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Determination of the energy dissipated during peel testing
Materials Science and Engineering A302 (2001) 168– 180 www.elsevier.com/locate/msea
Determination of the energy dissipated during peel testing Edith Breslauer a,*, Tom Troczynski b b
a Rafael, P.O. Box 2250, Haifa, Israel Metals and Materials Engineering, The Uni6ersity of British Columbia, 309 – 6350 Stores Rd, Vancou6er, BC, Canada V6T 1Z4
Abstract During the experimental determination of adhesion strength through a peel test, the deformation energy of the peeled foil, plus frictional losses may constitute a significant portion of the total peel energy. When the true adhesion strength is desired, the energy dissipated in bending of the foil has to be found, whether experimentally or analytically, and deducted from the total peel energy. Several works were dedicated to the modeling and calculation of the energy dissipated through plastic deformation so that the net adhesion energy could be deduced. Recently, the present authors proposed a simple experimental technique for the determination of the net adhesion strength. The method is not limited to any particular material and can be used successfully for a strain hardening plastic as well as metallic foils. In this work the calculation of the various energy terms dissipated during the peel test is carried out, in order to validate the results of the experimental method and to study the effect of the test variables, such as geometry and mechanical properties of the flexible foil. The main parameter that influences the energy expended during the test is the radius of curvature of the flexible adherend. It is found that the ratio of the total energy to the adhesion energy versus this radius of curvature behaves in a complex manner. For a low strength material, the ratio increases with the radius of curvature. For high strength materials, the ratio exhibits a critical radius, at which the ratio is maximal. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Peel adhesion test; Metallic foils; Plastic deformation energy dissipation
1. Introduction The peel tests can be classified into free peeling, Fig. 1A, or peeling around a mandrel, Fig. 1B. Two of the common ASTM standard methods [1,2] belong to the second category. These mandrel-assisted methods are frequently used for comparative tests because the measured peel force includes not only the force necessary to separate the two bodies, but also the forces required to bend the flexible foil and to overcome friction. These additional forces are generally unknown (or neglected), so in order to measure absolute adhesion in the peel test it is necessary to first determine the energy dissipated in bending the foil and in friction. The first to analyze the peel test was Spies [3]. The mechanics of the test and calculation of the plastic energy dissipated during peeling were later carried out by several researchers [4 – 7]. Much of the previous * Corresponding author. E-mail addresses: [email protected] [email protected] (T. Troczynski).
(E.
Breslauer),
theoretical work related to adhesion measurement through the peel test has dealt with the calculation of the plastic energy dissipation (e.g. [4,5]) or its experimental determination [8] using the free peel test (Fig. 1A). In this test, the flexible foil is pulled with force F at a predetermined angle , with respect to the rigid substrate surface. The curvature of the foil is a function of 0, which is the angle at the root of the interfacial crack. Models for the calculation of 0 and the plastic energy had been proposed by several groups [6,7]. Due to the complex nature of these calculations, they have been carried out mainly for perfectly plastic materials [6]. Kinloch et al. have also presented a solution for bi-linear materials [7]. Recently, the peel test had been modified for the determination of the adhesion strength of brittle ceramic coatings, in particular thermal spray coatings [9,10]. The details of the test are specified in [9–11], so only a short description will be given. In the test arrangement, Fig. 2, a 0.1 –0.2 mm thick metallic foil is coated with a thermal spray coating. The coated face of the foil is then adhesively bonded to an aluminum
0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 3 7 0 - 8
E. Breslauer, T. Troczynski / Materials Science and Engineering A302 (2001) 168–180
plate. The foil is now peeled with force F at 90° from the coating using a mandrel, around which the foil is bent. An alignment weight D must be applied to keep the structure in a fixed orientation and to obtain good and repeatable conformance of the foil to the mandrel surface. It was established through modeling and experiment that the plastic energy of deformation of the metallic foil, Wp, was a significant fraction of the total peel work in the above test [11]. The plastic deformation energy dissipated during the peel test varies, depending on the test geometry and the mechanical properties of the peeled material. This may partially explain the large discrepancy in the reported values of the peel strength. For example, the reported peel strength of similar polymeric adhesives ranged from about 0.5 to 10 kJ/m2 [12 –14], depending on the peeling material, its thickness and whether or not plastic energy dissipated to deform the peeled material was accounted for. In order to be able to compare the results, the additional energy dissipated during the peel test (e.g. the plastic deformation energy for metallic foils) has to be determined. In the following section, the energy and force balance during the peel test is outlined and a simple experimental method is proposed for the determination of the net
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adhesion strength in the mandrel-assisted peel. This method was reported previously [15], so only the basics of the method will be outlined. Using the proposed methodology there is no need to calculate or otherwise determine the energy dissipation during the process of peeling, in particular the frictional effects or the energy of plastic deformation of the peeled foils. The method is equally valid for any material that can dissipate part of the total peel energy. In the fourth section of the paper, an analytical calculation of the various energy terms dissipated during the peel test, based on small displacement beam theory, is presented. The results allow examining the effect of test variables on the different terms of peel energy. Thus, an optimal test can be designed.
2. Experimental determination of peel adhesion strength The total energy dissipated during the peel test (Utot), when the system has reached a steady state, can be divided into three categories: The work to separate the two materials (UI), the work to overcome jig friction and alignment weight (Uf) and the work of bending the foil (Ub). This is stated as: Utot = UI + Uf + Ub
(1)
Peel strength is usually reported as the force (F) per unit width (w) required to separate the foil. Peel strength (F/w) is thus related to energy per unit area by: F Utot = w w·dx
Fig. 1. The free peel test (A) and the mandrel assisted peeling (B).
where w is the peeling arm width and dx represents an element along the foil, Fig. 2. In a typical test the total peel force F is the principal parameter monitored during the experiment. If the energy balance of Eq. (1) is replaced by a force balance for a test performed at an angle b= 90° and for a strip of width w, F is the sum of the force required to separate the interface wG (where G= UI/wdx is the adhesion fracture energy rate), the alignment weight D, the friction force f ( f= Uf/dx), and the force required to perform bending of the foil per length peeled (Fp = Ub/dx). Neglecting the elastic energy stored in the foil under the force F, the peel force balance can now be written: F=wG + D+ f+Fp
Fig. 2. The peel adhesion test applied to a thermal sprayed coating.
(2)
(3)
If a free, non-adhering foil is deformed around the same mandrel in a calibration experiment, Eq. (1) still holds, with the condition of G=0. The friction in the jig (including the friction between the foil and mandrel and around the bearings), f is assumed to be proportional to the total force, f= vF (where v is a proportionality coefficient or an average coefficient of friction
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Fig. 3. Variation of the force F measured in the peel test vs the alignment weight for zero interfacial adhesion in the calibration run (A) and in peeling for the case of finite interfacial adhesion (B).
for the system). The plastic work in bending Ub is proportional to the width of the sample w, so Fp = Ub/ dx = wWp. The deformation of a peeled foil can thus be described by a simple modification of Eq. (1), for the case of zero interfacial adhesion and finite interfacial adhesion G, respectively: F(1−v)= D1 + wWp
(4a)
F(1 − v)= D2 + w(G + Wp)
(4b)
When the total peel force F increases, due to higher values of G, D2 or v, the radius of curvature of the foil will decrease, causing an increase in the plastic energy Wp. The increase in Wp will cause a further increase in F and so on, until a steady state is reached. Under steady state conditions, for a constant G, Wp is a function of F only and it varies from zero for pure elastic deformation of the foil, to some limiting value Wp0(F), in accordance with the limiting strain imposed by the mandrel. Determination of the work Wp(F) necessary to plastically deform the peeled foil is one of the experimental difficulties of the peel test. The methodology proposed below allows to eliminate the uncertainty related to the magnitude of Wp(F) by analyzing the behavior of the deforming foil. If the calibration test described by Eq. (4a) is repeated for various alignment weights, the peel force F/w can be plotted versus the alignment weight D/w, as schematically shown in Fig. 3A. As mentioned above, the amount of energy dissipated by plastic deformation of the bent foil, Wp(F), increases as a function of the total force F. Accordingly, the curve in Fig. 3A illustrates three distinctive regions related to different magnitudes of plastically dissipated work Wp(F). In region I, the bent foil undergoes purely elastic deformation and there is no energy dissipated by plastic deformation of the foil, i.e. Fp =wWp(F) = 0. In this region, the peel force F is a sum of only the alignment weight D and the friction force f. The curve is linear and the slope depends on the average friction coefficient only, therefore, the slope of F/w line has a constant value of
(1− v) − 1. At the beginning of region II, parts of the bent foil that experience the highest strain (i.e. the surface of the foil) exceed the yield point and start to deform plastically. The total peel force FT at the point of transition from region I to region II can be estimated by considering simple bending of the foil. The details were given previously [15]. Wp(F) starts to increase with increase of the alignment weight D/w. The slope of the F/w vs D/w curve increases due to additional energy required for plastic deformation through the thickness of the foil. In this region the shape of the F/w curve depends on the shape of the stress –strain curve of the material (i.e. the yield strength and the strain hardening coefficients) and the foil thickness. Region II would be less extensive, as a function of D, for thinner foils and more extensive for thicker foils. For thinner foils (thickness B 0.1 mm) or for low yield stress metals (e.g. Al) regions I and II may be difficult to discriminate on the F/w curve because of the relatively small contribution of the plastic deformation energy to the total energy. The measured force at the transition from region I to region II is characteristic of the material and the test geometry. At the end of region II the radius of curvature of the foil reaches the value of the radius of the mandrel and the amount of energy dissipated plastically becomes constant as a function of total peel force, i.e. Wp(F)= constant= Wp0(F). Consequently, the slope of F/w line returns to the constant value of (1− v) − 1. The intercept of the extrapolated region III line with F/w axis at D/w=0 gives the value of Wp0 /(1− v). The friction coefficient v can be determined from this slope and thus Wp0 can be determined. For thin (or low yield stress) foils the three regions will merge into an almost straight line (with a minimal Wp0/(1− v) cutoff length). Fig. 3B represents the measured force in an actual peel test, i.e. for the case of non-zero adhesion, Eq. (4b) [the curve for Eq. (5)a is also included in this figure, for comparison]. The peel force curve now shifts upward by the value of G/(1− v), or equivalently to the left by the value of G. Similarly as in Fig. 3A, the intercept of the extrapolated region III line with F/w
E. Breslauer, T. Troczynski / Materials Science and Engineering A302 (2001) 168–180
axis at D/w =0 determines (G + Wp0)/(1 −v), while the intercept of region I line with F/w =0 line determines the net interfacial adhesion energy G. The constant slope (1− v) − 1 of the lines in regions I and III determines the friction coefficient v. In order to be able to extrapolate the peel force curve F/w to low loads, a calibration curve must be obtained, using a free foil (i.e. for G =0), as illustrated in Fig. 3A. The peel force curve F/w during peeling with non-zero adhesion (G \ 0) will be similar in shape, but
Fig. 4. The steady state deformation of the peeling adherend.
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shifted by the value of the interfacial adhesion energy G, as shown in Fig. 3B. The confirmation that the values of the peel strength obtained using the above procedure are indeed the true net interfacial adhesion strength can be achieved by testing foils of the same material (and same interfacial adhesion strength) but of different thickness. The plastic deformation energy will, obviously be different for each thickness, whereas the interfacial adhesion strength should be similar, within experimental error. Experimental results illustrating this procedure were presented in a previous publication [15]. As the amount of the plastic deformation energy spent to deform the foil of constant geometry is a function of the total peel force only F, subtraction of Eq. (4a) from Eq. (4b) at constant peel force F, renders wG = D1 − D2. The two alignment weights D1 and D2 are those necessary to produce the identical peel force F to separate the adhesion interface and to deform the foil with adhesion (wG = D1 − D2 \ 0) and with no adhesion (G= 0) respectively. Consequently, horizontal shift between the two curves at any value of force F/w can be used to determine the net interfacial adhesion strength G (Fig. 3B), with no need to know the friction coefficient or the amount of energy dissipated by plastic deformation. It should be ensured, however, that the calibration test on non-adhering sample, Fig. 3A, is performed under the same friction conditions and using a foil with the same width, thickness and properties as those for the peeling of adhering foil, Fig. 3B. This method is suitable for determination of the net interfacial adhesion energy G when the peel test is performed under one alignment weight only, or for the calibration test where a range of alignment weights have been used.
3. Analytical calculation of the energy dissipated in a peel test
Fig. 5. The stress– strain curve for an element dz, at depth z, with elastic reverse bending (A) and plastic reverse bending (B).
In order to study the effect of the peel test variables, such as the thickness and mechanical properties of the flexible adherend or the angle, at which the adherend is pulled, it is beneficial to calculate the energy dissipated during the test. It should be noted that the following treatment applies generally, for a free peel as well as for a mandrel assisted test. Fig. 4 shows the configuration of the deformed adherend, during a steady state peel test. The radius of curvature r, during the test is a function of the total peel force and the properties of the adherend. For a large radius of curvature, all elements of the foil experience elastic strain only. At some critical radius, the outer surface of the adherend starts to strain plastically, and gradually (i.e. if the radius further decreases) the whole cross section of the adherend will experience plastic strain. Fig. 5 presents schematically the stress –strain curve for an element dz, at depth
E. Breslauer, T. Troczynski / Materials Science and Engineering A302 (2001) 168–180
172
z (z is 0 at the neutral axis and it equals H/2 at the foil surface). This stress – strain curve is dependent on whether the reverse bending is elastic (Fig. 5A) or plastic (Fig. 5B). The various points marked on the curve correspond to the points marked on the adherend in Fig. 4. The element dz is loaded up to the maximum strain or minimum radius of curvature, which corresponds to point B and then unloaded, when the adherend straightens up, to point D, where the strain is zero, passing through point C. At point D, there is a residual elastic strain energy stored in the foil. If the force is released, for the case of plastic reverse bending, the corresponding point on the curve will be point E (Fig. 5B). The energy balance during the peel test, assuming a rigid substrate, can be expressed as follows: Wt = G+Wp +Wr
(5)
where Wt = total expended energy; G = adhesion energy; Wp =plastic energy; Wr =residual elastic strain energy. The dissipated energy, Wp for each element, is found from the area under the stress – strain curve, i.e. area OBCD. The total energy, Wt for the same element includes the dissipated energy, the residual elastic strain energy and the complementary energy, which is found from area OAB, i.e. the total energy is found from area OABCD [7]: wWt =F[1−cos( − 0)] =area [OABCD]
(7)
& & n z
m
|dm dz
0
zy = zmy = z|y/E Where |y is the yield strength and E is the elastic modulus. If zy ] H/2 (H is the foil thickness), then the full thickness is under elastic stress and there is no plastic energy dissipation. If zy B H/2, the element dz for which z \zy will go into plastic deformation and energy will be dissipated. Elements for which z5 zy do not contribute to the plastic energy dissipation. For elements for which z\ zy, the plastic energy dissipated can be calculated by integrating the stress – strain curve over the appropriate limits, as seen in Fig. 5. The integration can be done numerically or by correlating the curve with a suitable expression. For work hardening materials a power-law model can be m used [16], where for \ 1: my | = Emy
m my
N
(9)
Alternatively, a bi-linear model could be used [7], havm ing for \ 1: my
|=Emy (1−h)+h
n m my
(10)
Where aE is the slope of the stress –strain line, beyond the yield strength. The material can also be modeled as a perfectly plastic material. For elements for which z \zy, the total dissipated energy is:
& & & H/2
Wp = 2
|dm dz
zy
H/2
=
(U1 + U2 − U3 + U4 + U5)dz
zy
Relating to Fig. 4, the equations for the energy terms will be derived as a function of z and the radius of curvature, z. Integration over z is then performed to get the final values of the various energy terms. The total energy dissipated may be expressed by the area under the stress – strain curve: Wp =2
The z at which the stress corresponds to the yield strength is:
(6)
F is the peel force. and 0 were defined in Fig. 1. The adhesion energy, the test geometry and the mechanical properties of the foil determine the required peeling force and a corresponding radius of curvature, z at the root of the peeled material. This radius determines the amount of plastic energy dissipated in bending. In order to find the adhesion energy, the dissipated energy has to be calculated and deducted from the total energy expended. After integration over z, the adhesion energy, G, can then be found from Eq. (5) as a function of the radius of curvature, z: G(z)=Wt(z)−Wp(z) − Wr(z)
m= z/z
(8)
0
Note that the area is multiplied by 2, as z is measured from the neutral axis of the foil (z =0) to the surface (z =H/2). Where the strain, o at z is:
= 2(Wp1 + Wp2 − Wp3 + Wp4 + Wp5)
(11)
The different energy terms will be derived below: Wp1 =
& &
H/2
U1dz =
zy
Wp2 =
H/2
zy
U2dz =
& & &
m 2yE m 2E H dz = y − zy 2 2 2 H/2
zy
m
|dm dz
(12) (13)
my
The specific expression for | can be chosen as described above, i.e. according to power law, bi-linear etc. The strain mr for which the stress goes from tensile to compressive can be expressed by:
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| z | mr =m − = − E z E
Wt =
& & & & & & & H/2
H/2
U3dz =
Wp3 =
zy
mr
zy
mr
zy
H/2
(m−mr)Edm dz =
zy
(m − mr)2E dz 2
|2 dz 2E
H/2
=
zy
(14)
When there is no plastic deformation on reverse bending, for a foil with a thickness smaller then 2zc, where zc is the z for which mr =myc, then zc must fulfill the equation z | − −myc = 0 z E
(15)
H So when mr 5 myc and zc ] , the expression for U4 will 2 be: U4 =
& &
mr
mEdm=
0
Wp4 =
m 2r E 2
m E dz = 2
H/2 2 r
zy
& H/2
zy
z | − z E
2
&
+ Wp4p − Wp3 + Wp5
|dm dz
m
H/2
=
m
Ez 3y Ez 2y H + − zy + 3z 2 z 2 2
(16) E dz 2
(17)
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H/2
|(m− my)dz + Wp4
zy
(24)
In practical peel adhesion test the total energy Wt is measured, whereas the radius of curvature is difficult to determine and monitor. The calculation procedure may start with selection of the first approximation of z, then calculation of the various energy terms and comparison of the resulting Wt to the measured value of total energy. The procedure is re-iterated until the calculated value corresponds to the measured one, and at this point G can be obtained. A Matlab [17] program was written and used to calculate the above terms, as a function of z, the mechanical properties of the adherend and its thickness. The results of these calculations can serve to study the effect of the test parameters on the total energy expended in peel, Wt, the plastic energy, Wp and the ratio of the total energy to the adhesion energy, Wt/G. It should be noted that although the Matlab program can be executed using any stress –strain curve approximation and any peel angle, all calculations presented below were performed using the bi-linear approximation and an angle = 90°.
Wp5 = 0 For reverse plastic bending mr \myc and zy Bzc B additional terms are required for z \zc:
& & & &
myc
Up4p =
|dm =
0
)
m 2E myc m 2ycE = 2 0 2
H/2
Up4p dz =
Wp4p =
zc
U5 =
mr
|dm
m yc
H/2
U5 dz =
Wp5 =
zc
m 2ycE H −zc 2 2
(18)
(19) (20)
& & mr
H/2
zc
m yc
H two 2
|dm dz
&
0 =
H/2
|mdz −Wp3 +Wp4 +Wp4p +Wp5
0
n
(22)
The expression for Wt corresponds to area OABCD in Fig. 5. In foils for which H/2 5zy the total work will be:
&
H/2
0
|mdz =
&
H/2
0
m 2Edz =
& H/2
0
In foils for which H/2 \ zy
First, in order to validate the results obtained from the above equations, the model of Kinloch et al. [7] was also programmed using Matlab. Kinloch et al. modeled the region below the peel front as an elastic beam on an elastic foundation, which allows root rotation at the peel front. The deformation process in their work was assumed to be one of predominantly bending with large displacements. A key parameter in the model is the angle between the foil and the substrate, at the peel front, 0:
(21)
It should be noted that Wp5 includes Wr, the residual elastic strain energy. The total work Wt, expended in the test can now be expressed as a function of z, by: Wt = 2
4. Results and discussion
z z
2
Edz=
H3 E 24z 2
(23)
2H 3 R0
(25)
Where R0 is the actual radius of curvature at the peel front (similar to r in the present analysis). The applied energy to the peel front is consequently affected by the root rotation, as the angle of the applied peeling force changes from to (− 0), thus the energy transmitted to the peel front changes by a factor of (1 − cos )/[1 −cos (− 0)]. The results of the model proposed by Kinloch et al. [7] and the model developed in this work are compared in Fig. 6. According to Eq. (6) and the subsequent derivations: G= f(z)
(26)
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Fig. 6. Comparison of the model of Kinloch et al. [7] and the present work for adhesion energy. sy =125 MPa, E =70 GPa, H =0.1 mm.
For fixed test parameters (|y =125 MPa, E =70 GPa, H= 0.1 mm), Eq. (26) can be plotted and calibrated against the previous data [7]. Fig. 6 shows that the results of both models agree very well. Consequently, our model is explored further to identify the relationships between total energy consumed in the peel test as a function of the test parameters. For example, the effect of the foil thickness on the total expended energy Wt and on the Wt/G ratio for a pure nickel foil (|y = 95 MPa, E= 212 GPa and h = 0.0009) is shown in Fig. 7 and Fig. 8, respectively. This results demonstrate that increasing the thickness from 0.1 to 0.2 mm increases the total expended energy, at a small radius of curvature, by a factor of 10. However, the Wt/G ratio is increased only by a factor of less than 2. At large radius of curvature, the difference in the total expended energy between the 0.1 mm and the 0.2 mm foil is relatively small, yet the difference in the Wt/G ratio increases. It is interesting to note that for the nickel foil, which is almost perfectly plastic, the Wt/G ratio increases monotonically as a function of the radius of curvature, for the range of curvatures studied. The smallest ratio is achieved for a thicker foil and for a small radius of curvature. Similar calculations were performed for a stainless steel foil (|y = 850 MPa, E =202 GPa and h = 0.032). The results are presented in Fig. 9 (Wt vs z) and Fig. 10 (Wt/G vs z). As can be expected, all energy terms are higher than those for the nickel foil, but the main difference is seen in the behavior of the Wt/G ratio vs r
curve. At each thickness, there is a critical radius of curvature, for which the Wt/G ratio is maximal. This radius is smaller for thinner foils. This behavior prevents simple interpretation of the peel test results, since the plastic energy dissipated, expressed as a fraction of the total energy, can be similar at two extremes of the bending behavior, that is for relatively small as well as for relatively large curvatures. It is evident that either calculating the plastic energy or using the experimental methodology outlined in Section 3, could resolve this difficulty. It should be noted that for 0.1 mm thickness, the value of the Wt/G energy ratio at a radius of curvature of 2.3 mm is less than 6 for stainless steel foil (Fig. 10), while for nickel it is about 11 at the same curvature. At first glance, this result may suggest advantageous use of a higher strength material, in order to avoid high energy losses during the test. Apparently, this interpretation is incorrect, since both the values of the total energy and the energy ratio should be compared. By looking at Figs. 7 and 9, it can be seen that the curvature for a 0.2 mm thick nickel foil and a total energy of 2*104 J/m2 is about 0.4 mm. For the same total energy and thickness, the curvature of the stainless steel foil is about 1.2 mm. The corresponding energy ratios for the nickel and steel are 2 and 4.7, respectively. Thus, in order to avoid high energy losses, a low strength material should be used. This result was also indicated by Moidu et al. [12].
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Figs. 11 and 12 show the energy ratio Wt/G versus the adhesive energy G, for foils having elastic modulus of 200 GPa, with different values of yield strength, for different thickness. It can be seen that
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the yield strength has a pronounced effect. Again, there is a critical value of G, for which the energy ratio is maximal. It is seen that for a low yield strength material (Fig. 11), maximal energy ratio is
Fig. 7. Total expended energy versus radius of curvature for a nickel foil. sy =95 MPa, E =212 GPa, a =0.009.
Fig. 8. Energy ratio versus radius of curvature for a nickel foil. sy =95 MPa, E= 212 GPa, a =0.009.
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Fig. 9. Total expended energy versus radius of curvature for stainless steel foil. sy =850 MPa, E =202 GPa, a =0.032.
Fig. 10. Energy ratio versus radius of curvature for stainless steel foil. sy =850 MPa, E =202 GPa, a =0.032.
received for G values lower then 10 J/m2, while the maximum energy ratio for the high strength material (Fig. 12) is received at G values around 1000 J/m2.
For a material with both low yield strength and low elastic modulus (Fig. 13), the maximum energy ratio is received at low G values, but the decline of the energy
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ratio versus G is very slow. The energy ratios are also affected by the thickness of the foil. Lower ratios are found for thinner foils. The critical G shifts to higher values with increasing thickness. Moidu et al. [12] demonstrated similar results for materials with elastic
177
modulus of 2.5 GPa and yield strength between 50 and 500 MPa. The influence of h, the coefficient of the second slope of the bi-linear representation of the stress –strain curve (Eq. (10)) is examined in Figs. 14 and 15, showing the
Fig. 11. Energy ratio versus adhesion energy. sy =95 MPa, E= 212 GPa, a =0.003.
Fig. 12. Energy ratio versus adhesion energy. sy =850 MPa, E =202 GPa, a =0.003.
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Fig. 13. Energy ratio versus adhesion energy. sy =100 MPa, E =70 GPa, a =0.003.
Fig. 14. Energy ratio versus adhesion energy. sy =95 MPa, E =212 GPa, H =0.2 mm.
results for yield strength of 95 and 850 MPa, respectively. It is seen that the energy ratio Wt/G increases as h decreases, for both materials. The maximum ratio for the low strength material with a small h is much higher
than that of the high strength material. However, for the higher value of h, the maximum energy ratio is similar for both materials. The variation in the energy ratio is more pronounced for the low strain hardening
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Fig. 15. Energy ratio versus adhesion energy. sy =850 MPa, E =202 GPa, H =0.2 mm.
material (low h). It is also evident that for higher values of G, the energy ratio is less sensitive to the strain-hardening coefficient or to the yield strength. From the results outlined above, it is evident that interpretation of the total energy measured during a peel test can be misleading. Moreover, the true adhesion energy can constitute only a few percent of the total energy measured during a peel test. In order to find the required true value of the adhesion energy, the plastic energy has to be found, whether experimentally, as outlined in Section 2, or analytically, as demonstrated in the above calculations. .
5. Conclusions In order to correctly interpret the peel test results, one must determine the energy dissipated in bending, whether experimentally or analytically, as demonstrated in this work. The amount of the total expended energy, Wt and the plastic energy, Wp, consumed during the test, is determined mainly by the yield strength of the material and by its thickness. The higher the yield strength or the thickness of the peeled foil, the higher is the energy expended (dissipated) in the test. In peel tests performed with a mandrel, the strain will be limited by the radius of the mandrel, and the plastic
energy dissipated will become constant, once the curvature of the peeled foil conforms to that radius. The ratio of the total energy to the adhesion energy behaves in a complex manner. For low strength materials, this ratio increases with the foil radius of curvature. In high strength materials, this ratio exhibits a critical radius at which the ratio is maximal. In order to limit the dissipation of plastic energy, a material with low strength and thickness should be used.
References [1] ASTM-D 3167, Standard Test Method for Peel Resistance of Adhesives. [2] ASTM-D 1781, Standard Test Method for Climbing Drum Peel Test for Adhesives. [3] G.J. Spies, J. Aircraft Eng. 25 (1953) 64 – 70. [4] N. Aravas, K.-S. Kim, M.J. Loukis, Mater. Sci. Eng. A107 (1989) 159– 168. [5] J.G. Williams, J. Adhesion 41 (1993) 225– 239. [6] K.-S. Kim, J. Kim, J. Eng. Mater. Technol Trans. 110 (1988) 266– 273. [7] A.J. Kinloch, C.C. Lau, J.G. Williams, Int. J. Fracture 66 (1994) 45 – 70. [8] A.R.J. Farris, J.L. Goldfarb, J. Adhesion Sci. Technol 7 (1993) 853– 868. [9] M. Sexsmith, T. Troczynski, J. Thermal Spray Technol. 3 (1994) 404– 411. [10] M. Sexsmith, T. Troczynski, J. Thermal Spray Technol. 5 (1996) 196– 206.
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[11] M. Sexsmith, T. Troczynski, E. Breslauer, J. Adhesion Sci. Technol. 11 (1997) 141–154. [12] A.K. Moidu, A.N. Sinclair, J.K. Spelt, J. Testing Evaluation 23 (1995) 241– 253. [13] R.A. Pike, Int. J. Adhesion Adhesives 5 (1985) 3–6. [14] J.D. Clark, I.J. McGregor, J. Adhesion 42 (1994) 227– 248.
.
[15] E. Breslauer, T. Troczynski, J. Adhesion Sci. Technol. 12 (1998) 367– 381. [16] K.S. Kim, N. Aravas, Int. J. Solids Structures 24 (1988) 417– 435. [17] Matlab, registered trade mark of MathWorks Inc. MA, USA.
Determination of the energy dissipated during peel testing Edith Breslauer a,*, Tom Troczynski b b
a Rafael, P.O. Box 2250, Haifa, Israel Metals and Materials Engineering, The Uni6ersity of British Columbia, 309 – 6350 Stores Rd, Vancou6er, BC, Canada V6T 1Z4
Abstract During the experimental determination of adhesion strength through a peel test, the deformation energy of the peeled foil, plus frictional losses may constitute a significant portion of the total peel energy. When the true adhesion strength is desired, the energy dissipated in bending of the foil has to be found, whether experimentally or analytically, and deducted from the total peel energy. Several works were dedicated to the modeling and calculation of the energy dissipated through plastic deformation so that the net adhesion energy could be deduced. Recently, the present authors proposed a simple experimental technique for the determination of the net adhesion strength. The method is not limited to any particular material and can be used successfully for a strain hardening plastic as well as metallic foils. In this work the calculation of the various energy terms dissipated during the peel test is carried out, in order to validate the results of the experimental method and to study the effect of the test variables, such as geometry and mechanical properties of the flexible foil. The main parameter that influences the energy expended during the test is the radius of curvature of the flexible adherend. It is found that the ratio of the total energy to the adhesion energy versus this radius of curvature behaves in a complex manner. For a low strength material, the ratio increases with the radius of curvature. For high strength materials, the ratio exhibits a critical radius, at which the ratio is maximal. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Peel adhesion test; Metallic foils; Plastic deformation energy dissipation
1. Introduction The peel tests can be classified into free peeling, Fig. 1A, or peeling around a mandrel, Fig. 1B. Two of the common ASTM standard methods [1,2] belong to the second category. These mandrel-assisted methods are frequently used for comparative tests because the measured peel force includes not only the force necessary to separate the two bodies, but also the forces required to bend the flexible foil and to overcome friction. These additional forces are generally unknown (or neglected), so in order to measure absolute adhesion in the peel test it is necessary to first determine the energy dissipated in bending the foil and in friction. The first to analyze the peel test was Spies [3]. The mechanics of the test and calculation of the plastic energy dissipated during peeling were later carried out by several researchers [4 – 7]. Much of the previous * Corresponding author. E-mail addresses: [email protected] [email protected] (T. Troczynski).
(E.
Breslauer),
theoretical work related to adhesion measurement through the peel test has dealt with the calculation of the plastic energy dissipation (e.g. [4,5]) or its experimental determination [8] using the free peel test (Fig. 1A). In this test, the flexible foil is pulled with force F at a predetermined angle , with respect to the rigid substrate surface. The curvature of the foil is a function of 0, which is the angle at the root of the interfacial crack. Models for the calculation of 0 and the plastic energy had been proposed by several groups [6,7]. Due to the complex nature of these calculations, they have been carried out mainly for perfectly plastic materials [6]. Kinloch et al. have also presented a solution for bi-linear materials [7]. Recently, the peel test had been modified for the determination of the adhesion strength of brittle ceramic coatings, in particular thermal spray coatings [9,10]. The details of the test are specified in [9–11], so only a short description will be given. In the test arrangement, Fig. 2, a 0.1 –0.2 mm thick metallic foil is coated with a thermal spray coating. The coated face of the foil is then adhesively bonded to an aluminum
0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 3 7 0 - 8
E. Breslauer, T. Troczynski / Materials Science and Engineering A302 (2001) 168–180
plate. The foil is now peeled with force F at 90° from the coating using a mandrel, around which the foil is bent. An alignment weight D must be applied to keep the structure in a fixed orientation and to obtain good and repeatable conformance of the foil to the mandrel surface. It was established through modeling and experiment that the plastic energy of deformation of the metallic foil, Wp, was a significant fraction of the total peel work in the above test [11]. The plastic deformation energy dissipated during the peel test varies, depending on the test geometry and the mechanical properties of the peeled material. This may partially explain the large discrepancy in the reported values of the peel strength. For example, the reported peel strength of similar polymeric adhesives ranged from about 0.5 to 10 kJ/m2 [12 –14], depending on the peeling material, its thickness and whether or not plastic energy dissipated to deform the peeled material was accounted for. In order to be able to compare the results, the additional energy dissipated during the peel test (e.g. the plastic deformation energy for metallic foils) has to be determined. In the following section, the energy and force balance during the peel test is outlined and a simple experimental method is proposed for the determination of the net
169
adhesion strength in the mandrel-assisted peel. This method was reported previously [15], so only the basics of the method will be outlined. Using the proposed methodology there is no need to calculate or otherwise determine the energy dissipation during the process of peeling, in particular the frictional effects or the energy of plastic deformation of the peeled foils. The method is equally valid for any material that can dissipate part of the total peel energy. In the fourth section of the paper, an analytical calculation of the various energy terms dissipated during the peel test, based on small displacement beam theory, is presented. The results allow examining the effect of test variables on the different terms of peel energy. Thus, an optimal test can be designed.
2. Experimental determination of peel adhesion strength The total energy dissipated during the peel test (Utot), when the system has reached a steady state, can be divided into three categories: The work to separate the two materials (UI), the work to overcome jig friction and alignment weight (Uf) and the work of bending the foil (Ub). This is stated as: Utot = UI + Uf + Ub
(1)
Peel strength is usually reported as the force (F) per unit width (w) required to separate the foil. Peel strength (F/w) is thus related to energy per unit area by: F Utot = w w·dx
Fig. 1. The free peel test (A) and the mandrel assisted peeling (B).
where w is the peeling arm width and dx represents an element along the foil, Fig. 2. In a typical test the total peel force F is the principal parameter monitored during the experiment. If the energy balance of Eq. (1) is replaced by a force balance for a test performed at an angle b= 90° and for a strip of width w, F is the sum of the force required to separate the interface wG (where G= UI/wdx is the adhesion fracture energy rate), the alignment weight D, the friction force f ( f= Uf/dx), and the force required to perform bending of the foil per length peeled (Fp = Ub/dx). Neglecting the elastic energy stored in the foil under the force F, the peel force balance can now be written: F=wG + D+ f+Fp
Fig. 2. The peel adhesion test applied to a thermal sprayed coating.
(2)
(3)
If a free, non-adhering foil is deformed around the same mandrel in a calibration experiment, Eq. (1) still holds, with the condition of G=0. The friction in the jig (including the friction between the foil and mandrel and around the bearings), f is assumed to be proportional to the total force, f= vF (where v is a proportionality coefficient or an average coefficient of friction
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Fig. 3. Variation of the force F measured in the peel test vs the alignment weight for zero interfacial adhesion in the calibration run (A) and in peeling for the case of finite interfacial adhesion (B).
for the system). The plastic work in bending Ub is proportional to the width of the sample w, so Fp = Ub/ dx = wWp. The deformation of a peeled foil can thus be described by a simple modification of Eq. (1), for the case of zero interfacial adhesion and finite interfacial adhesion G, respectively: F(1−v)= D1 + wWp
(4a)
F(1 − v)= D2 + w(G + Wp)
(4b)
When the total peel force F increases, due to higher values of G, D2 or v, the radius of curvature of the foil will decrease, causing an increase in the plastic energy Wp. The increase in Wp will cause a further increase in F and so on, until a steady state is reached. Under steady state conditions, for a constant G, Wp is a function of F only and it varies from zero for pure elastic deformation of the foil, to some limiting value Wp0(F), in accordance with the limiting strain imposed by the mandrel. Determination of the work Wp(F) necessary to plastically deform the peeled foil is one of the experimental difficulties of the peel test. The methodology proposed below allows to eliminate the uncertainty related to the magnitude of Wp(F) by analyzing the behavior of the deforming foil. If the calibration test described by Eq. (4a) is repeated for various alignment weights, the peel force F/w can be plotted versus the alignment weight D/w, as schematically shown in Fig. 3A. As mentioned above, the amount of energy dissipated by plastic deformation of the bent foil, Wp(F), increases as a function of the total force F. Accordingly, the curve in Fig. 3A illustrates three distinctive regions related to different magnitudes of plastically dissipated work Wp(F). In region I, the bent foil undergoes purely elastic deformation and there is no energy dissipated by plastic deformation of the foil, i.e. Fp =wWp(F) = 0. In this region, the peel force F is a sum of only the alignment weight D and the friction force f. The curve is linear and the slope depends on the average friction coefficient only, therefore, the slope of F/w line has a constant value of
(1− v) − 1. At the beginning of region II, parts of the bent foil that experience the highest strain (i.e. the surface of the foil) exceed the yield point and start to deform plastically. The total peel force FT at the point of transition from region I to region II can be estimated by considering simple bending of the foil. The details were given previously [15]. Wp(F) starts to increase with increase of the alignment weight D/w. The slope of the F/w vs D/w curve increases due to additional energy required for plastic deformation through the thickness of the foil. In this region the shape of the F/w curve depends on the shape of the stress –strain curve of the material (i.e. the yield strength and the strain hardening coefficients) and the foil thickness. Region II would be less extensive, as a function of D, for thinner foils and more extensive for thicker foils. For thinner foils (thickness B 0.1 mm) or for low yield stress metals (e.g. Al) regions I and II may be difficult to discriminate on the F/w curve because of the relatively small contribution of the plastic deformation energy to the total energy. The measured force at the transition from region I to region II is characteristic of the material and the test geometry. At the end of region II the radius of curvature of the foil reaches the value of the radius of the mandrel and the amount of energy dissipated plastically becomes constant as a function of total peel force, i.e. Wp(F)= constant= Wp0(F). Consequently, the slope of F/w line returns to the constant value of (1− v) − 1. The intercept of the extrapolated region III line with F/w axis at D/w=0 gives the value of Wp0 /(1− v). The friction coefficient v can be determined from this slope and thus Wp0 can be determined. For thin (or low yield stress) foils the three regions will merge into an almost straight line (with a minimal Wp0/(1− v) cutoff length). Fig. 3B represents the measured force in an actual peel test, i.e. for the case of non-zero adhesion, Eq. (4b) [the curve for Eq. (5)a is also included in this figure, for comparison]. The peel force curve now shifts upward by the value of G/(1− v), or equivalently to the left by the value of G. Similarly as in Fig. 3A, the intercept of the extrapolated region III line with F/w
E. Breslauer, T. Troczynski / Materials Science and Engineering A302 (2001) 168–180
axis at D/w =0 determines (G + Wp0)/(1 −v), while the intercept of region I line with F/w =0 line determines the net interfacial adhesion energy G. The constant slope (1− v) − 1 of the lines in regions I and III determines the friction coefficient v. In order to be able to extrapolate the peel force curve F/w to low loads, a calibration curve must be obtained, using a free foil (i.e. for G =0), as illustrated in Fig. 3A. The peel force curve F/w during peeling with non-zero adhesion (G \ 0) will be similar in shape, but
Fig. 4. The steady state deformation of the peeling adherend.
171
shifted by the value of the interfacial adhesion energy G, as shown in Fig. 3B. The confirmation that the values of the peel strength obtained using the above procedure are indeed the true net interfacial adhesion strength can be achieved by testing foils of the same material (and same interfacial adhesion strength) but of different thickness. The plastic deformation energy will, obviously be different for each thickness, whereas the interfacial adhesion strength should be similar, within experimental error. Experimental results illustrating this procedure were presented in a previous publication [15]. As the amount of the plastic deformation energy spent to deform the foil of constant geometry is a function of the total peel force only F, subtraction of Eq. (4a) from Eq. (4b) at constant peel force F, renders wG = D1 − D2. The two alignment weights D1 and D2 are those necessary to produce the identical peel force F to separate the adhesion interface and to deform the foil with adhesion (wG = D1 − D2 \ 0) and with no adhesion (G= 0) respectively. Consequently, horizontal shift between the two curves at any value of force F/w can be used to determine the net interfacial adhesion strength G (Fig. 3B), with no need to know the friction coefficient or the amount of energy dissipated by plastic deformation. It should be ensured, however, that the calibration test on non-adhering sample, Fig. 3A, is performed under the same friction conditions and using a foil with the same width, thickness and properties as those for the peeling of adhering foil, Fig. 3B. This method is suitable for determination of the net interfacial adhesion energy G when the peel test is performed under one alignment weight only, or for the calibration test where a range of alignment weights have been used.
3. Analytical calculation of the energy dissipated in a peel test
Fig. 5. The stress– strain curve for an element dz, at depth z, with elastic reverse bending (A) and plastic reverse bending (B).
In order to study the effect of the peel test variables, such as the thickness and mechanical properties of the flexible adherend or the angle, at which the adherend is pulled, it is beneficial to calculate the energy dissipated during the test. It should be noted that the following treatment applies generally, for a free peel as well as for a mandrel assisted test. Fig. 4 shows the configuration of the deformed adherend, during a steady state peel test. The radius of curvature r, during the test is a function of the total peel force and the properties of the adherend. For a large radius of curvature, all elements of the foil experience elastic strain only. At some critical radius, the outer surface of the adherend starts to strain plastically, and gradually (i.e. if the radius further decreases) the whole cross section of the adherend will experience plastic strain. Fig. 5 presents schematically the stress –strain curve for an element dz, at depth
E. Breslauer, T. Troczynski / Materials Science and Engineering A302 (2001) 168–180
172
z (z is 0 at the neutral axis and it equals H/2 at the foil surface). This stress – strain curve is dependent on whether the reverse bending is elastic (Fig. 5A) or plastic (Fig. 5B). The various points marked on the curve correspond to the points marked on the adherend in Fig. 4. The element dz is loaded up to the maximum strain or minimum radius of curvature, which corresponds to point B and then unloaded, when the adherend straightens up, to point D, where the strain is zero, passing through point C. At point D, there is a residual elastic strain energy stored in the foil. If the force is released, for the case of plastic reverse bending, the corresponding point on the curve will be point E (Fig. 5B). The energy balance during the peel test, assuming a rigid substrate, can be expressed as follows: Wt = G+Wp +Wr
(5)
where Wt = total expended energy; G = adhesion energy; Wp =plastic energy; Wr =residual elastic strain energy. The dissipated energy, Wp for each element, is found from the area under the stress – strain curve, i.e. area OBCD. The total energy, Wt for the same element includes the dissipated energy, the residual elastic strain energy and the complementary energy, which is found from area OAB, i.e. the total energy is found from area OABCD [7]: wWt =F[1−cos( − 0)] =area [OABCD]
(7)
& & n z
m
|dm dz
0
zy = zmy = z|y/E Where |y is the yield strength and E is the elastic modulus. If zy ] H/2 (H is the foil thickness), then the full thickness is under elastic stress and there is no plastic energy dissipation. If zy B H/2, the element dz for which z \zy will go into plastic deformation and energy will be dissipated. Elements for which z5 zy do not contribute to the plastic energy dissipation. For elements for which z\ zy, the plastic energy dissipated can be calculated by integrating the stress – strain curve over the appropriate limits, as seen in Fig. 5. The integration can be done numerically or by correlating the curve with a suitable expression. For work hardening materials a power-law model can be m used [16], where for \ 1: my | = Emy
m my
N
(9)
Alternatively, a bi-linear model could be used [7], havm ing for \ 1: my
|=Emy (1−h)+h
n m my
(10)
Where aE is the slope of the stress –strain line, beyond the yield strength. The material can also be modeled as a perfectly plastic material. For elements for which z \zy, the total dissipated energy is:
& & & H/2
Wp = 2
|dm dz
zy
H/2
=
(U1 + U2 − U3 + U4 + U5)dz
zy
Relating to Fig. 4, the equations for the energy terms will be derived as a function of z and the radius of curvature, z. Integration over z is then performed to get the final values of the various energy terms. The total energy dissipated may be expressed by the area under the stress – strain curve: Wp =2
The z at which the stress corresponds to the yield strength is:
(6)
F is the peel force. and 0 were defined in Fig. 1. The adhesion energy, the test geometry and the mechanical properties of the foil determine the required peeling force and a corresponding radius of curvature, z at the root of the peeled material. This radius determines the amount of plastic energy dissipated in bending. In order to find the adhesion energy, the dissipated energy has to be calculated and deducted from the total energy expended. After integration over z, the adhesion energy, G, can then be found from Eq. (5) as a function of the radius of curvature, z: G(z)=Wt(z)−Wp(z) − Wr(z)
m= z/z
(8)
0
Note that the area is multiplied by 2, as z is measured from the neutral axis of the foil (z =0) to the surface (z =H/2). Where the strain, o at z is:
= 2(Wp1 + Wp2 − Wp3 + Wp4 + Wp5)
(11)
The different energy terms will be derived below: Wp1 =
& &
H/2
U1dz =
zy
Wp2 =
H/2
zy
U2dz =
& & &
m 2yE m 2E H dz = y − zy 2 2 2 H/2
zy
m
|dm dz
(12) (13)
my
The specific expression for | can be chosen as described above, i.e. according to power law, bi-linear etc. The strain mr for which the stress goes from tensile to compressive can be expressed by:
E. Breslauer, T. Troczynski / Materials Science and Engineering A302 (2001) 168–180
| z | mr =m − = − E z E
Wt =
& & & & & & & H/2
H/2
U3dz =
Wp3 =
zy
mr
zy
mr
zy
H/2
(m−mr)Edm dz =
zy
(m − mr)2E dz 2
|2 dz 2E
H/2
=
zy
(14)
When there is no plastic deformation on reverse bending, for a foil with a thickness smaller then 2zc, where zc is the z for which mr =myc, then zc must fulfill the equation z | − −myc = 0 z E
(15)
H So when mr 5 myc and zc ] , the expression for U4 will 2 be: U4 =
& &
mr
mEdm=
0
Wp4 =
m 2r E 2
m E dz = 2
H/2 2 r
zy
& H/2
zy
z | − z E
2
&
+ Wp4p − Wp3 + Wp5
|dm dz
m
H/2
=
m
Ez 3y Ez 2y H + − zy + 3z 2 z 2 2
(16) E dz 2
(17)
173
H/2
|(m− my)dz + Wp4
zy
(24)
In practical peel adhesion test the total energy Wt is measured, whereas the radius of curvature is difficult to determine and monitor. The calculation procedure may start with selection of the first approximation of z, then calculation of the various energy terms and comparison of the resulting Wt to the measured value of total energy. The procedure is re-iterated until the calculated value corresponds to the measured one, and at this point G can be obtained. A Matlab [17] program was written and used to calculate the above terms, as a function of z, the mechanical properties of the adherend and its thickness. The results of these calculations can serve to study the effect of the test parameters on the total energy expended in peel, Wt, the plastic energy, Wp and the ratio of the total energy to the adhesion energy, Wt/G. It should be noted that although the Matlab program can be executed using any stress –strain curve approximation and any peel angle, all calculations presented below were performed using the bi-linear approximation and an angle = 90°.
Wp5 = 0 For reverse plastic bending mr \myc and zy Bzc B additional terms are required for z \zc:
& & & &
myc
Up4p =
|dm =
0
)
m 2E myc m 2ycE = 2 0 2
H/2
Up4p dz =
Wp4p =
zc
U5 =
mr
|dm
m yc
H/2
U5 dz =
Wp5 =
zc
m 2ycE H −zc 2 2
(18)
(19) (20)
& & mr
H/2
zc
m yc
H two 2
|dm dz
&
0 =
H/2
|mdz −Wp3 +Wp4 +Wp4p +Wp5
0
n
(22)
The expression for Wt corresponds to area OABCD in Fig. 5. In foils for which H/2 5zy the total work will be:
&
H/2
0
|mdz =
&
H/2
0
m 2Edz =
& H/2
0
In foils for which H/2 \ zy
First, in order to validate the results obtained from the above equations, the model of Kinloch et al. [7] was also programmed using Matlab. Kinloch et al. modeled the region below the peel front as an elastic beam on an elastic foundation, which allows root rotation at the peel front. The deformation process in their work was assumed to be one of predominantly bending with large displacements. A key parameter in the model is the angle between the foil and the substrate, at the peel front, 0:
(21)
It should be noted that Wp5 includes Wr, the residual elastic strain energy. The total work Wt, expended in the test can now be expressed as a function of z, by: Wt = 2
4. Results and discussion
z z
2
Edz=
H3 E 24z 2
(23)
2H 3 R0
(25)
Where R0 is the actual radius of curvature at the peel front (similar to r in the present analysis). The applied energy to the peel front is consequently affected by the root rotation, as the angle of the applied peeling force changes from to (− 0), thus the energy transmitted to the peel front changes by a factor of (1 − cos )/[1 −cos (− 0)]. The results of the model proposed by Kinloch et al. [7] and the model developed in this work are compared in Fig. 6. According to Eq. (6) and the subsequent derivations: G= f(z)
(26)
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Fig. 6. Comparison of the model of Kinloch et al. [7] and the present work for adhesion energy. sy =125 MPa, E =70 GPa, H =0.1 mm.
For fixed test parameters (|y =125 MPa, E =70 GPa, H= 0.1 mm), Eq. (26) can be plotted and calibrated against the previous data [7]. Fig. 6 shows that the results of both models agree very well. Consequently, our model is explored further to identify the relationships between total energy consumed in the peel test as a function of the test parameters. For example, the effect of the foil thickness on the total expended energy Wt and on the Wt/G ratio for a pure nickel foil (|y = 95 MPa, E= 212 GPa and h = 0.0009) is shown in Fig. 7 and Fig. 8, respectively. This results demonstrate that increasing the thickness from 0.1 to 0.2 mm increases the total expended energy, at a small radius of curvature, by a factor of 10. However, the Wt/G ratio is increased only by a factor of less than 2. At large radius of curvature, the difference in the total expended energy between the 0.1 mm and the 0.2 mm foil is relatively small, yet the difference in the Wt/G ratio increases. It is interesting to note that for the nickel foil, which is almost perfectly plastic, the Wt/G ratio increases monotonically as a function of the radius of curvature, for the range of curvatures studied. The smallest ratio is achieved for a thicker foil and for a small radius of curvature. Similar calculations were performed for a stainless steel foil (|y = 850 MPa, E =202 GPa and h = 0.032). The results are presented in Fig. 9 (Wt vs z) and Fig. 10 (Wt/G vs z). As can be expected, all energy terms are higher than those for the nickel foil, but the main difference is seen in the behavior of the Wt/G ratio vs r
curve. At each thickness, there is a critical radius of curvature, for which the Wt/G ratio is maximal. This radius is smaller for thinner foils. This behavior prevents simple interpretation of the peel test results, since the plastic energy dissipated, expressed as a fraction of the total energy, can be similar at two extremes of the bending behavior, that is for relatively small as well as for relatively large curvatures. It is evident that either calculating the plastic energy or using the experimental methodology outlined in Section 3, could resolve this difficulty. It should be noted that for 0.1 mm thickness, the value of the Wt/G energy ratio at a radius of curvature of 2.3 mm is less than 6 for stainless steel foil (Fig. 10), while for nickel it is about 11 at the same curvature. At first glance, this result may suggest advantageous use of a higher strength material, in order to avoid high energy losses during the test. Apparently, this interpretation is incorrect, since both the values of the total energy and the energy ratio should be compared. By looking at Figs. 7 and 9, it can be seen that the curvature for a 0.2 mm thick nickel foil and a total energy of 2*104 J/m2 is about 0.4 mm. For the same total energy and thickness, the curvature of the stainless steel foil is about 1.2 mm. The corresponding energy ratios for the nickel and steel are 2 and 4.7, respectively. Thus, in order to avoid high energy losses, a low strength material should be used. This result was also indicated by Moidu et al. [12].
E. Breslauer, T. Troczynski / Materials Science and Engineering A302 (2001) 168–180
Figs. 11 and 12 show the energy ratio Wt/G versus the adhesive energy G, for foils having elastic modulus of 200 GPa, with different values of yield strength, for different thickness. It can be seen that
175
the yield strength has a pronounced effect. Again, there is a critical value of G, for which the energy ratio is maximal. It is seen that for a low yield strength material (Fig. 11), maximal energy ratio is
Fig. 7. Total expended energy versus radius of curvature for a nickel foil. sy =95 MPa, E =212 GPa, a =0.009.
Fig. 8. Energy ratio versus radius of curvature for a nickel foil. sy =95 MPa, E= 212 GPa, a =0.009.
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Fig. 9. Total expended energy versus radius of curvature for stainless steel foil. sy =850 MPa, E =202 GPa, a =0.032.
Fig. 10. Energy ratio versus radius of curvature for stainless steel foil. sy =850 MPa, E =202 GPa, a =0.032.
received for G values lower then 10 J/m2, while the maximum energy ratio for the high strength material (Fig. 12) is received at G values around 1000 J/m2.
For a material with both low yield strength and low elastic modulus (Fig. 13), the maximum energy ratio is received at low G values, but the decline of the energy
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ratio versus G is very slow. The energy ratios are also affected by the thickness of the foil. Lower ratios are found for thinner foils. The critical G shifts to higher values with increasing thickness. Moidu et al. [12] demonstrated similar results for materials with elastic
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modulus of 2.5 GPa and yield strength between 50 and 500 MPa. The influence of h, the coefficient of the second slope of the bi-linear representation of the stress –strain curve (Eq. (10)) is examined in Figs. 14 and 15, showing the
Fig. 11. Energy ratio versus adhesion energy. sy =95 MPa, E= 212 GPa, a =0.003.
Fig. 12. Energy ratio versus adhesion energy. sy =850 MPa, E =202 GPa, a =0.003.
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Fig. 13. Energy ratio versus adhesion energy. sy =100 MPa, E =70 GPa, a =0.003.
Fig. 14. Energy ratio versus adhesion energy. sy =95 MPa, E =212 GPa, H =0.2 mm.
results for yield strength of 95 and 850 MPa, respectively. It is seen that the energy ratio Wt/G increases as h decreases, for both materials. The maximum ratio for the low strength material with a small h is much higher
than that of the high strength material. However, for the higher value of h, the maximum energy ratio is similar for both materials. The variation in the energy ratio is more pronounced for the low strain hardening
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Fig. 15. Energy ratio versus adhesion energy. sy =850 MPa, E =202 GPa, H =0.2 mm.
material (low h). It is also evident that for higher values of G, the energy ratio is less sensitive to the strain-hardening coefficient or to the yield strength. From the results outlined above, it is evident that interpretation of the total energy measured during a peel test can be misleading. Moreover, the true adhesion energy can constitute only a few percent of the total energy measured during a peel test. In order to find the required true value of the adhesion energy, the plastic energy has to be found, whether experimentally, as outlined in Section 2, or analytically, as demonstrated in the above calculations. .
5. Conclusions In order to correctly interpret the peel test results, one must determine the energy dissipated in bending, whether experimentally or analytically, as demonstrated in this work. The amount of the total expended energy, Wt and the plastic energy, Wp, consumed during the test, is determined mainly by the yield strength of the material and by its thickness. The higher the yield strength or the thickness of the peeled foil, the higher is the energy expended (dissipated) in the test. In peel tests performed with a mandrel, the strain will be limited by the radius of the mandrel, and the plastic
energy dissipated will become constant, once the curvature of the peeled foil conforms to that radius. The ratio of the total energy to the adhesion energy behaves in a complex manner. For low strength materials, this ratio increases with the foil radius of curvature. In high strength materials, this ratio exhibits a critical radius at which the ratio is maximal. In order to limit the dissipation of plastic energy, a material with low strength and thickness should be used.
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