Single Lap Joints With Rounded Adherend Corners

Journal of Adhesion Science and Technology 25 (2011) 819ndash836brillnljast

Single Lap Joints with Rounded Adherend CornersStress and Strain Analysis

X Zhao a R D Adams ab and L F M da Silva clowast

a Department of Mechanical Engineering University of Bristol Queenrsquos Building University WalkBristol BS8 1TR UK

b Department of Engineering Science University of Oxford Parks Road Oxford UKc Departamento de Engenharia Mecacircnica Faculdade de Engenharia Universidade do Porto

Rua Dr Roberto Frias 4200-465 Porto Portugal

Received in final form 3 June 2010

AbstractOne of the major difficulties in designing adhesive lap joints is the stress singularity present at the adherendcorners at the ends of the overlap One way to overcome this problem is to assume that the corners havea certain degree of rounding The objective of the present study was to better understand the effect of thechange in the geometry of the adherend corners on the stress distribution and therefore on the joint strengthVarious degrees of rounding were studied and two different types of adhesives were used one very brittleand another which could sustain a large plastic deformation The study gives a detailed stress and straindistribution around the rounded adherends using the finite element method The major finding is that thestresses or strains in the adhesive layer of a joint with rounded adherend corners are finite In real jointsadherends generally have small rounded corners Consequently the model with small radius corners may beused to represent real adherendscopy Koninklijke Brill NV Leiden 2011

KeywordsSingle lap joint epoxy finite element stress analysis adherend rounding

1 Introduction

It is well known that there are stress singularities at the re-entrant corners at theends of the overlap of adhesive lap joints [1] Although analytical methods can pre-dict stresses inside the adhesive layer accurately for most of the overlap they fail togive the stresses at the ends of the overlap accurately Analytical and experimentalanalyses show that it is at the ends of the overlap where failure starts As a result

To whom correspondence should be addressed Tel +351225081706 Fax +351225081445 e-maillucasfeuppt

copy Koninklijke Brill NV Leiden 2011 DOI101163016942410X520871

820 X Zhao et al J Adhesion Sci Technol 25 (2011) 819ndash836

the stresses at the ends of the overlap are critical for joint failure These stressesare affected by the material properties of both the adherends and the adhesive Ithas been shown [2ndash4] that the stiffer the adherends the lower are the stresses forlsquosquare-endedrsquo joints Also the stresses are reduced if the adhesive is more flexible[5] The critical stresses may also be affected by the local geometry at the ends of theoverlap [6 7] In general adhesively-bonded joints have some adhesive squeezedout from the adhesive layer forming fillets at the ends of the overlap Because thesefillets are near the critical regions in the joints it has been shown that they affectthe local stresses near the ends of the overlap significantly [6 8ndash18] This effectcan only be analysed by the finite element (FE) method The bondline thicknesshas been shown to have only a little effect on the joint strength in the range of01ndash04 mm [1 19] Another factor affecting the critical stresses at the ends of theoverlap is the geometry of the adherend corners If the corners are sharp the stressesare singular (infinite) In that case a fracture mechanics approach is more appro-priate such as that proposed by Groth [20] based on the strength of the singularityHowever it has been shown that stresses are significantly reduced by rounding theadherend corners Adams and Harris [6] demonstrated theoretically and experimen-tally that the strength of single lap joints with rounded adherends with a toughenedadhesive increased substantially compared with joints with sharp adherend cornersFurthermore the adherend corners are usually not sharp in practice There is ingeneral a small amount of rounding at the adherend corner due to the productionprocess This may affect the stress distribution in the region of the adherend cornerand therefore the joint strength because stresses in this area are very sensitive tothe change in the geometry of the adherend corners In order to have a better under-standing of the effect of the change in the geometry of the adherend corners on thestresses and therefore on joint strength this paper is a further study on the effectof rounded adherend corners on the stresses and strains in a single lap joint

2 Geometry and Material Properties

21 Geometry

The geometry of the single lap joints analysed in this study is shown in Fig 1where the width of the joints is 25 mm The adhesive thickness was 025 mm andthe overlap length was 25 mm The adherend length outside the overlap was 75 mmFour geometries of adherend corners were analysed sharp small radius mediumradius and large radius Sharp corners exist only in theory In practice the radiusvaries but it is not larger than the lsquolargersquo radius studied here

22 Adherends

All the adherends used were aluminium alloys As can be seen from Fig 2 thestressndashstrain curve is almost linear after yielding For modelling adherend plasticdeformation a straight line was drawn beyond yielding which was believed to be

X Zhao et al J Adhesion Sci Technol 25 (2011) 819ndash836 821

Figure 1 Aluminiumepoxy single lap joints with different degrees of rounding (a) Joint geometry(all dimensions in mm) (b) Sharp corner (c) Small radius (d) Medium radius (e) Large radius

accurate enough to represent the stress hardening part of the aluminium for thecurrent purpose

The Youngrsquos modulus determined from the stressndashstrain curve is close to 68 GPaTo verify this test a dynamic test was also carried out to measure the Youngrsquosmodulus Dynamic tests to measure Youngrsquos modulus and Poissonrsquos ratio have beenwidely used and shown to be accurate [21 22] This dynamic test gave a slightlyhigher value of Youngrsquos modulus of 723 GPa so an intermediate value of 70 GPawas used in the analysis A value of 033 was used for Poissonrsquos ratio [23] Themain properties of the aluminium alloy used are summarised in Table 1

23 Adhesives

Two adhesives were used in the present work Ciba-Geigy MY750 (a diglycidylether of bisphenol A) with hardener HY906 (anhydride) and the same adhesive butrubber-toughened with carboxyl terminated butadiene nitrile (CTBN) rubber Theseadhesives are designated as MY750 and CTBN respectively in this work and Fig 3shows their uniaxial tensile behaviour It should be noted that adhesive MY750 withhardener HY906 is brittle so it has been treated as a linear elastic material whileCTBN is a ductile adhesive The main properties are summarised in Table 1


Figure 2 Tensile stressndashstrain curve for aluminium adherends

Table 1Material mechanical properties

Adhesive Adhesive AluminiumMY750HY906 CTBN

Youngrsquos modulus (GPa) 28 25 70Poissonrsquos ratio 04 037 033Yield stress (MPa) 40 250Failure stress (MPa) 84 62 323Failure strain () 3 148

3 Analysis Methods

In this study the FE method was used for both linear and non-linear analyses Tosimulate the test conditions both ends of the joints were constrained so as not to ro-tate as shown in Fig 1(a) Provided the adherends are long enough these boundaryconditions have little effects on the stress distribution within the adhesive with largedisplacement models A 2D plane strain model was used with 8-noded isoparamet-ric elements for both the adherends and the adhesive Two steps were involvedin analysing the corner areas to obtain accurate stress or strain distributions Thefirst run was based on a quite coarse mesh for the whole joint followed by a finermesh around the adherend corner with the displacements from the first analysis


Figure 3 Tensile stressndashstrain curves for adhesives MY750 and CTBN

used as boundary conditions In the linear analysis a large displacement model wasused with both adherends and adhesive being linearly elastic In the elasticndashplasticanalysis both the adherends and the adhesive were allowed to yield as well as to ex-perience large displacements Analysis was performed for the four different degreesof adherend rounding shown in Fig 1(dndashe) In the analyses with a coarse mesh sixuniform elements were used across the adhesive layer thickness and across each ofthe adherends In the subsequent refined analyses ten graded elements were gen-erated across the adhesive layer To avoid large aspect ratios in the adhesive layer50 elements were used along the overlap in the refined analysis However such anumber of elements along the overlap caused the aspect ratios in the adherends tobe high Fortunately there was no yield in the adherends near the corners so thatthe large aspect ratios were still acceptable with the 8-noded isoparametric elementsused The refined mesh for the stress analysis in the sharp adherend corner region isshown in Fig 4 Similar meshes were used for the other rounded adherend cornersIncremental solutions with a series of loads were performed for both linearly-elasticand elasticndashplastic analyses In the elasticndashplastic analysis a modified von Mises[24] yield criterion was included which takes into account the effect of hydrostaticstress components on the yielding of the adhesive

4 Results

41 Linear Elastic Analyses with Adhesive MY750

The peel (σyy) shear (σxy) longitudinal (σxx) and maximum principal stresses inthe adhesive were plotted at the Gauss points close to the unloaded adherend at adistance of 0001 mm from the adherend as shown in Figs 5ndash8 for the four types


Figure 4 A typical fine mesh around the adherend corner

of joints It may be seen that almost no difference exists for most of the overlapexcept near the adherend corners for joints with a small radius and sharp adherendcorners This is because the stresses are not defined at the sharp corners and thesmall rounding is very localised to the adherend corner The stresses should in the-ory be infinite at the sharp corners [6 25] However stresses from this analysiswere finite because the mesh used here was of finite size To verify the existenceof the stress singularity Fig 9 shows a logarithmic plot of the stresses against thelogarithmic distance from the adherend corner across the adhesive layer thicknessThe almost linear relationship in the figure indicates the singular nature of the dis-tribution of the stresses However it should be noted that the stresses in the finalelement adjacent to the singular point are not reliable because a singular elementwas not used there The singular nature of the stresses covers almost the wholeadhesive layer as seen from Fig 9 Another feature of the stresses with the smallrounded corner is that a larger area with high stresses exists than with sharp cor-ners in which the stresses are high only in the very small area close to the sharpcorner (see Figs 5ndash8) The stresses around the adherend corner for the small radius


Figure 5 Peel stresses in adhesive MY750 with a 20 kN applied load close to the unloaded adherend

are finite and much smoother than those around the sharp corner Furthermore thepositions with the highest stresses move inside the overlap for the small radius ad-herend corner Very smooth stress variations exist for the large degrees of roundingand the area with the highest stresses is much larger than that with small or sharpradius corners This clearly shows that the stress concentration is not so severe forthe large degrees of rounding It should be noted that the longitudinal stresses aresignificant (Fig 7) In fact the longitudinal stresses for all the radii and for the sharpadherend corners are much higher than either the peel or the shear stresses plotted atthe same positions In lap joints with fillets the longitudinal stresses near the endsof the overlap are as important as the peel stresses because the fillets transfer partof the load and this results in large longitudinal stresses near the adherend ends Therounding of the adherend corners enhances the longitudinal stresses as can be seenfrom Fig 7 which shows that the longitudinal stress increases dramatically in theregion of the rounded adherend corner but remains almost constant in the middleof the overlap The stress state in the fillet is predominantly in tension at an angleof approximately 45 to the applied load Consequently the longitudinal stressesplay an important role in the failure of joints with fillets and rounded corners Themaximum principal stresses at the Gauss points close to the unloaded adherend areplotted in Fig 8 It may be seen that the distribution of the maximum principalstresses is similar to that of the other stresses especially the longitudinal stressesplotted at the same positions and these stresses increase sharply inside the rounded


Figure 6 Shear stresses in adhesive MY750 with a 20 kN applied load close to the unloaded ad-herend

adherend corners which is a typical feature of the longitudinal stresses plotted atthe same positions This also means that the longitudinal stresses contribute signif-icantly to the maximum principal stresses

The maximum principal stresses inside the fillets are plotted in Fig 10 It may beseen that stresses inside the fillets near the fillet edges hardly change for differentdegrees of rounding It should be noted that the stresses near the edge of the fillet(point E in Fig 10) are high where a weak singularity exists The stresses at theedges of the fillet increase sharply and the magnitudes are difficult to determinewhen the geometry consists of sharp edges It may be seen that the stresses at theedges of the fillet are higher than those near the unloaded adherend corners formedium and large radius adherends Therefore for large radius adherends the filletedges may be important for failure predictions In reality there is usually a wettingmeniscus which smoothes the 45 angle (rounded edge) at the edge of the fillet tozero and effectively removes the stress concentration [10]

The peel stresses at the Gauss points close to the loaded adherend are plottedin Fig 11 It may be seen that all the stresses peak in the region opposite to theunloaded adherend corner although there is no re-entrant corner close to the loadedadherend Figure 11 shows that there is little difference in the peel stresses betweensharp and small radius adherend corners This is easy to understand because nore-entrant corner exists close to the loaded adherend and the change in adherend


Figure 7 Longitudinal stresses in adhesive MY750 with a 20 kN applied load close to the unloadedadherend

geometry is a significant distance away from this region It may be seen by com-paring Figs 11 and 5 that the peel stresses near the loaded adherend are much lowerthan those near the unloaded adherend This means that the area around the un-loaded adherend corner is more important than the adjacent area close to the loadedadherend for joint strength predictions for elastic adherends The other stresses havesimilar features

To examine further the nature of the maximum principal stresses their direc-tions are drawn in Fig 12 for the small radius adherend corner It may be seen thatthe directions of the maximum principal stresses are approximately normal to theinterface between the unloaded adherend and the adhesive In comparison the di-rections of the maximum principal stresses around the unloaded adherend cornerfor a joint with sharp adherend corners are oriented about 45 to the interfaces as iswell documented in the literature [8] In general adhesives are weakest in tensionAt the interface close to the small radius adherend corner the maximum principalstresses act approximately normal to the interface This direction is the worst asfar as the joint strength is concerned In a joint with sharp adherend corners how-ever the tensile forces acting at the interface are either peel or longitudinal stresseswhose magnitudes are much smaller than those of the maximum principal stressesAs a result the direction of the maximum principal stresses for a joint with sharpcorners is less critical than that for a joint with small radius corners although the


Figure 8 Maximum principal stresses in adhesive MY750 with a 20 kN applied load close to theunloaded adherend

magnitudes of the former stresses are much larger than the latter As a result thejoint strength does not improve much in contrast to the reduction in the peak valuesof stresses for the rounded adherend corners

42 ElasticndashPlastic Analyses

A B-spline curve fitting procedure was used to represent the behaviour of the mate-rials For the CTBN rubber toughened epoxy the modified von Mises yield criterion[24] was used to take into account the hydrostatic stress components which can beexpressed as

J1(S minus 1) + J 21 (S minus 1)2 + 12J2

2S= Y 2

T (1)

where S is ratio of the yield stress in compression to the yield stress in tension YT isthe yield stress derived from the uniaxial tensile test on the material and

J1 = σx + σy + σz

J2 = 1

2

(σx minus J1

3

)2

+(

σy minus J1

3

)2

+(

σz minus J1

3

)2

+ τ 2xy + τ 2

xz + τ 2yz


Figure 9 Stresses from the sharp corner across the adhesive thickness for adhesive MY750 with a20 kN applied load

where σi (i = x y z) and τj (j = xy xz yz) are stress components For the highstrength aluminium alloy the von Mises yield function

f (σ ) = radic3J2 minus YT = 0 (2)

was used where J2 is defined as aboveA load of 20 kN was applied throughout the analysis Again the analysis involves

two steps a coarse mesh analysis followed by a refined mesh analysis The resultsare presented below

First the plastic energy density is plotted close to the unloaded adherend asshown in Fig 13 A similar trend of the plastic energy density exists as that of thestresses in the linear elastic analysis A high plastic energy density was apparentaround the corner this should in theory be infinite for a sharp corner as discussedby Adams and Harris [6] The plastic energy density with the small radius cornerwas finite and the position of the peak value moved inside the overlap occurringafter the edge of the rounding (see Fig 13) As for the stresses in the linear elasticanalysis the area with a large plastic energy density for the small rounded cornerswas much larger than that in the sharp corner For joints with medium and large ra-dius corners the magnitudes of the plastic energy density were much reduced andthe positions of the peak values moved further inside the overlap occurring afterthe edge of the rounding (see Fig 13) Again the larger the rounding the larger the


Figure 10 Maximum principal stress comparison inside the fillet for adhesive MY750 with a 20 kNapplied load

area in which the plastic energy density reached its peak values and the further theposition moved inside the overlap

The plastic energy density inside the fillet is shown in Fig 14 It can be seen thatthe majority of the fillet did not yield much with applied load However yieldinghappened at the edge of the fillet (point E in Fig 14) where there was a stressconcentration in the stress analysis for linear elastic materials This shows that theregions around the unloaded adherend corners and the fillet edges are importantareas to examine for failure analysis However as discussed above there is usuallya wetting meniscus which smoothes the 45 angle (rounded edge) at the edge of thefillet to zero and effectively removes the stress concentration [10]

Strains were also studied to analyse the failure mechanisms Because the straindistribution is very similar to that of the plastic energy density no plot with re-gards to the strain magnitudes has been made here However the directions of themaximum principal strains with a small rounded corner are presented in Fig 15which gives more information on the failure modes It may be seen that as forthe directions of the maximum principal stresses in the linear elastic analyses thedirections of the maximum principal strains were approximately normal to the in-terface between the adherend and the adhesive for small radius corners It can alsobe seen that the magnitudes of the maximum principal strains close to the unloadedadherend were much larger than those at other places in the adhesive layer The


Figure 11 Peel stresses in the adhesive MY750 with a 20 kN applied load close to the loaded ad-herend

magnitudes of the other principal strains in the same plane were also significantbut they were in compression This means that the shear strains at these points werevery significant compared with elongational strains As discussed in the linear elas-tic analysis stresses and strains are the highest near the interface so the joint ismost likely to fail there The action of the maximum principal strains is mainly re-sponsible for the joint failure The directions of the maximum principal strains withsharp adherend corners were almost the same as those with small radius corners andwere approximately normal to the interface between the adhesive and adherend Itshould be noted that the maximum principal strains were not actually normal to theinterface between the adherend and adhesive for joints bonded with sharp adherendcorners The direction of the maximum principal stresses for a joint with sharpcorners is less critical than that for a joint with small radius corners however themagnitude of the peak value of the maximum principal strain with sharp adherendcorners is much higher than that with small radius corners As a result like the caseof the elastic analysis the joint strength may not improve much in contrast to thereduction in the peak values of stresses for the rounded adherend corners

Finally the shear and peel strains across the adhesive layer are shown in Fig 16It may be seen from the figure that the shear strains are much larger than the peelstrains close to the unloaded adherend The strains remain reasonably constant forthe majority of the adhesive layer However the shear strains peak sharply close to


Figure 12 Pattern of the principal stresses in the adhesive MY750 for a 20 kN applied load aroundthe small (025 mm) rounded adherend corner

the unloaded adherend and are much larger than the peel strains The change in peelstrains is not as sharp as that in the shear strains

5 Conclusions

The following conclusions can be drawn from the above analyses

1 The stresses or strains in the adhesive layer of a joint with rounded adherendcorners are finite and can be determined In real joints adherends generally havesmall rounded corners Consequently the model with a small radius corner maybe used to represent real adherends However the magnitude of the rounding ofthe adherend needs to be determined

2 Rounding the adherend corners reduces the magnitudes of the stresses or strainsaround these corners The larger the rounding the larger the reduction in themagnitudes of the stresses Furthermore the peak values of the stresses orstrains move inside the overlap for joints with large rounded adherend cornersthe larger the rounding the further inside the overlap the peak values move

3 Rounding of the adherend corners worsens the stress conditions around the ad-herend corners because the direction of the maximum principal stress is normalto the adherend although the peak values of the stresses and strains are much


Figure 13 Plastic energy density in the CTBN adhesive with a 20 kN applied load close to theunloaded adherend

reduced Increasing the radius of the adherend corners results in a large area ofthe interface between the adherend and the adhesive which experiences largestresses or strains acting approximately normal to the interface between theadherend and the adhesive It also reduces the parallel portion (no rounding)between adherends which concentrates the load For example a 32 mm round-ing with a 25 mm overlap leaves a 19 mm parallel portion

4 Rounding the adherend corners is restricted in that the stresses or strains at theedges of the fillets do not change much for different degrees of rounding For alarge degree of rounding the edges of the fillets may become the critical pointsfor joint failure

5 The stress or strain concentrations are very much localised to the unloaded ad-herend corners while the stresses and strains away from the corners are quitesmooth This means that the strength does not change much for joints with sharpand small radius adherend corners

6 Finally the longitudinal stresses (or strains) along the overlap which are gener-ally ignored by closed-form solutions are more significant than the peel stresses(or strains) around the rounded adherend corners and they should be taken intoaccount in failure studies However it is the maximum principal stresses whichare the most responsible for joint failure because they take into account all the


Figure 14 Plastic energy density inside the fillet for the CTBN adhesive with a 20 kN applied load

Figure 15 Pattern of the principal strains in the CTBN adhesive with a 20 kN applied load around thesmall rounded adherend corner


Figure 16 Shear strain and peel strain for the CTBN adhesive with a 20 kN applied load across theadhesive layer

stress components Shear strains for the elasticndashplastic analyses are much largerthan either peel or longitudinal strains around the rounded adherend corners

References

1 R D Adams J Comyn and W C Wake Structural Adhesive Joints in Engineering 2nd ednChapman amp Hall London (1997)

2 O Volkersen Luftfahrtforschung 15 41 (1938)3 M Goland and E Reissner J Appl Mech 66 A17 (1944)4 L J Hart-Smith NASA Contract Report NASA CR-112236 (1973)5 L F M da Silva and R D Adams Int J Adhesion Adhesives 27 362 (2007)6 R D Adams and J A Harris Int J Adhesion Adhesives 7 69 (1987)7 A Oumlchsner L F M da Silva and R D Adams in Modeling of Adhesively Bonded Joints

L F M da Silva and A Oumlchsner (Eds) pp 131ndash154 Springer Heidelberg (2008)8 R D Adams and N A Peppiatt J Strain Anal 9 185 (1974)9 A D Crocombe and R D Adams J Adhesion 13 141 (1981)

10 R D Adams R W Atkins J A Harris and A J Kinloch J Adhesion 20 29 (1986)11 L Dorn and W Liu Int J Adhesion Adhesives 13 21 (1993)12 M Y Tsai and J Morton Composite Struct 32 123 (1995)13 T P Lang and P K Mallick Int J Adhesion Adhesives 18 167 (1998)14 G Belingardi L Goglio and A Tarditi Int J Adhesion Adhesives 22 273 (2002)15 M K Apalak and A Engin J Adhesion Sci Technol 18 529 (2004)


16 L Andreassi R Baudille and M E Biancolini Int J Adhesion Adhesives 27 458 (2007)17 L F M da Silva and R D Adams Int J Adhesion Adhesives 27 227 (2007)18 J Deng and M M K Lee Composites Part B 39 731 (2008)19 L F M da Silva T N S S Rodrigues M A V Figueiredo M F S F de Moura and

J A G Chousal J Adhesion 82 1091 (2006)20 H L Groth Int J Adhesion Adhesives 5 19 (1985)21 R D Adams and P Cawley in Research Techniques in Non-destructive Testing R S Sharp (Ed)

pp 303ndash360 Academic Press London (1985)22 L F M da Silva and R D Adams J Adhesion Sci Technol 19 109 (2005)23 J M Gere Mechanics of Materials 5th edn BrookesCole Pacific Grove CA (2001)24 R S Raghava R Cadell and G S Y Yeh J Mater Sci 8 225 (1973)25 V L Hein and F Erdogan Int J Fracture Mech 7 317 (1971)


the stresses at the ends of the overlap are critical for joint failure These stressesare affected by the material properties of both the adherends and the adhesive Ithas been shown [2ndash4] that the stiffer the adherends the lower are the stresses forlsquosquare-endedrsquo joints Also the stresses are reduced if the adhesive is more flexible[5] The critical stresses may also be affected by the local geometry at the ends of theoverlap [6 7] In general adhesively-bonded joints have some adhesive squeezedout from the adhesive layer forming fillets at the ends of the overlap Because thesefillets are near the critical regions in the joints it has been shown that they affectthe local stresses near the ends of the overlap significantly [6 8ndash18] This effectcan only be analysed by the finite element (FE) method The bondline thicknesshas been shown to have only a little effect on the joint strength in the range of01ndash04 mm [1 19] Another factor affecting the critical stresses at the ends of theoverlap is the geometry of the adherend corners If the corners are sharp the stressesare singular (infinite) In that case a fracture mechanics approach is more appro-priate such as that proposed by Groth [20] based on the strength of the singularityHowever it has been shown that stresses are significantly reduced by rounding theadherend corners Adams and Harris [6] demonstrated theoretically and experimen-tally that the strength of single lap joints with rounded adherends with a toughenedadhesive increased substantially compared with joints with sharp adherend cornersFurthermore the adherend corners are usually not sharp in practice There is ingeneral a small amount of rounding at the adherend corner due to the productionprocess This may affect the stress distribution in the region of the adherend cornerand therefore the joint strength because stresses in this area are very sensitive tothe change in the geometry of the adherend corners In order to have a better under-standing of the effect of the change in the geometry of the adherend corners on thestresses and therefore on joint strength this paper is a further study on the effectof rounded adherend corners on the stresses and strains in a single lap joint

2 Geometry and Material Properties

21 Geometry

The geometry of the single lap joints analysed in this study is shown in Fig 1where the width of the joints is 25 mm The adhesive thickness was 025 mm andthe overlap length was 25 mm The adherend length outside the overlap was 75 mmFour geometries of adherend corners were analysed sharp small radius mediumradius and large radius Sharp corners exist only in theory In practice the radiusvaries but it is not larger than the lsquolargersquo radius studied here

22 Adherends

All the adherends used were aluminium alloys As can be seen from Fig 2 thestressndashstrain curve is almost linear after yielding For modelling adherend plasticdeformation a straight line was drawn beyond yielding which was believed to be





23 Adhesives







3 Analysis Methods





4 Results
























2S= Y 2

T (1)



J2 = 1

2

(σx minus J1

3

)2

+(

σy minus J1

3

)2

+(

σz minus J1

3

)2

+ τ 2xy + τ 2

xz + τ 2yz




















5 Conclusions

















References













23 Adhesives







3 Analysis Methods





4 Results
























2S= Y 2

T (1)



J2 = 1

2

(σx minus J1

3

)2

+(

σy minus J1

3

)2

+(

σz minus J1

3

)2

+ τ 2xy + τ 2

xz + τ 2yz




















5 Conclusions

















References














3 Analysis Methods





4 Results
























2S= Y 2

T (1)



J2 = 1

2

(σx minus J1

3

)2

+(

σy minus J1

3

)2

+(

σz minus J1

3

)2

+ τ 2xy + τ 2

xz + τ 2yz




















5 Conclusions

















References












4 Results
























2S= Y 2

T (1)



J2 = 1

2

(σx minus J1

3

)2

+(

σy minus J1

3

)2

+(

σz minus J1

3

)2

+ τ 2xy + τ 2

xz + τ 2yz




















5 Conclusions

















References






























2S= Y 2

T (1)



J2 = 1

2

(σx minus J1

3

)2

+(

σy minus J1

3

)2

+(

σz minus J1

3

)2

+ τ 2xy + τ 2

xz + τ 2yz




















5 Conclusions

















References



























2S= Y 2

T (1)



J2 = 1

2

(σx minus J1

3

)2

+(

σy minus J1

3

)2

+(

σz minus J1

3

)2

+ τ 2xy + τ 2

xz + τ 2yz




















5 Conclusions

















References
























2S= Y 2

T (1)



J2 = 1

2

(σx minus J1

3

)2

+(

σy minus J1

3

)2

+(

σz minus J1

3

)2

+ τ 2xy + τ 2

xz + τ 2yz




















5 Conclusions

















References



















2S= Y 2

T (1)



J2 = 1

2

(σx minus J1

3

)2

+(

σy minus J1

3

)2

+(

σz minus J1

3

)2

+ τ 2xy + τ 2

xz + τ 2yz




















5 Conclusions

















References















2S= Y 2

T (1)



J2 = 1

2

(σx minus J1

3

)2

+(

σy minus J1

3

)2

+(

σz minus J1

3

)2

+ τ 2xy + τ 2

xz + τ 2yz




















5 Conclusions

















References




























5 Conclusions

















References





















5 Conclusions

















References
















5 Conclusions

















References












5 Conclusions

















References





















References















References












References













Documents

Single Lap Joints With Rounded Adherend Corners