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The root rotation in double cantilever beam and peel tests
Mechanics of Materials 38 (2006) 571–584 www.elsevier.com/locate/mechmat
The root rotation in double cantilever beam and peel tests B. Cotterell
a,d,*
, K. Hbaieb a, J.G. Williams
b,d
, H. Hadavinia c, V. Tropsa
b
a
Institute of Materials Research and Engineering (IMRE), MSCL, 3 Research Link, Singapore 117602, Singapore b Department of Mechanical Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK c Faculty of Engineering, Kingston University, London SW15 3DW, UK d School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia Received 10 August 2004
Abstract In a peel test the plastic work has to be separated from the total work of fracture to obtain the cohesive energy. The root rotation determines the plastic work in a peel test and needs to be estimated accurately if the peel test is to be analysed successfully. Approximate solutions for the root rotation factor, based on the engineers’ theory of bending, are compared with finite element results to establish a reliable simple method of the estimation of the root rotation factor. 2005 Elsevier Ltd. All rights reserved. Keywords: Rotation factor; Peel test; Cantilever beam; Cohesive zone; Bending theory
1. Introduction Cantilever beam specimens of various types are widely used as specimens for the measurement of fracture toughness. In the simplest analysis of the energy release rate for cantilever beam specimens, the beam at the crack tip is considered built in so that there is no rotation of the beam at the crack tip. Such analyses are adequate providing the beam is isotropic or the crack length, a, to beam height, h, is large. However if the beam is short, or anisotropic with the transverse modulus very much less than the longitudinal modulus, as in the case of laminar composites, then the rotation at the tip of the crack can *
Corresponding author. Address: School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia. Tel./fax: +61 2 99491908. E-mail address: [email protected] (B. Cotterell).
be significant (Williams, 1993; Williams et al., 2005). In the peel test, widely used for determining the adhesion between flexible laminates, the effective a/h is small and the root rotation can be a controlling parameter (Blackburn et al., 2003; Georgiou et al., 2003; Kim and Aravas, 1988; Kinloch et al., 1994; Williams and Hadavinia, 2002a,b). The basic problem in the test is that usually only the force per unit width (P) is measured which gives the total energy release rate per unit area, G, dissipated in creating the new surface area during peeling. There are two components to the total energy dissipated during peeling: the adhesive energy, C0, and the plastic energy dissipated in deforming the peel arm, Gp. For 90 peeling, the usual case, we have, P ¼ G ¼ C0 þ Gp .
ð1Þ
The adhesive energy, C0 is a quasi material constant that can depend upon the constraint. The plastic
0167-6636/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2005.11.001
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B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
energy dissipated depends upon the film thickness. Thus the fundamental problem is to find Gp which is a challenge since it is a large proportion (up to 90%) of the total fracture work, G. There are various strategies available ranging from a fully analytical approach in which Gp is computed (Georgiou et al., 2003; Kim and Aravas, 1988; Kinloch et al., 1994) to a fully experimental method in which it is measured (Kawashita et al., 2004). There are also intermediate possibilities in which some measurements, such as the radius of curvature of the peel arm, are made and the simpler calculation performed (Georgiou et al., 2003). An accurate analysis may be obtained via finite element or volume codes, but it is difficult since it involves the large displacement of slender beams in the elastic–plastic regime. This difficulty has been circumvented to some extent by using approximate solutions which for the peel arm are straight forward (Wei and Hutchinson, 1998). The advantage of the approximate solutions is that simple programmes can be written for PCs such as ICPEEL.1 A limiting factor in such solutions is the modelling of the bonded region near the tip of the peel. When both the free and bonded sections are elastic, the results have been worked out in some detail and some results were given for plasticity effects (Kinloch et al., 1994). If, as is usual, there is considerable plastic deformation both immediately prior to and after debonding the accuracy of the peeling solution depends largely on the accuracy of the calculation of the rotation in the film at the tip of the film. ICPEEL does allow plastic deformation, but the estimate of the root rotation used is based on elastic deformation and has not been quantified except experimentally. This paper reports results from a collaborative study which looked at refining the approximate approach by a new detailed study of the bonded section including the effects of constraint on plasticity and limiting cohesive stresses in the bonded section of the film. A simple estimate of the root rotation factor based on geometrical considerations is given in Section 2 for loading by an end moment. Bounding solutions based on these geometrical considerations are given in Section 2.1 for large cohesive bond strengths (Section 2.1.1) and large cohesive bond strengths (Section 2.1.2). The results of Sections 2.1.1 and 2.1.2 are summarized in Table 2. Bounding solutions for elastic-perfectly plastic films
1
http://www.me.ic.ac.uk/AACgroup/index.htm
are considered in Section 2.1.3 and summarized in Table 3. The constraint on plastic bending in the bonded section of the film is discussed in Section 2.2 where elastic-perfectly plastic films that only yield on their top surfaces are considered in Section 2.2.1, elastic-perfectly plastic films yielding on both surfaces are considered in Section 2.2.2, and the effect of strain hardening is discussed in Section 2.2.3. The results of Sections 2.2.1 and 2.2.2 are summarised in Table 4. In Section 2.3 a better approximation for the root rotation is obtained by integration of the equations of bending. The root rotation for an end force is discussed in Section 3. The results have been checked by numerical results from finite element (FE) analyses. Finally the overall conclusions on the accuracy of the approximate calculations of the root rotation factor are given in Section 4. 2. Approximate calculation of the root rotation factor due to a moment In peeling a moment M0 and a shear force S0 act at the tip of the peel. It is shown later that the rotation h0 at the root of the peel is a strong function of the moment and generally only a weak function of the shear force. Hence in this section the rotation due to a simple end moment M0 is calculated. Various approximate solutions based on the engineers’ theory of bending are presented. Since in this simple theory, plane sections remain plane, the root rotation is simply the rotation of this plane and the radius of curvature, R0, is the radius of the mid plane. A non-dimensional root rotation factor is defined as R0 / ¼ h0 . ð2Þ h 2.1. Approximate bounding solutions A simple correction to allow for root rotation is to consider the film built in not at the bond root but at a small distance D along the bond line (Williams and Hadavinia, 2002b). Fig. 1 shows the deformations which occur around the peel point O. The free film of thickness, h, has a radius of curvature R0 arising from a bending moment M0 per unit width of the film. Because of constraint, the curvature in the bonded region is different. Over the small distance D the radius of curvature in the bonded region can be assumed to be constant and
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
mined from the stiffness of the film. On the other hand if the cohesive strength is small , the cohesive zone will be large and v = v2 is controlled by the ^. cohesive strength, r
R0 M0
2.1.1. Large cohesive strength For large cohesive strength the lateral stiffness, S = S1, can be determined from a very simple model (Williams and Hadavinia, 2002b), where the stiffness comes from the elastic deformation across half the film thickness so that, ignoring Poisson’s effect
R1
0
h
v0
o
S ¼ S1 ¼
Fig. 1. Deformation of a film near the tip of the bond.
2E ; h
ð7Þ
and
equal to R1. Thus the root rotation, can be approximated by D h h0 ¼ ¼v ; ð3Þ R1 R1 where the factor v ¼ Dh . Thus the root rotation factor is given by R0 k1 /¼v ¼v ; ð4Þ R1 k0 where k is the non-dimensional curvature defined by pffiffiffi k = Rp/R and Rp ¼ h=ð 3eY Þ is the radius of curvature that just causes yielding in the outer surfaces of the film, eY being the tensile yield strain. In the elastic foundation model it is assumed that the deflection of the neutral axis ahead of the root of the peel is constrained by elastic stresses developed between the film and its substrate. Thus the stress acting on neutral axis, ry, is given by ry ¼ Sv;
573
ð5Þ
where S is the lateral stiffness per unit width of the film and v is the lateral displacement of the neutral axis. The elastic correction D obtained from the elastic foundation model is (Williams and Hadavinia, 2002b), 4 D E 1 ; ð6Þ ¼ h 3h S where E ¼ E=ð1 m2 Þ is the plane strain elastic modulus, E being the Young’s modulus and m the Poisson’s ratio. To avoid unnecessary complications when considering plastic bending, it is assumed that Poisson’s ratio is 0.5 so that E ¼ 4E=3. The effective lateral stiffness can be obtained in ^, is large, then two ways. If the cohesive strength, r the cohesive zone will be small and v = v1 is deter-
D 1 ¼ 1=4 0:639. h 6
v1 ¼
ð8Þ
Thus
k1 / ¼ v1 . k0
ð9Þ
2.1.2. Small cohesive strength For small cohesive strength the effective lateral stiffness, S = S2, can be obtained from ^ ¼ S 2 v0 ; r
ð10Þ
where v0 is the displacement at the peel root and is given in terms of the correction term D by v0 ¼
D2 . 2R1
Thus using Eq. (6) 1=2
1=2 E h 2 rY ¼ pffiffiffi ; v2 ¼ k1 ^ 6^ r R1 3 3 r where rY is the tensile yield strength and
1=2 3=2 k1 2 rY k1 / ¼ v2 ¼ pffiffiffi . ^ k0 k0 3 3 r
ð11Þ
ð12Þ
ð13Þ
For the maximum cohesive stress to be the same whether the stiffness is given by Eq. (7) or Eq. (10) pffiffiffi ^ 2 2 r k 1 ¼ 0:943k 1 . ¼ ð14Þ 3 rY Thus a high cohesive strength can be defined by ^=rY > 0:943k 1 . r
ð15Þ
Thus for an approximation to the root rotation it is only necessary to estimate the radius of curvature, R1. For an elastic beam, it can be assumed that
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R1 = R0 and for high cohesive strengths / = v1 = 0.639 (a more complete solution including shear effects gives / = 0.67, Williams and Hadavinia, 2002b). This rotation factor has in the past been used to analyse the peel test (Kinloch et al., 1994). If the film yields then equating R1 to R0 is only a lower bound.
2.5
upper bound lower bound
0.1
0.5
2
1.0 2.0
1.5
ˆ
0.5
2.1.3. Bounding solutions for elastic-perfectly plastic materials Because the film bonded to the substrate is constrained, its radius of curvature is greater than that in the free film for the same bending moment. The maximum constraint occurs when there is complete suppression of plastic deformation and the film remains elastic. The end moment M0 applied to the free film must be equal to the moment M1 in the bonded film to the left of point O in Fig. 1. It is convenient to use a non-dimensional moment, pffiffiffi m = M/Mc, where M c ¼ rY h2 =2 3 is the collapse moment for an elastic-perfectly plastic film. Thus for an elastic film 2 m ¼ k; 3
ð16Þ
and for an elastic-perfectly plastic film 2 m ¼ k; 3 m¼1
for k < 1; 1 ; 3k 2
ð17Þ for k > 1.
Thus a lower bound to the root rotation factor is obtained by equating the elastic–plastic moment to the elastic one to give k 1 ¼ k 0 ; for k 0 < 1; " # 2 1 k 1 ¼ 1 2 ; for k 0 > 1; 3 3k 0
ð18Þ
and the upper bound to k1 is k0 for all k0. Upper and lower bounds to the rotation factor, /, for an elastic-perfectly plastic film are shown in Fig. 2 using Eq. (9) or Eq. (13) depending on whether the cohesive strength is high or low as defined by the inequality (15). For k0 < 1 the upper and lower bounds are the same. Constraint has a large effect on the rotation factor. The rotation factors obtained from finite element analysis (see Appendix B) are also shown in Fig. 2. Apart from the results for k0 < 3 the rotation factor is bounded by the two approximate solutions and lies very close to the lower bound where plasticity is completely constrained.
0
Y
0.1 0.5 1.0 2.0
1
0.1 0.5 1.0 2.0 0
2
4
6
8
10
12
k0 Fig. 2. Upper and lower bounds to the rotation factors (markers indicate FE results).
2.2. Modelling the constraint For simplicity the elastic-perfectly plastic case is considered here and the general solution for a power law strain hardening material is given in Appendix A. Previous work has shown that except for a small plastic region associated with the root of the peel, the plastic deformation is restricted to the top half of the film except for very large end moments (Williams and Hadavinia, 2002a). 2.2.1. Yielding at the top of the film only for an elastic-perfectly plastic material It is assumed that the film is elastic over the region y < c2 with a linear variation in rx as shown in Fig. 3. At y = c2 the film is on the point of yielding and assuming that the transverse pstress, ry, is ffiffiffi small, the bending stress, rx ¼ 2rY = 3 for plane strain. Writing the longitudinal stress in terms of a parameter a such that at y = 0, rx ¼ p2ffiffi3 arY , then the stress is given by
2 3
Y
h
c2
y
Fig. 3. Elastic and plastic regimes for yielding confined to the upper section of the film.
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
2rY y p ffiffi ffi ; rx ¼ a ð1 þ aÞ c2 3 2rY rx ¼ pffiffiffi ; for y > c2 . 3
for y < c2 ; ð19Þ
The depth of yielding on the top surface, c2, can be found from the condition that the total force in the x direction must be zero. Thus h 1þa ¼ . ð20Þ c2 2 The moment on the section is given by
Z h rY h2 ð1 þ aÞ c2 2 M1 ¼ rx y dy ¼ pffiffiffi 1 ; 3 h 3 0
M1 ð1 þ aÞ c2 2 ¼2 1 . m1 ¼ 3 Mc h
Because the transverse stress ry, is not necessarily small near the bottom of the film, the strain cannot strictly be given by Eq. (22) across the whole thickness of the film. However, if the spirit of the engineers’ theory of bending is to be retained and in particular if the increment in work done, dW, by the stresses, rx, per unit length of the film can be written as Z h dW ¼ ðrx dex Þ dy Z
h
rx ðy cn Þ dð1=RÞ dy ¼ M dð1=RÞ;
ð23Þ
0
where R is the radius of curvature of the neutral axis and cn is the height of the neutral axis, we must assume that Eq. (22) applies for all y. With the above assumption the strain can be written in terms of the radius of curvature, R1, of the neutral axis as pffiffiffi ! c2 y 3e Y ex ¼ ð24Þ . R1 R1 2 Comparing Eqs. (22) and (24), gives pffiffiffi 3eY ð1 þ aÞ 1 ¼ ; R1 c2 2 and the non-dimensional curvature is given by
ð27Þ
Substituting Eq. (27) into Eq. (21) gives the constrained non-dimensional moment as
2 m1 ¼ 2 1 pffiffiffiffiffi . ð28Þ 3 k1
1 3k 2 þ 1 pffiffiffiffiffi ¼ 0 2 . k1 4k 0
Since m0 = m1 6 1, 0.75 6 (c2/h) 6 1. It is assumed that the transverse stress, ry, above the mid plane of the film is negligibly small and hence pffiffiffi
3e Y rx y ex ¼ ¼ a ð1 þ aÞ . ð22Þ c2 2 E
¼
and from Eq. (20) 2 2 h 1þa k1 ¼ ¼ . c2 2
ð26Þ
The moment on the bonded film must be equal to the moment in the free film, hence
ð21Þ
0
h 1 h 1þa p ffiffi ffi ; ¼ k1 ¼ 2 3 e Y R1 c 2
575
ð25Þ
ð29Þ
Note that the maximum possible value of k1 without yielding on the bottom surface of the film is 16/9 = 1.778. The rotation factor is given by /¼v
k1 16k 30 ¼v ; k0 ð1 þ 3k 20 Þ2
ð30Þ
where if the cohesive stress is high v = v1 is defined by Eq. (8) and if the cohesive stress is low v = v2 is defined by Eq. (12). The above solution applies providing there is no yielding on the bottom edge of the film. According to von Mises yield criterion, yielding will not occur on the bottom edge if, pffiffiffi pffiffiffi ^ ^ 3 r 3 r a> ; and a 6 1 þ ; 2 rY 2 rY ð31Þ pffiffiffi pffiffiffi ^ ^ 3 r 3 r or a < ; and a P 1. 2 rY 2 rY From Eqs. (26) and (31) it is seen that yielding cannot occur on the bottom edge if i ^ 4 pffiffiffiffiffi 4 hpffiffiffiffiffi r pffiffiffi k 1 P P pffiffiffi k1 1 ; rY 3 3 " # " # ð32Þ ^ 4 4k 20 4 k 20 1 r or pffiffiffi P pffiffiffi P . rY 3 3k 20 þ 1 3 3k 20 þ 1 Yielding on the bottom edge cannot occur if pffiffiffi pffiffiffi ^=rY < 2:3Þ for any ^=rY 6 4= 3 ð0:77 < r 4=3 3 6 r value of k0. Since the high cohesive strength solution ^=rY > 0:943k 1 and the minimum value of is for r k1 for yielding is unity, yielding on the bottom edge of the film only has to be considered for the low
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B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
^=rY < 2:3. The limits for cohesive strength case for r yielding to be completely prevented on the bottom surface of the film for both an elastic-perfectly plastic film and a strain hardening film with n = 0.2 (see Appendix A) are shown in Fig. 4. While for the elastic-perfectly plastic film yielding cannot occur at all on the bottom surface of the film for ^=rY < 2:3, yielding of the bottom surface 0:77 < r is possible for a strain hardening film at large curvatures. 2.2.2. Yielding on both surfaces of the film for an elastic-perfectly plastic material If yielding does occur at the bottom edge of the film, it is assumed that the bending stress is as shown in Fig. 5 where the stress on the bottom surface is assumed to be 2 ^. rx1 ¼ pffiffiffi rY þ r 3
ð33Þ
There are now two elastic–plastic interfaces at c1 and c2 and the stress distribution is given by
5
yield only on top surface n=0
3
ˆ
Y
2 1 0 0
2
4
6
8
10
12
k0 Fig. 4. Limits on k0 and r ^=rY for yielding to be confined to the upper section of the film.
2 3
where a is now simply a parameter that gives the stress level. By equating the stress in the two regions at y = c1 a c1 ¼ c2
pffiffiffi ! ^ 3 r 1þ 2 rY ð1 þ aÞ
.
ð35Þ
Using the condition that the total force in the x direction is zero gives pffiffiffi ! ^ c1 3 r 1þ 2 r Y c2
ð1 c1 =c2 Þ c1 ða 1Þ ð1 þ aÞ þ . 2 c2
h ¼1þ c2
The expression for k1 and hence " pffiffiffi 3 h k1 ¼ 1þ 4 ðc2 c1 Þ
ð36Þ
ð37Þ
given in Eq. (26) still holds # ^ h2 r . ¼ 2 rY ðc2 c21 Þ
ð38Þ
The non-dimensional moment is given by " pffiffiffi ! 2 # ^ 2 c1 þ c1 c2 þ c22 3 r 1þ m1 ¼ 2 1 . 3 4 rY h2 ð39Þ
c2
y
c1 Y
ð34Þ
for y > c2 ;
Y
h
2 3
2rY rx ¼ pffiffiffi 3
Substituting for a from Eq. (35) yields pffiffiffi ^ h 3 r ¼1þ . c1 þ c2 4 rY
yield only on top surface n=0.2
4
2rY ^ for y < c1 ; rx ¼ pffiffiffi þ r 3
2rY y rx ¼ pffiffiffi a ð1 þ aÞ for c1 < y < c2 ; c2 3
ˆ
Fig. 5. Elastic and plastic regimes when yielding occurs on both surfaces of the film.
The constrained moment can then be written as pffiffiffi ! ^ 3 r 1þ pffiffiffi !3 2 rY ^ 3 r 1 m1 ¼ pffiffiffi ! 2 1 þ 4 rY 3k 1 ^ 3 r 1þ 4 rY ¼ m0 ¼ 1
1 ; 3k 20
ð40Þ
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
and
3
pffiffiffi !2 ^ 3 r 1þ 4 rY k1 ¼" #1=2 ; p ffiffi ffi k0 ^ 3 r 1þ ð1 þ 3k 20 Þ 4 rY
Yield top surface Yield both surfaces
ˆ
Y
0.1 0.2 0.5 1.0 2.0
2
ð41Þ
ˆ Y 0.1
1
and the rotation factor is given by
1=2 k1 2 rY k1 / ¼ v2 ¼ pffiffiffi k1 ^ k0 k0 3 3 r pffiffiffi #3
1=2 " ^ 3 r 2 rY pffiffiffi 1þ k0 ^ 4 rY 3 3 r ¼ " #3=4 ; pffiffiffi ^ 3 r 1þ ð1 þ 3k 20 Þ 4 rY 2 pffiffiffi !#1=2 pffiffiffi !, ^ ^ 3 3 r 3 r ; for k 0 > 4 1 þ 1 4 rY 4 rY and
577
^ r 4 < pffiffiffi . rY 3 3
0.2
0
10.5 2
0
1
2
3
4
5
k0
6
7
8
9
3
0.1
0.2
2
ˆ
Yield top surface Yield both surfaces
1 2
0
For a constrained film the moment–curvature relationship is given in Appendix A. The rotation factors for both elastic-perfectly plastic and strain hardening for n = 0.2 (see Appendix A) films are shown in Figs. 6 and 7. The rotation factor for the strain hardening film is larger than that for the elastic-perfectly plastic film. The approximate estimation of the root rotation is only ^=rY > 1. accurate for r
Y
0.1 0.2 0.5 1.0 2.0
0.5
ð42Þ
and e are the von Mises equivalent stress where r and strain and n is the strain hardening index. For an unconstrained film the relationship between the non-dimensional moment and curvature is given by (Kim and Aravas, 1988)
2 1n 1 n m¼ k . ð44Þ 2þn 3 k2
11
Fig. 6. Rotation factors for elastic-perfectly plastic (n = 0) film under an end moment using Eq. (30) or Eq. (42) depending upon whether the top or both surfaces of the film yield (markers indicate FE results).
1
2.2.3. Power law hardening material For a strain hardening film the stress–strain relationship is modelled by the power law e r ¼ ; for e < eY ; rY eY n ð43Þ e r ¼ ; for e > eY ; rY eY
10
0
1
2
3
4
5
6
7
8
9
10
11
k0 Fig. 7. Rotation factors for a strain hardening (n = 0.2) film under an end moment using results of Appendix A (markers indicate FE results).
2.3. The root rotation factor obtained by integration of the equations of bending In the estimates of the root rotation factor given in Sections 2.1 and 2.2 it is assumed that the curvature is constant over the cohesive zone. A more exact estimate can be obtained by numerical integration of the equations of bending. For the purpose of calculating the root rotation, h0, the bonded section of the film is divided into two regions (see Fig. 8): (a) Region A far from the tip of the bond where the transverse stress on the bottom of the film ^ and the curvature k < 1. Depending r0 < r upon the cohesive strength either of these inequalities is the critical one that defines region A. In this region x is measured to the right as shown in Fig. 8.
578
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
Region B x
v
If initially k < 1 in region B, the integration could be performed analytically until k = 1, but it is found more convenient always to use numerical integration in region B. Two assumptions are made for the elastic–plastic bending when k > 1:
Region A x
ˆh ˆh and k 0 1 or k0 1 v < 2E 2E
M0
(a) The film is unconstrained and the non-dimensional moment m is given by Eq. (44). (b) The film is constrained as described in Section 2.2 and Appendix A.
v
root of delamination
Fig. 8. Schematic displacement of the centre line of the film.
(b) Region B near the tip of the bond where either of these inequalities is violated. In this region x is measured to the left (see Fig. 8). The general equation of bending in both regions A and B is
A Fortran programme performs the numerical integration of Eq. (45) in region B using the values va, ha, ka, and sa obtained from the solution of region A and calculates /. Dimensional analysis shows that the root rotation, h0, is given by the functional relationship r=rY Þ. h0 ¼ eY f ½k 0 ; n; ð^ Hence the rotation factor
2
dM ¼ r0 ; dx2 pffiffiffi r0 d2 m ¼ 2 3 ; 2 dx rY
/ ¼ h0 ð45Þ
where x ¼ x=h. The transverse stiffness of the film is assumed to be given by Eq. (7) and the transverse stress, r0, acting on the bottom of the film is given by v v ^; r0 ¼ 2E ; for 2E < r h h ð46Þ v ^. ^; for 2E > r r0 ¼ r h In region A the beam is elastic and the equation of bending becomes
1.6
v ¼ expðbxÞ½A cos bx þ B sin bx;
ð48Þ
0.8
and x ¼ x=h is the non-dimensional h2i d v distance. At x ¼ 0 the curvature k a ¼ d and x2 x¼0 h3i d v its derivative sa ¼ d are specified giving B = x3
0.6
where b = 6
x¼0
ka/2b3, A = sa/2b2 B. The deflection and slope at x ¼ 0 are then given by va ¼ ½vx¼0 ¼ A; ha ¼ v d ¼ ðB AÞb. These values are the starting dx x¼0 values for a numerical integration into region B.
R0 R 0 Rp 1 ¼ h0 ¼ pffiffiffi f ½k 0 ; n; ð^ r=rY Þ; h Rp h 3k 0
ð50Þ
is independent of the yield strain, but in the programme it was more convenient to introduce eY though the result is not dependent on it. The rotation factor, /, estimated by the constrained and unconstrained yield models is shown as a function of k0 for a range of non-dimensional ^=rY , in Figs. 9 and 10 together cohesive strengths, r with finite element results (see Appendix B). As is expected the values for the unconstrained yielding are greater than those for constrained yielding, but the difference is not that great. The unconstrained yielding model, which agrees best with the finite ele-
d4v ¼ 24v; ð47Þ dx4 where v ¼ v=h. The general solution of Eq. (47) for the non-dimensional deflection, is given by 1/4
ð49Þ
Unconstrained yield Constrained yield
1.4 1.2 1
0.4 0.2 0 0
ˆ
0.1 Y
0.1 0. 2 0.5 1.0 2. 0
0.10.2 1 0.20.5 20.5 12
2
4
6
8
10
12
k0 Fig. 9. Rotation factors for an elastic-perfectly plastic (n = 0) film, with an end moment, from integration of the bending equation compared with FE results shown by markers.
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584 1.6
1.2
0.1 0.1
1
a = 3h 0.2
0.8
P 0.2
0.6 0.4 0.2 0 0
M Mmax
Unconstrained yield Constrained yield
1.4
579
x
0.5
ˆ
Y
1 2 0.5 1 2
0.1 0.2 0.5 1.0 2.0
2
4
6
8
10
12
k0 Fig. 10. Rotation factors for strain hardening (n = 0.2) film, with an end moment, from integration of the bending equation compared with FE results shown by markers.
5
ˆ
ˆ
Y
0.1 0.2 0.5 1.0 2.0
4 3
0.5
1
0
Y
0.2
1.0 2.0
0
moment even if shear deformation in the film is neglected. The difference arises because unlike in the end moment case, the bending moment at first increases along the bonded portion of the film. If an end force, P, is applied at a distance a from the tip of the crack, the bending moment, M, in the cohesive zone is given by
0.1
2
2
4
6
8
10
Fig. 12. Bending moment distribution near the delamination tip under an end moment.
^ M ¼ P ða þ xÞ r 12
k0 Fig. 11. Rotation factors for an elastic film (n = 1) under an end moment from integration of the bending equation (markers indicate FE results).
x2 ; 2
ð51Þ
and the moment reaches a maximum, Mmax, at x ¼ P =^ r, given by
P2 M0 ¼ M0 1 þ M max ¼ Pa þ ; 2^ r 2^ ra " # ð52Þ 2 m0 h rY . or m1 ¼ mmax ¼ m0 1 þ pffiffiffi ^ r 4 3 a
ment results, overestimates and the constrained model underestimates the rotation factor for k0 > 4. For 1 < k0 < 4 the rotation factor is underestimated by the unconstrained model. As the strain hardening index increases so the relative difference between the rotation factors calculated assuming that the bending on the bonded section of the film is constrained or not decreases until at n = 1, an elastic film, there is no difference. The rotation factors for an elastic film are shown in Fig. 11 where the concept of artificial yield strength has been retained so that direct comparison can be made with Figs. 9 and 10. The approximate rotation factors agree well with the finite element results except for small k0.
In the absence of constraint, an elastic-perfectly plastic film cannot sustain a bending moment greater than the collapse moment, Mc, and hence the rotation factor becomes infinite when mmax = 1. The values of the non-dimensional curvature at the crack tip at which plastic collapse occurs in the film, for a/h = 3, is given in Table 1. Yield constraint in the bonded region prevents plastic collapse. If the cohe^, is large then Mmax is only slightly sive strength, r greater than M0 and the effect of the shear force is small for a constrained film. However, if the cohesive
3. Shear force effects
Table 1 Non-dimensional curvature, k0, at which plastic collapse occurs in an unconstrained film for a/h = 3
The root rotation factor is somewhat different for an end force on the film (see Fig. 12) than an end
^=rY r k0 for plastic collapse
0.1 1.64
0.2 2.19
0.5 3.32
1.0 4.63
2.0 6.50
580
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584 4
3 end force end moment 0.1
2
0.1
3.5
0.1 0.2
ˆ
Unconstrained yield Constrained yield
3 2.5
0.2
ˆ
1
0
Y
0.1
0
1
2
3
4
5
k0
6
7
8
9
0.2 0.5 2 1 10 11
Fig. 13. Comparison of the rotation factors for an end force using Eq. (52) and for an end moment using Eq. (30) or Eq. (42) depending upon whether the top or both surfaces of the film yield for an elastic-perfectly plastic film.
Y
0.1 0.2 0.5 1.0 2.0
0.5
2 1.5 1
1
2
0.5
0.2 0.5 1 2
0 0
2
4
6
8
10
12
k0 Fig. 14. Rotation factors for an elastic-perfectly plastic (n = 0) film, with an end force (with a/h = 3), from integration of the bending equation compared with FE results shown by markers.
strength is small then the shear force effect can be significant. The root rotation factor, for a constrained film, estimated from the curvature near to the crack tip would be underestimated from the curvature at the crack tip. An upper bound to the root rotation factor can be obtained if the curvature corresponding to the curvature for the maximum moment, Mmax, is used that enables the effect of the shear forces to be judged. The difference between the rotation factor for an end force (with a/h = 3) and an end moment is shown in Fig. 13 for an elasticperfectly plastic film. Although there are very signif^=rY < 0:5, there are only icant differences when r ^=rY P 0:5. slight differences for r
Fig. 15. Rotation factors for strain hardening (n = 0.2) film, with an end force (with a/h = 3), from integration of the bending equation compared with FE results shown by markers.
3.1. The root rotation factor obtained by integration of the equations of bending
4. Conclusions
The root rotation factor for an end force (with a/h = 3) has been obtained by integration of the equations of bending as explained in Section 2.3. This root rotation factor is compared in Figs. 14 and 15 with the FE results. For a strain hardening exponent n = 0.2 the unconstrained model generally overestimates the root rotation factor and the constrained model underestimates it much the same as in the end moment case with the unconstrained model generally giving the more accurate result. For an elastic-perfectly plastic film, the constrained yield model is best which is not surprising since the unconstrained model predicts infinite rotation factors as k0 approaches the values for plastic collapse given in Table 1, but again the constrained model still underestimates the rotation factor.
The root rotation factor for an end moment acting on an isotropic elastic film bonded to a rigid substrate with an infinite cohesive strength, which from finite element analysis is given as / = 0.66 (Williams and Hadavinia, 2002b), is given by / = 0.639 from the approximate bending theory. The cohesive zone becomes longer as the cohesive strength decreases and the end moment increases causing the root rotation factor to increase—for low cohesive strength the increase is large (see Fig. 11). The numerical integration of the equations of bending (see Section 2.3) gives a good approximation to the elastic root rotation factor obtained from finite element analysis (see Fig. 11). The estimates of the rotation factor obtained from simple geometric considerations are summa-
3.5
ˆ
0.1
3
Unconstrained yield Constrained yield
Y
0.1 0.2 0.5 1.0 2.0
2.5
0.2
2 1.5 1
0.5
0.1
1 0.2 2 0.5 1 2
0.5 0 0
2
4
6
8
10
12
k0
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584 Table 2 Rotation factors obtained from simple geometric considerations Normalized cohesive ^=rY strength r
Root rotation factor / k1 0:64 k 0 1=2 3=2 k1 ^ r 0:62 rY k0
>0.943 <0.943
Equation number (9) (13)
rized in Table 2 in terms of the ratio of the curvatures in the constrained and free film at the root. Upper and lower bounds to the ratio of the curvatures are summarized in Table 3. Estimates of the ratio of the curvatures for an elastic-perfectly plastic film are summarized in Table 4. Plastic deformation reduces the root rotation factor (see Figs. 6, 7, 9 and 10). The assumption that the curvature in the cohesive zone can be considered constant and the constrained yield model applied, does not predict the root rotation factor for an end moment well except for high cohesive strength (see Figs. 6 and 7). However, numerical integration of the equations of bending does predict the root rotation factor for an end moment reasonably well, but the unconstrained model is better than the constrained model (see Figs. 9 and 10). The rotation factor for an end load (with a = 3h) is very similar to that of an end moment if the coheTable 3 Upper and lower bounds to the k1/k0 ratio (see Eq. (18)) Bound
k0 range
k1/k0
Lower
k0 < 1
1
Upper
k0 > 1
" # 1:5 1 1 2 k0 3k 0
All k0
1
Table 4 Constrained yield estimates of the k1/k0 ratio for elastic perfectly plastic films Yield condition
Top surface only pffiffiffiffiffi ^ r k 1 P 0:433 r Y pffiffiffiffiffi P k1 1 Both surfaces pffiffiffiffiffi ^ r k 1 < 0:433 ; rY pffiffiffiffiffi r ^ or k 1 > 0:433 þ1 rY
k1/k0 16k 30
Equation number
(30)
2 1 þ 3k 20 ^ 2 r 1 þ 0:433 rY
(42) 1=2 ^ r 1 þ 0:433 1 þ 3k 20 rY
581
sive strength is high ð^ r=rY P 0:5Þ, but is much larger for low cohesive strength (see Fig. 13). Numerical integration of the equations of bending predicts the end load root rotation factor well when the unconstrained yield model is used for n = 0.2 (see Fig. 15). However for an elastic–plastic film, since there is plastic collapse away from the tip of the delamination at high curvatures, the constrained model is best (see Fig. 14). In summary, a reasonable accurate estimate of the root rotation factor can be obtained by the integration of the equation of bending for an end moment or load. Generally the unconstrained yield model gives the best result, but if the strain hardening exponent is small or the ratio a/h is small then the constrained yield model is best. Since in peeling a/h is small, the constrained yield model is likely to be the more accurate. In a future paper the approximate theory developed here will be used to model peeling. Appendix A. Modelling the constraint for a strain-hardening film The strain in the elastic and upper plastic region is assumed to be the same as in the elastic-perfectly plastic case and is still given by Eq. (22) and the non-dimensional curvature, k1, is given by Eq. (26). Neglecting transverse stress in the upper section of the film, the equivalent von Mises strain e is given by
y e ¼ eY a ð1 þ aÞ . ðA:1Þ c2 If there is no yielding on the bottom surface the stresses are given by
2 y rx ¼ pffiffiffi rY a ð1 aÞ ; for y < c2 ; c2 3
n 2 y rx ¼ pffiffiffi rY ð1 aÞ a ; for y > c2 . c2 3 ðA:2Þ Using the condition that the totals longitudinal force must be zero yields " # 1
nþ1 h 1 nþ1 2 aþ 1þ ¼ ða 1Þ . c2 ð1 þ aÞ 2 ðA:3Þ Integrating the moment of the stresses about the bottom of the film gives the non-dimensional moment m1
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
( " # c 2 ð2 aÞ ðh=c2 Þ½ð1 þ aÞðh=c2 Þ a1þn 1 2 m1 ¼ 4 þ h 6 ð1 þ nÞð1 þ aÞ " #) 2þn ½ð1 þ aÞðh=c2 Þ a 1 . ðA:4Þ ð1 þ nÞð2 þ nÞð1 þ aÞ2
This moment must equal m0, the non-dimensional moment on the free unconstrained film which is given by Eq. (44). The constrained curvature, k1, cannot be found explicitly, but can be calculated from Eqs. (26), (44), (A.3), (A.4) in terms of k0. The rotation factor can then be found from Eqs. (9) or (13) depending on condition (15). The bottom surface will not yield if the conditions (31) are met. Unlike the case for an elastic-perfectly plastic film, yielding of the bottom surface of the film can occur for any cohesive stress, ^, provided the applied moment is large enough (see r Fig. 4). If the film yields on its bottom surface to a depth c2 then the stresses above the lower yielded section y > c1, are given by Eq. (A.2).h Because the itransverse stress is neglected, e ¼ eY a ð1 þ aÞ cc12 > eY
at y = c2. To avoid this inconsistency the somewhat arbitrary assumption that the variation in the equivalent strain within the bottom plastic zone is the same as in the upper section of the film but, for continuity in the stress, the equivalent strain at the elastic–plastic boundary is the yield strain eY. Hence it is assumed
ð1 þ aÞðc1 yÞ e ¼ eY 1 þ ðA:5Þ ; for y < c1 ; c2 and the stress given by 2 r ^ rx ¼ pffiffiffi þ r 3
n 2rY ð1 þ aÞðc1 yÞ p ffiffi ffi ^; ¼ þr 1þ c2 3
for y < c1 . ðA:6Þ
The ratio of the depths of yielding c1/c2 is still given by Eq. (35). The condition that the total force must be zero gives ((
ð1þnÞ h 1 c1 1 þ ð1 þ aÞ ¼ c2 1 þ a c2 ð1 þ aÞð1 þ nÞ c1 c1 1 þ a 1 ð1 þ aÞ 2 c2 c2 )
)ð1=1þnÞ pffiffiffi r ^ c1 þ 3 þa . ðA:7Þ rY c2
Integration of the moment of the stresses gives ( 2 m1 h ¼ 4 c2
þ
)
ð1þnÞ c1 h h þ ð1 þ aÞ a 1 c2 c2 c2
(
ð1 þ aÞð1 þ nÞ
ð2þnÞ )
ð2þnÞ c1 h 2 1 þ ð1 þ aÞ ð1 þ aÞ a c2 c2
2
ð1 þ aÞ ð1 þ nÞð2 þ nÞ " 3 # 2 #) 1 c1 c1 2ð1 þ aÞ 1 þ 3a 1 6 c2 c2 pffiffiffi 2 ^ c1 3 r . ðA:8Þ 4 rY c2 (
"
This moment must be equal to the applied moment m0 given by Eq. (44) and k1 can be found as a function of k0. The expression for the rotation factor if the bottom surface of the film yields can then be found. Appendix B. Finite element analysis The finite element analysis was applied to a film of thickness 1 mm. The free section of the film was 3 mm long. Four nodes plane strain elements were used with incompatible mode to improve the bending behaviour. The elements in the free section of the film and for a distance of 3 mm from the root were 0.1 · 0.1 mm2 elements with larger elements further from the root (see Fig. B1). The Young’s modulus was taken as 100 GPa, Poisson’s ratio 0.3 and the yield strength of 100 MPa. Three material behaviours were considered: elastic, elastic-perfectly plastic, and elastic-strain hardening plastic as described by Eq. (43) with n = 0.2. Five values of ^=rY ¼ 0:1, 0.2, 0.5, 1.0, 2.0 were used. the ratio r A Tvergaard/Hutchinson type cohesive zone (Tvergaard and Hutchinson, 1992) is inserted to represent the bond between the film and the substrate. The substrate was constrained in the vertical direction but no restraint was imposed on the horizontal 0. 1mm x 0.1 mm mesh 3 mm
3 mm
1 mm
582
cohesive zone
Fig. B1. Schematic arrangement of FE model.
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
583
order term. A second method was examined for the end moment case which makes use of the equivalent work concept. If M0 is the end moment and the dh0 is the increment in the root rotation for an increment dM0 in the end moment, then the increment in work done must be given by Z h ½r0x du0 þ s0xy dv0 dy; ðB:1Þ M 0 dh0 ¼
σ
σˆ
0
δ -5
10 mm Fig. B2. Traction separation relationship.
direction (see Fig. B1). Hence the problem can be viewed as two identical films bonded back to back with symmetrical loading. Because of the freedom in the horizontal direction, there is no shear stress in the cohesive zone and the traction–separation relationship, shown in Fig. B2, can be defined in terms of the normal stress and normal displacement alone. Note that in this paper we were not concerned with fracture so that the right hand end of the usual traction–separation relationship was not defined. The traction separation relationship is slightly different to that used in the approximate analyses since it has an ‘‘elastic’’ loading portion for computational ease. However this loading portion has been minimised and does not affect the results. Two load cases were considered. In the first case the beam was subjected to an end moment, which is accomplished by giving the nodes of the left hand end of the film a rotation. Over the central part of the free section of the film the stress distribution is the same as for a pure bending moment. In the second case a vertical force, P, is applied to the left hand end of the beam which was distributed over the nodes so that all the nodes other than the outside nodes had an equal force and the two outside nodes had half of this value. The root rotation (h0) and the root curvature (1/R0) are directly obtainable from the approximate solution using the engineers’ theory of bending, but they are not directly obtainable from the FE analysis. Two methods of determining the root rotation (h0) were examined. In the first method, the displacement profile for the centre line of the film relative to the displacement at the root section, v(x) v(0), was obtained over a distance 0.5 on either side of the root section. A polynomial was fitted to this curve and the root rotation taken as the coefficient of the first
where the stresses and displacements are those at the root section. The root rotation is therefore given as # Z M 0Z h " 0 s0xy rx h0 ¼ du0 þ dv0 dy dM 0 . ðB:2Þ M0 M0 0 0 In the engineers’ theory of bending the shear stress s0xy is zero and this term is identically zero and it is interesting that in the FE solution the second term in Eq. (B.1) is negligible. The difference between the two methods was found to be slight and the method used to obtain the root rotation factor is the slope method. Since the purpose of obtaining the root rotation factor is to enable it to be used as a boundary condition in cases where the free film is modelled by the engineers’ theory of bending, the normalising curvature has been taken as the curvature in a free film. In the end moment case the curvature is found by plotting the strain, ex, as a function of y across a section 1.5h from the delamination root. The relationship is linear and the curvature, 1/R0, is its slope. In the end load case a separate FE problem is analysed where a free film of length 6h has a bending moment of 3Ph applied at both ends and the curvature at the centre of the film obtained from the strain, ex, across the central section. References Blackburn, B.R.K., Hadavinia, H., Kinloch, A.J., Williams, J.G., 2003. The use of cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints. Int. J. Fract. 119, 25–46. Georgiou, I., Hadavinia, H., Ivankovic, A., Kinloch, A.J., Tropsa, V., Williams, J.G., 2003. Cohesive zone models and the plastically-deforming peel test. J. Adhesion 79, 239–265. Kawashita, L.F., Moore, D.R., Williams, J.G., 2004. The development of a mandrel peel test for the measurement of adhesive fracture toughness of epoxy-metal laminates. J. Adhesion 80, 147–167. Kim, K.-S., Aravas, N., 1988. Elastic plastic analysis of the peel test. Int. J. Solids Struct. 24, 417–435. Kinloch, A.J., Lau, C.C., Williams, J.G., 1994. The peeling of flexible laminates. Int. J. Fract. 66, 45–70.
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Tvergaard, V., Hutchinson, J.W., 1992. The relationship between crack growth resistance and fracture process parameters in elastic–plastic solids. J. Mech. Phys. Solids 40, 1377–1397. Wei, Y., Hutchinson, J.W., 1998. Interface strength, work of adhesion and plasticity in the peel test. Int. J. Fract. 93, 315– 333. Williams, J.G., 1993. A review of the determination of energy release rates for strips in tension and bending. Part I—Static solutions. J. Strain Anal. 28, 237–246.
Williams, J.G., Hadavinia, H., 2002a. Analytical solutions for cohesive zone models. J. Mech. Phys. Solids 50, 809–825. Williams, J.G., Hadavinia, H., 2002b. Elastic and elastic–plastic correction factors for DCB specimens14th European Conference on Fracture—ECF 14 Fracture Mechanics Beyond 2000, vol. 3. EMAS Publishing, Cracow, Poland, pp. 573–592. Williams, J.G., Hadavinia, H., Cotterell, B., 2005. Anisotropic elastic and elastic bending solutions for edge constrained beams. Int. J. Solids Struct. 42, 4927–4946.
The root rotation in double cantilever beam and peel tests B. Cotterell
a,d,*
, K. Hbaieb a, J.G. Williams
b,d
, H. Hadavinia c, V. Tropsa
b
a
Institute of Materials Research and Engineering (IMRE), MSCL, 3 Research Link, Singapore 117602, Singapore b Department of Mechanical Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK c Faculty of Engineering, Kingston University, London SW15 3DW, UK d School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia Received 10 August 2004
Abstract In a peel test the plastic work has to be separated from the total work of fracture to obtain the cohesive energy. The root rotation determines the plastic work in a peel test and needs to be estimated accurately if the peel test is to be analysed successfully. Approximate solutions for the root rotation factor, based on the engineers’ theory of bending, are compared with finite element results to establish a reliable simple method of the estimation of the root rotation factor. 2005 Elsevier Ltd. All rights reserved. Keywords: Rotation factor; Peel test; Cantilever beam; Cohesive zone; Bending theory
1. Introduction Cantilever beam specimens of various types are widely used as specimens for the measurement of fracture toughness. In the simplest analysis of the energy release rate for cantilever beam specimens, the beam at the crack tip is considered built in so that there is no rotation of the beam at the crack tip. Such analyses are adequate providing the beam is isotropic or the crack length, a, to beam height, h, is large. However if the beam is short, or anisotropic with the transverse modulus very much less than the longitudinal modulus, as in the case of laminar composites, then the rotation at the tip of the crack can *
Corresponding author. Address: School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia. Tel./fax: +61 2 99491908. E-mail address: [email protected] (B. Cotterell).
be significant (Williams, 1993; Williams et al., 2005). In the peel test, widely used for determining the adhesion between flexible laminates, the effective a/h is small and the root rotation can be a controlling parameter (Blackburn et al., 2003; Georgiou et al., 2003; Kim and Aravas, 1988; Kinloch et al., 1994; Williams and Hadavinia, 2002a,b). The basic problem in the test is that usually only the force per unit width (P) is measured which gives the total energy release rate per unit area, G, dissipated in creating the new surface area during peeling. There are two components to the total energy dissipated during peeling: the adhesive energy, C0, and the plastic energy dissipated in deforming the peel arm, Gp. For 90 peeling, the usual case, we have, P ¼ G ¼ C0 þ Gp .
ð1Þ
The adhesive energy, C0 is a quasi material constant that can depend upon the constraint. The plastic
0167-6636/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2005.11.001
572
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
energy dissipated depends upon the film thickness. Thus the fundamental problem is to find Gp which is a challenge since it is a large proportion (up to 90%) of the total fracture work, G. There are various strategies available ranging from a fully analytical approach in which Gp is computed (Georgiou et al., 2003; Kim and Aravas, 1988; Kinloch et al., 1994) to a fully experimental method in which it is measured (Kawashita et al., 2004). There are also intermediate possibilities in which some measurements, such as the radius of curvature of the peel arm, are made and the simpler calculation performed (Georgiou et al., 2003). An accurate analysis may be obtained via finite element or volume codes, but it is difficult since it involves the large displacement of slender beams in the elastic–plastic regime. This difficulty has been circumvented to some extent by using approximate solutions which for the peel arm are straight forward (Wei and Hutchinson, 1998). The advantage of the approximate solutions is that simple programmes can be written for PCs such as ICPEEL.1 A limiting factor in such solutions is the modelling of the bonded region near the tip of the peel. When both the free and bonded sections are elastic, the results have been worked out in some detail and some results were given for plasticity effects (Kinloch et al., 1994). If, as is usual, there is considerable plastic deformation both immediately prior to and after debonding the accuracy of the peeling solution depends largely on the accuracy of the calculation of the rotation in the film at the tip of the film. ICPEEL does allow plastic deformation, but the estimate of the root rotation used is based on elastic deformation and has not been quantified except experimentally. This paper reports results from a collaborative study which looked at refining the approximate approach by a new detailed study of the bonded section including the effects of constraint on plasticity and limiting cohesive stresses in the bonded section of the film. A simple estimate of the root rotation factor based on geometrical considerations is given in Section 2 for loading by an end moment. Bounding solutions based on these geometrical considerations are given in Section 2.1 for large cohesive bond strengths (Section 2.1.1) and large cohesive bond strengths (Section 2.1.2). The results of Sections 2.1.1 and 2.1.2 are summarized in Table 2. Bounding solutions for elastic-perfectly plastic films
1
http://www.me.ic.ac.uk/AACgroup/index.htm
are considered in Section 2.1.3 and summarized in Table 3. The constraint on plastic bending in the bonded section of the film is discussed in Section 2.2 where elastic-perfectly plastic films that only yield on their top surfaces are considered in Section 2.2.1, elastic-perfectly plastic films yielding on both surfaces are considered in Section 2.2.2, and the effect of strain hardening is discussed in Section 2.2.3. The results of Sections 2.2.1 and 2.2.2 are summarised in Table 4. In Section 2.3 a better approximation for the root rotation is obtained by integration of the equations of bending. The root rotation for an end force is discussed in Section 3. The results have been checked by numerical results from finite element (FE) analyses. Finally the overall conclusions on the accuracy of the approximate calculations of the root rotation factor are given in Section 4. 2. Approximate calculation of the root rotation factor due to a moment In peeling a moment M0 and a shear force S0 act at the tip of the peel. It is shown later that the rotation h0 at the root of the peel is a strong function of the moment and generally only a weak function of the shear force. Hence in this section the rotation due to a simple end moment M0 is calculated. Various approximate solutions based on the engineers’ theory of bending are presented. Since in this simple theory, plane sections remain plane, the root rotation is simply the rotation of this plane and the radius of curvature, R0, is the radius of the mid plane. A non-dimensional root rotation factor is defined as R0 / ¼ h0 . ð2Þ h 2.1. Approximate bounding solutions A simple correction to allow for root rotation is to consider the film built in not at the bond root but at a small distance D along the bond line (Williams and Hadavinia, 2002b). Fig. 1 shows the deformations which occur around the peel point O. The free film of thickness, h, has a radius of curvature R0 arising from a bending moment M0 per unit width of the film. Because of constraint, the curvature in the bonded region is different. Over the small distance D the radius of curvature in the bonded region can be assumed to be constant and
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
mined from the stiffness of the film. On the other hand if the cohesive strength is small , the cohesive zone will be large and v = v2 is controlled by the ^. cohesive strength, r
R0 M0
2.1.1. Large cohesive strength For large cohesive strength the lateral stiffness, S = S1, can be determined from a very simple model (Williams and Hadavinia, 2002b), where the stiffness comes from the elastic deformation across half the film thickness so that, ignoring Poisson’s effect
R1
0
h
v0
o
S ¼ S1 ¼
Fig. 1. Deformation of a film near the tip of the bond.
2E ; h
ð7Þ
and
equal to R1. Thus the root rotation, can be approximated by D h h0 ¼ ¼v ; ð3Þ R1 R1 where the factor v ¼ Dh . Thus the root rotation factor is given by R0 k1 /¼v ¼v ; ð4Þ R1 k0 where k is the non-dimensional curvature defined by pffiffiffi k = Rp/R and Rp ¼ h=ð 3eY Þ is the radius of curvature that just causes yielding in the outer surfaces of the film, eY being the tensile yield strain. In the elastic foundation model it is assumed that the deflection of the neutral axis ahead of the root of the peel is constrained by elastic stresses developed between the film and its substrate. Thus the stress acting on neutral axis, ry, is given by ry ¼ Sv;
573
ð5Þ
where S is the lateral stiffness per unit width of the film and v is the lateral displacement of the neutral axis. The elastic correction D obtained from the elastic foundation model is (Williams and Hadavinia, 2002b), 4 D E 1 ; ð6Þ ¼ h 3h S where E ¼ E=ð1 m2 Þ is the plane strain elastic modulus, E being the Young’s modulus and m the Poisson’s ratio. To avoid unnecessary complications when considering plastic bending, it is assumed that Poisson’s ratio is 0.5 so that E ¼ 4E=3. The effective lateral stiffness can be obtained in ^, is large, then two ways. If the cohesive strength, r the cohesive zone will be small and v = v1 is deter-
D 1 ¼ 1=4 0:639. h 6
v1 ¼
ð8Þ
Thus
k1 / ¼ v1 . k0
ð9Þ
2.1.2. Small cohesive strength For small cohesive strength the effective lateral stiffness, S = S2, can be obtained from ^ ¼ S 2 v0 ; r
ð10Þ
where v0 is the displacement at the peel root and is given in terms of the correction term D by v0 ¼
D2 . 2R1
Thus using Eq. (6) 1=2
1=2 E h 2 rY ¼ pffiffiffi ; v2 ¼ k1 ^ 6^ r R1 3 3 r where rY is the tensile yield strength and
1=2 3=2 k1 2 rY k1 / ¼ v2 ¼ pffiffiffi . ^ k0 k0 3 3 r
ð11Þ
ð12Þ
ð13Þ
For the maximum cohesive stress to be the same whether the stiffness is given by Eq. (7) or Eq. (10) pffiffiffi ^ 2 2 r k 1 ¼ 0:943k 1 . ¼ ð14Þ 3 rY Thus a high cohesive strength can be defined by ^=rY > 0:943k 1 . r
ð15Þ
Thus for an approximation to the root rotation it is only necessary to estimate the radius of curvature, R1. For an elastic beam, it can be assumed that
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B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
R1 = R0 and for high cohesive strengths / = v1 = 0.639 (a more complete solution including shear effects gives / = 0.67, Williams and Hadavinia, 2002b). This rotation factor has in the past been used to analyse the peel test (Kinloch et al., 1994). If the film yields then equating R1 to R0 is only a lower bound.
2.5
upper bound lower bound
0.1
0.5
2
1.0 2.0
1.5
ˆ
0.5
2.1.3. Bounding solutions for elastic-perfectly plastic materials Because the film bonded to the substrate is constrained, its radius of curvature is greater than that in the free film for the same bending moment. The maximum constraint occurs when there is complete suppression of plastic deformation and the film remains elastic. The end moment M0 applied to the free film must be equal to the moment M1 in the bonded film to the left of point O in Fig. 1. It is convenient to use a non-dimensional moment, pffiffiffi m = M/Mc, where M c ¼ rY h2 =2 3 is the collapse moment for an elastic-perfectly plastic film. Thus for an elastic film 2 m ¼ k; 3
ð16Þ
and for an elastic-perfectly plastic film 2 m ¼ k; 3 m¼1
for k < 1; 1 ; 3k 2
ð17Þ for k > 1.
Thus a lower bound to the root rotation factor is obtained by equating the elastic–plastic moment to the elastic one to give k 1 ¼ k 0 ; for k 0 < 1; " # 2 1 k 1 ¼ 1 2 ; for k 0 > 1; 3 3k 0
ð18Þ
and the upper bound to k1 is k0 for all k0. Upper and lower bounds to the rotation factor, /, for an elastic-perfectly plastic film are shown in Fig. 2 using Eq. (9) or Eq. (13) depending on whether the cohesive strength is high or low as defined by the inequality (15). For k0 < 1 the upper and lower bounds are the same. Constraint has a large effect on the rotation factor. The rotation factors obtained from finite element analysis (see Appendix B) are also shown in Fig. 2. Apart from the results for k0 < 3 the rotation factor is bounded by the two approximate solutions and lies very close to the lower bound where plasticity is completely constrained.
0
Y
0.1 0.5 1.0 2.0
1
0.1 0.5 1.0 2.0 0
2
4
6
8
10
12
k0 Fig. 2. Upper and lower bounds to the rotation factors (markers indicate FE results).
2.2. Modelling the constraint For simplicity the elastic-perfectly plastic case is considered here and the general solution for a power law strain hardening material is given in Appendix A. Previous work has shown that except for a small plastic region associated with the root of the peel, the plastic deformation is restricted to the top half of the film except for very large end moments (Williams and Hadavinia, 2002a). 2.2.1. Yielding at the top of the film only for an elastic-perfectly plastic material It is assumed that the film is elastic over the region y < c2 with a linear variation in rx as shown in Fig. 3. At y = c2 the film is on the point of yielding and assuming that the transverse pstress, ry, is ffiffiffi small, the bending stress, rx ¼ 2rY = 3 for plane strain. Writing the longitudinal stress in terms of a parameter a such that at y = 0, rx ¼ p2ffiffi3 arY , then the stress is given by
2 3
Y
h
c2
y
Fig. 3. Elastic and plastic regimes for yielding confined to the upper section of the film.
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
2rY y p ffiffi ffi ; rx ¼ a ð1 þ aÞ c2 3 2rY rx ¼ pffiffiffi ; for y > c2 . 3
for y < c2 ; ð19Þ
The depth of yielding on the top surface, c2, can be found from the condition that the total force in the x direction must be zero. Thus h 1þa ¼ . ð20Þ c2 2 The moment on the section is given by
Z h rY h2 ð1 þ aÞ c2 2 M1 ¼ rx y dy ¼ pffiffiffi 1 ; 3 h 3 0
M1 ð1 þ aÞ c2 2 ¼2 1 . m1 ¼ 3 Mc h
Because the transverse stress ry, is not necessarily small near the bottom of the film, the strain cannot strictly be given by Eq. (22) across the whole thickness of the film. However, if the spirit of the engineers’ theory of bending is to be retained and in particular if the increment in work done, dW, by the stresses, rx, per unit length of the film can be written as Z h dW ¼ ðrx dex Þ dy Z
h
rx ðy cn Þ dð1=RÞ dy ¼ M dð1=RÞ;
ð23Þ
0
where R is the radius of curvature of the neutral axis and cn is the height of the neutral axis, we must assume that Eq. (22) applies for all y. With the above assumption the strain can be written in terms of the radius of curvature, R1, of the neutral axis as pffiffiffi ! c2 y 3e Y ex ¼ ð24Þ . R1 R1 2 Comparing Eqs. (22) and (24), gives pffiffiffi 3eY ð1 þ aÞ 1 ¼ ; R1 c2 2 and the non-dimensional curvature is given by
ð27Þ
Substituting Eq. (27) into Eq. (21) gives the constrained non-dimensional moment as
2 m1 ¼ 2 1 pffiffiffiffiffi . ð28Þ 3 k1
1 3k 2 þ 1 pffiffiffiffiffi ¼ 0 2 . k1 4k 0
Since m0 = m1 6 1, 0.75 6 (c2/h) 6 1. It is assumed that the transverse stress, ry, above the mid plane of the film is negligibly small and hence pffiffiffi
3e Y rx y ex ¼ ¼ a ð1 þ aÞ . ð22Þ c2 2 E
¼
and from Eq. (20) 2 2 h 1þa k1 ¼ ¼ . c2 2
ð26Þ
The moment on the bonded film must be equal to the moment in the free film, hence
ð21Þ
0
h 1 h 1þa p ffiffi ffi ; ¼ k1 ¼ 2 3 e Y R1 c 2
575
ð25Þ
ð29Þ
Note that the maximum possible value of k1 without yielding on the bottom surface of the film is 16/9 = 1.778. The rotation factor is given by /¼v
k1 16k 30 ¼v ; k0 ð1 þ 3k 20 Þ2
ð30Þ
where if the cohesive stress is high v = v1 is defined by Eq. (8) and if the cohesive stress is low v = v2 is defined by Eq. (12). The above solution applies providing there is no yielding on the bottom edge of the film. According to von Mises yield criterion, yielding will not occur on the bottom edge if, pffiffiffi pffiffiffi ^ ^ 3 r 3 r a> ; and a 6 1 þ ; 2 rY 2 rY ð31Þ pffiffiffi pffiffiffi ^ ^ 3 r 3 r or a < ; and a P 1. 2 rY 2 rY From Eqs. (26) and (31) it is seen that yielding cannot occur on the bottom edge if i ^ 4 pffiffiffiffiffi 4 hpffiffiffiffiffi r pffiffiffi k 1 P P pffiffiffi k1 1 ; rY 3 3 " # " # ð32Þ ^ 4 4k 20 4 k 20 1 r or pffiffiffi P pffiffiffi P . rY 3 3k 20 þ 1 3 3k 20 þ 1 Yielding on the bottom edge cannot occur if pffiffiffi pffiffiffi ^=rY < 2:3Þ for any ^=rY 6 4= 3 ð0:77 < r 4=3 3 6 r value of k0. Since the high cohesive strength solution ^=rY > 0:943k 1 and the minimum value of is for r k1 for yielding is unity, yielding on the bottom edge of the film only has to be considered for the low
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B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
^=rY < 2:3. The limits for cohesive strength case for r yielding to be completely prevented on the bottom surface of the film for both an elastic-perfectly plastic film and a strain hardening film with n = 0.2 (see Appendix A) are shown in Fig. 4. While for the elastic-perfectly plastic film yielding cannot occur at all on the bottom surface of the film for ^=rY < 2:3, yielding of the bottom surface 0:77 < r is possible for a strain hardening film at large curvatures. 2.2.2. Yielding on both surfaces of the film for an elastic-perfectly plastic material If yielding does occur at the bottom edge of the film, it is assumed that the bending stress is as shown in Fig. 5 where the stress on the bottom surface is assumed to be 2 ^. rx1 ¼ pffiffiffi rY þ r 3
ð33Þ
There are now two elastic–plastic interfaces at c1 and c2 and the stress distribution is given by
5
yield only on top surface n=0
3
ˆ
Y
2 1 0 0
2
4
6
8
10
12
k0 Fig. 4. Limits on k0 and r ^=rY for yielding to be confined to the upper section of the film.
2 3
where a is now simply a parameter that gives the stress level. By equating the stress in the two regions at y = c1 a c1 ¼ c2
pffiffiffi ! ^ 3 r 1þ 2 rY ð1 þ aÞ
.
ð35Þ
Using the condition that the total force in the x direction is zero gives pffiffiffi ! ^ c1 3 r 1þ 2 r Y c2
ð1 c1 =c2 Þ c1 ða 1Þ ð1 þ aÞ þ . 2 c2
h ¼1þ c2
The expression for k1 and hence " pffiffiffi 3 h k1 ¼ 1þ 4 ðc2 c1 Þ
ð36Þ
ð37Þ
given in Eq. (26) still holds # ^ h2 r . ¼ 2 rY ðc2 c21 Þ
ð38Þ
The non-dimensional moment is given by " pffiffiffi ! 2 # ^ 2 c1 þ c1 c2 þ c22 3 r 1þ m1 ¼ 2 1 . 3 4 rY h2 ð39Þ
c2
y
c1 Y
ð34Þ
for y > c2 ;
Y
h
2 3
2rY rx ¼ pffiffiffi 3
Substituting for a from Eq. (35) yields pffiffiffi ^ h 3 r ¼1þ . c1 þ c2 4 rY
yield only on top surface n=0.2
4
2rY ^ for y < c1 ; rx ¼ pffiffiffi þ r 3
2rY y rx ¼ pffiffiffi a ð1 þ aÞ for c1 < y < c2 ; c2 3
ˆ
Fig. 5. Elastic and plastic regimes when yielding occurs on both surfaces of the film.
The constrained moment can then be written as pffiffiffi ! ^ 3 r 1þ pffiffiffi !3 2 rY ^ 3 r 1 m1 ¼ pffiffiffi ! 2 1 þ 4 rY 3k 1 ^ 3 r 1þ 4 rY ¼ m0 ¼ 1
1 ; 3k 20
ð40Þ
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
and
3
pffiffiffi !2 ^ 3 r 1þ 4 rY k1 ¼" #1=2 ; p ffiffi ffi k0 ^ 3 r 1þ ð1 þ 3k 20 Þ 4 rY
Yield top surface Yield both surfaces
ˆ
Y
0.1 0.2 0.5 1.0 2.0
2
ð41Þ
ˆ Y 0.1
1
and the rotation factor is given by
1=2 k1 2 rY k1 / ¼ v2 ¼ pffiffiffi k1 ^ k0 k0 3 3 r pffiffiffi #3
1=2 " ^ 3 r 2 rY pffiffiffi 1þ k0 ^ 4 rY 3 3 r ¼ " #3=4 ; pffiffiffi ^ 3 r 1þ ð1 þ 3k 20 Þ 4 rY 2 pffiffiffi !#1=2 pffiffiffi !, ^ ^ 3 3 r 3 r ; for k 0 > 4 1 þ 1 4 rY 4 rY and
577
^ r 4 < pffiffiffi . rY 3 3
0.2
0
10.5 2
0
1
2
3
4
5
k0
6
7
8
9
3
0.1
0.2
2
ˆ
Yield top surface Yield both surfaces
1 2
0
For a constrained film the moment–curvature relationship is given in Appendix A. The rotation factors for both elastic-perfectly plastic and strain hardening for n = 0.2 (see Appendix A) films are shown in Figs. 6 and 7. The rotation factor for the strain hardening film is larger than that for the elastic-perfectly plastic film. The approximate estimation of the root rotation is only ^=rY > 1. accurate for r
Y
0.1 0.2 0.5 1.0 2.0
0.5
ð42Þ
and e are the von Mises equivalent stress where r and strain and n is the strain hardening index. For an unconstrained film the relationship between the non-dimensional moment and curvature is given by (Kim and Aravas, 1988)
2 1n 1 n m¼ k . ð44Þ 2þn 3 k2
11
Fig. 6. Rotation factors for elastic-perfectly plastic (n = 0) film under an end moment using Eq. (30) or Eq. (42) depending upon whether the top or both surfaces of the film yield (markers indicate FE results).
1
2.2.3. Power law hardening material For a strain hardening film the stress–strain relationship is modelled by the power law e r ¼ ; for e < eY ; rY eY n ð43Þ e r ¼ ; for e > eY ; rY eY
10
0
1
2
3
4
5
6
7
8
9
10
11
k0 Fig. 7. Rotation factors for a strain hardening (n = 0.2) film under an end moment using results of Appendix A (markers indicate FE results).
2.3. The root rotation factor obtained by integration of the equations of bending In the estimates of the root rotation factor given in Sections 2.1 and 2.2 it is assumed that the curvature is constant over the cohesive zone. A more exact estimate can be obtained by numerical integration of the equations of bending. For the purpose of calculating the root rotation, h0, the bonded section of the film is divided into two regions (see Fig. 8): (a) Region A far from the tip of the bond where the transverse stress on the bottom of the film ^ and the curvature k < 1. Depending r0 < r upon the cohesive strength either of these inequalities is the critical one that defines region A. In this region x is measured to the right as shown in Fig. 8.
578
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
Region B x
v
If initially k < 1 in region B, the integration could be performed analytically until k = 1, but it is found more convenient always to use numerical integration in region B. Two assumptions are made for the elastic–plastic bending when k > 1:
Region A x
ˆh ˆh and k 0 1 or k0 1 v < 2E 2E
M0
(a) The film is unconstrained and the non-dimensional moment m is given by Eq. (44). (b) The film is constrained as described in Section 2.2 and Appendix A.
v
root of delamination
Fig. 8. Schematic displacement of the centre line of the film.
(b) Region B near the tip of the bond where either of these inequalities is violated. In this region x is measured to the left (see Fig. 8). The general equation of bending in both regions A and B is
A Fortran programme performs the numerical integration of Eq. (45) in region B using the values va, ha, ka, and sa obtained from the solution of region A and calculates /. Dimensional analysis shows that the root rotation, h0, is given by the functional relationship r=rY Þ. h0 ¼ eY f ½k 0 ; n; ð^ Hence the rotation factor
2
dM ¼ r0 ; dx2 pffiffiffi r0 d2 m ¼ 2 3 ; 2 dx rY
/ ¼ h0 ð45Þ
where x ¼ x=h. The transverse stiffness of the film is assumed to be given by Eq. (7) and the transverse stress, r0, acting on the bottom of the film is given by v v ^; r0 ¼ 2E ; for 2E < r h h ð46Þ v ^. ^; for 2E > r r0 ¼ r h In region A the beam is elastic and the equation of bending becomes
1.6
v ¼ expðbxÞ½A cos bx þ B sin bx;
ð48Þ
0.8
and x ¼ x=h is the non-dimensional h2i d v distance. At x ¼ 0 the curvature k a ¼ d and x2 x¼0 h3i d v its derivative sa ¼ d are specified giving B = x3
0.6
where b = 6
x¼0
ka/2b3, A = sa/2b2 B. The deflection and slope at x ¼ 0 are then given by va ¼ ½vx¼0 ¼ A; ha ¼ v d ¼ ðB AÞb. These values are the starting dx x¼0 values for a numerical integration into region B.
R0 R 0 Rp 1 ¼ h0 ¼ pffiffiffi f ½k 0 ; n; ð^ r=rY Þ; h Rp h 3k 0
ð50Þ
is independent of the yield strain, but in the programme it was more convenient to introduce eY though the result is not dependent on it. The rotation factor, /, estimated by the constrained and unconstrained yield models is shown as a function of k0 for a range of non-dimensional ^=rY , in Figs. 9 and 10 together cohesive strengths, r with finite element results (see Appendix B). As is expected the values for the unconstrained yielding are greater than those for constrained yielding, but the difference is not that great. The unconstrained yielding model, which agrees best with the finite ele-
d4v ¼ 24v; ð47Þ dx4 where v ¼ v=h. The general solution of Eq. (47) for the non-dimensional deflection, is given by 1/4
ð49Þ
Unconstrained yield Constrained yield
1.4 1.2 1
0.4 0.2 0 0
ˆ
0.1 Y
0.1 0. 2 0.5 1.0 2. 0
0.10.2 1 0.20.5 20.5 12
2
4
6
8
10
12
k0 Fig. 9. Rotation factors for an elastic-perfectly plastic (n = 0) film, with an end moment, from integration of the bending equation compared with FE results shown by markers.
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584 1.6
1.2
0.1 0.1
1
a = 3h 0.2
0.8
P 0.2
0.6 0.4 0.2 0 0
M Mmax
Unconstrained yield Constrained yield
1.4
579
x
0.5
ˆ
Y
1 2 0.5 1 2
0.1 0.2 0.5 1.0 2.0
2
4
6
8
10
12
k0 Fig. 10. Rotation factors for strain hardening (n = 0.2) film, with an end moment, from integration of the bending equation compared with FE results shown by markers.
5
ˆ
ˆ
Y
0.1 0.2 0.5 1.0 2.0
4 3
0.5
1
0
Y
0.2
1.0 2.0
0
moment even if shear deformation in the film is neglected. The difference arises because unlike in the end moment case, the bending moment at first increases along the bonded portion of the film. If an end force, P, is applied at a distance a from the tip of the crack, the bending moment, M, in the cohesive zone is given by
0.1
2
2
4
6
8
10
Fig. 12. Bending moment distribution near the delamination tip under an end moment.
^ M ¼ P ða þ xÞ r 12
k0 Fig. 11. Rotation factors for an elastic film (n = 1) under an end moment from integration of the bending equation (markers indicate FE results).
x2 ; 2
ð51Þ
and the moment reaches a maximum, Mmax, at x ¼ P =^ r, given by
P2 M0 ¼ M0 1 þ M max ¼ Pa þ ; 2^ r 2^ ra " # ð52Þ 2 m0 h rY . or m1 ¼ mmax ¼ m0 1 þ pffiffiffi ^ r 4 3 a
ment results, overestimates and the constrained model underestimates the rotation factor for k0 > 4. For 1 < k0 < 4 the rotation factor is underestimated by the unconstrained model. As the strain hardening index increases so the relative difference between the rotation factors calculated assuming that the bending on the bonded section of the film is constrained or not decreases until at n = 1, an elastic film, there is no difference. The rotation factors for an elastic film are shown in Fig. 11 where the concept of artificial yield strength has been retained so that direct comparison can be made with Figs. 9 and 10. The approximate rotation factors agree well with the finite element results except for small k0.
In the absence of constraint, an elastic-perfectly plastic film cannot sustain a bending moment greater than the collapse moment, Mc, and hence the rotation factor becomes infinite when mmax = 1. The values of the non-dimensional curvature at the crack tip at which plastic collapse occurs in the film, for a/h = 3, is given in Table 1. Yield constraint in the bonded region prevents plastic collapse. If the cohe^, is large then Mmax is only slightly sive strength, r greater than M0 and the effect of the shear force is small for a constrained film. However, if the cohesive
3. Shear force effects
Table 1 Non-dimensional curvature, k0, at which plastic collapse occurs in an unconstrained film for a/h = 3
The root rotation factor is somewhat different for an end force on the film (see Fig. 12) than an end
^=rY r k0 for plastic collapse
0.1 1.64
0.2 2.19
0.5 3.32
1.0 4.63
2.0 6.50
580
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584 4
3 end force end moment 0.1
2
0.1
3.5
0.1 0.2
ˆ
Unconstrained yield Constrained yield
3 2.5
0.2
ˆ
1
0
Y
0.1
0
1
2
3
4
5
k0
6
7
8
9
0.2 0.5 2 1 10 11
Fig. 13. Comparison of the rotation factors for an end force using Eq. (52) and for an end moment using Eq. (30) or Eq. (42) depending upon whether the top or both surfaces of the film yield for an elastic-perfectly plastic film.
Y
0.1 0.2 0.5 1.0 2.0
0.5
2 1.5 1
1
2
0.5
0.2 0.5 1 2
0 0
2
4
6
8
10
12
k0 Fig. 14. Rotation factors for an elastic-perfectly plastic (n = 0) film, with an end force (with a/h = 3), from integration of the bending equation compared with FE results shown by markers.
strength is small then the shear force effect can be significant. The root rotation factor, for a constrained film, estimated from the curvature near to the crack tip would be underestimated from the curvature at the crack tip. An upper bound to the root rotation factor can be obtained if the curvature corresponding to the curvature for the maximum moment, Mmax, is used that enables the effect of the shear forces to be judged. The difference between the rotation factor for an end force (with a/h = 3) and an end moment is shown in Fig. 13 for an elasticperfectly plastic film. Although there are very signif^=rY < 0:5, there are only icant differences when r ^=rY P 0:5. slight differences for r
Fig. 15. Rotation factors for strain hardening (n = 0.2) film, with an end force (with a/h = 3), from integration of the bending equation compared with FE results shown by markers.
3.1. The root rotation factor obtained by integration of the equations of bending
4. Conclusions
The root rotation factor for an end force (with a/h = 3) has been obtained by integration of the equations of bending as explained in Section 2.3. This root rotation factor is compared in Figs. 14 and 15 with the FE results. For a strain hardening exponent n = 0.2 the unconstrained model generally overestimates the root rotation factor and the constrained model underestimates it much the same as in the end moment case with the unconstrained model generally giving the more accurate result. For an elastic-perfectly plastic film, the constrained yield model is best which is not surprising since the unconstrained model predicts infinite rotation factors as k0 approaches the values for plastic collapse given in Table 1, but again the constrained model still underestimates the rotation factor.
The root rotation factor for an end moment acting on an isotropic elastic film bonded to a rigid substrate with an infinite cohesive strength, which from finite element analysis is given as / = 0.66 (Williams and Hadavinia, 2002b), is given by / = 0.639 from the approximate bending theory. The cohesive zone becomes longer as the cohesive strength decreases and the end moment increases causing the root rotation factor to increase—for low cohesive strength the increase is large (see Fig. 11). The numerical integration of the equations of bending (see Section 2.3) gives a good approximation to the elastic root rotation factor obtained from finite element analysis (see Fig. 11). The estimates of the rotation factor obtained from simple geometric considerations are summa-
3.5
ˆ
0.1
3
Unconstrained yield Constrained yield
Y
0.1 0.2 0.5 1.0 2.0
2.5
0.2
2 1.5 1
0.5
0.1
1 0.2 2 0.5 1 2
0.5 0 0
2
4
6
8
10
12
k0
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584 Table 2 Rotation factors obtained from simple geometric considerations Normalized cohesive ^=rY strength r
Root rotation factor / k1 0:64 k 0 1=2 3=2 k1 ^ r 0:62 rY k0
>0.943 <0.943
Equation number (9) (13)
rized in Table 2 in terms of the ratio of the curvatures in the constrained and free film at the root. Upper and lower bounds to the ratio of the curvatures are summarized in Table 3. Estimates of the ratio of the curvatures for an elastic-perfectly plastic film are summarized in Table 4. Plastic deformation reduces the root rotation factor (see Figs. 6, 7, 9 and 10). The assumption that the curvature in the cohesive zone can be considered constant and the constrained yield model applied, does not predict the root rotation factor for an end moment well except for high cohesive strength (see Figs. 6 and 7). However, numerical integration of the equations of bending does predict the root rotation factor for an end moment reasonably well, but the unconstrained model is better than the constrained model (see Figs. 9 and 10). The rotation factor for an end load (with a = 3h) is very similar to that of an end moment if the coheTable 3 Upper and lower bounds to the k1/k0 ratio (see Eq. (18)) Bound
k0 range
k1/k0
Lower
k0 < 1
1
Upper
k0 > 1
" # 1:5 1 1 2 k0 3k 0
All k0
1
Table 4 Constrained yield estimates of the k1/k0 ratio for elastic perfectly plastic films Yield condition
Top surface only pffiffiffiffiffi ^ r k 1 P 0:433 r Y pffiffiffiffiffi P k1 1 Both surfaces pffiffiffiffiffi ^ r k 1 < 0:433 ; rY pffiffiffiffiffi r ^ or k 1 > 0:433 þ1 rY
k1/k0 16k 30
Equation number
(30)
2 1 þ 3k 20 ^ 2 r 1 þ 0:433 rY
(42) 1=2 ^ r 1 þ 0:433 1 þ 3k 20 rY
581
sive strength is high ð^ r=rY P 0:5Þ, but is much larger for low cohesive strength (see Fig. 13). Numerical integration of the equations of bending predicts the end load root rotation factor well when the unconstrained yield model is used for n = 0.2 (see Fig. 15). However for an elastic–plastic film, since there is plastic collapse away from the tip of the delamination at high curvatures, the constrained model is best (see Fig. 14). In summary, a reasonable accurate estimate of the root rotation factor can be obtained by the integration of the equation of bending for an end moment or load. Generally the unconstrained yield model gives the best result, but if the strain hardening exponent is small or the ratio a/h is small then the constrained yield model is best. Since in peeling a/h is small, the constrained yield model is likely to be the more accurate. In a future paper the approximate theory developed here will be used to model peeling. Appendix A. Modelling the constraint for a strain-hardening film The strain in the elastic and upper plastic region is assumed to be the same as in the elastic-perfectly plastic case and is still given by Eq. (22) and the non-dimensional curvature, k1, is given by Eq. (26). Neglecting transverse stress in the upper section of the film, the equivalent von Mises strain e is given by
y e ¼ eY a ð1 þ aÞ . ðA:1Þ c2 If there is no yielding on the bottom surface the stresses are given by
2 y rx ¼ pffiffiffi rY a ð1 aÞ ; for y < c2 ; c2 3
n 2 y rx ¼ pffiffiffi rY ð1 aÞ a ; for y > c2 . c2 3 ðA:2Þ Using the condition that the totals longitudinal force must be zero yields " # 1
nþ1 h 1 nþ1 2 aþ 1þ ¼ ða 1Þ . c2 ð1 þ aÞ 2 ðA:3Þ Integrating the moment of the stresses about the bottom of the film gives the non-dimensional moment m1
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
( " # c 2 ð2 aÞ ðh=c2 Þ½ð1 þ aÞðh=c2 Þ a1þn 1 2 m1 ¼ 4 þ h 6 ð1 þ nÞð1 þ aÞ " #) 2þn ½ð1 þ aÞðh=c2 Þ a 1 . ðA:4Þ ð1 þ nÞð2 þ nÞð1 þ aÞ2
This moment must equal m0, the non-dimensional moment on the free unconstrained film which is given by Eq. (44). The constrained curvature, k1, cannot be found explicitly, but can be calculated from Eqs. (26), (44), (A.3), (A.4) in terms of k0. The rotation factor can then be found from Eqs. (9) or (13) depending on condition (15). The bottom surface will not yield if the conditions (31) are met. Unlike the case for an elastic-perfectly plastic film, yielding of the bottom surface of the film can occur for any cohesive stress, ^, provided the applied moment is large enough (see r Fig. 4). If the film yields on its bottom surface to a depth c2 then the stresses above the lower yielded section y > c1, are given by Eq. (A.2).h Because the itransverse stress is neglected, e ¼ eY a ð1 þ aÞ cc12 > eY
at y = c2. To avoid this inconsistency the somewhat arbitrary assumption that the variation in the equivalent strain within the bottom plastic zone is the same as in the upper section of the film but, for continuity in the stress, the equivalent strain at the elastic–plastic boundary is the yield strain eY. Hence it is assumed
ð1 þ aÞðc1 yÞ e ¼ eY 1 þ ðA:5Þ ; for y < c1 ; c2 and the stress given by 2 r ^ rx ¼ pffiffiffi þ r 3
n 2rY ð1 þ aÞðc1 yÞ p ffiffi ffi ^; ¼ þr 1þ c2 3
for y < c1 . ðA:6Þ
The ratio of the depths of yielding c1/c2 is still given by Eq. (35). The condition that the total force must be zero gives ((
ð1þnÞ h 1 c1 1 þ ð1 þ aÞ ¼ c2 1 þ a c2 ð1 þ aÞð1 þ nÞ c1 c1 1 þ a 1 ð1 þ aÞ 2 c2 c2 )
)ð1=1þnÞ pffiffiffi r ^ c1 þ 3 þa . ðA:7Þ rY c2
Integration of the moment of the stresses gives ( 2 m1 h ¼ 4 c2
þ
)
ð1þnÞ c1 h h þ ð1 þ aÞ a 1 c2 c2 c2
(
ð1 þ aÞð1 þ nÞ
ð2þnÞ )
ð2þnÞ c1 h 2 1 þ ð1 þ aÞ ð1 þ aÞ a c2 c2
2
ð1 þ aÞ ð1 þ nÞð2 þ nÞ " 3 # 2 #) 1 c1 c1 2ð1 þ aÞ 1 þ 3a 1 6 c2 c2 pffiffiffi 2 ^ c1 3 r . ðA:8Þ 4 rY c2 (
"
This moment must be equal to the applied moment m0 given by Eq. (44) and k1 can be found as a function of k0. The expression for the rotation factor if the bottom surface of the film yields can then be found. Appendix B. Finite element analysis The finite element analysis was applied to a film of thickness 1 mm. The free section of the film was 3 mm long. Four nodes plane strain elements were used with incompatible mode to improve the bending behaviour. The elements in the free section of the film and for a distance of 3 mm from the root were 0.1 · 0.1 mm2 elements with larger elements further from the root (see Fig. B1). The Young’s modulus was taken as 100 GPa, Poisson’s ratio 0.3 and the yield strength of 100 MPa. Three material behaviours were considered: elastic, elastic-perfectly plastic, and elastic-strain hardening plastic as described by Eq. (43) with n = 0.2. Five values of ^=rY ¼ 0:1, 0.2, 0.5, 1.0, 2.0 were used. the ratio r A Tvergaard/Hutchinson type cohesive zone (Tvergaard and Hutchinson, 1992) is inserted to represent the bond between the film and the substrate. The substrate was constrained in the vertical direction but no restraint was imposed on the horizontal 0. 1mm x 0.1 mm mesh 3 mm
3 mm
1 mm
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cohesive zone
Fig. B1. Schematic arrangement of FE model.
B. Cotterell et al. / Mechanics of Materials 38 (2006) 571–584
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order term. A second method was examined for the end moment case which makes use of the equivalent work concept. If M0 is the end moment and the dh0 is the increment in the root rotation for an increment dM0 in the end moment, then the increment in work done must be given by Z h ½r0x du0 þ s0xy dv0 dy; ðB:1Þ M 0 dh0 ¼
σ
σˆ
0
δ -5
10 mm Fig. B2. Traction separation relationship.
direction (see Fig. B1). Hence the problem can be viewed as two identical films bonded back to back with symmetrical loading. Because of the freedom in the horizontal direction, there is no shear stress in the cohesive zone and the traction–separation relationship, shown in Fig. B2, can be defined in terms of the normal stress and normal displacement alone. Note that in this paper we were not concerned with fracture so that the right hand end of the usual traction–separation relationship was not defined. The traction separation relationship is slightly different to that used in the approximate analyses since it has an ‘‘elastic’’ loading portion for computational ease. However this loading portion has been minimised and does not affect the results. Two load cases were considered. In the first case the beam was subjected to an end moment, which is accomplished by giving the nodes of the left hand end of the film a rotation. Over the central part of the free section of the film the stress distribution is the same as for a pure bending moment. In the second case a vertical force, P, is applied to the left hand end of the beam which was distributed over the nodes so that all the nodes other than the outside nodes had an equal force and the two outside nodes had half of this value. The root rotation (h0) and the root curvature (1/R0) are directly obtainable from the approximate solution using the engineers’ theory of bending, but they are not directly obtainable from the FE analysis. Two methods of determining the root rotation (h0) were examined. In the first method, the displacement profile for the centre line of the film relative to the displacement at the root section, v(x) v(0), was obtained over a distance 0.5 on either side of the root section. A polynomial was fitted to this curve and the root rotation taken as the coefficient of the first
where the stresses and displacements are those at the root section. The root rotation is therefore given as # Z M 0Z h " 0 s0xy rx h0 ¼ du0 þ dv0 dy dM 0 . ðB:2Þ M0 M0 0 0 In the engineers’ theory of bending the shear stress s0xy is zero and this term is identically zero and it is interesting that in the FE solution the second term in Eq. (B.1) is negligible. The difference between the two methods was found to be slight and the method used to obtain the root rotation factor is the slope method. Since the purpose of obtaining the root rotation factor is to enable it to be used as a boundary condition in cases where the free film is modelled by the engineers’ theory of bending, the normalising curvature has been taken as the curvature in a free film. In the end moment case the curvature is found by plotting the strain, ex, as a function of y across a section 1.5h from the delamination root. The relationship is linear and the curvature, 1/R0, is its slope. In the end load case a separate FE problem is analysed where a free film of length 6h has a bending moment of 3Ph applied at both ends and the curvature at the centre of the film obtained from the strain, ex, across the central section. References Blackburn, B.R.K., Hadavinia, H., Kinloch, A.J., Williams, J.G., 2003. The use of cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints. Int. J. Fract. 119, 25–46. Georgiou, I., Hadavinia, H., Ivankovic, A., Kinloch, A.J., Tropsa, V., Williams, J.G., 2003. Cohesive zone models and the plastically-deforming peel test. J. Adhesion 79, 239–265. Kawashita, L.F., Moore, D.R., Williams, J.G., 2004. The development of a mandrel peel test for the measurement of adhesive fracture toughness of epoxy-metal laminates. J. Adhesion 80, 147–167. Kim, K.-S., Aravas, N., 1988. Elastic plastic analysis of the peel test. Int. J. Solids Struct. 24, 417–435. Kinloch, A.J., Lau, C.C., Williams, J.G., 1994. The peeling of flexible laminates. Int. J. Fract. 66, 45–70.
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Tvergaard, V., Hutchinson, J.W., 1992. The relationship between crack growth resistance and fracture process parameters in elastic–plastic solids. J. Mech. Phys. Solids 40, 1377–1397. Wei, Y., Hutchinson, J.W., 1998. Interface strength, work of adhesion and plasticity in the peel test. Int. J. Fract. 93, 315– 333. Williams, J.G., 1993. A review of the determination of energy release rates for strips in tension and bending. Part I—Static solutions. J. Strain Anal. 28, 237–246.
Williams, J.G., Hadavinia, H., 2002a. Analytical solutions for cohesive zone models. J. Mech. Phys. Solids 50, 809–825. Williams, J.G., Hadavinia, H., 2002b. Elastic and elastic–plastic correction factors for DCB specimens14th European Conference on Fracture—ECF 14 Fracture Mechanics Beyond 2000, vol. 3. EMAS Publishing, Cracow, Poland, pp. 573–592. Williams, J.G., Hadavinia, H., Cotterell, B., 2005. Anisotropic elastic and elastic bending solutions for edge constrained beams. Int. J. Solids Struct. 42, 4927–4946.