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Geometric Functions of Stress Intensity Factor Solutions for Spot Welds in Lap-Shear Specimens
D.-A. Wanga, P.-C. Linb and J. Panb*
aMechanical & Automation Engineering, Da-Yeh University, Chang-Hua 515, TaiwanbMechanical Engineering, The University of Michigan, Ann Arbor, MI 48109-2125, USA
January 18, 2005
Abstract
In this paper, the stress intensity factor solutions for spot welds in lap-shear specimens
are investigated by finite element analyses. Three-dimensional finite element models are
developed for lap-shear specimens to obtain accurate stress intensity factor solutions. In
contrast to the existing investigations of the stress intensity factor solutions based on the
finite element analyses, various ratios of the sheet thickness, the half specimen width, the
overlap length, and the specimen length to the nugget radius are considered in this
investigation. The computational results confirm the functional dependence on the
nugget radius and sheet thickness of the stress intensity factor solutions of Zhang (1997,
1999). The computational results provide some geometric functions in terms of the
normalized specimen width, the normalized overlap length, and the normalized specimen
length to the stress intensity factor solutions of Zhang (1997, 1999) for lap-shear
specimens. The computational results also indicate that when the spacing between spot
welds decreases, the mode I stress intensity factor solution at the critical locations
increases and the mode mixture of the stress intensity factors changes consequently.
____________
* Corresponding author: Tel.:+1-734-764-9404; fax:+1-734-647-3170
E-mail address: [email protected] (J. Pan).
2
Finally, based on the computational results, the dimensions of lap-shear specimens and
the corresponding approximate stress intensity factor solutions are suggested.
Keywords: Spot weld; Lap-shear specimen; Mixed mode; Stress intensity factor;
Geometric functions; Fracture; Fatigue
1. Introduction
Resistance spot welding is widely used to join sheet metals for automotive
components. The fatigue lives of spot welds have been investigated by many researchers
in various types of specimens, for example, see Zhang (1999). Since the spot weld
provides a natural crack or notch along the weld nugget circumference, fracture
mechanics has been adopted to investigate the fatigue lives of spot welds in various types
of specimens based on the stress intensity factor solutions at the critical locations of spot
welds (Pook, 1975, 1979; Radaj and Zhang, 1991a, 1991b, 1992; Swellam et al., 1994;
Zhang, 1997, 1999, 2001). The stress intensity factors usually vary point by point along
the circumference of spot welds in various types of specimens. Pook (1975, 1979) gave
the maximum stress intensity factor solutions for spot welds in lap-shear specimens,
coach-peel specimens, circular plates and other bending dominant plate and beam
specimens. Swellam et al. (1994) proposed a stress index iK by modifying their stress
intensity factor solutions to correlate their experimental results for various types of
specimens. Zhang (1997, 1999, 2001) obtained the stress intensity factor solutions for
spot welds in various types of specimens in order to correlate the experimental results of
spot welds in these specimens under cyclic loading conditions.
3
In order to obtain accurate stress and strain distributions and/or stress intensity
factor solutions for spot welds in lap-shear specimens, finite element analyses have been
carried out by various investigators (Radaj et al., 1990; Satoh et al., 1991; Deng et al.,
2000; Pan and Sheppard, 2002, 2003). Radaj et al. (1990) used a finite element model
where plate and brick elements are used for sheets and spot welds, respectively, to obtain
the stress intensity factor solutions along the nugget circumference for the main cracks in
various specimens. Satoh et al. (1991) conducted three-dimensional elastic and elastic-
plastic finite element analyses to investigate the stress and strain distributions in the
symmetry plane near spot welds in lap-shear specimens to identify the fatigue crack
initiation sites under high-cycle and low-cycle fatigue loading conditions. Deng et al.
(2000) conducted elastic and elastic-plastic three-dimensional finite element analyses to
investigate the stress fields in and near the nuggets in lap-shear and symmetrical coach
peel specimens to understand the effects of the nugget size and the thickness on the
interface and nugget pull out failure modes. Pan and Sheppard (2002) also used a three-
dimensional elastic-plastic finite element analysis to correlate the fatigue lives of spot
welds to the cyclic plastic strain ranges for the material elements near the main notch in
lap-shear specimens and modified coach-peel specimens. Pan and Sheppard (2003)
conducted a three-dimensional finite element analysis to investigate the critical local
stress intensity factor solutions for kinked cracks with a nearly elliptical shape emanating
from the main notch along the nugget circumference in lap-shear specimens and modified
coach-peel specimens.
Figure 1 schematically shows a lap-shear specimen used to investigate the
strength and fatigue lives of spot welds under shear dominant loading conditions. The
4
weld nugget is idealized as a circular cylinder as shown in the figure. The lap-shear
specimen has the thickness t , the width W2 , the nugget radius b , the overlap length V
of the upper and lower sheets, and the length L as shown in Figure 1. Note that two
spacers of the length S are attached to the both ends of the lap-shear specimen to induce
a pure shear to the interfacial plane of the nugget for the two sheets and to avoid the
initial realignment during testing.
In this paper, we investigate the stress intensity factor solutions for spot welds in
lap-shear specimens by a systematic finite element analysis. Three-dimensional finite
element models based on the finite element model for two circular plates with connection
(Wang et al., 2004) are used to obtain the stress intensity factor solutions for lap-shear
specimens. The stress intensity factor solutions as functions of the ratios bt , bW and
bV are investigated. The stress intensity factor solutions of our computational results
are compared with some existing computational and closed-form analytical stress
intensity factor solutions for lap-shear specimens. Geometric functions in terms of the
ratios bW , bL and bV based on our computational results are suggested to the stress
intensity factor solutions proposed by Zhang (1997, 1999).
2. Stress intensity factor solutions
Pook (1975) obtained stress intensity factor solutions at the critical locations
(point A and point B as shown in Figure 1) in lap-shear specimens of the thickness t and
the nugget radius b under the applied force P as
397.02/3 271.0 tb
bP
K I (1)
5
710.02/3 265.0282.0 tb
bP
K II (2)
Pook (1975) indicated that Equations (1) and (2) are applicable for 5tb or 2.0bt .
As discussed in Swellam et al. (1994), the mode I stress intensity factor solution
for spot welds was obtained from the solutions for two semi-infinite half spaces
connected by a circular patch under an axial force and a moment in Tada et al. (2000).
The mode II stress intensity factor solution for spot welds was approximated from the
mode III stress intensity factor solution for two semi-infinite half spaces connected by a
circular patch under a twisting moment in Tada et al. (2000). For lap-shear specimens,
the stress intensity factor solutions at the critical locations (point A and point B as shown
in Figure 1) were proposed as (Swellam et al., 1994)
2/5
061.12b
PtK I
(3)
2/3
354.02b
PK II
(4)
Zhang (1997, 1999) obtained the stress intensity factor IK and IIK solutions at
the critical locations (point A and point B as shown in Figure 1) and the stress intensity
factor IIIK solution at the critical locations (point C and point D as shown in Figure 1) for
lap-shear specimens as
tbP
K I 43
(5)
tbP
K II (6)
tbP
K III 2 (7)
6
Note that these solutions were obtained from the closed-form stress solutions for a rigid
inclusion under shear, center bending, counter bending and center twisting loading
conditions in an infinite plate and the J integral formulation. The functional forms of
Zhang’s IK , IIK and IIIK solutions should be valid for lap-shear specimens of thin
sheets with large ratios of bW and bV . Note that Zhang (1997, 1999) approximated
the counter bending for the rigid inclusion by a center bending, namely, a moment
applied to the rigid inclusion, to obtain the stress solutions for the derivation of the IK
solution.
In order to investigate the IK solution for lap-shear specimens with finite
geometry under counter bending, Lin et al. (2005a) recently derived the IK solution at
the critical locations of a spot weld connecting at the center of two square plates under an
applied counter bending moment 0M along the two edges of each plate per unit length as
1123 22
230 WbXXY
XYt
MK I
314121 862448 WWbWbbY (8)
where b is the radius of the spot weld, t is the thickness of the plate, W2 is the width of
the plate, and is the Poisson’s ratio of the plate material. Here, X and Y are defined
as
141 62244 WbWbX (9)
11 22 WbY (10)
7
In order to estimate the IK solution fora “square” lap-shear specimen under the applied
force P , the applied moment 0M can be taken as WPt 8 . The details of the validation
of the IK solution in Equation (8) are reported in Lin et al. (2005a). Equation (8) is used
as the basis for the benchmark of the IK solutions obtained from our computations.
3. Finite element analyses
In order to obtain accurate stress intensity factor solutions for spot welds in lap-
shear specimens and to systematically examine the validity of the stress intensity factor
solutions of Pook (1975), Swellam et al. (1994) and Zhang (1997, 1999), three-
dimensional finite element analyses are carried out. Note that finite element analyses
were carried out in Pook (1975), Cooper and Smith (1986), Radaj et al. (1990), Pan and
Sheppard (2003) and Zhang (2004) for specific ratios of half specimen width to nugget
radius, bW , specific ratios of overlap length to nugget radius, bV , and specific ratios
of specimen thickness to nugget radius, bt . No systematic investigation of the effects of
bt , bW and bV exists in the literature. The only systematic investigation of the
effects of bt by Pook (1975) appears to give significantly higher stress intensity factor
solutions when compared to the other solutions (Zhang, 1997, 1999, 2004). Therefore,
we emphasize our well-benchmarked finite element models used in this investigation.
Our finite element models for lap-shear specimens are evolved from the three-
dimensional finite element model for two circular plates with connection where a mesh
sensitivity study was carried out to benchmark to a closed-form analytical stress intensity
factor solution under axisymmetric loading conditions (Wang et al., 2005). The details to
8
select an appropriate three-dimensional mesh for obtaining accurate stress intensity factor
solutions for spot welds can be found in Wang et al. (2005).
Due to symmetry, only a half lap-shear specimen is considered. Figure 2(a)
shows a schematic plot of a half lap-shear specimen. The specimen has the thickness t
( 65.0 mm), the length L ( 3.77 mm), the half width W ( 9.18 mm), and the nugget
radius b ( 2.3 mm) according to the dimensions of the lap-shear specimens used in Lin
et al. (2004a). The overlap length V of the upper and lower sheets is 1.47 mm. The two
spacers have the length S ( 6.4 mm). Both the upper and lower sheets have the same
thickness. A Cartesian coordinate system is also shown in the figure. As shown in
Figure 2(a), a uniform displacement is applied in the x direction to the left edge
surface of the specimen, and the displacements in the x , y and z directions for the right
edge surface of the specimen are constrained to represent the clamped boundary
conditions in the experiment. The displacement in the y direction of the symmetry plane,
the zx plane, is constrained to represent the symmetry condition due to the loading
conditions and the geometry of the specimen. Figure 2(b) shows a mesh for a left half
finite element model. Figure 2(c) shows a close-up view of the mesh near the main crack
tip. Note that the main crack is modeled as a sharp crack here. The three-dimensional
finite element mesh near the weld nugget is evolved from the three-dimensional finite
element mesh for two circular plates with connection as discussed in Wang et al. (2005).
As shown in Figure 2(c), the mesh near the center of the weld nugget is refined to ensure
reasonable aspect ratios of the three-dimensional brick elements. The three-dimensional
finite element model for the half lap-shear specimen has 34,248 20-node quadratic brick
elements. The main crack surfaces are shown as bold lines in Figure 2(c).
9
In this investigation, the weld nugget and the base metal are assumed to be linear
elastic isotropic materials. The Young’s modulus E is taken as 200 GPa, and the
Poisson’s ratio is taken as 0.3. The commercial finite element program ABAQUS
(Hibbitt et al., 2001) is employed to perform the computations. Brick elements with
quarter point nodes and collapsed nodes along the crack front are used to model the r1
singularity near the crack tip.
First, the distributions of the stress intensity factor solutions along the
circumference of the nugget are investigated here. Figure 3 shows a top view of a nugget
with a cylindrical coordinate system centered at the nugget center. An orientation angle
θis measured counterclockwise from the critical location of point B. Figure 4 shows the
normalized IK , IIK and IIIK solutions as functions of θfor the crack front along the
nugget circumference based on our three-dimensional finite element computations for the
lap-shear specimen with 2.0bt , 91.5bW , 16.24bL and 72.14bV . Note
that the IK , IIK and IIIK solutions are normalized by the IIK solution at the critical
location of point A ( 180θ ). As shown in Figure 4, the maxima of the IK solution are
located at point B ( 0θ ) and point A ( 180θ ), the maximum and minimum of the
IIK solution are located at point A ( 180θ ) and point B ( 0θ ), respectively, and the
maximum and minimum of the IIIK solution are located at point D ( 90θ ) and point C
( 270θ ), respectively. The results shown in Figure 4 indicate that IIK is the dominant
stress intensity factor in lap-shear specimens under shear dominant loading conditions.
The distributions of the stress intensity factor solutions along the circumference of the
nugget as shown in Figure 4 are similar to those in Radaj et al. (1990).
10
Next, we investigate the effects of the ratio of the sheet thickness t to the nugget
radius b on the mode I and II stress intensity factor solutions at the critical locations
(point A and point B as shown in Figure 1) and the mode III stress intensity factor
solution at the critical locations (point C and point D as shown in Figure 1) for the
specimens of Lin et al. (2004a) with the ratio of 91.5bW and 72.14bV . Note that
the maximum magnitudes of the IK and IIK occur at point A and point B, whereas the
maximum magnitudes of the IIIK occur at point C and point D. We develop different
three-dimensional finite element models by changing the sheet thickness t with the
nugget radius b being fixed. Figure 5 shows the normalized IIK solutions at the critical
locations (point A and point B as shown in Figure 1) as functions of bt based on our
three-dimensional finite element computations and the analytical solutions of Pook (1975)
in Equation (2), Swellam et al. (1994) in Equation (4), and Zhang (1997, 1999) in
Equation (6) under the applied load P . The IIK solutions are normalized by Zhang’s
solution in Equation (6), denoted by zhangIIK . Note that for this case, the IIK solution at
point B is negative. But the magnitudes of the IIK solutions at points A and B are the
same. The range of the ratio bt is selected from 0.08 to 0.58 to represent the typical
values for spot welds used in the automotive industry. As shown in Figure 5, the
normalized IIK solution based on our three-dimensional finite element computations
appears to be relatively constant for the ratios of bt considered. The normalized IIK
solutions of Pook (1975) and Swellam et al. (1994) show significant deviations from the
solutions based on our three-dimensional finite element computations and the analytical
11
solution of Zhang (1997, 1999). Note that Pook’s solution is applicable for 2.0bt
(Pook, 1975).
The results of our three-dimensional finite element computations confirm the
functional dependence of Zhang’s analytical IIK solution on the sheet thickness t and
the nugget radius b . Zhang’s analytical IIK solution will be later used as the basis to
express the IIK solution for lap-shear specimens. Note that the IIK solution of Zhang
(1997, 1999) is obtained from the closed-form stress solution for a rigid inclusion under
shear and center bending loading conditions in an infinite plate and the J integral
formulation. For a lap-shear specimen with a finite width W2 , a finite specimen length
L and a finite overlap length V , a deviation from the closed-form analytical solution is
possible for finite values of bW , bL and bV . Therefore, the IIK solution for lap-
shear specimens should be written as
tb
PbVbLbWFK IIshearlapII
,, (11)
where IIF is a geometric factor which is a function of bW , bL and bV . Note that the
value of bL is usually very large for lap-shear specimens. For example, bL is 16.24
for the specimens of Lin et al. (2004a). The effects of large bL ’son the IIK and IIIK
solutions should be small and therefore not considered initially. Also, bV is 72.14 for
the specimens of Lin et al. (2004a). The effects of large bV ’s on the IIK and IIIK
solutions should be small and therefore not considered initially. Here, for the geometry
of the specimens of Lin et al. (2004a) with 91.5bW , 16.24bL , 72.14bV and
20.0bt , the geometric factor IIF is 04.1 .
12
Figure 6 shows the normalized IK solutions at the critical locations (point A and
point B as shown in Figure 1) for the specimens of Lin et al. (2004a) with 91.5bW ,
16.24bL and 72.14bV as functions of bt based on our three-dimensional finite
element computations and the analytical solutions of Pook (1975) in Equation (1),
Swellam et al. (1994) in Equation (3), and Zhang (1997, 1999) in Equation (5) for lap-
shear specimens under the applied load P . The IK solutions are normalized byZhang’s
IK solution in Equation (5), denoted by zhangIK . As shown in Figure 6, the normalized
IK solution based on our three-dimensional finite element computations appears to be
relatively constant for the range of the ratio bt considered. The IK solution based on
our three-dimensional finite element computations is nearly 50% higher than that
predicted by Zhang’s IK solution. The IK solutions of Pook (1975) and Swellam et al.
(1994) show significant deviations from our three-dimensional finite element
computational solutions and Zhang’s analytical IK solution. Note again that the IK
solution of Zhang (1997, 1999) is based on the closed-form stress solution for a rigid
inclusion under center bending in an infinite plate and the J integral formulation. The
IK solution based on our three-dimensional finite element computations confirms the
dependence of Zhang’s analytical IK solution on the sheet thickness t and the nugget
radius b . Therefore,Zhang’s analytical IK solution will be used as the basis to express
the IK solution for lap-shear specimens. For a lap-shear specimen with a finite width
W2 , a finite specimen length L and a finite overlap length V , the deviation of the
computational solutions from the closed-form analytical solution is possible. Therefore,
the IK solution for lap-shear specimens can be written as
13
tb
PbVbLbWFK IshearlapI 4
3,, (12)
where IF is a geometric factor which is a function of bW , bL and bV . For the
geometry of the specimens of Lin et al. (2004a) with 91.5bW , 16.24bL ,
72.14bV and 20.0bt considered in this investigation, the geometric factor is 49.1 .
Note that the IK solutions of our computational results are substantially higher than
those predicted by Zhang’s analytical IK solution. Therefore, the geometric factor IF is
needed.
Figure 7 shows the normalized IIIK solutions at the critical locations (point C and
point D as shown in Figure 1) for the specimens of Lin et al. (2004a) with 91.5bW ,
16.24bL and 72.14bV as functions of bt based on our three-dimensional finite
element computations and the analytical solution of Zhang (1997, 1999) in Equation (7)
for lap-shear specimens under the applied load P . The IIIK solutions are normalized by
Zhang’s IIIK solution in Equation (7), denoted by zhangIIIK . As shown in Figure 7, the
normalized IIIK solutions based on our three-dimensional finite element computations
appear to be relatively constant for the range of the ratio bt considered. The results of
our three-dimensional finite element computations confirm the functional dependence of
Zhang’s analytical IIIK solution on the sheet thickness t and the nugget radius b .
Zhang’s analytical IIIK solution will be used as the basis to express the IIIK solution for
lap-shear specimens. Note that the IIIK solution of Zhang (1997, 1999) is obtained from
the closed-form stress solution for a rigid inclusion under twisting conditions in an
infinite plate and the J integral formulation. For a lap-shear specimen with a finite
14
width W2 , a finite specimen length L and a finite overlap length V , a deviation from
the closed-form analytical solution is possible for finite values of bW , bL and bV .
Therefore, the IIIK solution for lap-shear specimens should be written as
tb
PbVbLbWFK IIIshearlapIII
2,,
(13)
where IIIF is a geometric factor which is a function of bW , bL and bV . For the
geometry of the specimens of Lin et al. (2004a) with 91.5bW , 16.24bL ,
72.14bV and 20.0bt considered in this investigation, the geometric factor IIIF is
02.1 .
As shown in Figures 5, 6 and 7, the IK , IIK and IIIK solutions from our
computations show a slight dependence on bt . This suggests that as bt increases, the
effects of the thickness increase, and the computational solutions will deviate from those
based on the thin plate theory. In order to validate the closed-form IK , IIK and IIIK
solutions based on the thin plate theory, we should pay more attention on the
computational solutions for small bt ’s. We continue the investigation based on the
assumption that the geometry factors IF , IIF and IIIF have a weak dependence on bt
for small values of bt .
As indicated in Equations (11), (12) and (13), IF , IIF and IIIF should be
functions of the three normalized length parameters bW , bL and bV . The number of
computations is too large for a full-scale numerical investigation of the three geometric
functions IF , IIF and IIIF with the complex coupling effects of the three parameters
bW , bL and bV . Therefore, we choose a large value of bL where the effects of the
15
applied load clamping boundary is supposedly minimized. We also choose a large value
of bV where the free surface boundary condition is supposedly minimized. We take
16.24bL , 72.14bV based on the specimens of Lin et al. (2004a). As indicated in
Lin et al. (2005a), IK is scaled by an applied counter bending moment 0M along the
plate edge per unit length. Therefore, the effects of the normalized specimen width bW
on IK will be most significant.
Now, the effects of bW on the geometric factors IF , IIF and IIIF are
investigated. Figure 8 shows the normalized IK and IIK solutions at the critical
locations (point A and point B as shown in Figure 1) and the normalized IIIK solution at
the critical locations (point C and point D as shown in Figure 1) as functions of bW
based on our three-dimensional finite element computations for 16.24bL ,
72.14bV and 20.0bt . The computational IK , IIK and IIIK solutions, denoted by
FEMIK , FEMIIK and FEMIIIK , are normalized by the IK , IIK and IIIK solutions of
Zhang (1997, 1999), denoted by zhangIK , zhangIIK and zhangIIIK , respectively.
Therefore, the normalized IK , IIK and IIIK solutions here represent the geometric
factors IF , IIF and IIIF for 72.14bV , 16.24bL and 20.0bt according to
Equations (12), (11) and (13), respectively. Note that we select a small value of
20.0bt to avoid the effects of the thickness. Six ratios of bW , namely, 05.2 , 70.4 ,
66.5 , 75.10 , 59.15 and 27.25 , are considered. Here, the nugget radius b , the nugget
thickness t , the specimen length L , and the overlap length V are fixed whereas the half
16
specimen width W is varied in the finite element models. Note that for the specimens of
Lin et al. (2004a), 91.5bW .
As shown in Figure 8, IIF decreases slightly as bW increases, and levels off as
bW is greater than 10 . As bW increases, IF decreases significantly and appears to
decrease slowly when bW is greater than 25 . IIIF increases slightly as bW increases,
and levels off as bW is greater than 10 . It is important to note that when bW is
greater than 10 , IIF is 1 and IIIF is nearly 1, which indicates the accuracy of Zhang’s
analytical IIK and IIIK solutions (Zhang, 1997, 1999). In contrast, when bW increases
to 25 , IF is nearly 71.0 . As bW further increases from 25 , IF seems to decrease
slightly based on the trend of IF as the function of bW . IF should continue to decrease
as bW increases based on the analytical solution in Equations (8) to (10). However, it
seems that the rate of the decrease slows down as bW increases further from bW equal
to 15. As also shown in Figure 8, when bW decreases from 5, IF and IIF increase
significantly and IIIF decreases slightly. For example, when bW equals 05.2 , the
values of the geometric factors IF , IIF and IIIF are 32.3 , 37.1 and 90.0 , respectively.
Figure 9 shows the normalized IK and IIIK solutions as functions of bW based
on our finite element computations. The IK and IIIK solutions are normalized by the
corresponding IIK solutions for the corresponding bW ’s. As shown in Figure 9, when
bW decreases from 5, the normalized IK and IIIK solutions change substantially. This
suggests that when the spacing of spot welds decreases, the mixture of the modes at the
17
critical locations changes significantly. Care should be taken for using the IK , IIK and
IIIK solutions for closely spaced spot welds.
Note that Zhang (1997, 1999) used the analytical stress solution for a rigid
inclusion under center bending in an infinite plate to approximate that under counter
bending for the IK solution in lap-shear specimens. The overestimation of the IK
solution for large ratios of bW ’sbased on Zhang’s analytical IK solution can be
attributed to this approximation. In order to estimate the IK solution for lap-shear
specimens, Lin et al. (2005a) derived the IK solution (Equation (8)) at the critical
locations for a spot weld at the center of two square plates under counter bending
conditions. Figure 10 shows the normalized IK solutions as functions of bW based on
our finite element computations, denoted by FEMIK and the analytical solution of Lin et
al. (2005a) in Equation (8), denoted by squareIK . The IK solutions are normalized by
Zhang’s IK solution, denoted by zhangIK . As shown in Figure 10, the general trends of
the IK solutions based on our finite element computations and the analytical solution of
Lin et al. (2005a) are similar. For bW ranging from 70.4 to 59.15 , the IK solutions of
our finite element computations and Lin et al. (2005a) are close to each other. As bW
increases from 59.15 , the solutions of our finite element computations and Lin et al.
(2005a) start to deviate from each other. Note that the IK solution of Lin et al. (2005a)
is obtained from the solution for a rigid inclusion in a square plate under counter bending,
whereas our finite element models are for lap-shear specimens with different ratios of
bW and specific ratios of bV ( 72.14 ) and bL ( 16.24 ). In order to investigate the
18
effects of LV on the IK solution, we performed a finite element computation for a
nearly square shaped lap-shear specimen with 85.80W mm, 54.151L mm,
30.141V mm, 20.3b mm and 65.0t mm. For this case, 27.25bW ,
16.44bV and 36.47bL . The normalized IK solution for this case is presented and
marked by squareFEMIK , in Figure 10. As shown in the figure, the result of this case is
very close to the analytical solution of Lin et al. (2005a) based on the solution for a rigid
inclusion in a square plate with 27.25bh . Note that h is the half width of the square
plates.
Note that when the ratio of bW is large, the normalized IK , IIK and IIIK
solutions do not change significantly as bW changes as shown in Figure 8. In order to
understand the effects of the ratio of bV on the IK , IIK and IIIK solutions, we develop
finite element models for specimens with the ratio of 27.25bW , 16.24bL and
20.0bt . Note that the number of computations is too high if we were to investigate
the effects of bV for different values of bW . Therefore, we choose a large value of
bW with minimum width effects to avoid the complex coupling effects of bW and
bV . We also choose a large value of bL with minimum clamped boundary effects to
avoid the complex coupling effects of bL and bV . These three-dimensional finite
element models have different values of the overlap length V whereas the specimen
length L , the nugget radius b , the nugget thickness t , and the half specimen width W
are fixed. Figure 11 shows the normalized IK and IIK solutions at the critical locations
(point A and point B as shown in Figure 1) and the normalized IIIK solution at the
19
critical locations (point C and point D as shown in Figure 1) as functions of bV based
on our three-dimensional finite element computations. The computational IK , IIK and
IIIK solutions, denoted by FEMIK , FEMIIK and FEMIIIK , are normalized by the IK ,
IIK and IIIK solutions of Zhang (1997, 1999), denoted by zhangIK , zhangIIK and
zhangIIIK , respectively. Therefore, the normalized IK , IIK and IIIK solutions here
represent the geometric factors IF , IIF and IIIF for 27.25bW , 16.24bL and
20.0bt according to Equations (12), (11) and (13), respectively. Six ratios of bV ,
namely, 07.5 , 42.9 , 56.12 , 72.14 , 26.21 and 43.23 , are considered. As shown in
Figure 11, IF , IIF and IIIF appear to be relatively constant when bV is greater than
72.14 . It should be noted that for the ratios of bV ranging from 72.14 to 43.23 , IIF is
nearly 1, and IIIF is nearly 05.1 , which indicates the accuracy of the analytical IIK and
IIIK solutions of Zhang (1997, 1999). In contrast, IF is nearly 71.0 for the ratios of
bV ranging from 72.14 to 43.23 . As also shown in Figure 11, when bV decreases
further from 72.14 , IF increases significantly, IIIF increases somewhat, and IIF
decreases slightly.
Figure 12 shows the normalized IK and IIIK solutions as functions of bV based
on our finite element computations. The IK and IIIK solutions are normalized by the
corresponding IIK solutions for the corresponding bV ’s. As shown in Figure 12, the
normalized IK and IIIK solutions stay relatively constant when bV is greater than
72.14 . When bV decreases from 72.14 , the normalized IK and IIIK solutions increase
substantially. When bV decreases to 07.5 , the IIIK solution becomes comparable to
20
the IIK solution, and the IK solution is nearly 65% of the IIK solution. Therefore, the
effects of the overlap length of lap-shear specimens can significantly affect the mixture of
the modes at the critical locations for small bV ’s.
We have examined the effects of bW and bV on the geometric factors IF , IIF
and IIIF . Now we examine the effects of bL on the geometric factors IF , IIF and IIIF .
Finite element models for specimens with different bL ’s are developed for 27.25bW ,
72.14bV and 20.0bt . Here, we take large values of bW and bV with minimum
width and overlap length effects to avoid the complex coupling effects of bL , bW and
bV . These three-dimensional finite element models have different values of the
specimen length L whereas the nugget radius b , the nugget thickness t , the half
specimen width W , and the overlap length V are fixed. Figure 13 shows the normalized
IK and IIK solutions at the critical locations (point A and point B as shown in Figure 1)
and the normalized IIIK solution at the critical locations (point C and point D as shown in
Figure 1) as functions of bL based on our three-dimensional finite element
computations. The computational IK , IIK and IIIK solutions, denoted by FEMIK ,
FEMIIK and FEMIIIK , are normalized by the IK , IIK and IIIK solutions of Zhang
(1997, 1999), denoted by zhangIK , zhangIIK and zhangIIIK , respectively. Therefore, the
normalized IK , IIK and IIIK solutions here represent the geometric factors IF , IIF and
IIIF for 27.25bW , 72.14bV and 20.0bt according to Equations (12), (11) and
(13), respectively. Five ratios of bL , namely, 15 , 20 , 16.24 , 30 and 80 , are
considered. As shown in Figure 13, IF , IIF and IIIF appear to be relatively constant
21
when bL is greater than 30 (or VL is larger than 2). It should be noted that for the
range of bL considered, IIF is nearly 1, and IIIF is nearly 06.1 . The solutions agree
with the analytical IIK and IIIK solutions of Zhang (1997, 1999) for spot welds in plates
of infinite size. However, as shown in Figure 13, as bL increases, IF decreases and the
decrease rate of IF becomes smaller for bL larger than 30 . IF approximately equals
63.0 for 30bL . This indicates that the effects of the length or the applied clamped
boundary condition fade away for bL larger than 30 . For 16.24bL , the effects of
the length is still there but not very large as shown in the figure. Therefore, the selection
of 16.24bL , 72.14bV based on the specimens of Lin et al. (2004a) seems to be a
good choice to avoid the effects of the clamped boundary and the free surface conditions.
Figure 14 shows the normalized IK and IIIK solutions as functions of bL based
on our finite element computations. The IK and IIIK solutions are normalized by the
corresponding IIK solutions for the corresponding bL ’s.As shown in Figure 14, the
normalized IK and IIIK solutions stay relatively constant when bL is greater than 30 .
When bL decreases from 30bL , the normalized IK solution increases substantially
and the normalized IIIK solution decreases slightly. When bL decreases to 15 , the IIIK
solution is nearly 74% of the IIK solution, and the IK solution is nearly 41% of the IIK
solution. Therefore, the effects of the specimen length of lap-shear specimens can
significantly affect the mixture of the modes at the critical locations for small bL ’s.
Based on the results of our finite element computations, the functional
dependence of the geometric factors IF , IIF and IIIF on bW , bL and bV are
22
investigated with minimum coupling effects of bW , bL and bV . As shown in Figure
8, the influence of bW ’s on the geometric factors IIF and IIIF is weak for 15bW .
As shown in Figure 11, the influence of bV ’s on the geometric factors IIF and IIIF is
weak for 15bV . As shown in Figure 13, the influence of bL ’s on the geometric
factors IF , IIF and IIIF is weak for 30bL . Therefore, for a lap-shear specimen with
25bW , 15bV and 30bL , the dependence of the geometric factors IF , IIF and
IIIF on bW , bV and bL is minimum. This can be used as a design guideline of lap-
shear specimen in the future.
In general, the values of bV and bL for specimens are large and 15bV and
30bL are recommended. Then the geometric functions presented in Figure 8 for
16.24bL , 72.14bV and 20.0bt can be used to estimate the geometric function
for specimens of different widths. Based on the results presented in Figure 8, for
engineering applications, we proposed that the lap-shear specimen should be made with
5bW , 15bV and 30bL . The stress intensity factor solution for specimens with
clamped loading conditions and the sizes given above are simply expressed as
1128
3 22
23 WbXXY
XYWt
PK I
314121 862448 WWbWbbY (14)
tbP
K II (15)
tbP
K III 2 (16)
23
where X and Y are defined in Equations (9) and (10). Note that Equation (14) is
obtained from the analytical solution in Equation (8). Equation (14) is shown to be a
good approximate solution for the computational results presented in Figure 10.
Now it is clear that the IK , IIK and IIIK solutions may be scaled by the nugget
radius b and the sheet thickness t as in Equations (12), (11) and (13) for large bW ’s,
bL ’s and bV ’s. For the specimen geometry with 91.5bW , 16.24bL and
72.14bV investigated here, the values of the geometric factors IF , IIF and IIIF are
49.1 , 04.1 and 02.1 , respectively. Recently, a two-dimensional elastic finite element
analysis of a rigid inclusion with a radius of b in a finite width specimen with a width of
W2 was performed by Lin et al. (2004b). The results indicate that when the ratio of
bW is greater than 10 , the radial stress at the critical locations of the nugget is close to
the analytical solution for a rigid inclusion in an infinite plate. However, when the ratio
of bW decreases from 10 , the radial stress at the critical locations of the nugget
increases. The radial stress at the critical locations of the rigid inclusion is the starting
point in the developmentof Zhang’s stress intensity factor solutions (Zhang, 1997, 1999).
It is clear that the analytical solutions of Zhang (1997, 1999) can characterize well the
trends of the IK , IIK and IIIK solutions based on our three-dimensional finite element
computations for large bW ’s, bL ’sand bV ’s. However, the geometric factors IF ,
IIF and IIIF are needed to account for the effects of the specimen width W2 , the
specimen length L and the overlap length V of the upper and lower sheets.
Now we attempt to make a comparison of the existing computational stress
intensity factor solutions with our computational results. We select the results from the
24
three-dimensional finite element computation of Pan and Sheppard (2003) and a
simplified finite element computation of Zhang (2004). First, we list the normalized IK ,
IIK and IIIK solutions at the critical locations by the analytical solutions of Zhang (1997,
1999) from our computations (listed as Wang and Pan (2004)), Pan and Sheppard (2003)
and Zhang (2004) without consideration of the effects of bt , bW , bL and bV in
Table 1. As listed in Table 1, we may not be able to judge the validity of the three
solutions. Next, we consider the effects of bW , bL and bV . Table 2 lists the values
of bt , bW , bL and bV for the specimens used in our computation, Pan and
Sheppard (2003) and Zhang (2004). Note that bL is 33 for specimens used in Pan and
Sheppard (2003) and bL for specimens of Zhang (2003) is not available.
The geometry factors IF , IIF and IIIF for the solutions of Pan and Sheppard
(2003) and Zhang (2004) are interpolated by a two-step approach. The interpolation
method is approximate in nature since we do not have the results to account for the
coupling effects of bW , bL and bV in this investigation. First, the initial values of
IF , IIF and IIIF are obtained from Figure 8 by linear interpolation to account for the
effects of bW . Note that the results in Figure 8 are for 16.24bL , 72.14bV and
2.0bt . Then, the initial values of IF , IIF and IIIF are multiplied by the ratios of IF ,
IIF and IIIF for a given bV interpolated from the results presented in Figure 11 for
27.25bW , 16.24bL and 2.0bt . Finally, the values of IF , IIF and IIIF are
multiplied by the ratios of IF , IIF and IIIF for a given bL interpolated from the results
presented in Figure 13 for 27.25bW , 72.14bV and 2.0bt . Since 33bL for
specimens used in Pan and Sheppard (2003), the ratios for IF , IIF and IIIF are nearly 1.
25
Since the value of bL for specimens used in Zhang (2004) is not available, the ratios for
IF , IIF and IIIF are taken as 1. Table 2 lists the values of IF , IIF and IIIF for the three
cases. As listed in this table, IF is much greater than 1 for the three cases, whereas IIF
and IIIF are close to 1 for the three cases.
Table 3 lists the normalized IK , IIK and IIIK solutions by zhangII KF ,
zhangIIII KF and zhangIIIIII KF , respectively. As listed in Table 3, the normalized IK ,
IIK and IIIK solutions of Zhang (2004) are quite close to 1 . This indicates that the
geometric functions IF , IIF and IIIF based on our computational results are indeed
needed and useful to scale the IK , IIK and IIIK solutions, and/or the solutions of Zhang
(2004) are quite accurate. On the other hand, the normalized IK and IIK solutions of
Pan and Sheppard (2003) are not so close to 1. Note that the effects of bt cannot be
accounted for by IF , IIF and IIIF , and the coupling effects of bW , bL and bV
cannot accounted for based on our interpolation method. The determination of IF , IIF
and IIIF is based on the computational results for the small ratio of 2.0bt . The
deviation from 1 of the normalized IK and IIK solutions of Pan and Sheppard (2003)
may be partly due to the large ratio of 47.0bt compared to the ratio of 2.0bt and
4.0 of Lin et al. (2004a) and Zhang (2004), respectively.
Here, we attempt to account for the effects of bt on the IK , IIK and IIIK
solutions based on the weak functional dependence of IK , IIK and IIIK solutions on bt
shown in Figures 5, 6 and 7 for 91.5bW , 16.24bL and 72.14bV . Figure 15
shows the normalized IIK solutions at the critical locations (point A and point B as
26
shown in Figure 1) based on our finite element computations, the three-dimensional finite
element computations of Pan and Sheppard (2003), and the simplified finite element
computation of Zhang (2004). Here, the IIK solutions are normalized by zhangIIII KF .
As shown in Figure 15, the normalized IIK solution based on the simplified finite
element computation of Zhang (2004) for 4.0bt , 8bW , and 8.12bV is still
nearly the same as that based on our finite element computations. The normalized IIK
solution based on the three-dimensional finite element computation of Pan and Sheppard
(2003) for 47.0bt , 50.5bW , 33bL and 10bV becomes lower than our
computational IIK solution.
Figure 16 shows the normalized IK solutions at the critical locations (point A and
point B as shown in Figure 1) based on our computational results, the three-dimensional
finite element computation of Pan and Sheppard (2003) and the simplified finite element
computation of Zhang (2004). Here, the IK solutions are normalized by zhangII KF . As
shown in Figure 16, the normalized IK solution based on the simplified finite element
computation of Zhang (2004) for 4.0bt , 8bW , and 8.12bV becomes slightly
higher than our computational solution. The normalized IK solution based on the three-
dimensional finite element computation of Pan and Sheppard (2003) for 47.0bt ,
50.5bW , 33bL and 10bV in fact becomes closer to our computational
solution. Figure 17 shows the normalized IIIK solutions at the critical locations (point C
and point D as shown in Figure 1) based on our finite element computations and the
simplified finite element computation of Zhang (2004). Here, the IIIK solutions are
27
normalized by zhangIIIIII KF . As shown in Figure 17, the normalized IIIK solution based
on the simplified finite element computation of Zhang (2004) 4.0bt , 8bW , and
8.12bV becomes slightly less than our computational solution.
Note that a mesh sensitivity study for the finite element model used in this
investigation has been performed in Wang et al. (2005). The stress intensity factor
solution based on the finite element model was well benchmarked to the analytical
solution for circular plates and cylindrical cup specimens under axisymmetric loading
conditions. At the other hand, Pan and Sheppard (2003) used the sub-modeling technique
in ABAQUS (Hibbitt et al., 2001) to obtain their stress intensity factor solutions. Pan and
Sheppard (2003) did not conduct a mesh sensitivity study to examine the accuracy of
their stress intensity factor solutions with respect to any known analytical solutions.
Therefore, more confidence can be placed on the accuracy of our stress intensity factor
solutions. The differences of the stress intensity factor solutions of this investigation and
Pan and Sheppard (2003) shown in Figures 15 and 16 may be due to the inaccuracy of the
stress intensity factor solutions of the sub-modeling technique in Pan and Sheppard (2003)
or the interpolation method from our limited sets of computational results.
4. Conclusions and Discussions
Three-dimensional finite element analyses are carried out to investigate the stress
intensity factor solutions for spot welds at the critical locations in lap-shear specimens.
The mode I and II stress intensity factor solutions at the critical locations (point A and
point B as shown in Figure 1) and the mode III stress intensity factor solution at the
critical locations (point C and point D as shown in Figure 1) for spot welds in lap-shear
28
specimens are obtained by three-dimensional finite element computations. The solutions
can be correlated with those based on Zhang’s solutions (Zhang, 1997, 1999) with new
geometric functions in terms of the normalized specimen width, the normalized specimen
overlap length and the normalized specimen length. The computational results confirm
the functional dependence on the nugget radius and sheet thickness of the stress intensity
factor solutions of Zhang (1997, 1999). The computational results indicate that as the
normalized specimen width changes, the geometric functions change. The results suggest
that when the spacing of spot welds decreases, the mode mixture of mode I, II and III
stress intensity factors at the critical locations can change substantially. As the
normalized overlap length decreases, the geometric function for mode I stress intensity
factor at the critical locations also changes significantly, and the mode mixture at the
critical locations therefore changes significantly. Finally, based on the computational
results, the dimensions of lap-shear specimens and the corresponding approximate stress
intensity factor solutions are suggested. The geometric functions can be complex
functions of the normalized sheet thickness, normalized specimen width, normalized
specimen length and normalized specimen overlap length. Therefore, consideration of
the geometric functions for stress intensity factor solutions can be useful to examine the
existing fatigue data for spot welds in lap-shear specimens of various dimensions. The
use of the stress intensity factor solutions to predict the fatigue lives of spot welds has
been addressed in Newman and Dowling (1998), Lin and Pan (2004), Lin et al. (2004a)
and Lin et al. (2005b) for fatigue life prediction of spot friction welds in lap-shear
specimens.
29
Acknowledgement
Partial support of this work from a Ford/Army IMPACT project and a Ford
University Research Program is greatly appreciated. Helpful discussions with S.-H. Lin
and P.-C. Lin of University of Michigan are greatly appreciated.
References
Cooper, J. F., Smith, R. A., 1986. Initial fatigue crack growth at spot-welds. International
Conference on Fatigue of Engineering Materials and Structures, Sheffield, England,
Mechanical Engineering Publication Ltd, London, England 2, 283-288.
Deng, X., Chen, W., Shi, G., 2000. Three-dimensional finite element analysis of the
mechanical behavior of spot welds. Finite Elements in Analysis and Design 35, 17-
39.
Hibbitt, H. D., Karlsson, B. I., Sorensen, E. P., 2001. ABAQUS user manual, Version 6-2.
Lin, S.-H., Pan, J., 2004. Fatigue life prediction for spot welds in coach-peel and lap-
shear specimens with consideration of kinked crack behavior. International Journal
of Materials and Product Technology 20, 31-50.
Lin, S.-H., Pan, J., Wung, P., Chiang, J., 2004a. A fatigue crack growth model for spot
welds in various types of specimens under cyclic loading conditions, submitted for
publication in International Journal of Fatigue.
Lin, P.-C., Lin, S.-H., Pan, J., 2004b. Modeling of plastic deformation and failure near
spot welds in lap-shear specimens. SAE Technical Paper 2004-01-0817, Society of
Automotive Engineers, Warrendale, Pennsylvania.
Lin, P.-C., Wang, D.-A., Pan, J., 2005a. Stress intensity factor solutions for spot welds
under counter bending conditions, to be submitted for publication.
Lin, P.-C., Pan, J., Pan, T., 2005b. Investigation of fatigue lives of spot friction welds in
lap-shear specimens of aluminum 6111-T4 sheet based on fracture mechanics. SAE
Technical Paper 2005-01-1250, Society of Automotive Engineers, Warrendale,
Pennsylvania.
Newman, J. A., Dowling, N. E., 1998. A crack growth approach to life prediction of spot-
30
welded lap joints. Fatigue & Fracture of Engineering Materials & Structures 21,
1123-1132.
Pan, N., Sheppard, S. D., 2002. Spot welds fatigue life prediction with cyclic strain range.
International Journal of Fatigue 24, 519-528.
Pan, N., Sheppard, S. D., 2003. Stress intensity factors in spot welds. Engineering
Fracture Mechanics 70, 671-684.
Pook, L. P., 1975. Fracture mechanics analysis of the fatigue behaviour of spot welds.
International Journal of Fracture 11, 173-176.
Pook, L. P., 1979. Approximate stress intensity factors obtained from simple plate
bending theory. Engineering Fracture Mechanics 12, 505-522.
Radaj, D., Zhang, S., 1991a. Stress intensity factors for spot welds between plates of
unequal thickness. Engineering Fracture Mechanics 39, 391-413.
Radaj, D., Zhang, S., 1991b. Simplified formulae for stress intensity factors of spot welds.
Engineering Fracture Mechanics 40, 233-236.
Radaj, D., Zhang, S., 1992. Stress intensity factors for spot welds between plates of
dissimilar materials. Engineering Fracture Mechanics 42, 407-426.
Radaj, D., Zhaoyun, Z., Mohrmann, W., 1990. Local stress parameters at the weld spot of
various specimens. Engineering Fracture Mechanics 37, 933-951.
Satoh, T., Abe., H., Nishikawa, K., Morita, M., 1991. On three-dimensional elastic-plastic
stress analysis of spot-welded joint under tensile shear load. Transactions of the
Japan Welding Society 22, 46-51.
Swellam, M. H., Banas, G., Lawrence, F. V., 1994. A Fatigue Design Parameter for Spot
Welds. Fatigue and Fracture of Engineering Materials & Structures 17, 1197-1204.
Tada, H., Paris, P. C., Irwin, G. R., 2000. The stress analysis of cracks handbook. ASME,
New York, New York.
Wang, D.-A., Lin, S.-H., Pan, J., 2005. Stress intensity factors for spot welds and
associated kinked cracks in cup specimens, International Journal of Fatigue, in press.
Zhang, S., 1997. Stress intensities at spot welds. International Journal of Fracture 88,
167-185.
Zhang, S., 1999. Approximate stress intensity factors and notch stresses for common
spot-welded specimens. Welding Journal 78, 173s-179s.
31
Zhang, S., 2001. Fracture mechanics solutions to spot welds. International Journal of
Fracture 112, 247-274.
Zhang, S., 2004. A simplified spot weld model for finite element analysis. SAE Technical
Paper 2004-01-0818, Society of Automotive Engineers, Warrendale, Pennsylvania.
32
Table 1. Normalized computational IK , IIK and IIIK solutions at the critical locationsby the analytical solutions of Zhang (1997, 1999) from our computations (listed as Wangand Pan (2004)), Pan and Sheppard (2003), and Zhang (2004).
zhangII KK zhangIIII KK zhangIIIIII KK
Wang and Pan (2004) 49.1 04.1 02.1
Pan and Sheppard (2003) 24.1 96.0 N/A
Zhang (2004) 19.1 02.1 01.1
Table 2. Values of IF , IIF and IIIF for different specimen geometries of Wang and Pan(2004), Pan and Sheppard (2003), and Zhang (2004).
bt bW bV bL IF IIF IIIF
Wang and Pan(2004) 20.0 91.5 72.14 16.24 49.1 04.1 02.1
Pan andSheppard
(2003)47.0 50.5 00.10 33 44.1 06.1 01.1
Zhang (2004) 40.0 00.8 80.12 NA 18.1 03.1 02.1
Table 3. Normalized computational IK , IIK and IIIK solutions at the critical locationsby zhangII KF , zhangIIII KF and zhangIIIIII KF , respectively, for different specimen
geometries of Wang and Pan (2004), Pan and Sheppard (2003), and Zhang (2004).
zhangIII KFK zhangIIIIII KFK zhangIIIIIIIII KFK
Wang and Pan (2004) 00.1 00.1 00.1
Pan and Sheppard (2003) 86.0 91.0 N/A
Zhang (2004) 01.1 99.0 99.0
33
Figure 1. A schematic plot of a lap-shear specimen and the applied force P shown as thebold arrows. The weld nugget is idealized as a circular cylinder and shown as a shadedcylinder. The critical locations with the maximum mode I and II stress intensity factorsare marked as point A and point B. The critical locations with the maximum mode IIIstress intensity factor are marked as point C and point D.
34
Figure 2. (a) A schematic plot of a half lap-shear specimen with a uniform displacementapplied to the left edge surface of the specimen shown as the bold arrows and theclamped boundary conditions for the right edge surface of the specimen. The shadowrepresents the half weld nugget. (b) A mesh for a left half finite element model. (c) Aclose-up view of the mesh near the main crack tip. A Cartesian coordinate system is alsoshown.
35
Figure 3. A top view of a nugget with an orientation angle θdefined as shown. SeeFigure 1 for the locations of points A, B, C and D.
36
Figure 4. The normalized IK , IIK and IIIK solutions as functions of θfor the crackfront along the nugget circumference based on our three-dimensional finite elementcomputations for the lap-shear specimen used in Lin et al. (2004a).
37
Figure 5. The normalized IIK solutions at the critical locations (point A and point B asshown in Figure 1) as functions of bt / based on our three-dimensional finite elementcomputations and the analytical solutions for lap-shear specimens. The IIK solutions arenormalized by Zhang’s solution in Equation (6), denoted by zhangIIK . The normalized
IIK solutions based on our finite element computations and the analytical solutions inEquation (2), Equation (4), and Equation (6) are marked as FEM, Pook, Swellam et al.,and Zhang, respectively.
38
Figure 6. The normalized IK solutions at the critical locations (point A and point B asshown in Figure 1) as functions of bt / based on our three-dimensional finite elementcomputations and the analytical solutions for lap-shear specimens. The IK solutions arenormalized by Zhang’s solution in Equation (5), denoted by zhangIK . The normalized
IK solutions based on our finite element computations and the analytical solutions inEquation (1), Equation (3), and Equation (5) are marked as FEM, Pook, Swellam et al.,and Zhang, respectively.
39
Figure 7. The normalized IIIK solutions at the critical locations (point C and point D asshown in Figure 1) as functions of bt / based on our three-dimensional finite elementcomputations and the analytical solution for lap-shear specimens. The IIIK solutions arenormalized by Zhang’s solution in Equation (7), denoted by zhangIIIK . The normalized
IIIK solutions based on our finite element computations and the analytical solution ofZhang in Equation (7) are marked as FEM and Zhang, respectively.
40
Figure 8. The normalized IK and IIK solutions at the critical locations (point A andpoint B as shown in Figure 1) and the normalized IIIK solution at the critical locations(point C and point D as shown in Figure 1) as functions of bW / based on our three-dimensional finite element computations.
41
Figure 9. The normalized IK solution at the critical locations (point A and point B asshown in Figure 1) and the normalized IIIK solution at the critical locations (point C andpoint D as shown in Figure 1) as functions of bW / based on our three-dimensional finiteelement computations.
42
Figure 10. The normalized IK solutions at the critical locations (point A and point B asshown in Figure 1) as functions of bW / based on our three-dimensional finite elementcomputations and the analytical solution of Lin et al. (2005a). The IK solutions arenormalized by the IK solution of Zhang (1997, 1999), respectively. The normalized IKsolutions based on our finite element computations and the analytical solution of Lin et al.(2005a) are marked as FEMIK and squareIK , respectively. The normalized IK solution
based on our finite element computation for the case of 27.25/ bW , 16.44/ bV and36.47/ bL is marked as squareFEMIK , .
43
Figure 11. The normalized IK and IIK solutions at the critical locations (point A andpoint B as shown in Figure 1) and the normalized IIIK solution at the critical locations(point C and point D as shown in Figure 1) as functions of bV / based on our three-dimensional finite element computations for 27.25/ bW .
44
Figure 12. The normalized IK solution at the critical locations (point A and point B asshown in Figure 1) and the normalized IIIK solution at the critical locations (point C andpoint D as shown in Figure 1) as functions of bV / based on our three-dimensional finiteelement computations for 27.25/ bW .
45
Figure 13. The normalized IK and IIK solutions at the critical locations (point A andpoint B as shown in Figure 1) and the normalized IIIK solution at the critical locations(point C and point D as shown in Figure 1) as functions of bL / based on our three-dimensional finite element computations for 27.25/ bW and 72.14/ bV .
46
Figure 14. The normalized IK solution at the critical locations (point A and point B asshown in Figure 1) and the normalized IIIK solution at the critical locations (point C andpoint D as shown in Figure 1) as functions of bL / based on our three-dimensional finiteelement computations for 27.25/ bW and 72.14/ bV .
47
Figure 15. The normalized IIK solutions at the critical locations (point A and point B asshown in Figure 3) as a function of bt / based on our three-dimensional finite elementcomputations, Pan and Sheppard (2003) and Zhang (2004). The IIK solutions arenormalized by zhangIIII KF . The normalized IIK solutions based on our finite element
computations, Pan and Sheppard (2003) and Zhang (2004) are marked as FEM, Pan andSheppard (2003), FEM, and Zhang (2004), FEM, respectively.
48
Figure 16. The normalized IK solutions at the critical locations (point A and point B asshown in Figure 1) as a function of bt / based on our three-dimensional finite elementcomputations, Pan and Sheppard (2003) and Zhang (2004). The IK solutions arenormalized by zhangII KF . The IK solutions based on our finite element computations,
Pan and Sheppard (2003) and Zhang (2004) are marked as FEM, Pan and Sheppard(2003), FEM, and Zhang (2004), FEM, respectively.
49
Figure 17. The normalized IIIK solutions at the critical locations (point C and point D asshown in Figure 1) as a function of bt / based on our three-dimensional finite elementcomputations and that of Zhang (2004). The IIIK solutions are normalized by
zhangIIIIII KF . The normalized IIIK solutions based on our finite element computations
and Zhang (2004) are marked as FEM and Zhang (2004), FEM, respectively.