3-1. Introduction
Basing on the preceding descriptions, the theoretical model is developed and the closed-form solutions also are found. In this present study both adhesively bonded adherends are subjected to a concentrated force and the peel and shear stress distributions in the adhesive layer joining the two adherends are examined as shown in Fig. 3.1. Such stress distributions are affected by geometric conditions, including the thicknesses and Young’s modulus adherends and the length, thickness, and Young’s modulus of the adhesive layer, as well as by the action point of the concentrated force. These stress distributions are investigated and the closed-form solutions are obtained by symbolic manipulation in the following sections.
Z
h
L L
X c
P d c
h
1h
2a
1 2
Adhesive layer Upper adherend
Lower adherend
Fig. 3.1The sketch showing two adherends bonded by an adhesive layer.
3-2. Mathematical Model
In this model the two adherends – the upper adherend and lower adherend – are bonded by an adhesive layer with the center coinciding with the origin of the coordinate system. (see Fig.
3.1). The thicknesses of the upper adherend, lower adherend, and adhesive layer are denoted
by h1, h2, and ha, respectively. Their lengths are represented, respectively, by 2c, (L1+L2), and 2c. The lower adherend is subjected to a concentrated force P under the pin-pin boundary conditions.
The governing equations for this study are based on the following assumptions:
(a) The transverse displacements of both the upper adherend and of the lower adherend subjected to the concentrated force P are much smaller than their dimensions, and their transverse displacements are presumed to be linear and small.
(b) The upper adherend and the lower adherend deform under a plane-stress condition; in other words, the plane section remains plane and the deformation of the cross sections is correspondingly normal to the neutral surfaces.
(c) The variations in both longitudinal and transverse displacements are linear in the adhesive layer.
(d) In the adhesive layer, the stress resulting from the longitudinal force is ignored when compared with stresses in the upper adherend and lower adherend. [14].
Based on the preceding assumptions, the governing equations are derived as follows. First, the lower adherend is divided into four segments whose ranges are −L1 ≤x≤−c ,
d , x c≤ ≤−
− −d ≤x≤c , and c≤x≤L2 , respectively on the x-axis. Next, the upper adherend is divided into two segments whose ranges are −c≤x≤−dand on the x-axis. Finally, the adhesive layer is also divided into two segments, each of which has the same range as the corresponding segment in the upper adherend.
c x d ≤ ≤
−
3-2-1. Bending moment, Shear force, and Longitudinal Force in the Upper Adherend and Lower adherend
The free-body diagram for the first segment (−L1 ≤x≤−c) is shown in Fig. 3.2– where NL, and FL represent the longitudinal force and reaction force, respectively, of the left-end support – and the bending moment, shear force, and longitudinal force of the first segment’s right-hand section are denoted by , and , in which the subscript refers to the first segment of the lower adherend. According to force and moment equilibrium equations, the bending moment , the shear force , and the longitudinal force can be derived in terms of , and as
x
x Q
M1 , 1 N1x 1x
M1x Q1x N1x
NL FL :
), ( 1
1 F L x
M x =− L + (3.1)
1x FL.
Q = (3.2)
and
1x NL,
N = (3.3)
Fig. 3.2 Free-body diagram for the first segment −L1 ≤ x≤−c, of lower adherend.
Similarly, in the free-body diagrams for the second, third, and fourth segments (displayed in Figs. 3.3, 3.4 and 3.5, respectively), the bending moment, shear force, and longitudinal
force of the section for the ith (i=2~4) segment, denoted by , and , respectively, can be written as shown below.
ix
ix Q
M , Nix
Fig. 3.3 Free body diagram for the second segment −c≤x≤−d, of lower adherend.
Fig. 3.4 Free body diagram for the third segment −d ≤ x≤c, of lower adherend
Fig. 3.5 Free body diagram for the fourth segment of c≤x≤L2, of lower adherend.
Specifically, the bending moment, shear force, and longitudinal force of the second segment’s right-hand section (−c≤ x≤−d ) are as follows:
2 , )
( 1 2 2 2
2
∫ ∫
− −
+ +
+
−
= x
c
x
c a a
L
x h dx
dx x x L F
M σ τ (3.4)
2 ,
Similarly, the bending moment, shear force, and longitudinal force of the third segment’s right-hand section (−d ≤x≤c) are
Lastly, the bending moment, shear force, and longitudinal force of the fourth segment’s left-hand section (c≤x≤L2) are
The upper adherend, whose range is −c≤ x≤c on the x-axis, must be divided into two segments whose ranges are−c≤x≤−d and −d ≤x≤c, respectively. Free-body diagrams of these two segments are presented in Figs. 3.6 and 3.7. The bending moment, shear force, and longitudinal force of the right section of the ith segment of the upper adherend, denoted as
, and , respectively, are as follows:
i
i Q
M , Ni
∫ ∫
− −
+
= x
c
x
c ai ai
i h dx
dx x
M σ τ
2
1 i=2 or 3, (3.13)
dx Q
x
c ai
i
∫
−
= σ i=2 or 3, (3.14)
dx N
x
c ai
i
∫
−
= τ i=2 or 3. (3.15)
When i = 2, the range of the upper adherend is −c≤ x≤−d(i.e. the first segment of the upper adherend). However, when i = 3, the range of the upper adherend is (i.e. the second segment of the upper adherend).
c x d ≤ ≤
−
Fig. 3.6 Free body diagram for the first segment −c≤x≤−d, of upper adherend.
Fig. 3.7 Free body diagram for the second segment −d≤x≤c, of upper adherend.
3-2-2. Relationship between Displacement and Stress
When the range of the adhesive layer for bonding the upper adherend to the lower adherend is −c≤ x≤c, the equations adopted from Ref. [5] are simplified by the small strain (i.e. the slope of the beam = 0) and are expressed as follows:
a
where and represent longitudinal and transverse displacements when i = 2 represents the first segment ( ) of the upper adherend and i = 3 represents its second segment ( ). In equations (3.16) – (3.17), when i = 2, transverse and longitudinal displacements for the second segment of the lower adherend are denoted by , and when i = 3, those for the third segment of the lower adherend are denoted by
These variables, which are either functions of both x and z or only a function of x, are
expressed as longitudinal displacement ui(h2a)of the upper adherend and the longitudinal displacement
) ( h2a
uix − of the lower adherend are then represented as a function of x and are expressed as either modulus, Young’s modulus, and the thickness of the adhesive layer.
a,
G Ea ha
The stresses of the upper adhered and lower adherend are expressed as follows:
dx
The stresses of the lower adherend are expressed in
dx duix
ix =
σ i = 1, 2, 3, or 4, (3.19)
when i =1, 2, 3 or 4 represents the first, second, third, or forth segments of the lower adherend.
3-2-3. Relationships among Displacement, Longitudinal Force, and Bending Moment
Following the beam theory, the transverse displacements of the upper adherend and of the lower adherend are written as shown below:
wi adherend in plane stress.
The longitudinal displacements of the upper adherend , and of the lower adherend can then be written as follows:
ui uix
To obtain the longitudinal displacements, transverse displacements, and slopes of the first and fourth segments in the lower adherend, Eqs. (3.1) – (3.3) and (3.10) – (3.12) are substituted into Eqs. (3.20) and (3.22) which are integrated over x to produce the following expressions:
where are constants. The subscripts i and k of represent the ith segment of the lower adherend and the index of the constants.
cik cik equations for the secondand third segments of the lower adherend:
⎭⎬
Using the previous procedure, the formulas for the first and second segments of the upper adherend can be obtained as follows:
⎭⎬
3-3. Non-dimensionalization and Symbolic manipulation
To regulate the magnitude of some parameters and illustrate clearly the detailed relationships among them, the parameters are non-dimensionalized and are listed in Table 1.
For the first and fourth segments of the lower adherend, Eqs. (3.24)–(3.29) may be non-dimensionalized and rearranged as follows:
,
,
Eqs. (3.30)–(3.33) can then be rewritten in the matrix form as:
⎥⎥
β , and other parameters are
a
The characteristic equation, det| AD |=0, of coupled differential equations (40) can then be derived as follows:
, transverse displacements wi of the upper adherend are written in the following form:
),
As the complete solutions of the model are extremely complex, this study employed Mathematica’s symbolic manipulation to solve ,~
u i ~ ,
and easily keyed in error, they taken from the output results of Mathematica package are pasted in Appendix A. To prove these analytical solutions are correct, they are once again substituted into the system differential equation (3.41), which shows and to be equal to zero.
4
ci ci5
The analytical solutions ~, u i ~ ,
uix , and (from Appendix: Eqs. A.1, A.2, A.15 and Eq.
3.40), which are substituted into Eqs. (3.16–3.17), and the adhesive layer’s non-dimensional
peel and shear stresses
wi wix
are revealed in Appendix B).
The analytical stress solutions, σai and τai, (from Appendix: Eqs. B1, B2)are substituted into Eqs. (3.4–3.9, 3.13–3.15). The shear force
P
Qi =Qi , the bending moment
Pc Mi Mi
=2 , and the longitudinal force
P
=2 , and the longitudinal force
P
Nix = Nix for the
lower adherend (all listed in Table 3.1 are also expressed in terms of cij, S, C, Ch, Sh, ,
Ch1 and Sh . These equations are shown in Appendix C. 1
The preceding bending moments , and longitudinal forces , are substituted
into Eqs (3.18)-(3.19). The non-dimensional stresses,
Mi Mix Ni Nix
and lower adherends can be found in terms of the coefficients of bending moments , and of longitudinal forces , .
Mi
Mix Ni Nix
For the first and fourth segments, Eqs. (3.1–3.3, 3.10–3.12) are rewritten and non-dimensionalized. The resulting non-dimensional bending moment, shear force, and longitudinal force (M1x, Q1x, N1x , M4x, Q4x, and N4x), are formulated as shown
),
Table 3.1 The non-dimensional terms and equations for upper adherend, adhesive layer and lower adherend.
Non-dimensional terms for upper adherend
Equation Non-dimensional terms for lower adherend
Equation
Thickness ratio
ha
h1
1 =
β Thickness ratio
ha
h2
2 = β
Thickness to length ratio
c h 2
1 1=
γ Thickness to length ratio
c
Elastic modulus
Ga
E E
* 1
1 = Elastic modulus
Ga
E E
* 2 2 = Shear force
P Longitudinal force
P
Ni = Ni Longitudinal force
P Nix = Nix
Normal stress
P
Non-dimensional terms for adhesive layer
Equation
Shear stress
P c ai
ai
τ = 2 τ
Elastic modulus
a
3-4. Constraint and Boundary Conditions
The constraint and boundary conditions for this study, shown in Fig. 3.1, can be identified and described in the following manner.
At the left-end pin support (x=−L1) of the lower adherend, there are two boundary conditions, i.e. zero transverse displacement and zero longitudinal displacement of the lower adherend. At x=−c , there are eight constraint conditions, six of which are continuity conditions for the lower adherend. That is, at junction point (x=−c) between the first and second segments of the lower adherend, both segments must have the same values of transverse displacement, slope, bending moment, shear force, longitudinal force, and longitudinal displacement. The other two conditions at x =−c are that both the bending moment and longitudinal force of the upper adherend must be equal to zero.
At junction point ( ) between the second and third segments, there are eleven conditions, eight of which are continuity conditions. First, in both upper and lower adherends, both segments must again have the same values of transverse displacement, slope, bending moment, and longitudinal displacement. Three other conditions are written as follows: (i) the total shear force in the left neighborhood of the junction point ( ) is
d x=−
− −
= d
x FL P~
~ /
, (ii) the total shear force in the right neighborhood of the junction point (x= d− +) is
(
FL P)
P/ ~
~ − ~ , and
(iii) the total longitudinal force has the same value at junction point ( ) for both second and third segments.
d x=−
The model also is subjected to eight constraint conditions at x= . At the junction point c ( ) between the third and fourth segments of the lower adherend, both segments must have the same values of transverse displacement, slope, bending moment, shear force, longitudinal force, and longitudinal displacement. In addition, the bending moment and longitudinal force of the upper adherend must be equal to zero.
c x =
At the right-end pin support (x=L2) of the lower adherend, there are again two boundary conditions, i.e. the transverse displacement and longitudinal displacement for the lower adherend must be zero.
Overall, the number of boundary and constraint conditions totals 31, equal to the number
of unknown constants. The unknown constants include , and
– where subscript i is equal to 2 or 3, k ranges from 1 to 3, and j ranges from 1 to 12 – but and (found in the preceding descriptions) equal zero. and are the unknown constants of the longitudinal displacements and result from substituting the analytical solutions
k k ai ai
ij c c c c
c , 1, 2, 1 , 4
NL 4
ci ci5 cai1 cai2
~, u i ~ ,
uix and (from Appendix: Eqs. A.1, A.2, A.15 and Eq. 3.46) into the integrated Eqs. (3.29) and (3.31).
wi wix
Imposing 31 constraint and boundary equations on the analytical solutions through symbolic manipulation produces 31 system equations expressed in the following matrix form.
[A][C]=[B] (3.49)
where matrix [A] has 31 rows and 31 columns, denoted by [A]31×31, and matrices [B] and [C]
have 31 rows and 1 column, denoted by [B]31×1 and [C]31×1, respectively. The elements in matrix[C]31×1 consist of 31 unknown constants, cij, cai1, cai2, c1k, c4k , and NL. [A]31×31, [B]31×1 and [C]31×1 are shown in the Appendix D
The matrix [C]31×1 is solved using Mathematica’s SVD algorithm because matrix [A] has a greater variation in the magnitude of the matrix elements . If one eigenvalue in the characteristic equation (3.39) is large, some elements of matrix [A] that involve
, 1
, Ch Sh
Sh , and Ch become much larger. However, the magnitude of those elements in 1
matrix [A] that do not involve Sh, Ch, Sh1 and Ch is much smaller. Thus, there is a 1
discrepancy in the magnitude of matrix [A] elements exceeding the exponential order of 10s.
In addition, because of computer truncation errors, the inverse of matrix [A] cannot be obtained by the adjoint method. Therefore, matrix [C]31×1 is solved by SVD algorithm and the non-dimensional peel stress and shear stress in the adhesive layer can be obtained by substituting matrix [C]31×1 into the expressions σai and τai.
3-5. Results and Discussion
3-5-1. Application of Closed-form Solutions
It is depicted below in more details that the preceding close-formed solutions are applied to cantilever beam strengthened by adhesively bonding [29] and a single lap joint [30].
Cornell [29] proposed the model of a cantilever beam strengthened by adhesively bonding. The sketch of Cornell’s model is shown in Fig. 3.8 and the symbols , , and are denoted as the thicknesses of the two adherends and the thickness of the adhesive. Fig. 3.9 shows the analytical solutions of this model employed to solve the problem proposed by Cornell, using the following values: (i)
h1 h2 ha
04 .
1 =0
h in (1.016mm), h2 =0.25 in (6.35mm), and in (2.54mm), 0.01 in (0.254mm), and 0.001 in (0.0254mm), (ii) Young’s modulus of two adherends and adhesive are respectively , , and , and (iii) shear modulus of adhesive layer is . Because Cornell’s Fig. 6 shown in the bottom diagram of Fig. 3.9 has used inch (in) as length units, Fig. 3.9 also uses inch as length units. All figures except Fig. 3.9 use mm as length units. It should be noted that the symbol in this study is synonymous with Cornell’s and the profiles of Fig. 3.9 are nearly consistent with those of Cornell’s Fig. 6. Single lap joint shown in Fig. 3.10 is one of Zou et al.’s examples [30] and the sketch of Zou et al.’s example shows the dimensions of the single lap joint. The Aluminum adherends bonded by adhesive are subjected to bending moment
1 .
=0 ha
6psi 10
30× 30×106psi 15×106 psi
6psi 10 5×
ha hb
100N.m in the Zou et al.’s example. Zou et al.’s example uses the following values: (i) Young’s modulus of Aluminum adherends and adhesive are 75GPa and 2.5GPa, and (ii) shear modulus of Aluminum adherends and adhesive are 28.846GPa and 1.0GPa. The numerical results obtained by employing the analytical solutions to solve the problem in Zou et al. are shown in Fig. 3.11. These data are almost consistent with those of Fig. 5 in the Zou et al.’s paper, except that for this study, the maximum shear stress is 4.38, while in Zou et al. it is 4.30 (MPa).
M
Fig. 3.8 The sketch of the Cornell’s model [29] showing cantilever beam strengthened by adhesively bonding
DISTANCE FROM TAB ENDING. (Unit: inch )(in)
Cornell [29]
Fig. 3.9 Comparison of the results between the present study (top) and figure 6 of the Cornell’s paper [29] (bottom).
100N.m
Fig. 3.10 The sketch of the Zou et al. model [30] showing a single lap joint
Fig. 3.11 Comparison of the results between figure 5 of the Zou et al.’s paper [30] and the present study.
3-5-2. Case Studies
The values E1 =6.0, E2 =6.0, E0 =2.75, P~
=1, and d = 0 are used as the following case 1-4. The symbols and (listed in Table 3.1) represent the ratios of the elastic modulus of the upper adherend, lower adherend, and adhesive layer, respectively, to the shear
1,
E E2, E0
modulus of the adhesive layer. The symbol d represents the distance from the center of the adhesive layer to the action point of the force.
Case 1: Upper adherend (h1) and lower adherend (h2) with the same thickness
Fig. 3.12 shows distributions of the non-dimensional peel stress and shear stress in the adhesive layer, whose thickness is ha =0.01mm . The thickness ratios 1 = 1 =10
ha
β h and
2 10
2 = =
ha
β h are defined as the thickness of the upper adherend and lower adherend
respectively relative to the adhesive layer’s thickness.
c
x = x is both the normalized axis and
the non-dimensional term of the adhesive layer, where −1≤x ≤1. The thickness to length ratio γ =γ1 =γ2 = 2hc is the ratio of the thickness h=h1 =h2 of the upper adherend to the length (2c) of the adhesive layer: γ =0.01667 and γ =0.06667 are used in this case.
Moreover, when γ =0.01667 , the length of the adhesive layer is four times that when 06667
.
=0
γ . Thus, the non-dimensional peel and shear stress distributions for an adhesive layer when γ =0.01667 are different from those when γ =0.06667. As the thickness to length ratio decreases –γ =0.06667 to γ =0.01667 – the non-dimensional peel and shear stresses in the adhesive layer become slightly less than 0.1.
Fig. 3.12 also illustrates that the non-dimensional maximum peel and shear stresses may occur either in the center or at the ends of the adhesive layer. Therefore, the values and positions of the non-dimensional maximum peel and shear stresses are the focus of the following paragraphs.
For the adhesive layer, as shown in Fig. 3.13, the non-dimensional peel stress occurs either in the center (x=0) or at the ends (x=±1), and the non-dimensional shear stress
occurs at the ends versus the thickness to length ratio γ . As γ becomes larger – i.e. the length (2c) of the adhesive layer becomes smaller in the same thickness ratiosβ =β1 =β2 – the peel stress in the center (x =0) is at first positive and smaller (i.e. tensile stress) but then becomes larger and then negative and even larger (i.e. compressive stress). As also shown in Fig. 3.13, the different thickness ratios
ha
= h
β produce the same results –β =β1=β2 =10, 20, or 30–
meaning that the thickness of the upper adherend as well as of the lower adherend can be 10, 20, or 30 times that of the adhesive layer. Thus, if both the upper adherend and lower adherend become thinner (i.e.β decreases from 30 to 10), the peel stress in the center becomes even larger as the thickness to length ratio γ increases. Moreover, since the maximum peel stress is always located either in the center (x =0) or at the ends (x =1), as the thickness to length ratio γ gradually becomes larger, the location of the maximum peel stresses in the adhesive layer changes from the ends to the center (see Fig. 3.13).
Fig. 3.12 Non-dimensional peel and shear stresses distributions in the adhesive layer (x = x/c) for the thickness to length ratio γ =γ1 =γ2 with the same thickness of both adherends β1 =β2 =10and (ha =0.01mm)
Fig. 3.13 Non-dimensional peel and shear stresses versus the thickness to length ratio
2
1 γ
γ
γ = = for the same thickness of the adherends as for Case 1 (ha =0.01mm).
Case 2: Upper adherend (h1) and lower adherend (h2) with different thicknesses
As Fig. 3.14(a) shows, in this case, the thickness of the upper adherend is three times that of the lower adherend, meaning that the thickness of the upper adherend in Fig. 3.14(a) is three times that in Fig. 3.12 even though the two figures have the same conditions otherwise. . For γ1 =0.05 and 0.01667
3 1
1
2 = γ =
γ in Fig. 3.14(a), the non-dimensional peel stress and
shear stress distributions are very similar to those in Fig. 3.12 (γ =0.01667). However, for 2
.
1 =0
γ and γ2 =0.06667, the non-dimensional peel stress in Fig. 3.14(a), in total contrast to the larger compressive peel stress in the center in Fig. 3.12 (γ =0.06667), vanishes in the center of the adhesive layer. In Fig. 3.14(a), the maximum peel stress at the ends is about one-and-a-quarter times that in Fig. 3.12, while the maximum shear stress at the ends in Fig.
3.14(a) is about 1.5 which is close to that in Fig. 3.12.
As Fig. 3.14(b) indicates, the thickness of the lower adherend is three times that of the lower adherend in Fig. 3.12, even though otherwise the two figures have the same conditions.
However, whether γ1 =0.06667or γ1 =0.01667, the non-dimensional peel stress vanishes in the center of the adhesive layer. Moreover, the maximum peel stress at the ends of the adhesive layer in Fig. 3.14(b) is about one-seventh of that in Fig. 3.14(a), while the maximum shear stress at the ends in Fig. 3.14(b) is about one-fifth of that in Fig. 3.14(a).
Fig. 3.15 shows the relationships among non-dimensional peel and shear stresses (at the
Fig. 3.15 shows the relationships among non-dimensional peel and shear stresses (at the