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Probabilistic analysis of failure in adhesive bonded joints
Mechanics of Materials 33 (2001) 77±84
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Probabilistic analysis of failure in adhesive bonded joints Reda Hadj-Ahmed *, Gilles Foret, Alain Ehrlacher Ecole Nationale des Ponts et Chauss ees, CERMMO-AM, 6-8 Avenue Blaise Pascal, Cit e Descartes, Champs-sur-Marne, 77455 Marne-La-Vallee, France Received 26 June 2000
Abstract We propose herein a strength probability law to estimate the shear strength of an adhesive joint. A simpli®ed model enables generating an analytical solution of the shear stress present in the adhesive. The strength probability law takes into account the scale eects experimentally established for the adhesive joint shear strength and leads to studying the in¯uence of both adhesive thickness and overlap length on joint strength. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Adhesive joint; Shear stress; Shear strength; Shear Lag model; Weibull's law; Failure
1. Introduction The use of adhesives in manufacturing in plane force-transmitting joints between various structural materials is now very widespread thanks to the advantages of bonding with regard to the other conventional techniques of material assembly. Several models, which enable analyzing the stresses in adhesive joints, exist in the literature (Volkersen, 1938; Goland and Reissner, 1944; Hart-Smith, 1973; Delale et al., 1981), yet failure prediction remains dicult due to a lack of sucient criteria. Elastic models show that adhesive joint shear strength increases as adhesive thickness increases. However, many tests show that joint behavior at failure depends on the adhesive. Indeed, some
* Corresponding author. Tel.: +33-1-64153736; fax: +33-164153741. E-mail address: [email protected] (R. Hadj-Ahmed).
adhesives are ecient when their thickness is small (Joubert et al., 1979). On the contrary, for other adhesives (Delmas, 1979), joint strength increases with adhesive thickness until an ``optimal adhesive thickness''. Beyond this thickness, joint strength decreases. We are thus confronted with a scale eect in the adhesive joint strength. With regard to the overlap length, elastic models (Jeandrau, 1985; Hadj-Ahmed et al., 1997) and test results (Hadj-Ahmed, 1999) have shown that adhesive joint strength increases with overlap length until a ``limit length'', beyond which the strength no longer increases. Scale eects are often related to the presence of random defects in the adhesive (see Fig. 1). We thus propose in this paper a probabilistic analysis that allows taking into account the defects present in the adhesive. We start by writing the probability law of a reference volume of a material subjected to a uniform stress ®eld. By supposing that the material breaks upon the initial damage, we can establish,
0167-6636/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 0 0 ) 0 0 0 5 1 - X
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R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
cording to the stress ®eld r applied on V0 . We denote by PnfV0 r the non-failure probability law of the volume V0 submitted to r. By assuming that the elementary volume strengths are independent, we can easily show (see Appendix A) that the strength probability law of an arbitrary volume V is given by h iV =V0 : 1 PnfV r PnfV0 r
Fig. 1. Defects in an adhesive.
from the reference volume probability law, the probability law of any volume. This relation then enables us to describe volume-scale eects. To correctly represent the strength probabilities of brittle materials, we have chosen to use Weibull's law (Weibull, 1951) with two parameters (Fitoussi et al., 1996; Calard and Lamon, 1998). We then generalized the probability law to a structure subjected to a non-uniform stress ®eld and composed of the material described above. In this work, we are interested in an adhesive joint. For the stress analysis we selected a simpli®ed model that yields a constant shear stress according to the adhesive thickness (Volkersen, 1938; Goland and Reissner, 1944; Hart-Smith, 1973; Delale et al., 1981). We show, in this case, that the strength probability corresponds to Weibull's law on the tensile force applied to the adhesive joint. This law enables us to study the in¯uence of both adhesive thickness and overlap length on the adhesive joint shear strength and to state the existence of an ``optimal adhesive thickness'' and a ``limit overlap length''. 2. Probability law of a material subjected to a uniform stress ®eld We consider a reference volume V0 of the studied material. By conducting a series of tests, we can determine its strength probability law ac-
The ratio V =V0 exhibits the volume-scale eect for brittle materials. From a practical point of view, we have chosen Weibull's law to accurately represent the strength probabilities of the brittle materials. For the sake of simplicity, we use a scalar criterion on r, de noted f r. Thus, Weibull's law for a volume V0 can be written: " ! # f r m V0 Pnf r exp ÿ ; 2 r0 where m and r0 are the test-determined Weibull parameters. As m increases, the strength dispersion decreases and vice versa. By taking into account the volume-scale eect (Eq. (1)), the strength probability of a volume V subjected to a uniform stress ®eld r is given by " !m !#V =V0 f r V 3 Pnf r exp ÿ r0 or PnfV r
"
V exp ÿ V0
f r
r0
!m # :
4
We again have a two parameters Weibull's law. The ®rst one (m) is unchanged, while the second one (r0 ) is replaced by r0 V0 =V 1=m . In Eq. (4), it is clear that the ratio V =V0 1=m represents the scale eect. As V increases, the joint strength decreases. A volume increase actually implies an increase in the number of defects. At the same time, the stress concentration near the ends of the adhesive thickness decreases. This competition between the ``number of defects'' eect and the stress concentration can explain the existence
R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
79
of an ``optimal adhesive thickness''. When the number of defects eect is predominant, the adhesive is ecient for thin adhesive layers. On the contrary, when the stress concentration eect is predominant, the joint strength as a function of adhesive thickness displays a maximum. 3. Probability law of a structure subjected to a nonuniform stress ®eld We are interested herein in a structure X composed of the material described above and submitted to a non-uniform stress ®eld r x. We start by cutting X into elementary volumes DVi on which we can assume that the stress ®eld r xi is uniform. From Eq. (1) we obtain the following: h iDVi =V0 : 5 PnfDVi r xi PnfV0 r xi
By supposing that all the elementary volumes are independent, we can ®rst approximate the strength probability of the structure X by Y V DV =V Pnf0 r xi i 0 : 6 PnfX r xi i
In the case of Weibull's law, we ®nd !m # " xi X DVi f r X : Pnf r x exp ÿ V0 r0 i
7
By having DVi tend to 0, the probability law expression obtained is " Z # ! f r x m 1 dV : 8 PnfX r x exp ÿ r0 X V0
4. Application to an adhesive joint We have chosen a simpli®ed model (Bazant and Desmorat, 1995; Desmorat, 1996) in which the criterion f r is written onto the longitudinal shear stress sxz in the adhesive. This shear stress is denoted by s in the following text and for simplicity we assume that s is constant with respect to
Fig. 2. Symmetrical adhesive bonded double lap joint.
z (see Fig. 2). We also suppose that the joint is in plane strain. Thus, s varies only as a function of x. Eq. (8) then becomes Z l m b s x PnfX r exp ÿ e x dx ; 9 s0 0 V0 where · b is the bond width; · e x is the adhesive thickness (eventually variable); · s0 replaces r0 as the criterion on shear; · s x is the shear stress in the adhesive determined by a simpli®ed model; · l is the overlap length. s x is a linear function of the tensile force F applied on the adhesive joint (see Fig. 2). By denoting s~ x as the shear stress under a unit force (F 1), we have s x F s~ x:
10
Eq. (8) becomes " PnfX r
exp ÿ F
m
Z
l 0
s~ x b e x s0 V0
!m
# dx : 11
Thus, the strength probability follows Weibull's law on the tensile force applied on the adhesive joint. The ®rst parameter m is unchanged; the second one is replaced by F0 , where " Z !m #ÿ1=m l s~ x b e x dx : 12 F0 s0 V0 0 We can then write m F X : Pnf r exp ÿ F0
13
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R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
The parameter F0 is related to the mean strength F of the adhesive joint (Vinh, 1988; Foret, 1995) by means of the following expression: 1 C is the function gamma: F F0 C 1 m 14 We must emphasize that F0 depends on the adhesive material through m, s~ x and s0 , and on the joint geometry through b, e x and s~ x. In the following section, we will study the in¯uence of adhesive thickness and overlap length on the joint strength.
5. Adhesive thickness in¯uence We have chosen a Shear Lag model in which the longitudinal shear stress s~ in the adhesive is proportional to the dierence between the displacement of the outer adherend and the displacement of half of the inner adherend as it is shown in Eq. (22) (Volkersen, 1938; Vidonne, 1995; Hadj-Ahmed, 1996). In the case of a symmetrical and balanced adhesive joint (see Appendix B), this model provides s~ by the following expression: s~ x
k cosh k x ÿ l=2 ; 2b sinh kl=2
15
where s 2Gc k Es es e
Table 1 Mechanical and geometrical characteristics of the adhesive joint in a Shear Lag model Adherends (steel)
Adhesive (epoxy)
Es (GPa)
es (mm)
Gc (GPa)
e (mm)
210
3
1.55
0:05±2
the overlap length l is set at 40 mm. The other mechanical and geometrical characteristics of the joint in a Shear Lag model are presented in Table 1. We note here that F0 is related to the mean strength of the adhesive joint. We can then focus on two values of the parameter m. The ®rst one is m 2; the low values of m imply more scale eect. The second one is m 4; in this case (high values of m), the scale eect is weak. 5.1. High dispersion In this part we demonstrate by means of F0 and PnfX that for adhesives with high dispersion, the joint is ecient when the adhesive thickness is small. Fig. 3 presents a plot of F0 against e. We can remark that the mean strength is a decreasing function of the adhesive thickness. Similarly, Fig. 4 displays the variation of the non-failure probability PnfX as a function of the tensile force F for many adhesive thickness values. On the one hand, for each adhesive thickness, we can determine the non-failure probability at a given force level. On the other hand, we remark that as adhesive thickness increases, the non-fail-
with · Es is the adherend Young's modulus; · es is the adherend thickness; · Gc is the adhesive shear modulus. Thus, 1=m sinh kl=2 mÿ1=m
F 0 s0 V 0 e
ÿ1=m
Z
k=2
0
l
b
m ÿ1=m l cosh k x ÿ dx : 16 2
To illustrate this result, let us study the dependence of F0 on e. For the numerical applications,
Fig. 3. Mean strength versus adhesive thickness for m 2.
R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
Fig. 4. Non-failure probability versus tensile force for m 2.
81
Fig. 5. Mean strength versus adhesive layer for m 4.
ure probability decreases. For low values of the parameter m, there are so many defects in the adhesive that a slight increase in adhesive thickness implies a decrease in strength. 5.2. Low dispersion In this case, there are fewer defects in the adhesive. When the adhesive layer is small, the maximum value of shear stress is very high; therefore, the joint strength is low. By increasing the adhesive layer, all of the models assert that the maximum value of shear stress decreases. This result implies that joint strength increases with adhesive thickness until an ``optimal adhesive thickness''. Beyond this thickness, the number of defects becomes signi®cant and the joint strength decreases. Fig. 5 shows that the mean strength as a function of e has a maximum at e 0:5 mm. Similarly, Fig. 6 shows that for a given tensile force the non-failure probability is maximal at e 0:5 mm. 6. Overlap length in¯uence By comparison with experimental results (Jeandrau, 1985; Hadj-Ahmed, 1999), elastic models produce a good prediction of the in¯uence of overlap length on joint strength. Figs. 7 and 8 reveal that regardless of the m value, the joint strength displays a ``limit overlap length'' (l 40
Fig. 6. Non-failure probability versus tensile force for m 4.
mm). With regard to the non-failure probability, the curves for l equal to 40, 60 and 80 mm are superposed. For the numerical applications, e has been set here at 0.1 mm. 7. Discussion We have seen above that the in¯uence of adhesive thickness on joint strength is directly related to the parameter m. To highlight this dependence we have plotted in Fig. 9 F0 versus m for three adhesive thicknesses: 0.05, 0.5 and 1 mm. These values are typically encountered in the literature. We notice in Fig. 9 that for low values of the parameter m (m 6 3:2), the adhesive is ecient for low adhesive thicknesses. In this case, there are
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R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
Fig. 7. In¯uence of overlap length on F0 and PnfX for m 2.
Fig. 8. In¯uence of overlap length on F0 and PnfX for m 4.
many defects in the adhesive and an increase in adhesive thickness certainly involves a decrease in joint strength. When m is very high (m P 5), failure occurs in the material (the adhesive) deterministically. The increase in adhesive thickness only involves the decrease in stress concentration, hence joint strength continues to increase with adhesive thickness. For intermediate values of m (3:2 6 m 6 5), the adhesive joint displays an optimal adhesive thickness. In this case, an increase in adhesive thickness initially implies an increase in
the joint strength. This joint strength cannot increase inde®nitely with adhesive thickness as the number of defects increases with adhesive thickness as well. 8. Conclusion In this paper, we have proposed a law for the shear strength probability of an adhesive joint. The use of a simpli®ed model has allowed estab-
R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
h i2 Pnf2V0 r PnfV0 r :
83
17
We can easily extend this reasoning to a volume V pV0 , where p is an integer, and write h ip 18 PnfpV0 r PnfV0 r :
By symmetrical reasoning, the strength probability law of a volume V V0 =q is h i1=q V =q Pnf0 r PnfV0 r : 19
Fig. 9. Mean strength versus m.
lishing an analytical solution. It was then possible to study the in¯uence of adhesive thickness and overlap length by taking into account the scale eects observed experimentally. We have shown that according to the m value (high or low dispersion), the adhesive behavior varies. In particular, we have seen that for very low values of m, the joint is ecient as long as the adhesive thickness remains small. In the case of very high values of m the joint strength increases with adhesive thickness. For intermediate values of m, the joint strength as a function of adhesive thickness exhibits a maximum. Concerning the overlap length, it seems that the dispersion character of the adhesive does not in¯uence the dependence of the joint strength on the overlap length, and the adhesive joint displays nearly the same ``limit overlap length''.
Appendix A. The strength probability law of a volume V To determine the strength probability law PnfV r of an arbitrary volume V submitted to a uniform stress ®eld r we initially consider a vol ume V 2V0 (with the strength probability law PnfV0 r of the volume V0 being known). By sup posing that the volume strengths are independent, we obtain under the same stress ®eld r the fol lowing:
In applying the two reasonings above, we can write, for a volume V p=qV0 , h ip=q p=qV 20 Pnf 0 r PnfV0 r :
The continuous passing leads to, for any value of V =V0 : h iV =V0 : 21 PnfV r PnfV0 r
Appendix B. The longitudinal shear stress in the adhesive layer For an adhesive joint, the Shear Lag model chosen in this work assumes that shear stress in the adhesive is proportional to the dierence between the outer adherend (layer 1) displacement u1 x and half of the inner adherend (layer 3) displacement u3 x by means of the following expression: s x
Gc 3 u x ÿ u1 x: e
22
The elastic behavior of the adherends leads to du1 x r1 x 1 ; dx E du3 x r3 x 3 : dx E Thus, Eq. (22) becomes ds x Gc r3 x r1 x ÿ 1 ; e dx E3 E
23
24
where r1 x and r3 x are the tensile stresses in the adherends 1 and 3, respectively; E1 and E3 are the
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R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
elastic moduli of the adherends 1 and 3, respectively. Moreover, the equilibrium is given by dr1 x ÿs x; dx 3 dr x s x; e3 dx
e1
25
where e1 and e3 are the thicknesses of adherends 1 and 3, respectively, and the boundary conditions are given by be1 r1 0 F ; be3 r3 l F :
26
By eliminating r1 x and r3 x from Eqs. (24) and (25), we obtain: ds x Gc 1 1 s x: 27 e e1 E 1 e3 E 3 dx Let us denote k Gc =e 1=e1 E1 1=e3 E3 . By using Eqs. (26), the resolution of Eq. (27) yields Gc 1 1 s x F ek e1 E1 tanh kl e3 E3 sinh kl 1 28 cosh kx ÿ 1 1 sinh kx : eE In the case of a symmetrical (e1 e3 es ) and balanced (e1 E1 e3 E3 es Es ) adhesive joint, this equation becomes s x F
k cosh k x ÿ l=2 : 2b sinh kl=2
29
References Bazant, Z.P., Desmorat, R., 1995. Softening slip and size eect in bond fracture. Interface Fracture and Bond, American Concrete Institute SP-156, Detroit, USA, pp. 11±23.
Calard, J., Lamon, J., 1998. Approches probabilistes de la rupture des composites a matrice ceramique en ¯exion 3 points et pseudo 4 points. 11emes Journees Nationales sur les Composites, Arcachon. Delale, F., Erdogan, F., Aydinoglu, M.N., 1981. Stresses in adhesively bonded joints: a closed-form solution. J. Compos. Mater. 15, 249. Delmas, J.P., 1979. Contribution a l'etude des collages structuraux: application au mortier plaque, Doctoral Dissertation, Universite de Reims. Desmorat, R., 1996. Methode d'analyse de la resistance de zones fortement sollicitees, Doctoral Dissertation, ENS Cachan. Fitoussi, J., Guo, G., Baptiste, D., 1996. Modelisation micromecanique statistique du comportement en dommage anisotrope d'un composite, 10emes Journees Nationales sur les Composites, Paris. Foret, G., 1995. Eets d'echelle dans la rupture des composites unidirectionnels, Doctoral Dissertation, Ecole Nationale des Ponts et Chaussees. Goland, M., Reissner, E., 1944. Stresses in cemented joints. J. Appl. Mech. 2, 17±27. Hadj-Ahmed, R., 1996. Optimisation du transfert des eorts par cisaillement dans un joint de colle, DEA Report, Ecole Nationale des Ponts et Chaussees. Hadj-Ahmed, R., Foret, G., Ehrlacher, A., 1997. Simulation and optimization of the strength of an adhesive joint. Mechanical behaviour of adhesive joints, Euromech 358, Nevers. Hadj-Ahmed, R., 1999. Modelisation des assemblages colles: application a l'optimisation du transfert des eorts par cisaillement, Doctoral Dissertation, Ecole Nationale des Ponts et Chaussees.. Hart-Smith, L.J., 1973. Adhesive bonded single lap joint, NASA CR 112236. Jeandrau, J.P., 1985. Calcul des joints colles, Etude bibliographique, CETIM Report no 1, Section 536, St-Etienne. Joubert, J.L., Benazet, D., Ancenay, H., 1979. Calcul des dimensions des joints colles, CETIM Final Report of the CCPR 13N050 study. Vidonne, M.P., 1995. Endommagements et ruptures des interfaces dans les multimateriaux, Doctoral Dissertation, ENS Cachan. Vinh, T., 1988. Les essais de rupture des composites, aspects probabilistes, vol. 1, ISMCM, CESMI, Saint-Ouen. Volkersen, O., 1938. Luftfahrtforschung 15, 41±47. Weibull, W.A., 1951. Statistical distribution function of wide applicability. J. Appl. Phys. 18, 293.
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Probabilistic analysis of failure in adhesive bonded joints Reda Hadj-Ahmed *, Gilles Foret, Alain Ehrlacher Ecole Nationale des Ponts et Chauss ees, CERMMO-AM, 6-8 Avenue Blaise Pascal, Cit e Descartes, Champs-sur-Marne, 77455 Marne-La-Vallee, France Received 26 June 2000
Abstract We propose herein a strength probability law to estimate the shear strength of an adhesive joint. A simpli®ed model enables generating an analytical solution of the shear stress present in the adhesive. The strength probability law takes into account the scale eects experimentally established for the adhesive joint shear strength and leads to studying the in¯uence of both adhesive thickness and overlap length on joint strength. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Adhesive joint; Shear stress; Shear strength; Shear Lag model; Weibull's law; Failure
1. Introduction The use of adhesives in manufacturing in plane force-transmitting joints between various structural materials is now very widespread thanks to the advantages of bonding with regard to the other conventional techniques of material assembly. Several models, which enable analyzing the stresses in adhesive joints, exist in the literature (Volkersen, 1938; Goland and Reissner, 1944; Hart-Smith, 1973; Delale et al., 1981), yet failure prediction remains dicult due to a lack of sucient criteria. Elastic models show that adhesive joint shear strength increases as adhesive thickness increases. However, many tests show that joint behavior at failure depends on the adhesive. Indeed, some
* Corresponding author. Tel.: +33-1-64153736; fax: +33-164153741. E-mail address: [email protected] (R. Hadj-Ahmed).
adhesives are ecient when their thickness is small (Joubert et al., 1979). On the contrary, for other adhesives (Delmas, 1979), joint strength increases with adhesive thickness until an ``optimal adhesive thickness''. Beyond this thickness, joint strength decreases. We are thus confronted with a scale eect in the adhesive joint strength. With regard to the overlap length, elastic models (Jeandrau, 1985; Hadj-Ahmed et al., 1997) and test results (Hadj-Ahmed, 1999) have shown that adhesive joint strength increases with overlap length until a ``limit length'', beyond which the strength no longer increases. Scale eects are often related to the presence of random defects in the adhesive (see Fig. 1). We thus propose in this paper a probabilistic analysis that allows taking into account the defects present in the adhesive. We start by writing the probability law of a reference volume of a material subjected to a uniform stress ®eld. By supposing that the material breaks upon the initial damage, we can establish,
0167-6636/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 0 0 ) 0 0 0 5 1 - X
78
R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
cording to the stress ®eld r applied on V0 . We denote by PnfV0 r the non-failure probability law of the volume V0 submitted to r. By assuming that the elementary volume strengths are independent, we can easily show (see Appendix A) that the strength probability law of an arbitrary volume V is given by h iV =V0 : 1 PnfV r PnfV0 r
Fig. 1. Defects in an adhesive.
from the reference volume probability law, the probability law of any volume. This relation then enables us to describe volume-scale eects. To correctly represent the strength probabilities of brittle materials, we have chosen to use Weibull's law (Weibull, 1951) with two parameters (Fitoussi et al., 1996; Calard and Lamon, 1998). We then generalized the probability law to a structure subjected to a non-uniform stress ®eld and composed of the material described above. In this work, we are interested in an adhesive joint. For the stress analysis we selected a simpli®ed model that yields a constant shear stress according to the adhesive thickness (Volkersen, 1938; Goland and Reissner, 1944; Hart-Smith, 1973; Delale et al., 1981). We show, in this case, that the strength probability corresponds to Weibull's law on the tensile force applied to the adhesive joint. This law enables us to study the in¯uence of both adhesive thickness and overlap length on the adhesive joint shear strength and to state the existence of an ``optimal adhesive thickness'' and a ``limit overlap length''. 2. Probability law of a material subjected to a uniform stress ®eld We consider a reference volume V0 of the studied material. By conducting a series of tests, we can determine its strength probability law ac-
The ratio V =V0 exhibits the volume-scale eect for brittle materials. From a practical point of view, we have chosen Weibull's law to accurately represent the strength probabilities of the brittle materials. For the sake of simplicity, we use a scalar criterion on r, de noted f r. Thus, Weibull's law for a volume V0 can be written: " ! # f r m V0 Pnf r exp ÿ ; 2 r0 where m and r0 are the test-determined Weibull parameters. As m increases, the strength dispersion decreases and vice versa. By taking into account the volume-scale eect (Eq. (1)), the strength probability of a volume V subjected to a uniform stress ®eld r is given by " !m !#V =V0 f r V 3 Pnf r exp ÿ r0 or PnfV r
"
V exp ÿ V0
f r
r0
!m # :
4
We again have a two parameters Weibull's law. The ®rst one (m) is unchanged, while the second one (r0 ) is replaced by r0 V0 =V 1=m . In Eq. (4), it is clear that the ratio V =V0 1=m represents the scale eect. As V increases, the joint strength decreases. A volume increase actually implies an increase in the number of defects. At the same time, the stress concentration near the ends of the adhesive thickness decreases. This competition between the ``number of defects'' eect and the stress concentration can explain the existence
R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
79
of an ``optimal adhesive thickness''. When the number of defects eect is predominant, the adhesive is ecient for thin adhesive layers. On the contrary, when the stress concentration eect is predominant, the joint strength as a function of adhesive thickness displays a maximum. 3. Probability law of a structure subjected to a nonuniform stress ®eld We are interested herein in a structure X composed of the material described above and submitted to a non-uniform stress ®eld r x. We start by cutting X into elementary volumes DVi on which we can assume that the stress ®eld r xi is uniform. From Eq. (1) we obtain the following: h iDVi =V0 : 5 PnfDVi r xi PnfV0 r xi
By supposing that all the elementary volumes are independent, we can ®rst approximate the strength probability of the structure X by Y V DV =V Pnf0 r xi i 0 : 6 PnfX r xi i
In the case of Weibull's law, we ®nd !m # " xi X DVi f r X : Pnf r x exp ÿ V0 r0 i
7
By having DVi tend to 0, the probability law expression obtained is " Z # ! f r x m 1 dV : 8 PnfX r x exp ÿ r0 X V0
4. Application to an adhesive joint We have chosen a simpli®ed model (Bazant and Desmorat, 1995; Desmorat, 1996) in which the criterion f r is written onto the longitudinal shear stress sxz in the adhesive. This shear stress is denoted by s in the following text and for simplicity we assume that s is constant with respect to
Fig. 2. Symmetrical adhesive bonded double lap joint.
z (see Fig. 2). We also suppose that the joint is in plane strain. Thus, s varies only as a function of x. Eq. (8) then becomes Z l m b s x PnfX r exp ÿ e x dx ; 9 s0 0 V0 where · b is the bond width; · e x is the adhesive thickness (eventually variable); · s0 replaces r0 as the criterion on shear; · s x is the shear stress in the adhesive determined by a simpli®ed model; · l is the overlap length. s x is a linear function of the tensile force F applied on the adhesive joint (see Fig. 2). By denoting s~ x as the shear stress under a unit force (F 1), we have s x F s~ x:
10
Eq. (8) becomes " PnfX r
exp ÿ F
m
Z
l 0
s~ x b e x s0 V0
!m
# dx : 11
Thus, the strength probability follows Weibull's law on the tensile force applied on the adhesive joint. The ®rst parameter m is unchanged; the second one is replaced by F0 , where " Z !m #ÿ1=m l s~ x b e x dx : 12 F0 s0 V0 0 We can then write m F X : Pnf r exp ÿ F0
13
80
R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
The parameter F0 is related to the mean strength F of the adhesive joint (Vinh, 1988; Foret, 1995) by means of the following expression: 1 C is the function gamma: F F0 C 1 m 14 We must emphasize that F0 depends on the adhesive material through m, s~ x and s0 , and on the joint geometry through b, e x and s~ x. In the following section, we will study the in¯uence of adhesive thickness and overlap length on the joint strength.
5. Adhesive thickness in¯uence We have chosen a Shear Lag model in which the longitudinal shear stress s~ in the adhesive is proportional to the dierence between the displacement of the outer adherend and the displacement of half of the inner adherend as it is shown in Eq. (22) (Volkersen, 1938; Vidonne, 1995; Hadj-Ahmed, 1996). In the case of a symmetrical and balanced adhesive joint (see Appendix B), this model provides s~ by the following expression: s~ x
k cosh k x ÿ l=2 ; 2b sinh kl=2
15
where s 2Gc k Es es e
Table 1 Mechanical and geometrical characteristics of the adhesive joint in a Shear Lag model Adherends (steel)
Adhesive (epoxy)
Es (GPa)
es (mm)
Gc (GPa)
e (mm)
210
3
1.55
0:05±2
the overlap length l is set at 40 mm. The other mechanical and geometrical characteristics of the joint in a Shear Lag model are presented in Table 1. We note here that F0 is related to the mean strength of the adhesive joint. We can then focus on two values of the parameter m. The ®rst one is m 2; the low values of m imply more scale eect. The second one is m 4; in this case (high values of m), the scale eect is weak. 5.1. High dispersion In this part we demonstrate by means of F0 and PnfX that for adhesives with high dispersion, the joint is ecient when the adhesive thickness is small. Fig. 3 presents a plot of F0 against e. We can remark that the mean strength is a decreasing function of the adhesive thickness. Similarly, Fig. 4 displays the variation of the non-failure probability PnfX as a function of the tensile force F for many adhesive thickness values. On the one hand, for each adhesive thickness, we can determine the non-failure probability at a given force level. On the other hand, we remark that as adhesive thickness increases, the non-fail-
with · Es is the adherend Young's modulus; · es is the adherend thickness; · Gc is the adhesive shear modulus. Thus, 1=m sinh kl=2 mÿ1=m
F 0 s0 V 0 e
ÿ1=m
Z
k=2
0
l
b
m ÿ1=m l cosh k x ÿ dx : 16 2
To illustrate this result, let us study the dependence of F0 on e. For the numerical applications,
Fig. 3. Mean strength versus adhesive thickness for m 2.
R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
Fig. 4. Non-failure probability versus tensile force for m 2.
81
Fig. 5. Mean strength versus adhesive layer for m 4.
ure probability decreases. For low values of the parameter m, there are so many defects in the adhesive that a slight increase in adhesive thickness implies a decrease in strength. 5.2. Low dispersion In this case, there are fewer defects in the adhesive. When the adhesive layer is small, the maximum value of shear stress is very high; therefore, the joint strength is low. By increasing the adhesive layer, all of the models assert that the maximum value of shear stress decreases. This result implies that joint strength increases with adhesive thickness until an ``optimal adhesive thickness''. Beyond this thickness, the number of defects becomes signi®cant and the joint strength decreases. Fig. 5 shows that the mean strength as a function of e has a maximum at e 0:5 mm. Similarly, Fig. 6 shows that for a given tensile force the non-failure probability is maximal at e 0:5 mm. 6. Overlap length in¯uence By comparison with experimental results (Jeandrau, 1985; Hadj-Ahmed, 1999), elastic models produce a good prediction of the in¯uence of overlap length on joint strength. Figs. 7 and 8 reveal that regardless of the m value, the joint strength displays a ``limit overlap length'' (l 40
Fig. 6. Non-failure probability versus tensile force for m 4.
mm). With regard to the non-failure probability, the curves for l equal to 40, 60 and 80 mm are superposed. For the numerical applications, e has been set here at 0.1 mm. 7. Discussion We have seen above that the in¯uence of adhesive thickness on joint strength is directly related to the parameter m. To highlight this dependence we have plotted in Fig. 9 F0 versus m for three adhesive thicknesses: 0.05, 0.5 and 1 mm. These values are typically encountered in the literature. We notice in Fig. 9 that for low values of the parameter m (m 6 3:2), the adhesive is ecient for low adhesive thicknesses. In this case, there are
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R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
Fig. 7. In¯uence of overlap length on F0 and PnfX for m 2.
Fig. 8. In¯uence of overlap length on F0 and PnfX for m 4.
many defects in the adhesive and an increase in adhesive thickness certainly involves a decrease in joint strength. When m is very high (m P 5), failure occurs in the material (the adhesive) deterministically. The increase in adhesive thickness only involves the decrease in stress concentration, hence joint strength continues to increase with adhesive thickness. For intermediate values of m (3:2 6 m 6 5), the adhesive joint displays an optimal adhesive thickness. In this case, an increase in adhesive thickness initially implies an increase in
the joint strength. This joint strength cannot increase inde®nitely with adhesive thickness as the number of defects increases with adhesive thickness as well. 8. Conclusion In this paper, we have proposed a law for the shear strength probability of an adhesive joint. The use of a simpli®ed model has allowed estab-
R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
h i2 Pnf2V0 r PnfV0 r :
83
17
We can easily extend this reasoning to a volume V pV0 , where p is an integer, and write h ip 18 PnfpV0 r PnfV0 r :
By symmetrical reasoning, the strength probability law of a volume V V0 =q is h i1=q V =q Pnf0 r PnfV0 r : 19
Fig. 9. Mean strength versus m.
lishing an analytical solution. It was then possible to study the in¯uence of adhesive thickness and overlap length by taking into account the scale eects observed experimentally. We have shown that according to the m value (high or low dispersion), the adhesive behavior varies. In particular, we have seen that for very low values of m, the joint is ecient as long as the adhesive thickness remains small. In the case of very high values of m the joint strength increases with adhesive thickness. For intermediate values of m, the joint strength as a function of adhesive thickness exhibits a maximum. Concerning the overlap length, it seems that the dispersion character of the adhesive does not in¯uence the dependence of the joint strength on the overlap length, and the adhesive joint displays nearly the same ``limit overlap length''.
Appendix A. The strength probability law of a volume V To determine the strength probability law PnfV r of an arbitrary volume V submitted to a uniform stress ®eld r we initially consider a vol ume V 2V0 (with the strength probability law PnfV0 r of the volume V0 being known). By sup posing that the volume strengths are independent, we obtain under the same stress ®eld r the fol lowing:
In applying the two reasonings above, we can write, for a volume V p=qV0 , h ip=q p=qV 20 Pnf 0 r PnfV0 r :
The continuous passing leads to, for any value of V =V0 : h iV =V0 : 21 PnfV r PnfV0 r
Appendix B. The longitudinal shear stress in the adhesive layer For an adhesive joint, the Shear Lag model chosen in this work assumes that shear stress in the adhesive is proportional to the dierence between the outer adherend (layer 1) displacement u1 x and half of the inner adherend (layer 3) displacement u3 x by means of the following expression: s x
Gc 3 u x ÿ u1 x: e
22
The elastic behavior of the adherends leads to du1 x r1 x 1 ; dx E du3 x r3 x 3 : dx E Thus, Eq. (22) becomes ds x Gc r3 x r1 x ÿ 1 ; e dx E3 E
23
24
where r1 x and r3 x are the tensile stresses in the adherends 1 and 3, respectively; E1 and E3 are the
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R. Hadj-Ahmed et al. / Mechanics of Materials 33 (2001) 77±84
elastic moduli of the adherends 1 and 3, respectively. Moreover, the equilibrium is given by dr1 x ÿs x; dx 3 dr x s x; e3 dx
e1
25
where e1 and e3 are the thicknesses of adherends 1 and 3, respectively, and the boundary conditions are given by be1 r1 0 F ; be3 r3 l F :
26
By eliminating r1 x and r3 x from Eqs. (24) and (25), we obtain: ds x Gc 1 1 s x: 27 e e1 E 1 e3 E 3 dx Let us denote k Gc =e 1=e1 E1 1=e3 E3 . By using Eqs. (26), the resolution of Eq. (27) yields Gc 1 1 s x F ek e1 E1 tanh kl e3 E3 sinh kl 1 28 cosh kx ÿ 1 1 sinh kx : eE In the case of a symmetrical (e1 e3 es ) and balanced (e1 E1 e3 E3 es Es ) adhesive joint, this equation becomes s x F
k cosh k x ÿ l=2 : 2b sinh kl=2
29
References Bazant, Z.P., Desmorat, R., 1995. Softening slip and size eect in bond fracture. Interface Fracture and Bond, American Concrete Institute SP-156, Detroit, USA, pp. 11±23.
Calard, J., Lamon, J., 1998. Approches probabilistes de la rupture des composites a matrice ceramique en ¯exion 3 points et pseudo 4 points. 11emes Journees Nationales sur les Composites, Arcachon. Delale, F., Erdogan, F., Aydinoglu, M.N., 1981. Stresses in adhesively bonded joints: a closed-form solution. J. Compos. Mater. 15, 249. Delmas, J.P., 1979. Contribution a l'etude des collages structuraux: application au mortier plaque, Doctoral Dissertation, Universite de Reims. Desmorat, R., 1996. Methode d'analyse de la resistance de zones fortement sollicitees, Doctoral Dissertation, ENS Cachan. Fitoussi, J., Guo, G., Baptiste, D., 1996. Modelisation micromecanique statistique du comportement en dommage anisotrope d'un composite, 10emes Journees Nationales sur les Composites, Paris. Foret, G., 1995. Eets d'echelle dans la rupture des composites unidirectionnels, Doctoral Dissertation, Ecole Nationale des Ponts et Chaussees. Goland, M., Reissner, E., 1944. Stresses in cemented joints. J. Appl. Mech. 2, 17±27. Hadj-Ahmed, R., 1996. Optimisation du transfert des eorts par cisaillement dans un joint de colle, DEA Report, Ecole Nationale des Ponts et Chaussees. Hadj-Ahmed, R., Foret, G., Ehrlacher, A., 1997. Simulation and optimization of the strength of an adhesive joint. Mechanical behaviour of adhesive joints, Euromech 358, Nevers. Hadj-Ahmed, R., 1999. Modelisation des assemblages colles: application a l'optimisation du transfert des eorts par cisaillement, Doctoral Dissertation, Ecole Nationale des Ponts et Chaussees.. Hart-Smith, L.J., 1973. Adhesive bonded single lap joint, NASA CR 112236. Jeandrau, J.P., 1985. Calcul des joints colles, Etude bibliographique, CETIM Report no 1, Section 536, St-Etienne. Joubert, J.L., Benazet, D., Ancenay, H., 1979. Calcul des dimensions des joints colles, CETIM Final Report of the CCPR 13N050 study. Vidonne, M.P., 1995. Endommagements et ruptures des interfaces dans les multimateriaux, Doctoral Dissertation, ENS Cachan. Vinh, T., 1988. Les essais de rupture des composites, aspects probabilistes, vol. 1, ISMCM, CESMI, Saint-Ouen. Volkersen, O., 1938. Luftfahrtforschung 15, 41±47. Weibull, W.A., 1951. Statistical distribution function of wide applicability. J. Appl. Phys. 18, 293.