125 126 127
The Von Mises based progressive failure model has been implemented into ABAQUS Explicit finite element code within shell elements. The code formulation is based on the updated Lagrangian formulation which is used in conjunction with the central difference time integration scheme for integrating the resultant set of
n +1
( ε ps )
{a}T [C ]{˙ε } t = {a}T [C ]{a} + H
with {a} = ∂ f n+1 (σ¯ , σY )/∂{ S } – Else, ε p = 0
(A.10)
128 129 130 131 132
JID:AESCTE AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.11 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
1
3) Check for interfacial failure at current time n + 1:
7) Update total equivalent plastic strain,
2 3
(ε ps )n+1 = (ε ps )n + ( ε ps )n+1
(A.11)
4 5 7 8 9 10 11
F
14
εnps+1 = max − 1 ≥ 0 ε ps
(A.12)
17 18
27 28 29 30 31
(A.14)
⎩
σII σIII ⎭
⎡
(A.13)
9) Update total stresses:
{σ }n+1 = { S }n+1 + P n+1 δi j
(33)
10) End of one direct integration cycle. 11) Compute new stable time increment t and strain increments { ε } using the central difference method integration scheme and return to step 1).
32 33 34
69 70 71
(B.6)
δ¯0 δ¯n+1
αn+1 , and update de
2 ¯ n +1 n +1 ¯ n +1 −3 1 + km 2km δ δ
Appendix B. Numerical implementation of the damage model for adhesively bonded interface
37 38 39
42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
1) Based on the current time step and strain increments, compute local relative displacement vector increment and update the local relative displacement vector at current time:
⎧ ⎫ ⎫ ⎧ ⎨ εzz ⎬ ⎨ w ⎬ γxz { δ} = u = w n + h0 ⎩ ⎭ ⎭ ⎩ v γ yz ⎫ ⎧ ⎫n+1 ⎧ ⎫n ⎧ ⎨w⎬ ⎨w⎬ ⎨ w ⎬ {δ}n+1 = u = u + u ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ v v v
64 65 66
(B.1)
(B.2)
where the subscripts n and n + 1 refer to the previous and current time step, respectively. h0 is the initial interface thickness associated with the undeformed configuration. 2) Compute total elastic stresses acting on the interface and the resultant relative displacement δ¯n+1 and resultant stress σ¯ n+1 at current time:
⎧ ⎫n+1 ⎨ σI ⎬ ⎩
62 63
80 81
= dnm
82 83 84 85 86
⎤
87 88 89 90 91
(B.8)
5) End of one direct integration cycle. 6) Compute new stable time increment t and strain increments { ε } using the central difference method integration scheme and return to step 1).
92 93 94 95 96 97 98 99
References
100 101
The damage model for adhesively bonded interface has been implemented into ABAQUS/Explicit FE code within single integration solid elements using VUMAT user-defined material model subroutine. Details on the model implementation are given as follows.
40 41
75
79
35 36
74
78
⎧ ⎫n+1 ⎨w⎬ u ⎩ ⎭ v
73
77
K w w (1 − dnm+1 (δ¯n+1 )) 0 0 ⎦ 0 0 K uu (1 − dnm+1 (δ¯n+1 )) =⎣ 0 0 K v v (1 − dnm+1 (δ¯n+1 ))
×
72
76
⎧ ⎫n+1 ⎨ σI ⎬
25 26
68
σII + σII0
– Else, 4) Compute damaged interfacial stress at current time n + 1:
dn+1 = dn
23
n +1 ¯ n +1 =1− dm δ
dnm+1 (δ¯n+1 )
– Else,
22
67
n+1 2
(B.7)
d(t 0 ) = 1.0
19
σ
– If F n+1 (σ I , σII , σIII ) ≥ 0 compute, β n+1 , damage variable:
(A.13)
δ = 0 for F εnps+1 ≥ 0 n +1 δ = 1 for F ε ps < 0
16
0 I n +1 2 III 0 III
2
σ + σ
where
15
24
ε
n +1 ps
dn+1 = dn − 0.01(1 − δ)
13
21
(σ I , σII , σIII ) =
max(0, σ In+1 )
– If F (εnps+1 ) ≥ 0 update the time step based damage parameter,
12
20
F
n +1
8) Compute failure index at current time step,
6
11
σII σIII ⎭
δ¯n+1 =
σ¯
n +1
=
⎡
Kww
=⎣ 0 0
u n +1
2
0 K uu 0
+ v n +1
max 0, σ
⎤ ⎧ ⎫n+1 0 ⎨w⎬ 0 ⎦ u ⎩ ⎭
n +1 I
2
v
Kvv
2
+ w n +1
+ σ
(B.3)
2
n +1 2 II
(B.4)
+ σ
n +1 2 III
(B.5)
[1] N. Chowdhury, W.K. Chiu, J. Wang, P. Chang, Static and fatigue testing thin riveted, bonded and hybrid carbon fiber double lap joints used in aircraft structures, Compos. Struct. 121 (2015) 315–323. [2] S. Azari, M. Papini, J.A. Schroeder, J.K. Spelt, Fatigue threshold behavior of adhesive joints, Int. J. Adhes. Adhes. 30 (3) (2010) 145–159. [3] S.K. Panda, B.N. Singh, Thermal post-buckling behaviour of laminated composite cylindrical/hyperboloidal shallow shell panel using nonlinear finite element method, Compos. Struct. 91 (3) (2009) 366–384. [4] S.K. Panda, B.N. Singh, Post-buckling analysis of laminated composite doubly curved panel embedded with SMA fibres subjected to thermal environment, Mech. Adv. Mat. Struct. 20 (10) (2013) 842–853. [5] S.K. Panda, B.N. Singh, Thermal post-buckling analysis of laminated composite shell panel using NFEM, Mech. Based Des. Struct. Mach. 41 (4) (2013) 468–488. [6] S.K. Panda, B.N. Singh, Thermal post-buckling analysis of laminated composite spherical shell panel embedded with SMA fibres using nonlinear FEM, Proc. Inst. Mech. Eng., Part C, J. Mech. Eng. Sci. 224 (4) (2010) 757–769. [7] P. Kuhn, J.P. Peterson, L.R. Levin, A summary of diagonal tension, pt I, Vol. I: Methods of analysis, NACA Tech. Note 2661, 1952. [8] B.L. Agarwal, Postbuckling behavior of composite shear webs, AIAA J. 19 (7) (1981) 933–939. [9] M.J. Shuart, J.A. Hagaman, Buckling and failure characteristics of graphite – polyimide shear panels, NASA Technical Paper 2153, 1983. [10] G.L. Farley, D.J. Baker, In-plane shear test of thin panels, Exp. Mech. 23 (1983) 81–88. [11] A. Murphy, M. Price, C. Lynch, A. Gibson, The computational post-buckling analysis of fuselage stiffened panels loaded in shear, Thin-Walled Struct. 43 (2005) 1455–1474. [12] D.R. Ambur, N. Jaunky, M.W. Hilburger, Progressive failure studies of stiffened panels subjected to shear loading, Compos. Struct. 65 (2004) 129–142. [13] M.A. Arbelo, S.F.M. de Almeida, M.V. Donadon, An experimental and numerical analysis for the post-buckling behavior of composite shear webs, Compos. Struct. 93 (2011) 465–473. [14] M.A. Arbelo, S.F.M. de Almeida, M.V. Donadon, A numerical model for postbuckling analysis of composite shear webs, Mech. Adv. Mat. Struct. 17 (2010) 1–7. [15] R. Degenhardt, A. Kling, K. Rohwer, A.C. Orifici, R.S. Thomson, Design and analysis of stiffened composite panels including post-buckling and collapse, Comput. Struct. 86 (2008) 919–929.
102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14
AID:3366 /FLA
[m5G; v1.157; Prn:8/07/2015; 13:39] P.12 (1-12)
A. de Paula Guedes Villani et al. / Aerospace Science and Technology ••• (••••) •••–•••
[16] A.C. Orifici, R.S. Thomson, R. Degenhardt, C. Bisagni, J. Bayandor, A finite element methodology for analysing degradation and collapse in postbuckling composite aerospace structures, J. Compos. Mater. 43 (26) (2009) 3239–3263. [17] L. Boni, A. Lanciotti, L. Lazzeri, Experimental tests and numerical analyses of fiber metal laminate panels under shear load with bonded window frame, J. Compos. Mater. 43 (16) (2009) 1729–1750. [18] Singer, J. Arbocz, T. Weller, J., Buckling Experiments: Experimental Methods in Buckling of Thin-Walled Structures: Basic Concepts, Columns, Beams and Plates – Volume 1, John Wiley & Sons, Inc., 1998. [19] M.A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures– vol. 1, John Wiley & Sons Ltd, England, UK, 1991, 345 pp. [20] W.C. Cui, M.R. Wisnson, M. Jones, Criteria to predict delamination of unidirectional glass/epoxi specimens waisted through the thickness, Composites 23 (1992) 158–166. [21] E.M. Wu, R.J.R.C., Crack extension in fiberglass reinforced plastics, T. and AM report N◦ 275, University of Illinois, 1965.
[22] M.L. Benzeggagh, M. Kenane, Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxi composites with mixed-mode bending apparatus, Compos. Sci. Technol. 56 (1996) 439–449. [23] D. Bürger, A.R. de Faria, S.F.M. de Almeida, F.C.L. de Melo, M.V. Donadon, Ballistic impact simulation of an armour-piercing projectile on hybrid ceramic/fiber reinforced composite armours, Int. J. Impact Eng. 43 (2012) 63–77. [24] P.A.A.E. Mendes, M.V. Donadon, Numerical prediction of compression after impact behavior of woven composite laminates, Compos. Struct. 113 (2014) 476–491. [25] ABAQUS v6. 11, ABAQUS Theory Manual, 2010. [26] United States, MIL-HDBK-5J: Metallic Materials and Elements for Aerospace Vehicle Structures, Department of Defense, Washington, DC, 2003. [27] E.F. Bruhn, Analysis and Design of Flight Vehicle Structures, Tri-State Offset Company, New York, 1973. [28] G. Gerard, H. Becker, Handbook of Structural Stability, Part I – Buckling of Flat Plates, NACA TN 3781, 1957.
67 68 69 70 71 72 73 74 75 76 77 78 79 80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63
129
64
130
65
131
66
132