Mechanical behaviour of FRP adhesive joints: A theoretical model

Composites: Part B 40 (2009) 116–124

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Mechanical behaviour of FRP adhesive joints: A theoretical model Francesco Ascione * Department of Civil Engineering, University of Rome ‘‘Tor Vergata”, Italy

a r t i c l e

i n f o

Article history: Received 3 March 2008 Accepted 9 November 2008 Available online 21 November 2008 Keywords: B. Cohesive fracture A. FRP B. Interface bilinear laws D. Double and single lap joints

a b s t r a c t In this paper a mathematical model for studying the equilibrium problem of adhesive joints between FRP adherents is presented. In particular, double and single-lap joints, both in the case of normal and shear/ flexure stresses are considered. The problem is non linear due to the cohesive constitutive law adopted for modeling the interface. On the contrary, the adherents are supposed to be indefinitely linear elastic. The possibility to uncouple the problem of shear/flexure from the extensional one, as well as disregard the mutual effects between the normal and tangential stresses acting at the joint interfaces is also assumed. As highlighted in literature, this hypothesis allows results which are sufficiently correct from a technical perspective when the mechanical characteristics of the adherents are almost the same, as supposed in the present paper. Within such a framework it is possible to trace back the examined equilibrium problems to those of simpler auxiliary structural schemes. A simple and efficient iterative procedure for solving the aforementioned problems is also proposed. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The use of adhesive lap joints for structural purposes is highly convenient in various fields of engineering, in particular mechanical and aeronautical. In fact, their use not only reduces realization time and costs, but also increases corrosion and fatigue resistance as well as strengthens against fracture. Such joints also have significant dissipation properties. Over the last few years, adhesive lap joints have also been used in Civil Engineering, for structural applications of fiber reinforced polymers (FRP). Such applications include both the restoration of existing structures, made mainly from concrete and masonry, as well as the realization of new structures, made entirely from FRP. The behaviour of adhesive joints depends on various factors, including the physical and mechanical properties of both the adherents and adhesive, the geometry of the joint, the length of the overlapping area, the total length of the joint as well as the thickness of the adherents and adhesive. Studies have shown that this behaviour can be sensitive to both geometric and mechanical imperfections as well as the load application mode. There are two traditional approaches to studying the mechanical behaviour of adhesive joints. The first is based on the analysis of the stress and deformation states of the joint in linear elasticity [1–7]. The second, more recent, is based on the fracture mechanic principles [8–15]. For a wider discussion see [38]. The stress and deformation analysis of adhesive joints has been * Corresponding author. E-mail address: [email protected] 1359-8368/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2008.11.006

developed over the last 60 years, through both theoretical and numerical models, in particular by using the finite element method [16–26][27]. The modeling of the adherents/adhesive interface through cohesive constitutive laws, as highlighted in Fig. 1.1a and b, is more realistic. They are generally expressed through relations, independent among themselves, which correlate, respectively, the normal or tangential interaction of the interface, r or s, to the relative corresponding displacement, d or s, between the joint bonded surfaces. The displacements d cause mode I of fracture (opening), while the displacements s are responsible for mode II of fracture (sliding). Current literature includes several significant papers which have analysed mode II assuming the interface law of Fig. 1.1b [28,29]. Mode II of fracture is the most common, given that joints are realised to principally transfer axial stresses. The possible co-presence of shear and flexural stresses, along with the normal ones, even if secondary respect to the latter, justifies the use of models which analyse the joint behaviour in relation to a mixed mode I/II of fracture. There are several significant papers dealing with this subject. They have highlighted the most important aspects of the problem as well as the efficacy of the interface model, through both numerical and experimental analysis [30–35]. Apart from assuming that the two laws, r(d) and s(s), are independent from each other, it can also be assumed that the total fracture energy, G, can be additionally broken down in the following way:

G ¼ GI þ GII :

ð1:1Þ

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F. Ascione / Composites: Part B 40 (2009) 116–124

GI

GII

u

u

e

se s

u

su

s

Fig. 1.1. Interface laws: (a) law r(d); (b) law s(s).

In Eq. (1.1), the two terms, GI and GII, have the following meaning:

GI ¼

Z

d

ðdÞ dd;

ð1:2aÞ

0

GII ¼

Z

s

sðsÞds:

ð1:2bÞ

allows results which are sufficiently correct from a technical point of view to be obtained, when the mechanical characteristics of the adherents are almost the same, as supposed in the present paper. Numerical applications and comparisons with results available in literature will be developed in a future paper.

0

They must satisfy the following fracture criterion (mixed mode I/II) [36,37]:

2. Balanced double-lap joints subject to shear and flexure

GI GII þ 6 1; GIO GIIO

This section deals with the equilibrium problem of a balanced double-lap joint (single-lap joints are dealt with in the Appendix). The adherents are FRPs, subject to shear and flexure. The analysis is limited to the case of a symmetrical load respect to the middle plane of the joint (Fig. 2.1). This hypothesis is frequent in technical applications and therefore justifies the interest in it. Both the external adherents as well as the intermediate one are modeled like an Eulero–Bernoulli beam. Furthermore, let be:

ð1:3Þ

where the denominators of the two fractions to the first member of in Eq. (1.3) represent, respectively, the subtended areas of the two diagrams of Fig. 1.1a and b, for  d ¼ du and s ¼ su . This study aims at developing an organic analysis of the behaviour of bonded joints adopting the interface laws presented in Fig. 1.1a and b. In particular, the bilinear interface laws (Fig. 1.2a and b), constituted from an elastic linear branch, followed by another similar one, corresponding to the softening behaviour of the adhesive. These interfacial laws, reported in the most recent bibliography, are sufficiently accurate for technical purposes. In this paper the non linear equilibrium problems of the joints, double and single-lap, between FRP adherents, subjected to shear/ flexural as well as normal loads, are formulated and discussed. Non-linearity depends exclusively on the type of constitutive laws used for the interface, while an indefinitely elastic linear behaviour is assumed for the adherents. An iterative algorithm is proposed in this paper for solving the aforementioned problems. It should be assumed that it is possible to uncouple the problem of shear/flexure from the extensional one, as well as disregard the mutual effects between the normal and tangential stresses acting at the interface. As highlighted in literature [7], this hypothesis

bp tp t0p ¼ 2tp Ip

the external adherents width; the external adherents thickness; the intermediate adherent thickness; the moment of inertia, with respect to the central x axis, of the external adherents cross section; Ap = bptp the cross section area of the external adherents; A0p ¼ bp t 0p the cross section area of the intermediate adherent; the elastic longitudinal modulus of the external and interEp mediate adherents; the adhesive thickness. ta The adhesive layer is modeled by a continuous distribution of independent springs, transversal to the interface, capable of contrasting the relative displacements between the adherents in the direction of the y axis. The constitutive law of the springs is represented in Fig. 1.2a, where r represents the adherents/adhesive interaction per surface unit, positive in the case of traction and d is the relative transversal

u u

I

e

II

se

II

s u se

u I

u u

e u

u

arctg

arctg

I

e

I

arctg

arctg

u

Fig. 1.2. Bilinear interface laws: (a) law r(d); (b) law s(s).

se

su

s

118

F. Ascione / Composites: Part B 40 (2009) 116–124

y

adherend

adhesive y

tp

2tp

t'p

V

M x

tp

bp

ta

bp z

ta

tp L/2

adherend

L/2

V

L

adherend

M

Fig. 2.1. Balanced double-lap joint symmetrically subject to shear and flexure.

The present equilibrium problem is non-linear due to the particular constitutive law of the springs. In addition, it does not admit solution for every load condition. In fact, if there exists a solution to this problem, there must be (Fig. 2.2a) a value of the coordinate z, where the beam deflection, as well as the reaction of the corresponding spring, must be equal to zero. It is also worth noting that the end of the beam, where V and M are applied, shows no deflection greater than du. In particular, if the entity of such a deflection is less than de, due to the well-known properties of the elastic beams resting on Winkler soils, then all the springs of the auxiliary model react in the elastic range. However, if the deflection presents a value between de and du, due to the continuity, there should be a section of the beam where the lateral deflection is equal to de. In Fig. 2.2a, the distance of this section from the end of the beam, where the active loads, V and M, are applied, is denoted

displacement between the adherents, positive if it corresponds to a separation of the latter. For obvious reasons of symmetry, the relative displacement, d, coincides with the lateral deflection of each of the external adherents, v. It is also easy to trace back the equilibrium problem of the auxiliary scheme presented in Fig. 2.2a. It consists of a beam which models the superior adherent, constrained on a continuous distribution of springs similar to the one previously described. In Fig. 2.2b, the positive stress resultants, V and M acting on an infinitesimal beam element of length dz, are shown. With the symbols of Fig. 2.2a and b, the constitutive law of the springs can be expressed as follows:

ðaÞ ð2:1Þ

y

y

V

M z

O

x

bp L-z

z L y

dz M

M+dM

V

tp

z

V+dV

(1)

=v -v

(2)

Fig. 2.2. (a) Auxiliary scheme; (b) infinitesimal element.

tp

ðbÞ ðcÞ

ta

8 if d 6 de ; > < aI d r ¼ bI ðdu  dÞ if de 6 d 6 du ; > : 0 if d > du :

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F. Ascione / Composites: Part B 40 (2009) 116–124

by z. It is worth assuming this section as the origin of the reference system. Furthermore, it is worth verifying that the springs on the left side of the origin work exclusively in the elastic range, while those on the right, which present deflections between de and du, work in the softening range of Fig. 1.2a. Subsequently, the equilibrium indefinite equation of the auxiliary scheme,

2

b a d c 6 ða þ bÞ ða  bÞ ðd  cÞ ðd þ cÞ 6 6 6 1 0 1 0 6 6 x x1I x1I x1 I 1I 6 6 6 0 2x21I 0 2x21I 6 6 2x3 2x31I 2x31I 2x31I 6 1I 6 4 0 0 0 0 0 0 0 0

0 0 1 x2I x22 I x32I

d d  bp rðdÞ ¼ 0 dz4

4

d d1  aI bp d1 ¼ 0 8z 2 ½L  z; 0; dz4 4 d d2 Ep Ip 4  bI bp ðdu  d2 Þ ¼ 0 8z 2 ½0; z; dz

Ep I p

ð2:2aÞ ð2:2bÞ

where, d1 and d2, are the restrictions of d on the intervals ½L  z; 0 and ½0; z, respectively. All the terms of the two Eq. (2.2a) and (2.2b) can be divided by the flexural rigidity of the external adherents, EpIp, and it is assumed:

aI bp ; 4Ep Ip

x42I ¼

bI bp ; Ep Ip

ð2:3a; bÞ

where the two quantities, x1I and x2I, have the dimensions of the inverse of a length. The general integrals of the two differential equation (2.2a) and (2.2b)can be expressed as follows:

d1 ðzÞ ¼ Aex1I z cosðx1I zÞ þ Bex1I z senðx1I zÞ þ Cex1I z cosðx1I zÞ þ Dex1I z senðx1I zÞ;

ð2:3bÞ

d2 ðzÞ ¼ du þ Eex2I z þ Fex2I z þ G cosðx2I zÞ þ Hsenðx2I zÞ

ð2:4bÞ

being A, B, C, D, E, F, G, H arbitrary constants to be determined by means of the corresponding, static or kinematic, boundary conditions. These are

 2 d d1   dz2 

¼ 0;

z¼ðLzÞ

 3 d d1   dz3 

¼ 0;

ð2:5a; bÞ

z¼ðLzÞ

d1 ð0Þ ¼ d2 ð0Þ;

ð2:5cÞ

  dd1  dd2  ¼ ; dz z¼0 dz z¼0

 2 d d1   dz2 

z¼0

 2 d d2  ¼  dz2 

;

z¼0

 3 d d1   dz3 

z¼0

 3 d d2  ¼  dz3 

;

z¼0

ð2:5d; e; fÞ Ep Ip

 2 d d2   dz2 

¼ M;

Ep Ip

z¼z

 3 d d2   dz3 

¼ V:

ð2:5g; hÞ

z¼z

In addition, in the origin of the reference system, for the above indicated considerations, the following equalities must also be satisfied:

d1 ð0Þ ¼ d2 ð0Þ ¼ de :

ð2:6Þ

By indicating with

a ¼ ex1I ðLzÞ cos½x1I ðL  zÞ;

b ¼ ex1I ðLzÞ sen½x1I ðL  zÞ; ð2:10a; bÞ

x1I ðLzÞ

c¼e

3 32 3 0 A 6 0 7 7 76 B 7 6 7 76 7 6 76 7 6 du 7 7 76 C 7 6 1 0 7 76 7 6 6 7 6 0 7 0 x2I 7 7 76 D 7 6 ¼ 7 6 7 6 0 7: 2 7 6 7 7 6 x2I 0 7 76 E 7 6 0 7 6 7 7 6 3 0 x2 76 F 7 6 7 76 7 6  M 2 7 EP Ip x2I 7 cosðx2I zÞ senðx2I zÞ 54 G 5 6 5 4 V H senðx2I zÞ cosðx2I zÞ E I p x3 0 0

0 0

P

cos½x1I ðL  zÞ;

x1I ðLzÞ

d¼e

sen½x1I ðL  zÞ; ð2:10c; dÞ

the relations which allow to determine the constants of integration, can be expressed in the following compact matrix form:

2I

ð2:20Þ

ð2:2Þ

can be then specialised in the two different equations identified below:

x41I ¼

1

x2I x22I x32I

ex2I z ex2I z ex2I z ex2I z

4

Ep I p

0 0

2

Eq. (2.20), with the objective function (2.6), allow to iteratively determine the position z, corresponding to assigned values of M and V, as well as the values of the constants of integration, given that there exists a solution to the equilibrium problem. In particular, for an assigned value of V (or M), it is possible to determine the corresponding values of M (or V) to which, at the end, z ¼ z, of the joint, the limit condition dðzÞ ¼ du is also reached. Such values of V and M represent a couple of ultimate limit values of the shear and bending moment which can be applied to the joint preserving the integrity of the interfaces between the adherents.

3. Balanced double-lap joints subject to normal stresses In Fig. 3.1 the equilibrium problem of a balanced double-lap joint with FRP adherents, subject to normal stresses is represented (single-lap joints are dealt with in the Appendix). Both the external and internal adherents are modeled according to the beam technical theory assuming the hypothesis of conservation of the plane sections. The same symbols introduced in the Section 2. are utilized here. The adhesive layer is modeled by a continuous set of independent springs, placed parallel to the axis of the joint in order to contrast the relative axial displacements of the adherents in contact. The constitutive law of the springs is represented in Fig. 1.2b, where s denotes the adherents/adhesive interaction per surface unit and s denotes the relative axial displacement between the adherents. It can be analytically expressed as follows:

8 if j s j6 se ; > < aII s sðsÞ ¼ bII ðsu  sÞ if se 6j s j6 su ; > : 0 if j s j> su :

ðaÞ ðbÞ ðcÞ

ð3:1Þ

The equilibrium problem here examined, in relation to the one studied in Section 2, presents several substantial particularities, attributed to the fact that the relative displacement, s, between the touching adherents, does not coincide with the absolute displacement of the external adherents, as occurs in the case of the joints subject to shear and flexure. For obvious reasons of symmetry, the external adherents present the same axial displacement, w(e). The axial displacement of the internal adherent is denoted by w(i); the two relative displacements between the external adherents and that of the internal one, which are equal to each other, be s = w(e)  w(i). The equilibrium indefinite equation in direction z of each of the external adherents, as well as the internal one, can be expressed as the following: 2

d wðeÞ  sðsÞbp ¼ 0; dz2 2 ðiÞ d w 2Ep Ap þ 2sðsÞbp ¼ 0: dz2

Ep Ap

ð3:2Þ ð3:3Þ

By subtracting the second of these equations from the first, through simple algebra, the following can be obtained:

F. Ascione / Composites: Part B 40 (2009) 116–124

y

adhesive

adherent

y

bp

t'p

tp

2T

2t p

T

ta

bp

z

ta L/2

adherent

x

tp

120

tp

T

L/2 L

adherent

Fig. 3.1. Balanced double-lap joint subject to normal stresses.

2

d s 2bp  sðsÞ ¼ 0: dz2 Ep Ap

ð3:4Þ

It is easy to note that, being the joint balanced, the relative displacements, s, as well as the interactions, s, are symmetrical with respect to the y axis (Fig. 3.1). Consequently, the boundary conditions to place next to the differential equation (3.4) can be expressed as follows:

E p Ap

 ds ¼ T: dzz¼L

ð3:5a; bÞ

2

Taking this into consideration, the boundary value problem constituted by Eq. (3.4) along with Eq. (3.5a,b), can be identified with the equilibrium problem of the auxiliary structural scheme represented in Fig. 3.2. This figure also indicates the positive resultant stresses acting on an infinitesimal beam element of length dz. Such a model consists of an elastic beam with extensional rigidity EpAp, and length L/2: only one end is loaded. Furthermore, the beam is constrained on a continuous distributions of independent springs, capable of contrasting the axial displacement, s, in accordance to the constitutive law represented in Fig. 1.2b. It is worth noting that the angular coefficients of the two branches of the constitutive law of the springs are doubled with respect to the corresponding values of the original adhesive layer.

2

d s1 2bp  aII s1 ¼ 0 8z 2 dz2 Ep Ap

   L   z ; 0 ; 2

2

being s1 and s2, the restrictions of s to the intervals ½0; z, respectively. It is assumed:

x21II ¼

2bp aII ; Ep Ap

x22II ¼

2bp bII ; Ep Ap y

z

T

O

x

bp z L/2

dz y

T

T+dT z

(1)

(2)

ta

s= w - w

Fig. 3.2. (a) Auxiliary scheme; (b) infinitesimal element.

ð3:6bÞ 



 2L  z ; 0 and

ð3:7a; bÞ

y

L/2-z

ð3:6aÞ

d s2 2bp  b ðsu  s2 Þ ¼ 0 8z 2 ½0; z dz2 Ep Ap II

tp

 ds ¼ 0; dzz¼0

tp

E p Ap

Subsequently, the reaction of the springs is double the corresponding reactions in the adhesive layer, while their elongations are equal to the corresponding relative displacements between each of the external adherents with the intermediate one. Regarding the equilibrium problem studied here (Figs. 3.1 and 3.2), it is worth taking into account the same considerations presented in Section 2, when dealing with the problem of shear and flexure. In particular, if there exists a solution to this problem, there should be a value of the coordinate z, where the beam displacement must be equal to zero. The corresponding point, O, is assumed as origin of the reference system. The Eq. (3.4) can therefore be specialised in the two differential equations identified below:

121

F. Ascione / Composites: Part B 40 (2009) 116–124

where the two quantities, x1II and x2II, have the dimensions of the inverse of a length. The Eq. (3.6a,b) can therefore be rewritten as

    d s1 L z ; 0 ;  x s ¼ 0 8 z 2   1II 1 2 dz2

3.1. Non balanced double-lap joint subject to normal stresses With respect to the case studied in the previous section, such a type of joint presents the only variation that t0p –2t p (Fig. 3.1). In other words the extensional rigidity of the intermediate adherent, Ep A0p , is not double that of the two external adherents, EpAp. Using the previous symbols, the equilibrium indefinite equations along the z axis of the two external adherents, as well as the intermediate adherent, can be expressed, respectively, as follows:

2

ð3:8aÞ

2

d ðs2  su Þ þ x2II ðs2  su Þ ¼ 0 8z 2 ½0; z: dz2

ð3:8bÞ

Their general integrals of (3.8) are of the type:

s1 ðzÞ ¼ A coshðx1II zÞ þ Bsenhðx1II zÞ;

ð3:9aÞ

s2 ðzÞ ¼ su þ C cosðx2II zÞ þ Dsenðx2II zÞ;

ð3:9bÞ

2

d wðeÞ  sðsÞbp ¼ 0; dz2 2 d wðiÞ Ep A0p þ 2sðsÞbp ¼ 0: dz2

Ep Ap

being A, B, C and D arbitrary constants to be determined with the corresponding static and kinematic, boundary conditions. The latter are the following:

 ds1  ¼ 0; dz z¼ðL zÞ s1 ð0Þ ¼ s2 ð0Þ;

E p Ap

ð3:10aÞ

2

d s 1 2  þ dz2 Ep Ap Ep A0p ð3:10b; cÞ

 ds2  ¼ T: dz z¼z

 ds 1 ¼ T; dzz¼L Ep A0p

In the origin of the reference system, the following equalities must also be satisfied:

ð3:11Þ

6 6 6 4







x1II senh x1II 2L  z







x1II cosh x1II 2L  z

1

0

0 0

x1II 0

sðsÞbp ¼ 0;

ð3:16Þ

 ds 1 ¼ T: dzz¼L Ep Ap

ð3:17a; bÞ

It is therefore possible to recognize that the boundary value problem constituted by Eqs. (3.16) and (3.17a,b) can be traced back to the equilibrium problem of the auxiliary scheme in Fig. 3.3.

By substituting Eq. (3.9a,b) in Eq. (3.10), the following matrix compact relation is obtained:

2

!

where s = w(e)  w(i). In the case of a not balanced joint, the boundary condition (3.5a), relative to the middle cross section, can no longer be used. The right conditions, both static, are the following:

ð3:10dÞ

s1 ð0Þ ¼ s2 ð0Þ ¼ se :

ð3:15Þ

By subtracting the second equation from the first, through further simple manipulations, the following is obtained:

2

  ds1  ds2  ¼ ;  dz z¼0 dz z¼0

ð3:14Þ

0

0

1

0

32

A

3

2

0

3

7 76 B 7 6 su 7 76 7 6 7: 76 7 ¼ 6 7 54 C 5 6 0 x2II 4 0 5 T D x2II senðx2II zÞ x2II cosðx2II zÞ Ep Ap

Eq. (3.13), with the objective function (3.11), allows to iteratively determine the position z for an assigned value of T, as well as the corresponding values of the constants of integration, given that there exists a solution to the equilibrium problem. Upon determining the unknown function s(z), the values of the interactions transferred to the joint interface, s(s), can be obtained starting from Eq. (3.1). In particular, it is possible to calculate the value of the axial force T, to which the limit condition, j sðzÞ j¼ su , is also reached. This represents the ultimate limit value which can be applied to the joint.

ð3:13Þ

Unlike the scheme used in Fig. 3.2a, the latter has a length L, equal to that  of the joint, and springs whose rigidity is amplified 2A by the factor 1 þ A0p p with respect to the rigidity (aII, bII) of the actual adhesive layer. The equilibrium problem of the auxiliary structural model in Fig. 3.3 can be discussed and dealt with in the same way as when discussing a balanced joint, apart from having to divide the interval L

 2 ; þ 2L into three parts with the introduction of two points, O and O0 , for which the displacements absolute values, s, exhibited by the corresponding springs are equal to se. These points have different distances from the end of the joint: in the interval between

y 1 T Ep A'p

1 Ep A p

T

L Fig. 3.3. Auxiliary structural scheme.

z

122

F. Ascione / Composites: Part B 40 (2009) 116–124

y

adhesive

y,v

adherent (1)

V

tp

M

M z,w

L/2

V

x

tp

t'p

bp

L/2

adherent (2)

L

y

dz M

M+dM

tp

z

V

V+dV

(1)

(2)

= v -v

ta Fig. 4.1. (a) Single-lap joint symmetrically subject to shear and flexure; (b) infinitesimal element.

O and O0 the springs of the auxiliary structural scheme work in the elastic range, beyond it they work in the softening range. The details of the iterative procedure are omitted because they do not introduce any further original consideration. 4. Conclusions The present paper represents the first part of a wider study dealing with the equilibrium problem of FRP double or single-lap joints subjected to shear and bending moment as well as normal stress. The adherents have been modeled following the classical hypotheses of the beam technical theory, while the adhesive layers have been modeled by means of two different interfacial cohesive bilinear laws, for mode I and II of fracture, respectively, independent among themselves. The mutual effects of normal and tangential stresses at the adherents interfaces have been disregarded, as shown in recent literature when the mechanical adherents properties are almost the same. Within this framework, the analysis of the aforementioned equilibrium problems has been traced back to that of simpler auxiliary structural schemes. Appendix A. Single-lap joint This appendix deals with the equilibrium problem of a singlelap joint of length L, realised from two FRP adherents, with a thickness, respectively, tp and t 0p , the same width (bp) as well as the same elastic modulus in the longitudinal direction, Ep. As in Sections 2 and 3., the adherents are modeled as beams which are non-shear deformable. The adhesive, with a thickness, ta, through laws of interface, independent for mode I and II of fracture, characterised by an initial linear elastic branch followed by a softening linear branch. It is worth considering the cases of shear/flexure (Fig. 4.1) and the case of normal stresses (Fig. 4.2), separately. Symbols v(i), w(i), denote, respectively, the lateral deflection and longitudinal displacement of the ith adherent (i = 1, 2): the first can be noted in the presence of shear/flexure, while the second in the

presence of normal stresses. The relative displacements between the two adherents are denoted by d = v(1)  v(2) and s = w(1)  w(2), respectively. The interface constitutive laws, r(d) and s(s), can be expressed through the relations previously introduced in (2.1a–c) and (3.1a– c). The equilibrium indefinite equations of the two adherents are the following for shear/flexure and normal stresses, respectively: 4

d v ð1Þ þ rðv ð1Þ  v ð2Þ Þbp ¼ 0; dz4 4 d v ð2Þ Ep Ip  rðv ð1Þ  v ð2Þ Þbp ¼ 0; dz4 Ep Ip

ð4:1a; bÞ

2

d wð1Þ  sðwð1Þ  wð2Þ Þbp ¼ 0; dz2 2 d wð2Þ Ep Ap þ sðwð1Þ  wð2Þ Þbp ¼ 0: dz2 Ep Ap

ð4:2a; bÞ

In Eqs. (4.1a,b) and (4.2a,b) the quantities EpIp andEpAp denote the flexural and extensional rigidities of the adherents, respectively (Ip and I0p are, in order, the central moments of inertia with respect to the x axis of the two adherents cross sections, (1) and (2); Ap eA0p are the corresponding cross sections areas). The following is assumed: Ep I0p ¼ lEp I p , Ep A0p ¼ kEp Ap . From Eqs. (4.1a,b) and (4.2a,b), subtracting term by term the equation (b) from (a), through several simple operations, the following differential equations in the unknowns, respectively, d(z) and s(z), are obtained:

  4 d d 1 bp þ 1 þ rðdÞ ¼ 0; dz4 l Ep I p   2 d s 1 bp  1 þ sðsÞ ¼ 0: dz2 k Ep Ap

ð4:3Þ ð4:4Þ

In the same way, the actual boundary conditions can be obtained by the difference of those corresponding to the adherents (1) and (2) at the coordinates z = ±L/2 (Figs. 4.1 and 4.2):

 2 d d  2 dz 

¼ z¼2L

M ; lEp Ip

 3 d d  3 dz 

¼ z¼2L

V ; kEp Ip

ð4:5a; bÞ

123

F. Ascione / Composites: Part B 40 (2009) 116–124 y

adhesive

adherent (1)

y,v

tp

T

t'p

x

tp

z,w

T

bp

adherent (2)

L/2

L/2 L

dz y

T

tp

T+dT z

(1)

(2)

s=w -w

ta

Fig. 4.2. (a) Single-lap joint symmetrically subject to normal stresses; (b) infinitesimal element.

 2 d d  2 dz 

¼

z¼2L

M ; Ep I p

 ds T ¼ ; dzz¼L kEp Ap 2

 3 d d  3 dz 

¼

z¼2L

V ; Ep I p

ð4:5c; dÞ

 ds T ¼ : dzz¼L Ep Ap

ð4:6a; bÞ

2

In conclusion, the boundary value problems constituted by Eqs. (4.3)–(4.5) and (4.4)–(4.6), relative to single-lap joints subject to either shear/flexure or normal stresses can be traced back to the equilibrium problems of the following two auxiliary structural schemes in Fig. 4.3. These schemes are characterised by springs with purposely amplified rigidity to that of the actual adhesive layer: the amplification factor is (1 + 1/l) in the case of shear/flexure and (1 + 1/k) in the case of normal stresses. The equilibrium problems of the joints in Figs. 4.1 and 4.2, as well as the auxiliary schemes associated to them (Fig. 4.3) are non-linear and do not ever admit solution. For a more detailed discussion on the conditions required to guarantee, under assigned loads, possible solutions of these problems, see Section 2. Appendix B. Symmetrical single-lap joints In addition to what has previously been discussed, in the case of symmetric joints (l = 1 and k = 1), it is possible to trace back to

even simpler auxiliary structural schemes. In such cases, the springs of the auxiliary schemes are characterised by values of the constitutive parameters which are double those of the actual adhesive layer. In particular, if the scheme of Fig. 4.3a is only loaded in shear, V, it results symmetrically loaded with respect to the same axis. Subsequently, it is possible to limit the equilibrium problem in only one half of the scheme, constraining the central section, X, with a double connecting rod with horizontal axes (Fig. 4.4a). Unlike what was previously observed, and in perfect analogy with what happens in the case of the law s (s), an elastic-softening behaviour can also be assumed in the case of the law r(d), even for the negative values of d. Furthermore the condition r(d) = r(d) it is assumed to hold. Within these hypotheses, the structural scheme of Fig. 4.3a, only loaded in flexure, M, at the ends, results symmetrical with respect to the y axis and asymmetrically loaded with respect to the same axis. Subsequently, it is still possible to limit the equilibrium problem in only half of the scheme, this time constraining the middle section, X, through a roller support with vertical axis (Fig. 4.4b). Finally, in being symmetrical with respect to the y axis and asymmetrically loaded with respect to the same axis, the scheme of Fig. 4.3b can be traced back to the scheme of Fig. 4.3c. In particular, in the case of Fig. 4.4a and b, if the length of the joint is such to be considered infinite, the presence of the double connecting rod and the roller support becomes superfluous and it

y,v

M

V

V

M z,w

L/2

L/2 y,v

T

z,w L/2

L/2

Fig. 4.3. Auxiliary structural schemes.

T

124

F. Ascione / Composites: Part B 40 (2009) 116–124

y,v

y,v

M

V z,w

z,w

y,v

T

z,w

L/2 Fig. 4.4. Simplified auxiliary structural schemes.

y,v V

M

z Fig. 4.5. Simplified auxiliary structural schemes for joints of infinite length.

is therefore possible to trace back to a unique scheme, represented in Fig. 4.5. It is possible to also combine on the same scheme, if both are present, shear and bending moment. References [1] Volkersen O. Die Niektraftvertleiling in Zugheanspruchten mit Konstanten Laschenquerschritten. Luftfartfoeshing 1938;15:41–7. [2] Goland M, Reissner E. The stresses in cemented joints. J Appl Mech 1944;11:A17–27. [3] Hart-Smith LJ. Adhesive-bonded single-lap joints. NASA 1973:CR-112236. [4] de Bruyne NA. The strength of glued joints. Aircraft Eng 1944;16:115–8. [5] Hart-Smith LJ. Adhesive. Bonded double-lap joints. NASA 1973:CR-112235. [6] Tsai MY, Oplinger DW, Morton J. Improved theoretical solutions for adhesive lap joints. Int J Solids Struct 1998;35(12):1163–85. [7] Bigwood DA, Crocombe AD. Elastic analysis and engineering design formulae for bonded joints. Int J Adhes Adhes 1989;9(4):229–42. [8] Anderson GP, Brinton SH, Ninow KJ, DeVries KL. A fracture mechanics approach to predicting bond strength. Advances and adhesively-bonded joints. New York: ASME; 1988. p. 93–101. [9] Chai H. Shear fracture. Int J Fract 1988;37:137–59. [10] Crocombe AD, Hua YX, Loh WK, Abdel Wahab MM, Ashcroft IA. Predicting the residual strength for environmentally degraded adhesive lap joints. Int J Adhes Adhes 2006;26:325–36. [11] Fernlund G, Papini M, McCammond D, Spelt JK. Fracture load predictions for adhesive joints. Compos Sci Technol 1994;51:587–600. [12] Papini M, Fernlund G, Spelt JK. The effects of geometry on the Fracture of Adhesive joints. Int J Adhes Adhes 1994;14:5–13. [13] Suo Z, Hutchinson JW. Interface crack between two elastic layers. Int J Fract 1990;43:1–18. [14] Tong L. Bond strength for adhesive bonded single lap joints. Acta Mech 1996;117:103–13. [15] Tsai MY, Morton J. An evaluation of analytical and numerical solutions to the single-lap joint. Int J Solid Struct 1994;31:2537–63. [16] Abdel Wahab MM, Ashcroft IA, Crocombe AD, Smith PA. Numerical prediction of fatigue crack propagation lifetime in adhesively bonded structures. Int J Fatigue 2002;24:705–9.

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