- Email: [email protected]
metal joints with shear-deformable adherends: Model validation with FE analysis
International Journal of Adhesion & Adhesives 37 (2012) 11–18
Contents lists available at SciVerse ScienceDirect
International Journal of Adhesion & Adhesives journal homepage: www.elsevier.com/locate/ijadhadh
On the thermomechanical behavior of two-dimensional foam/metal joints with shear-deformable adherends: Model validation with FE analysis Jaona Randrianalisoa a,n, Remy Dendievel b, Yves Bre´chet b, Paul-Marie Michaud c, Romain Filipi c a
GRESPI, Universite´ de Reims Champagne Ardenne, F-51687, Reims, France SIMAP, CNRS, Grenoble INP, Universite´ Joseph Fourier, F-38402, Saint Martin d’He res, France c EC2MS, 66 Bd Niels Bohr, F-69603,Villeurbanne, France b
a r t i c l e i n f o
abstract
Available online 21 January 2012
The stress distributions in metal/adhesive/foam planar joints subjected to biaxial tensile load and thermal load was investigated through a semi-analytical model. The shear deformation of adherends was accounted for according to a linear law in order to obtain closed-form solutions. For the model validation, a comparative study with a finite element (FE) simulation was carried out. A 2D behavior of stress fields is observed due especially to the Poisson’s ratio effects and the biaxial nature of loads. The through thickness shear stresses are comparable to normal stresses; therefore, the adherend shear deformation must be accounted for correct failure prediction. According to the comparison with FE results, the normal stress distributions at any location in the foam and the shear stresses in the foam regions close to the adhesive surface can be well predicted by the proposed model. The through thickness shear stresses, however, showed to vary according to a cubic law rather than a linear law. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Epoxy adhesive Aluminum and foams Finite element stress analysis Stress distribution
1. Introduction For space launch vehicles, the need to develop light structures has led to foam as insulating material while the structural function is carried out by a metallic shell. A typical example, which is studied here, is the cryogenic tanks of launchers. In these structures, the insulation system consists of polymer foam panels glued on the metallic shell. In this work, the main concern is to study the stress distributions within the insulation/metallic shell assembly. Such analysis is useful for predicting the assembly strength through stress based criteria. The combination foam and metal is very critical due to the incompatibility between their thermal dilatation and this situation may lead to failure either within the foam or the adhesive layer. In addition, the reservoir is often under pressure so that the mechanical loading of the foam–metal bilayer is a complex mixture of biaxial and shear loading. As discussed in the our recent study [1], the thermomechanical problem of insulation of a large cryogenic reservoir (halfillustrated in Fig. 1(a)) can be reduced to the problem of a single insulating panel glued on the metallic plate and subjected to thermal stresses due to the thermal expansion incompatibility of components, and axial and hoop stresses due to reservoir pressure effects (see in Fig. 1(b)). On the first hand, a finite element (FE) solution of this reduced problem is possible but it is not necessarily the best strategy when the ultimate goal is to provide an optimization of the material
n
Corresponding author. Tel.: þ33 3 26 91 32 51; fax: þ33 3 26 91 32 50. E-mail address: [email protected] (J. Randrianalisoa).
0143-7496/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijadhadh.2012.01.006
choice and the component dimensions. On the other hand, the simple formula often used for dimensioning tank insulation [2] is very useful but is questionable in some situations since it misses some important mechanics. There is thus a need to develop an effortless and less time-consuming approach enabling us to provide a transparent analysis of the influence of the variables of the problem (material properties and component dimensions). In the literature, adhesively bonded bi-materials are usually treated using the shear-lag theory. The recent literature review, conducted by da Silva et al. [3], gives an overview of adhesively bonded joints and the proposed solution approaches. The most studied joints are the single-lap and double-lap joints [4–7]. There are also some studies concerning stiffened joints [8], which seem close to the current configuration. It is interesting to note that almost all existing closed-form solutions neglect (i) the stresses in the normal direction to the loads caused by Poisson’s ratio strains in the adherends; and (ii) the through thickness shear adherend deformations. In many situation encountered in practice (such as structures repaired or reinforced by patches [9–11] and reservoirs insulated with glued panels as in the current problem), the systems are twodimensional (2D) and the loading is more complex. Attempts to include these complexities have been initiated by Adams and Pappiatt [12] through a simplified 2D stress model. A 2D strain– stress relationship, allowing the connection between the in-plane stresses due to the Poisson’s ratio effects, was considered. It has been, however, shown that this simplified model provides only accurate prediction for joints subjected to uniaxial loading [10,12]. Later, Mathias et al. [10] improved the Adams and Pappiatt’s works
12
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
z
R
θ
r
P
σ z∞ Ly
σ∞ y
ΔT
Lz
L/2 x
σ∞ y
ΔT
σ z∞ P
ei
es
Fig. 1. Illustration of the studied configurations. (a) Lower-half part of cryogenic tank recovered with foam panels and subjected to internal pressure P and temperature drop DT; (b) plane geometry approximation of an elementary unit.
by taking into account correctly the coupling between in-plane stresses and the effects of the Poisson’s ratio strains. Cases of joints subjected to biaxial tension or plane shear loadings were investigated. Deheeger et al. [11] extended the Mathias et al.’ model to study the problem of reinforcement of metallic structures with composite patches subjected to thermal stresses due to the difference of thermal expansion of adherends. In all the above theoretical models, the shear deformations of adherends have been neglected. However, when the shear stiffnesses of adherends are much lower than that of the adhesive, large shear deformations will also be present at the adherend surfaces adjacent to the adhesive layer for shear stress equilibrium at the interface. It is the case of some laminated composite adherends [13–15] and probably of polymer foam layer glued on metallic plate as studied here. Thus, the adherend shear deformations must be included in the theoretical models. In our recent works [1], a semianalytical model enabling to take into account the adherend shear deformations, the Poisson’s ratio effects, and the 2D nature of loadings was developed and was applied to explore the influence of material mechanical properties and component dimensions on the foam/metallic plate behavior. To get an idea about the validity limits of this model, a comparative study with a reference approach is necessary. In the current study, a comparative study of the semianalytical model against FE simulation is conducted. The effects of type of loading (biaxial stresses or thermal stresses) are explored independently. Also, the case of joints made of isotropic and orthotropic polymer foam adherends is considered. The paper is structured as follows. First, the semi-analytical model is recalled after evoking briefly the major simplifying hypotheses. Then, the comparative study of the current model against FE simulation is presented. The appropriateness of each simplification is then discussed.
(for the insulating plate), ea (for the adhesive) and es (for the substrate). The adherends were assumed to be subjected only to in-plane stresses (classical assumption of problems of thin reservoirs [16]). The adhesive layer was considered to be a shear spring allowing transferring the in-plane forces from an adherend to the other. The shear stresses in the adhesive layer were assumed constant through the thickness. The three materials involved were assumed to be linear elastic until they fail by brittle fracture (at least at the temperature level 20 K that we are interested here). The substrate and the adhesive are isotropic whereas the insulating foam may be considered as orthotropic. This orthotropic behavior of the insulation is inherited from the foaming process discussed in Section 3.2. The involved material parameters are thus the following: Es , Gs , us and as for the substrate; Eiy , Eiz , Giyx , Gizx , uiyz , aiy , and aiz for the insulating material (ignoring the x-components); and Ga for the adhesive. E, G, u, and a refer to the Young’s modulus, shear modulus, Poisson’s ratio, and thermal expansion coefficient, respectively. The effects of bending moments and peeling due to load-path eccentricity were neglected. The shear stresses vary linearly through the adherend thicknesses. This simplification is essential to obtain a semi-analytical solution of the current problem. Note that this simplification fulfils the boundary and continuity conditions (in terms of shear stresses and strains), i.e., the adherend shear stresses are zero at the outer insulation and inner substrate surfaces; and the adherend and adhesive shear stresses/strains are identical at interfaces.
2.2. Semi-analytical model According to the assumptions listed in Section 2.1, the unknowns of the insulation/adhesive/metallic plate bonded joint problem are the following stress components. 2 3 2 3 0 sjyx sjzx 0 sayx sazx 6 j 7 6 a j 6 syx sy s 0 0 7 ð1Þ 0 7 5 4 5 and 4 yx a j j s 0 0 zx szx 0 sz
2.1. Hypothesis
The left equation refers to the stress states in the adherends with j ¼i for the insulating material, and j ¼s for the substrate whereas the right equation refers to the stresses in the adhesive layer, which are only function of y and z. Fig. 2(a) depicts the x–z plane view of the joint. The inner substrate surface is at abscissa x ¼ 0, the adhesive/substrate interface is at abscissa xs , the insulation/adhesive interface is at abscissa xa , and the outer insulation surface is at abscissa xi . The equilibrium equations schematically shown in Fig. 2(b), the stress–strain relationship, the kinematic equations, the continuity of internal forces, and the through-thickness linear shear stress assumption enable us to derive the equations governing the stress distributions in Eq.(1). For the details of derivation, the readers are recommended to refer to reference [1]. It is shown that the normal stresses in the in-plane directions in the insulating material satisfy the second order differential Eqs. (2) and (3):
In the following, the indexes i, s, and a refer respectively to insulating plate, metallic shell, and adhesive layer.
@2 siz ¼ az siz þ bz siy þcz @z2
2. Theoretical model
As illustrated in Fig. 1(b), a parallelepipedic configuration
@2 siy
was considered (length, Lz, width, Ly, and the thicknesses ei
@y2
¼ ay siy þ by siz þ cy
ð2Þ
ð3Þ
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
13
Closed-form solutions of coupled differential Eqs. (2) and (3) are more practical than numerical method. The semi-analytical method based on Fourier series [10] gives the following closedform of normal (siy and siz ) and interfacial shear stresses (siyx and sizx at abscissa x ¼xa), for instance, in the insulating component. ( ) 1 1 X X i sy ¼ f mn,y sinðkm,y yÞsin½kn,y ðz þ jm,y Þ ð7Þ n¼1
siz ¼
m¼1
(
1 X n¼1
xa
s
Fig. 2. (a) x–z plane view of the current configuration indicating the abscissa of component boundaries; (b) local balance in the x–z plane. From left to right: the elements of foam insulation, adhesive layer, and metallic substrate.
xa
) f mn,z sinðkm,z zÞsin½kn,z ðyþ jm,z Þ
ð8Þ
m¼1
siyx ¼ ei i zx
1 X
¼ ei
1 X
(
n¼1 1 X
1 X
) km,y f mn,y cosðkm,y yÞsin½kn,y ðz þ jm,y Þ
ð9Þ
m¼1
(
n¼1
1 X
) km,z f mn,z cosðkm,z zÞsin½kn,z ðyþ jm,z Þ
ð10Þ
m¼1
with km,y ,km,z kn,y , kn,z f mn,y , f mn,z , jm,y , and jm,z parameters function only of material properties and/or dimensions. Their expressions are not reported here for shortness reasons but they can be obtained without difficulty by following the derivation given in [10].
3. Comparison with FE simulation With " # " # 1 1 1 1 1 1 ; a ; þ ¼ þ z dy ei Eiy es Es dz ei Eiz es Es " # " # 1 uiyz us 1 uiyz us by ¼ þ ¼ þ ; b ; z s s dy ei Eiy es E dz ei Eiz es E " # s i 1 s 1us us s1 z sy þ E ðay a ÞDT ; cy ¼ s 1us dy ei E " # s i 1 s 1us us s1 y sz þ E ðaz a ÞDT cz ¼ ; s 1us dz ei E ea es e ea es e þ i ; and dz ¼ a þ þ i : dy ¼ a þ 3Gs 3Giyx 3Gs 3Gizx G G ay ¼
ð4Þ
The a, b, c, and d coefficients in Eq. (4) are only function of material properties and external loads. On the first hand, when the adhesive is infinitely thin ea =Ga o o es =Gs þ ei =Gikx , we have dk ei =3Gikx þ es =3Gs (for k¼y and z). It means that the problem becomes independent of the adhesive properties and the stresses sayx and sazx define the shear stresses at the insulation/substrate interface. We can thus use the current model to study the problems of overlaminated joints in which the adherends are brought in direct contact [15]. On the other hand, for strongly rigid adherends ea =Ga 4 4 es =Gs þei =Gikx (for k¼ y and z), we have dy ¼ dz ea =Ga . The adherend shear deformations become negligible and the current approach reduces to the previous models, which disregard the adherend shear deformations [10,11]. The boundary conditions corresponding to the present problem are [1]:
For validation purposes, the results of the semi-analytical model are compared in this section with numerical results obtained via a FE model. The FE calculations were performed with ABAQUS 6.9 package. Thanks to the symmetry in the model, only one-quarter of the plate was simulated according to Fig. 3(a). The FE model consists of 70,000 tridimensional eight-noded solid elements (C3D8R). Small element aspect ratios were chosen near the borders of the plates (y¼0 and z¼0) and near the interfaces with the adhesive layer while higher element aspect ratios were used near the middle of the plates (in which the stresses are expected to be almost constant) and near the upper insulation surface (see Fig. 3(b)). Through a parametric study (which is not shown here for shortness reasons) where the type and size of elements were varied, we have checked that the above mentioned element type and mesh distribution enable us to obtain accurate results of the current problem. The integrity of the proposed semi-analytical solution was evaluated by considering successively the cases of isotropic and orthotropic insulations. Moreover, to assess the effects of the loading type on the joint behavior, we considered that the system
insulating panel
Lz/2
siy ¼ 0 and siz ¼
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi cz 1coshðLz az Þ pffiffiffiffiffi sinhðz az Þ1 coshðz az Þ þ sinhðLz az Þ az
∞ y
∞ z
adhesive layer
at y¼0 and y¼Ly,
Ly/2
aluminum plate x y
z
ð5Þ
at z¼0 and z ¼Lz, siz ¼ 0 and siy ¼
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 1coshðLy ay Þ cy coshðy ay Þ þ pffiffiffiffiffi sinhðy ay Þ1 ay sinhðLy ay Þ
ð6Þ
Fig. 3. Geometries considered in the FE model. (a) Illustration of a biaxial tensile loading and the displacement boundary conditions. (b) Mesh distribution.
14
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
is subjected either to thermal stresses or biaxial tensile stresses. The results in the case of combined loads can be found in reference [1]. In the following, the loads, the material properties, and the structure dimensions are chosen so that they are representative of cryogenic rocket reservoirs. The substrate was in aluminum alloy 2219 T87, commonly used for cryogenic tanks of space vehicles [17]. The properties of the aluminum are from references [18]. The properties of adhesive are those of EA 9394 epoxy film adhesive manufactured by Henkel Corporation, frequently recommended for cryogenic applications [19]. The insulating component was a low density (about 80 kg/m3) closed-cell polymer foam. The insulation dimensions were Ly ¼Lz ¼50 cm and ei ¼2 cm. The adhesive layer and substrate thicknesses are ea ¼0.15 mm and es ¼2 mm, respectively. The mechanical properties used in the calculations were all at cryogenic temperature (between 20 K and 30 K) retrieved from literature and summarized in Table 1. The thermal load was a negative differential temperature of DT¼ 270 K. This is typically the temperature drop of the metallic shell during the fueling with liquid hydrogen (LH2) where the walls are initially at room temperature. However, for simplicity reasons, we assumed that the temperature difference DT was uniform over the joint, i.e., there was no temperature variation through the component thicknesses. Note that when the temperature gradients through the adherends are important, this assumption tends to give conservative failure load predictions. The fuel pressure imposes biaxial stresses in the metallic substrate. Typical values of operating fuel pressure (about 3 bars) and reservoir dimensions (thickness of 2 mm and radius of 2.7 m) give approximately s1 y ¼ 400 MPa as hoop stress and s1 z ¼ 200 MPa as axial stress. The metallic shell has much higher strength than the other components. Consequently, the damage of the foam/adhesive/ metal assembly will probably occur within softer components, i.e., the insulating or the adhesive materials. Moreover, the shear deformation in the metallic component will be small due to its high stiffness. For these reasons, we will focus our attention on the stress distributions within the two vulnerable components. The normal stresses are almost thickness independent (i.e., identical on any y–z plane perpendicular to the insulation thickness). In the FE model, the normal stresses were taken at the middle plane of the insulation. The shear stress distributions on the insulation/adhesive interface are particularly interesting because the most critical shear stresses are expected to take place close to this plane.
1 through s1 y and sz . The distributions of normal stresses along the y- and z-directions of insulation are depicted in Fig. 4(a) and (c) according to the current model and in Fig. 4(b) and (d) according to the FE simulation. Note that the geometry orientation is highlighted in these figures. We can see that:
The distributions of stresses in y-direction, siy , and z-direction, siz , are qualitatively similar but the values of stresses are quite
The distributions of shear stresses siyx and sizx on the y–z plane at abscissa x¼ xa are depicted in Fig. 5(a) and (c) according to the semi-analytical model and in Figs. 5(b) and (d) according to the FE model. The following points can be noted:
In contrast to the normal stresses, the shear stresses are
3.1. Isotropic insulating foam We considered first that the insulating component, i.e., the foam, exhibits isotropic properties. The properties of the isotropic foam having a typical density of 80 kg/m3 are summarized in Table 1. 3.1.1. Discussion of stress distributions 3.1.1.1. Reservoir pressure effects. The reservoir pressure effects were modeled by a biaxial tensile loading in the y–z plane Table 1 Mechanical properties in the temperature range 20 K to 30 K used in calculations. Material
Ez, GPa Ey, GPa m yz
Aluminum 85 Adhesive 11.72 Isotropic foam 0.146 Orthotropic foam 0.073 0.183
Gzx, GPa Gyx, GPa a, 10 6 1/K
0.32 32.3 – 4.14 0.33 0.050 0.51 0.030 0.060
14.4 44 36.15 36.15
different. This is because the hoop stresses (along y-axis) are greater than the axial stresses (along z-axis). The normal stresses are nearly constant except near the borders where they decrease rapidly. Concerning siy (see Fig. 4(a) and (b)), it vanishes at the free borders y¼0 and y ¼Ly. The 2D description allows detecting also the difference between siy along the boundaries z ¼0 and the maximum value of siy due to the Poisson’s ratio effects. Note that if a 1D model was used, the results would be equal to siy along the boundaries z ¼0 because such very simple model does not account for the Poisson’s ratio effects. The same trends are observed for siz (Fig. 4(c) and (d)) by considering analogous boundaries. Moreover, the Poisson’s ratio effects are more noticeable on siz than on siy because of the large contribution of hoop effects.
approximately zero in the major part of the plate area; however, they increase rapidly toward the free borders (boundaries at y¼0 and y¼Ly for siyx ; and boundaries at z¼0 and z¼Lz for sizx ). Such stress distribution means that the load transferred by shear between the insulation and the substrate is restricted principally to these finite zones near the free borders. Along these borders, the stresses should vanish because of the stress-free end condition. The FE model fulfils this condition but the semi-analytical model does not verify it. This is a common drawback of all analytical or semi-analytical solution methods. Nevertheless, the incapability of semi-analytical solution to satisfy the end condition does not question its correctness as we will see below. As for normal stresses, the predominance of hoop stress over the axial stress results in high insulation stress values along ydirection siyx , and in a large Poisson’s ratio effects on the insulation stresses along z-direction sizx .
3.1.1.2. Thermal loading. The insulating component contracts more than the metallic shell since aiy DT and aiz DT o as DT o0. This situation is equivalent to the previous one, i.e., a foam/metal joint subjected to a biaxial tensile loading. As a consequence, the normal and shear stresses in the insulation resulting from the thermal stress have the same trend as that resulting from the reservoir pressure effects. However:
The stress distributions along y- and z-directions are now
identical because the thermal load is the same in any direction when the materials are isotropic. The maximum stress values in the insulation under a thermal load is approximately two times greater than that under the 1 biaxial tensile loads s1 y and sz . This means that the thermal effects predominate over the reservoir pressure effects in cryogenic insulation applications as considered here.
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
15
Fig. 4. Normal stress distributions due to the reservoir pressure effects for isotropic insulation. (a) siy and (c) siz results of the semi-analytical model. (b) siy and (d) siz results of the FE model.
Fig. 5. Shear stress distributions due to the reservoir pressure effects for isotropic insulation. (a) siyx and (c) sizx results from the semi-analytical model. (b) siyx and (d) sizx results from FE model.
3.1.2. Comparison between semi-analytical model and FE simulation The comparison of results from both approaches depicted in Fig. 4(a)–(d) to Fig. 5(a)–(d) enables us to show that the current solution method predicts qualitatively well the stress distributions
calculated by the FE model in the presence of reservoir pressure effects. In the case of a joint subjected to a thermal loading, the semi-analytical method also gives good predictions but the results are not reported here for shortness reasons.
16
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
For further comparison, the normal stresses siy and siz along typical in the insulation from the two approaches are depicted in Fig. 6(a) and (b) for both cases of loading. As it can be seen, the theoretical results match very well the numerical counterpart. This confirms the accuracy of the proposed method for predicting, for instance, the normal stress distributions in the y–z plane of isotropic foam insulation glued on a metallic shell subjected to thermal load or reservoir pressure effects. Similarly, the corresponding shear stresses siyx and sizx along the centerlines of the insulating plate at abscissa x¼xa are shown in Fig. 6(c) and (d). The semi-analytical model agrees well with the reference solution but it deviates slightly close to the border. Although the semi-analytical model does not satisfy the stress-free end condition contrary to the FE model, it predicts the values of maximum shear stress and their locations with an accuracy of 6%.
So far, we have shown that the insulation shear stresses close to the adhesive layer are accurately predicted by the proposed model. However, we have no idea concerning the shear stress variation through the insulation thickness. In order to do so, the variation of shear stress peaks along the thickness direction, x, are depicted in Fig. 7(a) (in the case of thermal effects) and Fig. 7(b) (in the case of reservoir pressure effects). In these figures, abscissas 0 and 1 refer, respectively to the insulation/adhesive interface and the outer insulation surface. We can note first that the semi-analytical approach enables us to predict the stress at boundaries and close to adhesive surfaces. However, the prediction of shear stress peaks within the insulation thickness is less satisfactorily because the shear stresses vary according to a cubic law rather than a linear law. The relative difference between the semi-analytical solution and the FE simulation can reach 50%.
1.4 z = Lz/2
1.2
Normal stress σz, MPa
Normal stress σy, MPa
1.4
thermal effects z=0 z = Lz/2 z=0 pressure effects
1.0 0.8 0.6 0.4 0.2
FE Semi-analytical
0.0
0.8 0.4 0.2
-0.4
thermal effects
-0.8 -1.0
FE Semi-analytical
-1.2 0.00
0.05
0.10
0.15
y=0 pressure effects 0.0 0.1 0.2 0.3 0.4 0.5 Dimensionless insulating plate length, z/Lz
Interfacial shear stress σzx, MPa
Interfacial shear stress σyx, MPa
-0.2 -0.6
FE Semi-analytical y = Ly/2
0.6
0.0
pressure effects
0.0
thermal effects y=0
1.0
0.0 0.1 0.2 0.3 0.4 0.5 Dimensionless insulating plate width, y/Ly
0.2
y = Lz/2
1.2
0.2
0.2
pressure effects
0.0 -0.2 -0.4 -0.6
thermal effects
-0.8 -1.0
FE Semi-analytical
-1.2 0.00
Dimensionless insulating plate width, y/Ly
0.05
0.10
0.15
0.2
Dimensionless insulating plate length, z/Lz
Fig. 6. Comparison of stress profiles along typical lines along an isotropic insulation calculated from the semi-analytical and FE models. (a) siy along the centerline z ¼Lz/2 and along the border z¼ 0; (b) siz along the centerline y¼ Ly/2 and along the border y¼ 0; (c) siyx along the centreline z¼ Lz/2; (d) sizx along the centreline y¼Ly/2.
0.7
FE Semi-analytical
1.2
Shear stress peaks, MPa
Shear stress peaks, MPa
1.4
1.0 yx=
0.8
zx
0.6 0.4 0.2
FE Semi-analytical
0.6 yx
0.5 0.4 0.3 0.2 0.1 zx
0.0 0.0
0.2 0.4 0.6 0.8 Dimensionless thickness, (x-x )/e
1.0
0.0
0.2 0.4 0.6 0.8 Dimensionless thickness, (x-x )/e
1.0
Fig. 7. Comparison of peaks of shear stresses through the insulation thickness calculated from the semi-analytical and FE models. (a) Reservoir pressure effects; (b) Thermal stress effects.
1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
1.0
thermal effects
z = Lz/2
Normal stress σz, MPa
Normal stress σy, MPa
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
z=0 z = Lz/2 z=0
pressure effects FE Semi-analytical
pressure effects
0.0 -0.2 -0.4 -0.6
thermal effects
-0.8 -1.0
FE Semi-analytical
-1.2 -1.4 0.00
0.05
0.10
0.15
0.20
Dimensionless insulating plate width, y/Ly
y = Lz/2
0.6
thermal effects y=0
0.4
y = Ly/2
0.2
y=0 pressure effects
0.0
0.0 0.1 0.2 0.3 0.4 0.5 Dimensionless insulating plate length, z/Lz Interfacial shear stress σzx, MPa
Interfacial shear stress σyx, MPa
0.2
FE Semi-analytical
0.8
0.0 0.1 0.2 0.3 0.4 0.5 Dimensionless insulating plate width, y/Ly
17
0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8
pressure effects
thermal effects FE Semi-analytical 0.00
0.05
0.10
0.15
0.20
Dimensionless insulating plate length, z/Lz
Fig. 8. Comparison of stress profiles along typical lines along an orthotropic insulation calculated from the semi-analytical and FE models. (a) siy along the centerline z¼ Lz/2 and along the border z¼ 0. (b) siz along the centerline y¼ Ly/2 and along the border y¼0. (c) siyx along the centerline z¼ Lz/2. (d) sizx along the centerline y¼ Ly/2.
Although this deviation is non-negligible, generally the locations of maximum stresses, which are actually near the adhesive interface, are of main importance for dimensioning purpose. This justifies the suitability of the proposed approach. Moreover, this drawback can be overcome by modeling the insulation as a multilayer and by applying the linear stress assumption in each layer. It is interesting to note that the shear stresses in the insulation, particularly near the adhesive surface, are of the same magnitude order as the normal stresses. Therefore, both stresses may affect the insulation failure while in the classical formula, only the normal stresses are assumed to be relevant [2]. Note also that the current FE model predicts quite insignificant adhesive peel stresses (the results are not shown for shortness reasons), which is consistent with prior analysis of stiffened joints subjected to tensile loading [8].
case. For comparison purposes, the stress profiles along typical lines of the insulating plate from the proposed method and the FE model are depicted hereafter. The normal stresses in y-direction siy and z-direction siz are shown in Fig. 8(a) and (b), respectively. The Poisson’s ratio strain effects given by the difference of stresses along the border (e.g., z¼ 0 in Fig. 8(a) and y¼ 0 in Fig. 8(b)) and along the centerline (e.g., z ¼Lz/2 in Fig. 8(a) and y¼Ly/2 in Fig. 8(b)) are important as for isotropic foam. A slight deviation of the semi-analytical solution (less than 6%) from the FE occurs at points where the normal stresses begin to decrease abruptly but elsewhere the agreement is remarkable. The shear stress profiles are shown in Fig. 8(c) and (d). The proposed approach predicts globally the shear stress profiles. However, the agreement is not better than that with isotropic insulation. Indeed, the deviation of shear stresses and particularly the stress peaks in the case of orthotropic insulating foam can reach 15%.
3.2. Orthotropic insulating foam 4. Conclusion The foams are generally anisotropic. The anisotropy considered here is due to the elongation of the cell in one direction. For polymer foams, the shape anisotropy occurs due to the rise process during the foam production. Therefore, the foam mechanical properties are direction dependent. Typical mechanical properties of orthotropic polymer foams are summarized in Table 1. As we can notice, the thermal expansion coefficients remain identical in all directions because they are mainly bulk polymer properties, i.e., independent of cell anisotropy [20]. The normal and shear stress distributions on the y–z planes, not reported here for clarity reasons, have similar trend as in the case of isotropic foam. Again, the thermal effects are more significant than the reservoir pressure effects. Due to the foam orthotropy, the normal and shear stress values in the y-direction are much greater than those in the z-direction for each loading
A semi-analytical approach was applied for analyzing the stress distributions in rectangular foam panel glued on metallic plate subjected to biaxial tensile loading or differential thermal expansion. A comparative study with the FE simulation is reported here for various test cases to validate the model. They include isotropic and orthotropic insulations, and loadings due to thermal stress and reservoir pressure effects. The following conclusions can be drawn. The stress distributions present a 2D behavior due essentially to the Poisson’s ratio strain effects and the biaxial-nature of loadings. Therefore, the use of 1D model is not realistic and a 2D modeling such as the solution presented here is required. The insulation shear stresses are comparable to the normal stresses. Both stresses should be considered when analyzing the joint
18
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
strength through a stress based failure criterion (e.g., the famous Deshpande and Fleck or Gibson and Ashby model for foams [20]). The present model predicts satisfactorily the results of the FE simulation. This shows the suitability of the model for studying the load transfer between shear deformable two-dimensional adherends such as insulation/metallic shell bonded joints. As a consequence, thanks to its semi-analytical nature, it is more practical than FE simulation, for example, for optimization purposes. The most critical hypothesis in the current model concerns the linear shear stress and the uniform temperature gradient assumptions through the insulation thickness. However, such crude simplifications can be improved by considering the insulating plate as multi-layer in which one the linear shear stress and a uniform temperature gradient (but may be different from other layers) are assumed. This improvement is currently in progress. Finally, it is interesting to note that when predicting the joint strength, the stress based criteria evoked above may be inaccurate in presence of stress singularities, which occur in the vicinity of geometrical discontinuities. In that case, others methods such as Fracture mechanic approach could be followed.
Acknowledgment The authors acknowledge gratefully the support of the French National Research Agency through the project No. ANR-08-MAPR0009. References [1] Randrianalisoa J, Dendievel R, Bre´chet Y. On the thermomechanical behavior of two-dimensional foam/metal joints with shear-deformable adherends – parametric study. Compos. Part B 2011;42:2055–66.
[2] Roark R, Young W. Formulas for Stress and Strain.4th edn. New York: McGraw-Hill; 1975. [3] da Silva LFM, das Neves PJC, Adams RD, Spelt JK. Analytical models of adhesively bonded joints – Part I: Literature survey. Int. J. Adhes. Adhes. 2009;29:319–30. [4] Volkersen O. Die Nietktraftverteilung in zugbeanspruchten Nietverbindungen mit konstanten Laschenquerschritten. Luftfahrtforschung 1938;15:41–7. [5] de Bruyne NA. The strength of glued joints. Aircr. Eng. 1944;16:115–8. [6] Goland M, Reissner E. The stresses in cemented joints. J. Appl. Mech. 1944;11:A17–27. [7] Hart-Smith LJ. Adhesive-bonded double-lap joints. NASA Contract Rep. NASA C 1973:R–112235. [8] Zou GP, Shahin K, Taheri F. An analytical solution for the analysis of symmetric composite adhesively bonded joints. Compos. Struct. 2004;65: 499–510. [9] Baker AA. Bonded composite repair of fatigue-cracked primary aircraft structure. Compos. Struct. 1999;47:431–43. [10] Mathias JD, Balandraud X, Gre´diac M. On the bidirectional stress distribution in rectangular bonded composite patches. Int. J. Solids Struct. 2006;43:6921–47. [11] Deheeger A, Mathias JD, Gre´diac M. A closed-form solution for the thermal stress distribution in rectangular metal/composite bonded jointsJ. Int. Adhes. Adhes. 2009;29:515–24. [12] Adams RD, Peppiatt NA. Effect of Poisson’s ratio strains in adherends on stresses of an idealized lap joint. J. Strain Anal. 1973;8:134–9. [13] Tsai MY, Morton J, Oplinger DW. Deformation and stress analyses of doublelap adhesive joints with laminated composite adherends. Nashville, TN, USA: The 8th International Congress on Experimental Mechanics; 1996. [14] Tsai MY, Oplinger DW, Morton J. Improved theoretical solutions for adhesive lap joints. Int. J. Solids Struct. 1998;35:1163–85. [15] Osnes H, Mc George D. Analysis of overlaminated double-lap joints. Compos. Part B 2005;36:544–58. [16] Timoshenko SP. Strength of Materials: Part II Advanced Theory and Problems.3rd edn. New York: D. van Nostrand Company; 1956. [17] Davis JR, editor. Materials Park, Beachwood, OH: ASM International; 1993. [18] Simon NJ, Drexler ES, Reed RP. Review of cryogenic mechanical and thermal properties of Al–Li alloys and Alloy 2219. National Institute of Standards and Technology 1991:91–3971. NISTIR-. [19] A. Bartoszyk, Design/analysis of metal/composite bonded joints for survivability at cryogenic temperatures, Finite Element Modeling Continuous Improvement Workshop FEMCI, Greenbelt, MD, USA (2004). [20] Gibson LJ, Ashby MF. Cellular Solids: Structure and Properties.2nd edn. Cambridge, UK: Cambridge University Press; 1997.
Contents lists available at SciVerse ScienceDirect
International Journal of Adhesion & Adhesives journal homepage: www.elsevier.com/locate/ijadhadh
On the thermomechanical behavior of two-dimensional foam/metal joints with shear-deformable adherends: Model validation with FE analysis Jaona Randrianalisoa a,n, Remy Dendievel b, Yves Bre´chet b, Paul-Marie Michaud c, Romain Filipi c a
GRESPI, Universite´ de Reims Champagne Ardenne, F-51687, Reims, France SIMAP, CNRS, Grenoble INP, Universite´ Joseph Fourier, F-38402, Saint Martin d’He res, France c EC2MS, 66 Bd Niels Bohr, F-69603,Villeurbanne, France b
a r t i c l e i n f o
abstract
Available online 21 January 2012
The stress distributions in metal/adhesive/foam planar joints subjected to biaxial tensile load and thermal load was investigated through a semi-analytical model. The shear deformation of adherends was accounted for according to a linear law in order to obtain closed-form solutions. For the model validation, a comparative study with a finite element (FE) simulation was carried out. A 2D behavior of stress fields is observed due especially to the Poisson’s ratio effects and the biaxial nature of loads. The through thickness shear stresses are comparable to normal stresses; therefore, the adherend shear deformation must be accounted for correct failure prediction. According to the comparison with FE results, the normal stress distributions at any location in the foam and the shear stresses in the foam regions close to the adhesive surface can be well predicted by the proposed model. The through thickness shear stresses, however, showed to vary according to a cubic law rather than a linear law. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Epoxy adhesive Aluminum and foams Finite element stress analysis Stress distribution
1. Introduction For space launch vehicles, the need to develop light structures has led to foam as insulating material while the structural function is carried out by a metallic shell. A typical example, which is studied here, is the cryogenic tanks of launchers. In these structures, the insulation system consists of polymer foam panels glued on the metallic shell. In this work, the main concern is to study the stress distributions within the insulation/metallic shell assembly. Such analysis is useful for predicting the assembly strength through stress based criteria. The combination foam and metal is very critical due to the incompatibility between their thermal dilatation and this situation may lead to failure either within the foam or the adhesive layer. In addition, the reservoir is often under pressure so that the mechanical loading of the foam–metal bilayer is a complex mixture of biaxial and shear loading. As discussed in the our recent study [1], the thermomechanical problem of insulation of a large cryogenic reservoir (halfillustrated in Fig. 1(a)) can be reduced to the problem of a single insulating panel glued on the metallic plate and subjected to thermal stresses due to the thermal expansion incompatibility of components, and axial and hoop stresses due to reservoir pressure effects (see in Fig. 1(b)). On the first hand, a finite element (FE) solution of this reduced problem is possible but it is not necessarily the best strategy when the ultimate goal is to provide an optimization of the material
n
Corresponding author. Tel.: þ33 3 26 91 32 51; fax: þ33 3 26 91 32 50. E-mail address: [email protected] (J. Randrianalisoa).
0143-7496/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijadhadh.2012.01.006
choice and the component dimensions. On the other hand, the simple formula often used for dimensioning tank insulation [2] is very useful but is questionable in some situations since it misses some important mechanics. There is thus a need to develop an effortless and less time-consuming approach enabling us to provide a transparent analysis of the influence of the variables of the problem (material properties and component dimensions). In the literature, adhesively bonded bi-materials are usually treated using the shear-lag theory. The recent literature review, conducted by da Silva et al. [3], gives an overview of adhesively bonded joints and the proposed solution approaches. The most studied joints are the single-lap and double-lap joints [4–7]. There are also some studies concerning stiffened joints [8], which seem close to the current configuration. It is interesting to note that almost all existing closed-form solutions neglect (i) the stresses in the normal direction to the loads caused by Poisson’s ratio strains in the adherends; and (ii) the through thickness shear adherend deformations. In many situation encountered in practice (such as structures repaired or reinforced by patches [9–11] and reservoirs insulated with glued panels as in the current problem), the systems are twodimensional (2D) and the loading is more complex. Attempts to include these complexities have been initiated by Adams and Pappiatt [12] through a simplified 2D stress model. A 2D strain– stress relationship, allowing the connection between the in-plane stresses due to the Poisson’s ratio effects, was considered. It has been, however, shown that this simplified model provides only accurate prediction for joints subjected to uniaxial loading [10,12]. Later, Mathias et al. [10] improved the Adams and Pappiatt’s works
12
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
z
R
θ
r
P
σ z∞ Ly
σ∞ y
ΔT
Lz
L/2 x
σ∞ y
ΔT
σ z∞ P
ei
es
Fig. 1. Illustration of the studied configurations. (a) Lower-half part of cryogenic tank recovered with foam panels and subjected to internal pressure P and temperature drop DT; (b) plane geometry approximation of an elementary unit.
by taking into account correctly the coupling between in-plane stresses and the effects of the Poisson’s ratio strains. Cases of joints subjected to biaxial tension or plane shear loadings were investigated. Deheeger et al. [11] extended the Mathias et al.’ model to study the problem of reinforcement of metallic structures with composite patches subjected to thermal stresses due to the difference of thermal expansion of adherends. In all the above theoretical models, the shear deformations of adherends have been neglected. However, when the shear stiffnesses of adherends are much lower than that of the adhesive, large shear deformations will also be present at the adherend surfaces adjacent to the adhesive layer for shear stress equilibrium at the interface. It is the case of some laminated composite adherends [13–15] and probably of polymer foam layer glued on metallic plate as studied here. Thus, the adherend shear deformations must be included in the theoretical models. In our recent works [1], a semianalytical model enabling to take into account the adherend shear deformations, the Poisson’s ratio effects, and the 2D nature of loadings was developed and was applied to explore the influence of material mechanical properties and component dimensions on the foam/metallic plate behavior. To get an idea about the validity limits of this model, a comparative study with a reference approach is necessary. In the current study, a comparative study of the semianalytical model against FE simulation is conducted. The effects of type of loading (biaxial stresses or thermal stresses) are explored independently. Also, the case of joints made of isotropic and orthotropic polymer foam adherends is considered. The paper is structured as follows. First, the semi-analytical model is recalled after evoking briefly the major simplifying hypotheses. Then, the comparative study of the current model against FE simulation is presented. The appropriateness of each simplification is then discussed.
(for the insulating plate), ea (for the adhesive) and es (for the substrate). The adherends were assumed to be subjected only to in-plane stresses (classical assumption of problems of thin reservoirs [16]). The adhesive layer was considered to be a shear spring allowing transferring the in-plane forces from an adherend to the other. The shear stresses in the adhesive layer were assumed constant through the thickness. The three materials involved were assumed to be linear elastic until they fail by brittle fracture (at least at the temperature level 20 K that we are interested here). The substrate and the adhesive are isotropic whereas the insulating foam may be considered as orthotropic. This orthotropic behavior of the insulation is inherited from the foaming process discussed in Section 3.2. The involved material parameters are thus the following: Es , Gs , us and as for the substrate; Eiy , Eiz , Giyx , Gizx , uiyz , aiy , and aiz for the insulating material (ignoring the x-components); and Ga for the adhesive. E, G, u, and a refer to the Young’s modulus, shear modulus, Poisson’s ratio, and thermal expansion coefficient, respectively. The effects of bending moments and peeling due to load-path eccentricity were neglected. The shear stresses vary linearly through the adherend thicknesses. This simplification is essential to obtain a semi-analytical solution of the current problem. Note that this simplification fulfils the boundary and continuity conditions (in terms of shear stresses and strains), i.e., the adherend shear stresses are zero at the outer insulation and inner substrate surfaces; and the adherend and adhesive shear stresses/strains are identical at interfaces.
2.2. Semi-analytical model According to the assumptions listed in Section 2.1, the unknowns of the insulation/adhesive/metallic plate bonded joint problem are the following stress components. 2 3 2 3 0 sjyx sjzx 0 sayx sazx 6 j 7 6 a j 6 syx sy s 0 0 7 ð1Þ 0 7 5 4 5 and 4 yx a j j s 0 0 zx szx 0 sz
2.1. Hypothesis
The left equation refers to the stress states in the adherends with j ¼i for the insulating material, and j ¼s for the substrate whereas the right equation refers to the stresses in the adhesive layer, which are only function of y and z. Fig. 2(a) depicts the x–z plane view of the joint. The inner substrate surface is at abscissa x ¼ 0, the adhesive/substrate interface is at abscissa xs , the insulation/adhesive interface is at abscissa xa , and the outer insulation surface is at abscissa xi . The equilibrium equations schematically shown in Fig. 2(b), the stress–strain relationship, the kinematic equations, the continuity of internal forces, and the through-thickness linear shear stress assumption enable us to derive the equations governing the stress distributions in Eq.(1). For the details of derivation, the readers are recommended to refer to reference [1]. It is shown that the normal stresses in the in-plane directions in the insulating material satisfy the second order differential Eqs. (2) and (3):
In the following, the indexes i, s, and a refer respectively to insulating plate, metallic shell, and adhesive layer.
@2 siz ¼ az siz þ bz siy þcz @z2
2. Theoretical model
As illustrated in Fig. 1(b), a parallelepipedic configuration
@2 siy
was considered (length, Lz, width, Ly, and the thicknesses ei
@y2
¼ ay siy þ by siz þ cy
ð2Þ
ð3Þ
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
13
Closed-form solutions of coupled differential Eqs. (2) and (3) are more practical than numerical method. The semi-analytical method based on Fourier series [10] gives the following closedform of normal (siy and siz ) and interfacial shear stresses (siyx and sizx at abscissa x ¼xa), for instance, in the insulating component. ( ) 1 1 X X i sy ¼ f mn,y sinðkm,y yÞsin½kn,y ðz þ jm,y Þ ð7Þ n¼1
siz ¼
m¼1
(
1 X n¼1
xa
s
Fig. 2. (a) x–z plane view of the current configuration indicating the abscissa of component boundaries; (b) local balance in the x–z plane. From left to right: the elements of foam insulation, adhesive layer, and metallic substrate.
xa
) f mn,z sinðkm,z zÞsin½kn,z ðyþ jm,z Þ
ð8Þ
m¼1
siyx ¼ ei i zx
1 X
¼ ei
1 X
(
n¼1 1 X
1 X
) km,y f mn,y cosðkm,y yÞsin½kn,y ðz þ jm,y Þ
ð9Þ
m¼1
(
n¼1
1 X
) km,z f mn,z cosðkm,z zÞsin½kn,z ðyþ jm,z Þ
ð10Þ
m¼1
with km,y ,km,z kn,y , kn,z f mn,y , f mn,z , jm,y , and jm,z parameters function only of material properties and/or dimensions. Their expressions are not reported here for shortness reasons but they can be obtained without difficulty by following the derivation given in [10].
3. Comparison with FE simulation With " # " # 1 1 1 1 1 1 ; a ; þ ¼ þ z dy ei Eiy es Es dz ei Eiz es Es " # " # 1 uiyz us 1 uiyz us by ¼ þ ¼ þ ; b ; z s s dy ei Eiy es E dz ei Eiz es E " # s i 1 s 1us us s1 z sy þ E ðay a ÞDT ; cy ¼ s 1us dy ei E " # s i 1 s 1us us s1 y sz þ E ðaz a ÞDT cz ¼ ; s 1us dz ei E ea es e ea es e þ i ; and dz ¼ a þ þ i : dy ¼ a þ 3Gs 3Giyx 3Gs 3Gizx G G ay ¼
ð4Þ
The a, b, c, and d coefficients in Eq. (4) are only function of material properties and external loads. On the first hand, when the adhesive is infinitely thin ea =Ga o o es =Gs þ ei =Gikx , we have dk ei =3Gikx þ es =3Gs (for k¼y and z). It means that the problem becomes independent of the adhesive properties and the stresses sayx and sazx define the shear stresses at the insulation/substrate interface. We can thus use the current model to study the problems of overlaminated joints in which the adherends are brought in direct contact [15]. On the other hand, for strongly rigid adherends ea =Ga 4 4 es =Gs þei =Gikx (for k¼ y and z), we have dy ¼ dz ea =Ga . The adherend shear deformations become negligible and the current approach reduces to the previous models, which disregard the adherend shear deformations [10,11]. The boundary conditions corresponding to the present problem are [1]:
For validation purposes, the results of the semi-analytical model are compared in this section with numerical results obtained via a FE model. The FE calculations were performed with ABAQUS 6.9 package. Thanks to the symmetry in the model, only one-quarter of the plate was simulated according to Fig. 3(a). The FE model consists of 70,000 tridimensional eight-noded solid elements (C3D8R). Small element aspect ratios were chosen near the borders of the plates (y¼0 and z¼0) and near the interfaces with the adhesive layer while higher element aspect ratios were used near the middle of the plates (in which the stresses are expected to be almost constant) and near the upper insulation surface (see Fig. 3(b)). Through a parametric study (which is not shown here for shortness reasons) where the type and size of elements were varied, we have checked that the above mentioned element type and mesh distribution enable us to obtain accurate results of the current problem. The integrity of the proposed semi-analytical solution was evaluated by considering successively the cases of isotropic and orthotropic insulations. Moreover, to assess the effects of the loading type on the joint behavior, we considered that the system
insulating panel
Lz/2
siy ¼ 0 and siz ¼
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi cz 1coshðLz az Þ pffiffiffiffiffi sinhðz az Þ1 coshðz az Þ þ sinhðLz az Þ az
∞ y
∞ z
adhesive layer
at y¼0 and y¼Ly,
Ly/2
aluminum plate x y
z
ð5Þ
at z¼0 and z ¼Lz, siz ¼ 0 and siy ¼
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 1coshðLy ay Þ cy coshðy ay Þ þ pffiffiffiffiffi sinhðy ay Þ1 ay sinhðLy ay Þ
ð6Þ
Fig. 3. Geometries considered in the FE model. (a) Illustration of a biaxial tensile loading and the displacement boundary conditions. (b) Mesh distribution.
14
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
is subjected either to thermal stresses or biaxial tensile stresses. The results in the case of combined loads can be found in reference [1]. In the following, the loads, the material properties, and the structure dimensions are chosen so that they are representative of cryogenic rocket reservoirs. The substrate was in aluminum alloy 2219 T87, commonly used for cryogenic tanks of space vehicles [17]. The properties of the aluminum are from references [18]. The properties of adhesive are those of EA 9394 epoxy film adhesive manufactured by Henkel Corporation, frequently recommended for cryogenic applications [19]. The insulating component was a low density (about 80 kg/m3) closed-cell polymer foam. The insulation dimensions were Ly ¼Lz ¼50 cm and ei ¼2 cm. The adhesive layer and substrate thicknesses are ea ¼0.15 mm and es ¼2 mm, respectively. The mechanical properties used in the calculations were all at cryogenic temperature (between 20 K and 30 K) retrieved from literature and summarized in Table 1. The thermal load was a negative differential temperature of DT¼ 270 K. This is typically the temperature drop of the metallic shell during the fueling with liquid hydrogen (LH2) where the walls are initially at room temperature. However, for simplicity reasons, we assumed that the temperature difference DT was uniform over the joint, i.e., there was no temperature variation through the component thicknesses. Note that when the temperature gradients through the adherends are important, this assumption tends to give conservative failure load predictions. The fuel pressure imposes biaxial stresses in the metallic substrate. Typical values of operating fuel pressure (about 3 bars) and reservoir dimensions (thickness of 2 mm and radius of 2.7 m) give approximately s1 y ¼ 400 MPa as hoop stress and s1 z ¼ 200 MPa as axial stress. The metallic shell has much higher strength than the other components. Consequently, the damage of the foam/adhesive/ metal assembly will probably occur within softer components, i.e., the insulating or the adhesive materials. Moreover, the shear deformation in the metallic component will be small due to its high stiffness. For these reasons, we will focus our attention on the stress distributions within the two vulnerable components. The normal stresses are almost thickness independent (i.e., identical on any y–z plane perpendicular to the insulation thickness). In the FE model, the normal stresses were taken at the middle plane of the insulation. The shear stress distributions on the insulation/adhesive interface are particularly interesting because the most critical shear stresses are expected to take place close to this plane.
1 through s1 y and sz . The distributions of normal stresses along the y- and z-directions of insulation are depicted in Fig. 4(a) and (c) according to the current model and in Fig. 4(b) and (d) according to the FE simulation. Note that the geometry orientation is highlighted in these figures. We can see that:
The distributions of stresses in y-direction, siy , and z-direction, siz , are qualitatively similar but the values of stresses are quite
The distributions of shear stresses siyx and sizx on the y–z plane at abscissa x¼ xa are depicted in Fig. 5(a) and (c) according to the semi-analytical model and in Figs. 5(b) and (d) according to the FE model. The following points can be noted:
In contrast to the normal stresses, the shear stresses are
3.1. Isotropic insulating foam We considered first that the insulating component, i.e., the foam, exhibits isotropic properties. The properties of the isotropic foam having a typical density of 80 kg/m3 are summarized in Table 1. 3.1.1. Discussion of stress distributions 3.1.1.1. Reservoir pressure effects. The reservoir pressure effects were modeled by a biaxial tensile loading in the y–z plane Table 1 Mechanical properties in the temperature range 20 K to 30 K used in calculations. Material
Ez, GPa Ey, GPa m yz
Aluminum 85 Adhesive 11.72 Isotropic foam 0.146 Orthotropic foam 0.073 0.183
Gzx, GPa Gyx, GPa a, 10 6 1/K
0.32 32.3 – 4.14 0.33 0.050 0.51 0.030 0.060
14.4 44 36.15 36.15
different. This is because the hoop stresses (along y-axis) are greater than the axial stresses (along z-axis). The normal stresses are nearly constant except near the borders where they decrease rapidly. Concerning siy (see Fig. 4(a) and (b)), it vanishes at the free borders y¼0 and y ¼Ly. The 2D description allows detecting also the difference between siy along the boundaries z ¼0 and the maximum value of siy due to the Poisson’s ratio effects. Note that if a 1D model was used, the results would be equal to siy along the boundaries z ¼0 because such very simple model does not account for the Poisson’s ratio effects. The same trends are observed for siz (Fig. 4(c) and (d)) by considering analogous boundaries. Moreover, the Poisson’s ratio effects are more noticeable on siz than on siy because of the large contribution of hoop effects.
approximately zero in the major part of the plate area; however, they increase rapidly toward the free borders (boundaries at y¼0 and y¼Ly for siyx ; and boundaries at z¼0 and z¼Lz for sizx ). Such stress distribution means that the load transferred by shear between the insulation and the substrate is restricted principally to these finite zones near the free borders. Along these borders, the stresses should vanish because of the stress-free end condition. The FE model fulfils this condition but the semi-analytical model does not verify it. This is a common drawback of all analytical or semi-analytical solution methods. Nevertheless, the incapability of semi-analytical solution to satisfy the end condition does not question its correctness as we will see below. As for normal stresses, the predominance of hoop stress over the axial stress results in high insulation stress values along ydirection siyx , and in a large Poisson’s ratio effects on the insulation stresses along z-direction sizx .
3.1.1.2. Thermal loading. The insulating component contracts more than the metallic shell since aiy DT and aiz DT o as DT o0. This situation is equivalent to the previous one, i.e., a foam/metal joint subjected to a biaxial tensile loading. As a consequence, the normal and shear stresses in the insulation resulting from the thermal stress have the same trend as that resulting from the reservoir pressure effects. However:
The stress distributions along y- and z-directions are now
identical because the thermal load is the same in any direction when the materials are isotropic. The maximum stress values in the insulation under a thermal load is approximately two times greater than that under the 1 biaxial tensile loads s1 y and sz . This means that the thermal effects predominate over the reservoir pressure effects in cryogenic insulation applications as considered here.
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
15
Fig. 4. Normal stress distributions due to the reservoir pressure effects for isotropic insulation. (a) siy and (c) siz results of the semi-analytical model. (b) siy and (d) siz results of the FE model.
Fig. 5. Shear stress distributions due to the reservoir pressure effects for isotropic insulation. (a) siyx and (c) sizx results from the semi-analytical model. (b) siyx and (d) sizx results from FE model.
3.1.2. Comparison between semi-analytical model and FE simulation The comparison of results from both approaches depicted in Fig. 4(a)–(d) to Fig. 5(a)–(d) enables us to show that the current solution method predicts qualitatively well the stress distributions
calculated by the FE model in the presence of reservoir pressure effects. In the case of a joint subjected to a thermal loading, the semi-analytical method also gives good predictions but the results are not reported here for shortness reasons.
16
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
For further comparison, the normal stresses siy and siz along typical in the insulation from the two approaches are depicted in Fig. 6(a) and (b) for both cases of loading. As it can be seen, the theoretical results match very well the numerical counterpart. This confirms the accuracy of the proposed method for predicting, for instance, the normal stress distributions in the y–z plane of isotropic foam insulation glued on a metallic shell subjected to thermal load or reservoir pressure effects. Similarly, the corresponding shear stresses siyx and sizx along the centerlines of the insulating plate at abscissa x¼xa are shown in Fig. 6(c) and (d). The semi-analytical model agrees well with the reference solution but it deviates slightly close to the border. Although the semi-analytical model does not satisfy the stress-free end condition contrary to the FE model, it predicts the values of maximum shear stress and their locations with an accuracy of 6%.
So far, we have shown that the insulation shear stresses close to the adhesive layer are accurately predicted by the proposed model. However, we have no idea concerning the shear stress variation through the insulation thickness. In order to do so, the variation of shear stress peaks along the thickness direction, x, are depicted in Fig. 7(a) (in the case of thermal effects) and Fig. 7(b) (in the case of reservoir pressure effects). In these figures, abscissas 0 and 1 refer, respectively to the insulation/adhesive interface and the outer insulation surface. We can note first that the semi-analytical approach enables us to predict the stress at boundaries and close to adhesive surfaces. However, the prediction of shear stress peaks within the insulation thickness is less satisfactorily because the shear stresses vary according to a cubic law rather than a linear law. The relative difference between the semi-analytical solution and the FE simulation can reach 50%.
1.4 z = Lz/2
1.2
Normal stress σz, MPa
Normal stress σy, MPa
1.4
thermal effects z=0 z = Lz/2 z=0 pressure effects
1.0 0.8 0.6 0.4 0.2
FE Semi-analytical
0.0
0.8 0.4 0.2
-0.4
thermal effects
-0.8 -1.0
FE Semi-analytical
-1.2 0.00
0.05
0.10
0.15
y=0 pressure effects 0.0 0.1 0.2 0.3 0.4 0.5 Dimensionless insulating plate length, z/Lz
Interfacial shear stress σzx, MPa
Interfacial shear stress σyx, MPa
-0.2 -0.6
FE Semi-analytical y = Ly/2
0.6
0.0
pressure effects
0.0
thermal effects y=0
1.0
0.0 0.1 0.2 0.3 0.4 0.5 Dimensionless insulating plate width, y/Ly
0.2
y = Lz/2
1.2
0.2
0.2
pressure effects
0.0 -0.2 -0.4 -0.6
thermal effects
-0.8 -1.0
FE Semi-analytical
-1.2 0.00
Dimensionless insulating plate width, y/Ly
0.05
0.10
0.15
0.2
Dimensionless insulating plate length, z/Lz
Fig. 6. Comparison of stress profiles along typical lines along an isotropic insulation calculated from the semi-analytical and FE models. (a) siy along the centerline z ¼Lz/2 and along the border z¼ 0; (b) siz along the centerline y¼ Ly/2 and along the border y¼ 0; (c) siyx along the centreline z¼ Lz/2; (d) sizx along the centreline y¼Ly/2.
0.7
FE Semi-analytical
1.2
Shear stress peaks, MPa
Shear stress peaks, MPa
1.4
1.0 yx=
0.8
zx
0.6 0.4 0.2
FE Semi-analytical
0.6 yx
0.5 0.4 0.3 0.2 0.1 zx
0.0 0.0
0.2 0.4 0.6 0.8 Dimensionless thickness, (x-x )/e
1.0
0.0
0.2 0.4 0.6 0.8 Dimensionless thickness, (x-x )/e
1.0
Fig. 7. Comparison of peaks of shear stresses through the insulation thickness calculated from the semi-analytical and FE models. (a) Reservoir pressure effects; (b) Thermal stress effects.
1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
1.0
thermal effects
z = Lz/2
Normal stress σz, MPa
Normal stress σy, MPa
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
z=0 z = Lz/2 z=0
pressure effects FE Semi-analytical
pressure effects
0.0 -0.2 -0.4 -0.6
thermal effects
-0.8 -1.0
FE Semi-analytical
-1.2 -1.4 0.00
0.05
0.10
0.15
0.20
Dimensionless insulating plate width, y/Ly
y = Lz/2
0.6
thermal effects y=0
0.4
y = Ly/2
0.2
y=0 pressure effects
0.0
0.0 0.1 0.2 0.3 0.4 0.5 Dimensionless insulating plate length, z/Lz Interfacial shear stress σzx, MPa
Interfacial shear stress σyx, MPa
0.2
FE Semi-analytical
0.8
0.0 0.1 0.2 0.3 0.4 0.5 Dimensionless insulating plate width, y/Ly
17
0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8
pressure effects
thermal effects FE Semi-analytical 0.00
0.05
0.10
0.15
0.20
Dimensionless insulating plate length, z/Lz
Fig. 8. Comparison of stress profiles along typical lines along an orthotropic insulation calculated from the semi-analytical and FE models. (a) siy along the centerline z¼ Lz/2 and along the border z¼ 0. (b) siz along the centerline y¼ Ly/2 and along the border y¼0. (c) siyx along the centerline z¼ Lz/2. (d) sizx along the centerline y¼ Ly/2.
Although this deviation is non-negligible, generally the locations of maximum stresses, which are actually near the adhesive interface, are of main importance for dimensioning purpose. This justifies the suitability of the proposed approach. Moreover, this drawback can be overcome by modeling the insulation as a multilayer and by applying the linear stress assumption in each layer. It is interesting to note that the shear stresses in the insulation, particularly near the adhesive surface, are of the same magnitude order as the normal stresses. Therefore, both stresses may affect the insulation failure while in the classical formula, only the normal stresses are assumed to be relevant [2]. Note also that the current FE model predicts quite insignificant adhesive peel stresses (the results are not shown for shortness reasons), which is consistent with prior analysis of stiffened joints subjected to tensile loading [8].
case. For comparison purposes, the stress profiles along typical lines of the insulating plate from the proposed method and the FE model are depicted hereafter. The normal stresses in y-direction siy and z-direction siz are shown in Fig. 8(a) and (b), respectively. The Poisson’s ratio strain effects given by the difference of stresses along the border (e.g., z¼ 0 in Fig. 8(a) and y¼ 0 in Fig. 8(b)) and along the centerline (e.g., z ¼Lz/2 in Fig. 8(a) and y¼Ly/2 in Fig. 8(b)) are important as for isotropic foam. A slight deviation of the semi-analytical solution (less than 6%) from the FE occurs at points where the normal stresses begin to decrease abruptly but elsewhere the agreement is remarkable. The shear stress profiles are shown in Fig. 8(c) and (d). The proposed approach predicts globally the shear stress profiles. However, the agreement is not better than that with isotropic insulation. Indeed, the deviation of shear stresses and particularly the stress peaks in the case of orthotropic insulating foam can reach 15%.
3.2. Orthotropic insulating foam 4. Conclusion The foams are generally anisotropic. The anisotropy considered here is due to the elongation of the cell in one direction. For polymer foams, the shape anisotropy occurs due to the rise process during the foam production. Therefore, the foam mechanical properties are direction dependent. Typical mechanical properties of orthotropic polymer foams are summarized in Table 1. As we can notice, the thermal expansion coefficients remain identical in all directions because they are mainly bulk polymer properties, i.e., independent of cell anisotropy [20]. The normal and shear stress distributions on the y–z planes, not reported here for clarity reasons, have similar trend as in the case of isotropic foam. Again, the thermal effects are more significant than the reservoir pressure effects. Due to the foam orthotropy, the normal and shear stress values in the y-direction are much greater than those in the z-direction for each loading
A semi-analytical approach was applied for analyzing the stress distributions in rectangular foam panel glued on metallic plate subjected to biaxial tensile loading or differential thermal expansion. A comparative study with the FE simulation is reported here for various test cases to validate the model. They include isotropic and orthotropic insulations, and loadings due to thermal stress and reservoir pressure effects. The following conclusions can be drawn. The stress distributions present a 2D behavior due essentially to the Poisson’s ratio strain effects and the biaxial-nature of loadings. Therefore, the use of 1D model is not realistic and a 2D modeling such as the solution presented here is required. The insulation shear stresses are comparable to the normal stresses. Both stresses should be considered when analyzing the joint
18
J. Randrianalisoa et al. / International Journal of Adhesion & Adhesives 37 (2012) 11–18
strength through a stress based failure criterion (e.g., the famous Deshpande and Fleck or Gibson and Ashby model for foams [20]). The present model predicts satisfactorily the results of the FE simulation. This shows the suitability of the model for studying the load transfer between shear deformable two-dimensional adherends such as insulation/metallic shell bonded joints. As a consequence, thanks to its semi-analytical nature, it is more practical than FE simulation, for example, for optimization purposes. The most critical hypothesis in the current model concerns the linear shear stress and the uniform temperature gradient assumptions through the insulation thickness. However, such crude simplifications can be improved by considering the insulating plate as multi-layer in which one the linear shear stress and a uniform temperature gradient (but may be different from other layers) are assumed. This improvement is currently in progress. Finally, it is interesting to note that when predicting the joint strength, the stress based criteria evoked above may be inaccurate in presence of stress singularities, which occur in the vicinity of geometrical discontinuities. In that case, others methods such as Fracture mechanic approach could be followed.
Acknowledgment The authors acknowledge gratefully the support of the French National Research Agency through the project No. ANR-08-MAPR0009. References [1] Randrianalisoa J, Dendievel R, Bre´chet Y. On the thermomechanical behavior of two-dimensional foam/metal joints with shear-deformable adherends – parametric study. Compos. Part B 2011;42:2055–66.
[2] Roark R, Young W. Formulas for Stress and Strain.4th edn. New York: McGraw-Hill; 1975. [3] da Silva LFM, das Neves PJC, Adams RD, Spelt JK. Analytical models of adhesively bonded joints – Part I: Literature survey. Int. J. Adhes. Adhes. 2009;29:319–30. [4] Volkersen O. Die Nietktraftverteilung in zugbeanspruchten Nietverbindungen mit konstanten Laschenquerschritten. Luftfahrtforschung 1938;15:41–7. [5] de Bruyne NA. The strength of glued joints. Aircr. Eng. 1944;16:115–8. [6] Goland M, Reissner E. The stresses in cemented joints. J. Appl. Mech. 1944;11:A17–27. [7] Hart-Smith LJ. Adhesive-bonded double-lap joints. NASA Contract Rep. NASA C 1973:R–112235. [8] Zou GP, Shahin K, Taheri F. An analytical solution for the analysis of symmetric composite adhesively bonded joints. Compos. Struct. 2004;65: 499–510. [9] Baker AA. Bonded composite repair of fatigue-cracked primary aircraft structure. Compos. Struct. 1999;47:431–43. [10] Mathias JD, Balandraud X, Gre´diac M. On the bidirectional stress distribution in rectangular bonded composite patches. Int. J. Solids Struct. 2006;43:6921–47. [11] Deheeger A, Mathias JD, Gre´diac M. A closed-form solution for the thermal stress distribution in rectangular metal/composite bonded jointsJ. Int. Adhes. Adhes. 2009;29:515–24. [12] Adams RD, Peppiatt NA. Effect of Poisson’s ratio strains in adherends on stresses of an idealized lap joint. J. Strain Anal. 1973;8:134–9. [13] Tsai MY, Morton J, Oplinger DW. Deformation and stress analyses of doublelap adhesive joints with laminated composite adherends. Nashville, TN, USA: The 8th International Congress on Experimental Mechanics; 1996. [14] Tsai MY, Oplinger DW, Morton J. Improved theoretical solutions for adhesive lap joints. Int. J. Solids Struct. 1998;35:1163–85. [15] Osnes H, Mc George D. Analysis of overlaminated double-lap joints. Compos. Part B 2005;36:544–58. [16] Timoshenko SP. Strength of Materials: Part II Advanced Theory and Problems.3rd edn. New York: D. van Nostrand Company; 1956. [17] Davis JR, editor. Materials Park, Beachwood, OH: ASM International; 1993. [18] Simon NJ, Drexler ES, Reed RP. Review of cryogenic mechanical and thermal properties of Al–Li alloys and Alloy 2219. National Institute of Standards and Technology 1991:91–3971. NISTIR-. [19] A. Bartoszyk, Design/analysis of metal/composite bonded joints for survivability at cryogenic temperatures, Finite Element Modeling Continuous Improvement Workshop FEMCI, Greenbelt, MD, USA (2004). [20] Gibson LJ, Ashby MF. Cellular Solids: Structure and Properties.2nd edn. Cambridge, UK: Cambridge University Press; 1997.