design of adhesive on peak elastic stress
Design of adhesive joints based on peak elastic stresses
L.Goglio a,?,M.Rossetto a ,E.Dragoni b
a Dipartimento di Meccanica,Politecnico di Torino,Corso Duca degli Abruzzi 24,10129Torino,Italy
b
Dipartimento di Scienze e Metodi dell’Ingegneria,Universita
`di Modena e Reggio Emilia,Via Fogliani 2,42100Reggio Emilia,Italy a r t i c l e i n f o
Available online 6April 2008Keywords:
Structural acrylics Stress analysis Joint design
a b s t r a c t
The paper is focused on the static strength of adhesively bonded structural joints and seeks a simple calculation rule that can assist the designer in everyday engineering practice.The work encompasses three steps.In the ?rst step,an experimental campaign is carried out on an assortment of customized bonded joints (single lap and T-peel)made of steel strips bonded by an acrylic structural adhesive.The dimensions of the joints are chosen so as to produce a wide range of combinations of shear and peel stresses in the adhesive layer.In the second step,the stress analysis of the joints is performed by means of a sandwich model that describes the variability of shear and peel stresses over the overlap length but disregards the stress singularities at the corners.In the third step,a design rule is inferred by noting that,in a chart having as axes the peak values of the peel and shear components in the adhesive at failure,the points —calculated for each joint at the 2%(deviation from linearity)proof load —de?ne a limit zone.The inferred design rule is that the adhesive withstands the load if the representative point of the stress state lies inside this zone.For the tested case,the envelope of the limit zone has an approximately rectangular shape.This criterion predicts the failure load of the joints far better than the simplistic approach based on the nominal stress calculated as the ratio of the load to the bonded area.
The paper also discusses the response which is obtained by applying,to the same experimental data,the traditional calculation based on the mean stress (force to area ratio),and the more sophisticated approach based on the stress intensity factor,which accounts for the singularity of the stress ?eld.Applied to our experimental data,the performance of both has been unsatisfactory.
&2008Elsevier Ltd.All rights reserved.
1.Introduction
In spite of sustained effort devoted to engineering research on adhesives over the years [1,2],widespread use of adhesive technology for structural applications is still far from being achieved.One of the obstacles to the popularity of structural adhesive bonding is the lack of simple and reliable methods for joint strength assessment which can assist the designer in everyday practice;methods —for instance —well established in case of bolted or welded joints.This shortcoming not only affects the more complex situations of fatigue or creep loading combined with material and geometrical nonlinearities,but also arises in the simplest case when loading is monotonic (quasi-static or impact)and remains within the linear range of the adhesive behaviour.As a result,adhesives are often disregarded as a loadworthy joining solution to the advantage of purely mechanical design options (bolting,welding,friction).
Foreseeing the strength of an adhesive joint presents intrinsic dif?culties.From the theoretical standpoint,a major problem lies in the singularity of the stress ?eld,associated with the sharp re-entrant corners of most joint geometries and with the abrupt
material discontinuity at the adherend/adhesive interface.In addition,even in the areas free from those singularities (e.g.the midsurface of the adhesive layer),the stress distribution is strongly irregular,with severe peaks and multiaxial states of stress taking place at the ends of the bondline.This explains why,apart from the case of rubbery or toughened adhesives [3,4],the simplistic calculation of the mean stress over the bonded area gives only a rough (and often misleading)estimate of the joint strength [5].
In order to cope rationally with the complexity of the stresses within the adhesive of bonded joints,basically three lines of thought are available from the technical literature [6]:
approaches based on the elastic ?eld in the neighbourhood of the singularity (including linear elastic fracture mechanics); approaches based on the conceptual tools of the damage mechanics,with different de?nitions of the damage parameter;
approaches based on the concept of effective stress.
The elastic ?eld approaches address directly the stress singula-rities arising either at the corner point of a re-entrant bimaterial wedge [7–10]or at the tip of an assumed ‘‘inherent crack’’[11].The singular stress ?eld is represented by a stress intensity factor (SIF)and the related failure criterion assumes that failure starts
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International Journal of Adhesion &Adhesives
0143-7496/$-see front matter &2008Elsevier Ltd.All rights reserved.doi:10.1016/j.ijadhadh.2008.04.001
?Corresponding author.Tel.:+390115646934;fax:+390115646999.
E-mail address:luca.goglio@polito.it (L.Goglio).
International Journal of Adhesion &Adhesives 28(2008)427–435
when this factor reaches a critical value.As discussed and experimentally con?rmed in[10]for a butt joint,the approach is valid as long as the size of the plastic zone around the singularity is smaller than the extent of the singular stress?eld.Unfortunately, the sophistication of this approach represents also its main short-coming.Evaluation of the SIF requires exact numerical calculations or very re?ned?nite element models[9],approximate solutions being available only for the ideal case(even if not too far from qreality in many cases)of rigid adherends bonded by a deformable adhesive[8].An intrinsic complication is due to the fact that the order of the singularity depends also on the tiny geometrical features of the lap end(spew?llet,etc.),which are dif?cult to predict at the design stage and poorly controlled during fabrication. This makes the construction of a database for the critical SIF values a dif?cult and disputable achievement,because the critical value depends on the singularity order.It is worth noting that if linear fracture mechanics is applied by assuming a?ctitious inherent crack,the order of the singularity is forced to beà0.5because in the Westergaard formula the coordinate r appears under square root in the denominator.
The damage mechanics approaches are,schematically, of two types.One type(see,for instance,[12])assumes as damage index the accumulated plastic strain in the adhesive,and failure occurs when a threshold is attained.The other type[13–15] is based on the elasto-plastic study of the nucleation and propagation of the fracture,in the sense that cracking of the adhesive is related to overall(also plastic)deformation by means of a traction-separation law.This overcomes the limit of the linear fracture mechanics,allowing for the description of non-linear behaviour from the?rst damage until complete failure.In both cases,accurate knowledge of the non-linear stress–strain relationship of the adhesive is required and the de?nition of the damage parameter or law implies a certain degree of arbitrariness.
In order to avoid the complications related to singular stress ?eld and damage mechanics methods,approaches based on the concept of effective stresses provided by structural models(based on beam-or plate-like solutions as the pioneering work of Volkersen[16]and Goland and Reissner[17])or the local stresses calculated by?nite elements at a distance from the singularity, have been reconsidered[6,18].The underlying assumption is that the peak stresses calculated in this way do represent the severity of the singular stress?eld without the burdens of the fracture mechanics and of the damage mechanics approaches.
Compared to the SIF,the structural stress approach allows for a simpler calculation but is less sensitive to the detail geometrical features(spew angle or/and radius,etc.).On the other side,the limit stress can be sensitive to some parameters.For instance,a consequence of disregarding—or not describing exactly—the stress variation through the adhesive thickness is that the apparent limit stress is a function of the thickness[18].
Following this line of thought,the present paper explores the merits of a stress-based design rule for the prediction of the design load of an adhesive joint.Although simple,the rule is precise enough to be useful at an early dimensioning stage.
The analysis is limited to linear behaviour;this is coherent with the usual approach to the design of the structural members, which in service are usually loaded within the elastic range. Moreover,even though the adhesives,in general terms,are affected by non-linear phenomena,modern high-strength adhe-sives exhibit an appreciable linear range.
The presentation of the material and the identi?cation of the rule are organized in three steps:
testing of custom(non-standard)specimens,designed to attain adhesive failure under different combinations of peel and shear stresses;
calculation of peel and shear stresses in the adhesive;
analysis of the results and synthesis of the design rule.
The identi?cation procedure applies to the broad family of joints between thin sheets,bonded by a thin layer of structural adhesive and subject to quasi-static loading.For joints involving massive parts with zero-thickness bonding layers,other failure criteria apply[19].
2.Steps of the study
2.1.Testing
A test campaign has been carried out,based on an assortment of specimens belonging either to the family of single lap(Fig.1a) or to the family of T-peel(Fig.1b)tensile joints.The adhesive utilized was a one-part acrylic liquid product for structural bonding(Loctite330by Henkel,Du¨sseldorf,Germany)to be used with a spray activator.
The overall design comprises the14arrangements(12single-lap,two T-peel)of Table1,resulting from the combination of four adherend thicknesses,h,two adhesive thicknesses,t,and four overlap lengths,L.Joint architecture and levels of the variables were chosen so as to generate in the adhesive the broadest spectrum of stress conditions that could be achieved with reasonable effort.These stress conditions range from the state of ‘‘nearly pure shear’’of the shortest single laps to the state of‘‘pure peeling’’of the T-peels and include several states of‘‘shear and peel combined’’,arising in the longer single laps.Each run of Table1was replicated four times under nominally identical test conditions.
The adherends were fabricated by cutting blanks of appro-priate length out of20-mm-wide strips,made of wrought structural(S355EN10028)steel with a minimum yield strength of350MPa.In the case of the T-peel specimens,the blanks were also bent with an inner radius of curvature of5mm.Preparation of the test joints was carried out in compliance with the following check list:
grinding of the surfaces with abrasive cloth(grit P100);
cleaning of the surfaces by means of Loctite7063cleaner;
application of the activator(Loctite7388)on the face of one adherend;
Fig.1.Geometries and loading conditions of the tested joints:(a)single lap and(b) T-peel.
L.Goglio et al./International Journal of Adhesion&Adhesives28(2008)427–435 428
application of the adhesive(Loctite330)on the face of the mating adherend;
placement on the adhesive of calibrated steel wires to achieve the desired thickness(0.10or0.25mm);
closure of the joint under steady pressure,exerted by soft clamps;
curing(under clamping)at room temperature for at least36h. Care was taken to place the calibrated wires far from the edges of the bondline(X2.5mm)so as to affect neither the peak stresses nor the strength of the joints.
Loading of the specimens was performed on a screw-driven, 100-kN testing machine(Schenk Trebel),at a crosshead speed of about1mm/min.For the lap joints,additional pads were inserted within the holding grips to ensure alignment between the load and the mid-plane of the adhesive.During the tests,applied load and displacement of the cross-head were recorded up to complete debonding of the joint for later elaboration.
2.2.Stress analysis
For the calculation of the stresses in the adhesive,the category of structural models is favored here.The same results could be obtained by means of?nite element method(FEM)models of the considered joints,but the analytical approach(thanks to the simplicity of the involved geometries)is easier to implement and
allows for exploring rapidly the effect of changing the geometrical parameters.
Among the several structural models available in the literature, the one proposed by Bigwood and Crocombe[20]is preferred for its simplicity and generality.According to this model,the over-lapping portion of the joint(Fig.2)is idealized as a three-layered sandwich beam(Fig.2a)in which adherends and adhesive interact by mutual shear(t xy)and peel(s y)stresses(Fig.2b).At both ends of the sandwich,each adherend undergoes a set of membrane,shear and bending forces(Fig.2a),transmitted by the parts of the joint external to the overlap.Stresses t xy and s y are assumed constant through the thickness of the adhesive layer. This assumption makes these stresses unaffected by the singula-rities that arise near the corners by the adherends in the continuum approach of the classical theory of elasticity.
The model can be applied to many geometries(Fig.3)and loadings by simply adjusting the boundary conditions[20,21]. Furthermore,it offers closed-form stress solutions(see below) that are easily implemented on spreadsheets or by in-house software.
By enforcing equilibrium and compatibility of the layers of the elemental sandwich of Fig.2b,a pair of coupled differential equations are obtained,which govern the distribution of stresses over the bondline.If the adherends have the same geometric and elastic properties(as the specimens tested in this work),the two equations decouple as follows:
d3t xy
àK2
6
d t xy
?0(1a)
d4s y
d x4
t4K4
5
s y?0(1b)
where K5and K6are constants depending on geometry and elastic properties of adherends and adhesive.The solutions of Eq.(1a,b) are:
t xy?B1cos heK6xTtB2sin heK6xTtB3(2a) s y?A1coseK5xTcosheK5xTtA2coseK5xTsinheK5xT
tA3sineK5xTcosheK5xTtA4sineK5xTsinheK5xT(2b)
Table1
Experimental plan with specimen dimensions(width of all joints20mm)
Series Joint type(Fig.1)h(mm)t(mm)L(mm)
1Single lap 2.00.105
2Single lap 2.00.1015
3Single lap 2.00.1025
4Single lap 2.00.255
5Single lap 2.00.2515
6Single lap 2.00.2525
7Single lap 4.00.105
8Single lap 4.00.1015
9Single lap 4.00.1025
10Single lap 4.00.255
11Single lap 4.00.2515
12Single lap 4.00.2525
13T-peel 3.00.10150
14T-peel 1.50.10150
N
N
1
1
+dN2
y
x
Fig.2.Global(a)and elemental(b)idealization of the general adhesive joint between two thin adherends[20].
Fig.3.Examples of joints that can be reduced to the model of Fig.1.
L.Goglio et al./International Journal of Adhesion&Adhesives28(2008)427–435429
The constants A i,B i are calculated from the boundary condi-tions,i.e.loading of the overlap ends(traction,shear,moment). Explicit formulae are given in[20],however,in this work the end loads have been evaluated taking into account the Goland–Reiss-ner correction for joint rotation[17].These stresses agree with the ?nite element values calculated on the midplane of the adhesive [22],as implied by the assumption of constant value through the thickness.Eq.(2a)locates the shear stress peaks exactly at (instead of near to)the free edges of the adhesive layer(where the shear stress must be zero),however the peak values are correct.
2.3.Results
Examples of the theoretical distributions of peel and shear adhesive stresses calculated by means of Eq.(2a,b)are displayed in Fig.4for three specimens:a single lap joint with short overlap (Fig.4a),a single lap joint with longer overlap(Fig.4b)and a T-peel joint(Fig.4c).The dimensions of the joints and the values of the applied loads are provided in the legend.The values of the elastic constants adopted in the calculations for the adherends are those typical of steel(Young’s modulus206GPa,Poisson’s ratio 0.29).For the adhesive,the adopted values are Young’s modulus 880MPa and Poisson’s ratio0.15.The latter values have been taken from[23],in which strain gauges have been applied on a specimen of adhesive Loctite330in bulk.It can also be deduced from[23]that the adhesive is substantially brittle.The stress ?elds in Fig.4show that the short overlap undergoes mainly shear stress,the longer overlap experiences peel and shear stresses of comparable magnitude and the T-peel exhibits a state of pure peel stress.The achievement of this variety of stress states was the aim of the tests.
Examples of load–displacement diagrams recorded during testing are displayed in Fig.5for a single lap(Fig.5a)and a T-peel(Fig.5b)joint.All tested specimens showed predominantly cohesive failure of the adhesive layers.No gross plastic strain was observed in the adherends after failure of the joints,although the maximum elastic stress(sum of axial and bending components) calculated in the adherends occasionally exceeded the nominal yield strength of the steel.
For the elaboration of a design rule,from each experimental curve,two critical load values were selected:the ultimate load and a proof load(Fig.6).The ultimate load,P u,is de?ned as the load corresponding to the uppermost point of the curve.The proof load,P p2,is de?ned as the load corresponding to a difference of 0.02P u between the measured load–displacement curve and the linear regression carried out on the part of the diagram up to 50–75%of the ultimate load.The tolerance value0.02P u has been chosen to reduce the effect of the small oscillations of the recorded load–displacement curve,oscillations mainly due to the approximate measurement of the elongation(cross-head displa-cement).Due to these oscillations the results in terms of the proof load P p2of different curves are affected by higher standard deviation.
The mean ultimate load of each joint type(Pˉu),average of four replicates per joint,and the corresponding standard deviation, s(P u)are provided in the second and in the third columns of Table2.The last four columns of Table2contain the ultimate mean shear stress(t u mean),the ultimate mean peel stress(s u mean), the ultimate peak shear stress(t u peak)and the ultimate peak peel stress(s u peak)of the joints.The ultimate mean stresses were calculated as the ratio of Pˉu to the bonded area.The ultimate peak stresses were calculated as the maximum stresses given by Eq.(2a,b) for an external load Pˉu applied to the joint.
Similarly to Table2,Table3collects the stress values of the joints,corresponding to the proof loads,P p2,instead of the ultimate load.The meaning of the symbols is similar to that in Table2with the replacement of the subscript u with the subscript p2.
Fig.7displays the experimental points in terms of peak shear and peak peel stresses at failure,with the peak stresses retrieved from the last two columns of Table2.The vertical and horizontal bars across the points show the standard deviation ranges.The points lying on the vertical axis(s u peak)correspond to T-peel joints,which are free from shear stresses.The points in the right zone of the diagram correspond to single-lap joints.Among them, the effect of the overlap length is clearly noticeable:the points close to the horizontal axis(t u peak)correspond to short single-lap joints,in which peel stresses are low,the points in the top-right region of the diagram correspond to long single-lap joints, characterized by like values of shear and peel stresses.
In a similar fashion,the pattern of critical stresses based on the proof load,P p2,is shown in Fig.8.
4.Discussion
4.1.Analysis of the results in a stress space
Examination of the fourth and?fth columns of Table2,listing the mean stresses shows that these values are not constant through the series,even considering separately the different adhesive thicknesses.For instance,the mean shear stress ranges from approximately15to40MPa,when the adhesive thickness is t?0.1mm,and from approximately17to43MPa,when t?0.25 mm.This result contradicts the traditional calculation of the joints (based on the load to area ratio),which relies on the stress leveling at failure due to plastic?ow of the adhesive.Obviously, the situation does not improve by examining the fourth and?fth columns of Table3,in which stress levelling is even less likely since the proof load level is lower than failure.
The calculation of the peak stress values in Table2by means of Eq.(2a,b)relies on the assumption of linear elastic behaviour, which is one of the hypotheses of the model[20].By examining the experimental load–displacement diagrams(as those in Fig.5), it comes out that this assumption is not too far from reality,and this justi?es the stress calculation at ultimate load,at least as?rst approximation.However,the use of the proof load is more rigorous and will be preferred in the following.
The observation of the peak stresses,arranged graphically as shown in Fig.7for the case of ultimate loads(Table2),suggests the existence of an admissible zone in the ts diagram,bounded by an envelope that intercepts both axes at approximately 40–50MPa.The behaviour becomes clearer by considering the peak stresses at2%proof load of Fig.8(Table3),which are with good approximation internally bounded by a rectangular envelope whose sides lie approximately at t p2?19MPa and s p2?38MPa. The position of the sides is in?uenced by the adhesive thickness, as it can be noticed by considering separately the empty and grey symbols.
As an attempt to interpret the behaviour observed experimen-tally,the classical failure hypotheses for structural materials(ideal stresses),in the special case of a single normal and a single shear stress,have been applied to the results.This is presented in Fig.9, in which the envelopes corresponding to the different cases are plotted.Table4reports the formulae for each hypothesis,the last two(Mohr and Drucker-Prager with exponent2)have been considered to account for the potential effect of the(hydrostatic) pressure on the adhesive failure,typical of polymers.A tensile limit stress of40MPa(see Fig.8)and,for the pressure sensitive hypotheses,a tentative value of1.3for the compressive stress limit to tensile stress limit ratio m,have been considered.
L.Goglio et al./International Journal of Adhesion&Adhesives28(2008)427–435 430
-100
1020304050
60-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Position along the overlap (mm)
S t r e s s (M P a )
-20-10010
2030405060Position along the overlap (mm)
S t r e s s (M P a )
-10010
20
30405060-75
-60
-45
-30-1501530456075
Position along the overlap (mm)
S t r e s s (M P a )
Fig.4.Distribution of peel and shear stresses for a short single lap joint (a),a long single lap joint (b)and a T-peel joint (c).
L.Goglio et al./International Journal of Adhesion &Adhesives 28(2008)427–435
431
Comparing Figs.8and 9it is interesting to notice that practically all hypotheses are not successful,especially in describing the nearly vertical side of the envelope of the experimental points.Thus,even if not (or not yet)supported by a theoretical background,the practical assumption of a rectangular admissible zone is more effective in reproducing the experimental evidence.
The bondline thickness values typical of this kind of adhesive (tenths of mm)moderately exhibit the in?uence of the gap on the strength.From Table 2it can be seen that in terms of ultimate load (or mean stress)at failure the usual effect of the thickness [24](‘‘the thinner the stronger’’)is not veri?ed.
On the contrary,considering the proof load or the correspond-ing mean and peak stresses (Table 3)the usual effect of the thickness is veri?ed.In Fig.8the right end side of the envelope shifts from 22to 29MPa as the adhesive thickness changes from 0.25to 0.1mm and,in general,the points corresponding to the lower adhesive thickness lie to the right of the points for the higher thickness.This could be explained by the ?ndings of other
investigations [18],in the sense that in a thicker adhesive layer the stress ?eld on the mid-plane (i.e.the values calculated by Eq (2a,b))is lower,but it is higher at the interface with the adherends.Thus,the thicker adhesive is in a more critical condition and the related limit stresses evaluated on the mid-plane are lower.
4.2.Stress intensity factor
With the aim of enlarging the analysis of the case,also the SIF approach has been applied to the results.The underlying assumption is that in the neighbourhood of a vertex of the adhesive layer the solution for a generic stress component is of the form (?rst term of a series,however,suf?cient to describe the main features of the stress ?eld in the adhesive):s ij ?
K f ij
(3)
where K is the SIF,f ij is an angular function,r is the distance from the vertex and l is the singularity order.The singularity order has been calculated as the root of the following equation,given in [8](case of square-edged adhesive layer):
e3à4n Tcos ep e1àl TTà2e1àl T2t8n 2à12n t5?0
(4)
Again,as proposed in [8],the SIF has been calculated as K ??s n A en Ttt n B en T t l
(5)
where s ?and t ?are the peel and shear stresses evaluated by means of the structural solution,t is the adhesive thickness,A (n )and B (n )are combination coef?cients (functions of the adhesive Poisson’s ratio only)given by the following formulae:A en T?0:836à2:23n t6:29n 2à9:64n 3B en T?3:12à15:8n t40:1n 2à37:6n 3
(6a,b)
The results of the procedure are reported graphically in Fig.10(l ?0.1788,A ?0.61049,B ?1.52535,stress values as in Fig.8).It is evident that the SIF at proof load does not remain constant through the different cases:for the lap specimens,it ?uctuates for the different lap lengths,whilst in the case of the T-peel specimens (rightmost bars in the diagram)the values are much lower.This unsatisfactory behaviour is likely due to the fact that the small values of the adhesive thickness (typical for this type of adhesive)do not ful?l the condition for the applicability of the SIF approach [10].Akisanya and Meng state that the plastic zone around the corner must be small with respect to the zone dominated by the elastic singular solution,and this implies that the thickness of the adhesive layer must be greater than a minimum value.
5.Conclusions
The paper has investigated the possibility of establishing a simple rule for the calculation of bonded joints under static load,suitable especially at a preliminary stage of the design.The work consisted of three steps.In the ?rst step,an assortment of ad hoc bonded joints (single lap and T-peel),chosen to produce a wide range of combinations of shear and peel stresses in the adhesive,have been tested statically to fracture.In the second step,the stress analysis of the joints has been performed by means of a structural sandwich model that describes the variability of shear and peel stresses over the overlap length but does not include the stress singularities.In the third step,a design rule is inferred by noting that the envelope of the peak elastic stresses (shear and peel)—calculated in the adhesive of each joint and corresponding to the 2%proof load —is approximately rectangular.The obtained
0.01.02.03.04.05.06.07.08.09.010.011.0
Displacement (mm)
L o a d (k N )
0.00.20.40.60.81.01.21.41.61.8
Displacement (mm)
L o a d (k N )
Fig.5.Load–displacement diagrams for a single lap joint (a)and a T-peel joint (b).
L.Goglio et al./International Journal of Adhesion &Adhesives 28(2008)427–435
432
design rule is that the peak elastic stresses must lie inside the envelope to ensure success.
The adopted approach relies on the assumptions that the failure is cohesive and the adherends remain in elastic regime.The ?rst requirement can usually be ful?lled if,for the adhesive/adherend under examination,the joining procedure is performed at its best (surface treatment,primer application,etc.).About the second requirement,the adhesive is regarded as the ‘‘weakest
Table 2
Test results in terms of ultimate load and related stresses Series P ˉu (N)s (P u )(N)t u
mean
(MPa)s u
mean
(MPa)t u
peak
(MPa)s u
peak
(MPa)
1352010435.20.039.818.92567011218.90.034.837.63751013615.00.039.340.5436619036.60.038.59.0562*******.80.028.626.86841117716.80.030.228.27398013939.80.042.5 6.58783015626.10.039.443.7910,05015320.10.042.647.71042989643.00.044.1 3.41188488829.50.035.928.41211,55817923.10.035.035.51313791080.00.50.042.114
930
141
0.00.30.047.7
Table 3
Test results in terms of 2%proof load and related stresses Series P ˉp2(N)s (P p2)(N)t p2
mean
(MPa)s p2mean
(MPa)t p2
peak
(MPa)s p2
peak
(MPa)
1282639028.30.031.915.12470955815.70.029.131.83535222810.70.028.730.54262237226.20.027.6 6.45437495914.60.020.219.56522744810.40.019.219.0728*******.60.030.3 4.68570269913.60.028.732.19613429812.30.026.329.91026646022.00.027.4 2.011529042917.60.021.517.212684363613.70.020.921.7131265890.00.50.038.614
815
153
0.00.3
0.041.8
τu peak (MPa)
σu p e a k (M P a )
Fig.7.Failure envelope in terms of peak shear and peel stresses based on ultimate load (bars show standard deviation ranges).
L o a d
Displacement
P p2
P u Fig.6.De?nition of ultimate load and proof load.
L.Goglio et al./International Journal of Adhesion &Adhesives 28(2008)427–435
433
link’’of the structure,and the rule can be used to assess the maximum load sustainable by a given overlap length or,vice versa,to assume the proper overlap length required to sustain a given load.Clearly if,for instance,the overlap length is extended to a level that the joint becomes stronger than the adherends,the latter must be veri?ed separately (by means of the same structural solution used for the joint,or in different way, e.g.FEM).The approach based on the SIF,although theoretically rigorous,has not given positive results in our case,likely because
the layer of adhesive (typical of the acrylic product)is too thin to ful?ll the condition for the applicability.
Beyond the case of the considered adhesive,for which quantitative values are presented in the paper,the proposed approach can be applied with limited effort to the particular case of the structural adhesive needed by the designer:it is enough to carry out a limited set of experiments,using lap-shear and T-peel specimens of different length,to obtain a reasonable outline of the admissible zone in the stress space.The thickness of the adherends is relatively unimportant,since the model accounts for its effect on the stiffness,whereas the thickness of the adhesive should be adjusted to the value to be used in service.In the experimentation,particular attention should be paid to avoid the additional effects of details (spew,etc.)which increase the load carrying capacity,but are dif?cult to predict by a simple model and therefore alter the results with respect to the basic behaviour.
Acknowledgements
The ?nancial support of Italy’s Ministero dell’Universita
`e della Ricerca (Ministry of University and Research)is gratefully acknowledged.References
[1]Adams RD,Comyn J,Wake WC.Structural adhesive joints in engineering.
London:Chapman &Hall;1997.
[2]Kinloch AJ.Adhesion and adhesives.London:Chapman &Hall;1987.
[3]Van Straalen IJ,Botter H,Adams RD.In:Proceedings of the SAE VII,Bristol,
2004.p.36.
[4]Adams RD.In:Proceedings of the Euromech Colloquium 358—mechanical
behaviour of adhesive joints:analysis,testing and design,1997.p.3.
[5]Goglio L,Rossetto M,Dragoni E.In:Proceedings of the SAE VII.Bristol,2004.
p.76.
τp2 peak (MPa)
σp 2 p e a k (M P a )
Fig.8.Failure envelope in terms of peak shear and peel stresses based on 2%proof load (bars show standard deviation ranges).
τp2 peak (MPa)
σp 2 p e a k (M P a )
Fig.9.Plots of the failure hypotheses (see Table 4)applied to the shear and peel stresses based on 2%proof load.
10203040
5060Adherend thickness - h / Lap length - L (mm)
S t r e s s i n t . f a c t o r (M P a m m λ)
2 / 5
2 / 15 2 / 25 1.5/150
3 / 150
4 / 254 / 154 / 5Fig.10.Values of the stress intensity factor for the different series of tests (shear and peel stresses based on 2%proof load,bars show standard deviation ranges).
Table 4
Formulae of the failure hypotheses plotted in Fig.9.
Tresca ??????????????????
?s 2t4t 2p ?C
Von Mises ??????????????????
?s 2t3t 2p ?C
Maximum strain 1=2?e1àn Ts te1tn T???????????????????
s 2t4t 2p ?C
Maximum stress
s =2t????????????????????????es =2T2tt 2q ?C
Mohr
em à1=m Tes =2Ttem t1=m T????????????????????????
es =2T2tt 2q ?C Drucker–Prager (exponent 2)
s 2+3t 2+(m à1)C s ?mC 2
Where s peel stress,t shear stress (subscripts have been dropped for sake of brevity),C tensile stress limit,n Poisson’s ratio,m compressive stress limit to tensile stress limit ratio.
L.Goglio et al./International Journal of Adhesion &Adhesives 28(2008)427–435
434
[6]McGrath GC.Int J Adhes Adhes1997;17(4):339.
[7]Bogy DB.J Appl Mech1971;38(2):377.
[8]Wang CH,Rose LRF.Int J Adhes Adhes2000;20(2):145.
[9]Lazzarin P,Quaresimin M,Ferro P.J Strain Anal2002;37(5):385.
[10]Akisanya AR,Meng CS.J Mech Phys Sol2003;51(1):27.
[11]Abdel Wahab MM.J Adhes Sci Technol2000;14(6):851.
[12]Sheppard A,Kelly D,Tong L.Int J Adhes Adhes1998;18(6):385.
[13]Apalak KM,Engin A.J Adhes Sci Technol2003;17(14):1889.
[14]Du J,Salmon FT,Pocius AV.J Adhes Sci Technol2004;18(3):287.
[15]Jin ZH,Sun CT.Eng Fract Mech2005;72(12):1805.
[16]Volkersen O.Constr Me′tall1965;4:3.[17]Goland M,Reissner E.J Appl Mech1944;11:A17.
[18]Gleich DM,Van Tooren MJL,Beukers A.J Adhes Sci Technol2001;15(9):
1091.
[19]Dragoni E,Mauri P.Proc Inst Mech Eng L2002;216(L1):9.
[20]Bigwood DA,Crocombe AD.Int J Adhes Adhes1989;9(4):229.
[21]Gay D.Mate′riaux Composites.Paris:Herme`s;1997.
[22]Belingardi G,Goglio L,Tarditi A.Int J Adhes Adhes2002;22(4):273.
[23]Pirondi A,Nicoletto G.In:Proceedings of the XV conference of the Italian
Group on Fracture,Bari,2000.
[24]Da Silva LFM,Rodrigues TNSS,Figuereido MAV,De Moura MFSF,Chousal JAG.
J Adhes2006;82(11):1091.
L.Goglio et al./International Journal of Adhesion&Adhesives28(2008)427–435435