International Journal of Mechanical Sciences 78 (2014) 154–166

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International Journal of Mechanical Sciences

journal homepage: www.elsevier.com/locate/ijmecsci

Failure analysis for resistance spot welding in lap-shear specimens

Fengxiang Xu a, Guangyong Sun a, Guangyao Li a,n, Qing Li b
a
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China
b
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia

art ic l e i nf o a b s t r a c t

Article history: Based on the elasticity theory and finite element method, this paper aims to explore the failure incidence
Received 4 April 2012 of resistance spot welding in dual-phase lap-shear specimens. The stress function approach is adopted to
Received in revised form derive an analytical solution to a lap-shear specimen containing a spot weld nugget subjected to the
14 October 2013
uniformly distributed loading condition, which provides a means to exploring the stress distributions
Accepted 15 November 2013
near the spot weld nugget. The normalized effective stress obtained indicates that the initial yielding
Available online 21 November 2013
failures likely occur at four specific angles of 38.021, 141.981, 218.021, and 321.981 along the spot weld
Keywords: nugget in the lap-shear plate. In addition, the contours of normalized stress are also plotted in the polar
Spot welds system to understand the surrounding stress distributions, which reveals that the locations of the
Airy stress function
maximum and minimum values of normalized radial, hoop, and shear stresses are located at angles 01/
Finite element analysis
1801 and 901/2701, 901/2701 and 01/1801, 1351/3151 and 451/2251, respectively, as the normalized radial
Advanced high strength steel (AHSS)
Plastic deformation distance r/a goes to infinity. The elasto-plastic finite element analysis (FEA) is also conducted to analyze
Failure mechanism the initial necking or thinning phenomenon. It is found that the angular locations of the maximum
equivalent plastic strain or initial necking failure points are located at four angular intervals for the
advanced high strength steel (AHSS) plate with a spot weld nugget. The derived stress distributions allow
predicting failure behavior and evaluating damage evolution on many engineering structures jointed
with spot welds.
& 2013 Elsevier Ltd. All rights reserved.

1. Introduction is higher, and the interfacial failure usually carries lower mechan-
ical loading and absorbs less energy than the nugget pullout
Resistance spot welding is considered one of main joint failure [2]. Since the spot weld may generate an inherent crack
techniques that have been extensively used in the automotive along the weld nugget, it is of a critical importance to the study on
industry with typically each vehicle containing several thousand failure modes.
spot welds. With rapidly increased requirements in lightweight, There have been some studies on the stress distributions
many automotive companies have widely been utilizing various around a spot weld recently [3–6]. However, reliable evaluation
advanced materials, such as aluminum alloy and advanced high of fatigue life remains challenging mainly because the stress field
strength steel (AHSS), to stamp into final products such as close to the spot region is rather complex due to the notch effect
automotive body panels and other major structural frames. AHSSs, along with the edge of the spot weld [7]. In addition, these spot
as one class of promising engineering materials used in car body welds are often subjected to multiaxial loading. As a result, various
structures, have drawn increasing attention recently, which allow types of specimens such as U-tensile specimens, cross-tension
reducing structural weight but enhancing the performance under specimens, lap shear specimens and coach peel specimens have
operational and crashing conditions. Nevertheless, challenge been used to determine the strengths and fatigue lives of spot
remains in integration of AHSS into automotive structure through welds under different loading conditions [8–10]. As one of most
quality welds. Indeed, the weldability of AHSS signifies a critical typical joint structures the lap-shear specimens have been com-
issue in vehicle production. To make better use of such advanced monly used to examine the fatigue life, failure mechanism, impact
materials as dual-phase steels for welding process, it is important and static strengths of spot welds [5,11–20]. In this joint scenario
to characterize and quantify their spot welding behaviors properly the tension and shear represent the critical loading conditions. The
[1]. In practical automobile applications, the nugget pullout is a fatigue strength of such joints has been analyzed and measured on
preferable failure test approach because the load-carrying capacity a basis of peak stresses at the weld root of the joint [21–23].
Note that stress distribution near the spot welds is crucial to
evaluate the fatigue strength of such joints. Note that although the
n
Corresponding author. Tel.: þ 86 731 8882 1717; fax: þ 86 731 8882 2051. plastic stress/strain around the nugget plays a more direct role on
E-mail address: gyli@hnu.edu.cn (G. Li). the failure model and mechanism of spot welds, it is very difficult

0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijmecsci.2013.11.011
 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166 155

to derive the analytical elasto-plastic solutions to the stresses and coordinate system to understand the locations of the maximum
strains in the lap-shear specimens based on the elasto-plasticity and minimum values of the normalized radial, hoop and shear
theory [6,24,25]. On contrast, elasticity-based analytical solution is stress. Nevertheless, these elastic analytical solutions only provide
still deemed fairly effective to gain preliminary insights into the some preliminary understanding of the stress and strain status
stress and strain distributions near the spot weld. For example, near the spot weld. In order to explore the failure mechanism due
Zhang [9,26] obtained the approximate elastic solutions to the to the resistance spot welds, an elasto-plastic finite element
nominal stress fields near the weld nuggets that were considered analysis is also conducted here for the spot weld nugget welded
as the rigid inclusions in the infinite plates under general loading by two different advanced high strength steels (AHSSs) (DP600
conditions. The analytical solutions of stress with rigid inclusions and DP980). In the elasto-plastic FEA, the lap-shear specimen is
in the plates under shear, bending and opening loading conditions subjected to uniformly distributed loading. The angular locations
were also derived by Muskhelishvili and Radok [27] and of the maximum equivalent plastic strain or initial necking failure
Timoshenko et al. [28]. In addition, a number of researchers also points are determined to understand the development of plastic
examined the stresses and strains near the nugget and their flow. Thus, the derived equivalent von Mises stress and plastic
relations to failure modes based on the elasticity theory. For strain solutions allow us to predict failure behavior and damage
example, Kan [12] showed that initial yielding occurs on in the evolution in some AHSS structures jointed by spot welds.
middle of the nugget circumference around the spot weld in a
lap-shear specimen by simplifying the problem to a 2D sheet
subjected to a uniformly distributed shear force. However, the
obtained results were mainly based on the assumption of the 2. Analytical solutions
uniform shear stress distributions along with the weld nugget.
Salvini et al. [7,24] proposed a simplified spot weld model and 2.1. Two dimensional analytical model
obtained an analytical solution based on the elasticity theory, in
which the weld nugget was also considered as a rigid inclusion in To derive an analytical solution, a simplified two dimensional
a finite plate subjected to shear, bending and orthogonal loading (2D) model with a spot weld joint subjected to a uniformly
conditions. They attempted to evaluate the local stiffness and distributed load is presented in Fig. 1. Similar configuration of the
stress of spot joints by introducing the new criteria for predicting lap-shear specimen has also been adopted by other researchers
fatigue behavior and damage evolution in some special mechan- [5,11,41]. In the figure, the doublers that play a role on supporting
ical structures joined in different spot welds [29]. pieces are used to align the applied load, thereby avoiding the initial
In addition to the analytical solution, some other approaches have realignment of the specimen under lap-shear loading conditions. The
also been adopted to the understanding of mechanical behaviors shaded cylinder represents a rigid inclusion of diameter 2a that can
of spot welds. Some researchers investigated the failure mecha- be considered as an approximation to the spot weld in a lap-shear
nism under shear-tension condition by using experimental methods specimen. According to the superposition principle in elastic theory,
[30–35]. Besides, the numerical methods were also widely adopted the load applied to the lap-shear specimen can be decomposed into
to study the plastic strains/stresses or failure of the weld joints under four loading components, namely counter bending, central bending,
shear-tension or impact conditions [36,37]. A simplified model of a shear, and tension, respectively. As a primary load in the lap-shear
lap-shear specimen was presented and a 2D elasto-plastic finite specimens, the shear tension is considered in this study and the 2D
element (FE) analysis based on the cyclic stress–strain curves was analytical model is illustrated in Fig. 2.
used to obtain the local strain distribution for characterizing the
fatigue failure mode [12]. Satoh et al. [38] and Deng et al. [39]
developed the 3D elastic and elasto-plastic FE solution to the stress Doubler
and strain near spot welds in the lap-shear specimens, thereby
2a
exploring the mechanical behavior of spot welds, in which the
q
former [38] indentified the fatigue crack initiation sites under high- q
cyclic and low-cyclic fatigue loading conditions, while the latter [39] Spot weld nugget
understood the effects of the nugget size and sheet thickness on the t
interfacial and pull out failure modes. Lin et al. [11] not only obtained Doubler
the analytical stress solutions for an infinite plate containing a rigid Fig. 1. A schematic plot of a lap-shear specimen subjected to a uniformly distributed
inclusion, but also suggested that the location of the initial necking load q and the spot weld nugget is idealized as a circular cylindrical weld nugget.
failure should occur near the middle of the nugget circumference in
the sheet based on the elasto-plastic finite element analysis. And the
failure of the sheet was determined by the forming limit diagram
(FLD) which has been widely adopted in the sheet metal forming
the infinite plate
engineering. Recently, Asim et al. [40] conducted an experimental
study on the failure mechanism and strength of laser welds in lap-
shear specimens of high strength alloy steel. They pointed out that
the laser welds failed in a ductile necking/shearing mode that was y
Spot weld nugget
initiated at a distance away from the crack tip near the boundary of r=b
the base metal and heat affected zone.
In this paper, an analytical solution will be derived to explore q θ
x q
the stress distributions near the spot weld nugget in a lap-shear
specimen subjected to a uniformly distributed load. Based on the 2a
analytical solutions from the stress function approach in the
elasticity theory, the locations of the initial yielding are estimated
from the angular and radial distributions of the normalized
effective stress near the nugget in the lap-shear specimen. Fig. 2. A 2D analytical model of a rigid inclusion of radius a in an infinite plate
Furthermore, the obtained stress contours are plotted in the polar subjected to a uniformly distributed load q.
156 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166

From Fig. 2, the rigid inclusion represents the spot weld nugget Thus, the compatibility equation turns into the following form
and the infinite plate represents the upper sheet of a large lap- in terms of φðr; θÞ,
shear specimen. In the present investigation, a polar coordinate
2 
2 
∂ 1 ∂ 1 ∂2 ∂ φ 1 ∂φ 1 ∂2 φ
system is considered with the origins from the center of rigid þ  þ 2 2  þ  þ 2  2 ¼0 ð4Þ
∂r 2 r ∂r r ∂θ ∂r 2 r ∂r r ∂θ
inclusion.
As such, the system is represented by a single fourth order
differential equation as Eq. (4). Further, the strain–displacement
2.2. Basic formulation with respect to a polar coordinate system relation can be expressed as follows

Fig. 3 shows a top view of an infinitesimal element ABCD from εrr ¼ ∂u∂rrr
the plate with associated radial stress srr , shear stress srθ , hoop urr 1 ∂uθθ
εθθ ¼ þ
stress sθθ , and body forces f rr ; f θθ in the polar system. The r r ∂θ
equilibrium equation is formulated as, 1 ∂urr ∂uθθ uθθ
εrθ ¼ þ  ð5Þ
∂srr r ∂θ ∂r r
∂r þ r
1
 ∂s∂θrθ þ srr r sθθ þ f r ¼ 0
In the plane stress condition, the strain components, according
1 ∂sθθ ∂srθ 2srθ to Kirchhoff and Love theory [42], can be also obtained from the
 þ þ þf θ ¼ 0 ð1Þ
r ∂θ ∂r r Hooke's law as follows,
εrr ¼ 1Eðsrr  υsθθ Þ
The compatibility condition is given as,
1

2  εθθ ¼ ðsθθ υsrr Þ
∂ 1 ∂ 1 ∂2 E
þ þ ðsrr þ sθθ Þ ¼ 0 ð2Þ 1 2ð1 þ υÞ
∂r 2 r ∂r r 2 ∂θ2 εrθ ¼ srθ ¼ srθ ð6Þ
G E

By using the Airy function φ ¼ φðr; θÞ, the corresponding stres-

2.3. The analytical solutions subjected to a uniformly distributed
ses can be expressed as
load q
srr ¼ 1r ∂φ 1 ∂ φ
2

∂r þ r 2 ∂θ2
The Airy stress function approach allows us to solve the elastic
∂2 φ boundary value problem. Unfortunately, it is relatively difficult to
sθθ ¼ 2
∂r directly obtain the Airy stress function when considering a rigid
!
∂ 1 ∂φ inclusion in an infinite plate. For this reason, the 2D analytical model
srθ ¼  ð3Þ can be separated into two simplified models based on the super-
∂r r ∂θ
position principle: an axisymmetric model and a symmetric model,
whose Airy stress functions could be readily determined, as illustrated
in Fig. 4. From the figure, the axisymmetric model is subjected to
uniform tension q/2 along with all four edges (Fig. 4(b)). Another
singly symmetric model is subjected to uniform tension q/2 in the left
and right edges, and uniformly compression q/2 in the upper and low
side edges (Fig. 4(c)). For the convenience, the boundaries of these
two models are named as mechanical boundary condition (I) and
mechanical boundary condition (II), respectively. The total solutions
∂ will be obtained by adding up these separate results.
∂

2.3.1. Stress solution under boundary condition (I)

∂ ∂
∂
The elastic solution to the axisymmetric model (Fig. 4
∂ ∂
∂ (b) boundary condition (I)) in all four sides is derived here. Note
that the stress status in the bigger far-field circle (the dot line) is
the same as the plate without the spot weld nugget according to
Fig. 3. Stresses in the differential element of a plate in a polar coordinate system. the St Venant principle. Thus, the stress boundary condition can be

q q
2 2

y y y
r=b r=b
r=b
q q q
q =q
θ θ + θ
q x x 2 2 x 2
2
2a 2a 2a

(I) (II)

q q
2 2

Fig. 4. The schematic of the superposition principle; (a) the original 2D model, (b) the first simplified model (mechanical boundary condition (I)), (c) the second simplified
model (mechanical boundary condition (II)).
 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166 157

approximated as combining Eq. (3) and Eqs. (13)–(15).

h   3ð1 þ υÞa4 i
q þ υÞ a 2
ðsrr Þr ¼ b ¼ ; ðsrθ Þr ¼ b ¼ 0 ð7Þ s2rr ¼ 2q 1 þ 4ð1
3υ r  3υ r cos 2θ
2
  
The displacement boundary condition and displacement single q 3ð1 þ υÞ a 4
s2θθ ¼  1  cos 2θ
value condition along the circular boundary are given as follows, 2 3υ r
  
respectively, q 2ð1 þ υÞ a 2 3ð1 þυÞa 4
s2rθ ¼  1  þ sin 2θ ð16Þ
ðurr Þr ¼ a ¼ 0; ðuθθ Þr ¼ a ¼ 0 ð8Þ 2 3 υ r 3υ r

(
urr ðr; θ þ 2πÞ  urr ðr; θÞ ¼ 0 Finally, the total stress solutions to a lap-shear specimen under
: ð9Þ
uθθ ðr; θ þ2πÞ  uθθ ðr; θÞ ¼ 0 uniformly distributed load q can be obtained by adding Eqs. (11)
and (16) as per the superposition principle.
Thus, the Airy stress function can be obtained from the boundary
conditions (Eqs. (7)–(9)) and strain and stress relationship (Eqs. (5) and srr ¼ s1rr þ s2rr
 
(6)), as q 1 þ υ 1  υa 2 4ð1 þ υÞa 2 3ð1 þ υÞa 4
¼ þ þ 1þ  cos 2θ
1 ð1 þ υÞq 2 2 2 2 r 3υ r 3υ r
φ1 ¼ ð1  υÞa2 qInr þ r þD1 ð10Þ
4 8 sθθ ¼ s1θθ þ s2θθ
 
q 1 þ υ 1  υ a 2 3ð1 þ υÞa 4

Since constant D1 has no effects on the elastic stress/strain, it ¼   1  cos 2θ

2 2 2 r 3υ r
will not be considered.
srθ ¼ s1rθ þ s2rθ
Combining Eq. (3) with Eq. (10), the stress solution under  
q 2ð1 þ υÞa 2 3ð1 þ υÞa 4
Condition (I) can be obtained as follows. ¼  1 þ sin 2θ ð17Þ
h i 2 3υ r 3υ r
s1rr ¼ 2q 1 þ υ 1υ a 2
2 þ 2 ðr Þ
 
q 1þυ 1υ a 2 Eq. (17) defines the stress distributions around the spot weld
s1θθ ¼  ð Þ nugget which can be a key indicator to the fatigue life of the
2 2 2 r
s1rθ ¼ s1θr ¼ 0 ð11Þ welded structure. Often, the following effective (von Mises) stress
se is used in failure analysis
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Note that the elastic stresses are the functions of load q, se ¼ s2rr  srr sθθ þ s2θθ þ 3s2rθ ð18Þ
Poisson's ratio υ and normalized radial distance r/a, but are not
related to angular coordinate θ and materials properties.
Also, the normalized stress components s^ rr ; s^ θθ and s^ rθ can be
expressed as
2.3.2. Stress solution under boundary condition (II)  h   i
srr υ 1υ a 2 þ υÞ a 2 þ υÞ a 4
As in Fig. 4(c), the primary difference between Condition (I) and s^ rr ¼ q=2 ¼ 1þ
2 þ 2 r þ 1 þ 4ð1
3υ r  3ð1
3υ r cos 2θ
Condition (II) is that the latter is subjected to uniform tensile load q/    
sθθ 1 þ υ 1  υ a 2 3ð1 þ υÞ a 4
2 in the left and right edges, and uniform compressive load q/2 in s^ θθ ¼ ¼   1 cos 2θ
the upper and lower edges. So the stress boundary condition can be q=2 2 2 r 3υ r
   
presented as, srθ 2ð1 þ υÞ a 2 3ð1 þ υÞ a 4
s^ rθ ¼ ¼  1 þ sin 2θ ð19Þ
q q q=2 3υ r 3υ r
ðsrr Þr ¼ b ¼ cos 2θ; ðsrθ Þr ¼ b ¼  sin 2θ ð12Þ
2 2 To determine the initial onset of failure in a spot weld, the
The displacements along with the circular boundary are similar normalized von Mises effective stress s^ e can be expressed in a form
to those under Condition (I) (Eqs. (8) and (9)). From Eqs. (8), similar to Eq. (18) but with the components formulated in Eq. (19).
(9) and (12), and the strain and stress relationship in Eqs. (5) and
(6), the Airy stress function can be obtained as 2.4. Analysis of stress solutions

φ2 ¼ ðA2 r 4 þ B2 r 2 þC 2 þ D2 r  2 Þ cos 2θ ð13Þ

Note that the normalized effective stress s^ e at the normalized
where radial distance r/a is important to determine the locations of the
 h a2 i initial yielding failure near the rigid spot weld nugget, it is thus
q a 2
A2 ¼  2 1 selected to explore the locations of possible initial yielding here.
Tb b b
" # The visualized angular change of initial yielding in different
q 3υ 4 a
ð3 þ υÞ2 a 6
B2 ¼  3 þ4 normalized radial distances r/a is presented in Fig. 5. It can be
2T 1 þ υ b ð1 þ υÞ2 b
  seen that the angular locations of the initial yielding along the
qa 2 
3 υ a 6
rigid spot weld nugget (i.e. at the normalized radial distance r/
C2 ¼  1þ
T 1 þυ b a¼ 1) are located at the four special angles of 38.021, 141.981,
 
qa4 3 υa 4 218.021, and 321.981, respectively. When the radius r increases, for
D2 ¼  1þ ð14Þ example, the normalized radial distances r/a equals 1.05, the
2T 1 þυ b
angular locations of initial yielding become 33.091, 146.911,
213.091, and 326.911, respectively. As the radius r further increases
In Eq. (14), T can be read as follows and reaches 1.10a, for instance, the initial yielding locates at four
3υ a 2 a 4 4ð3 þ υÞ2 a 6 3  υ a 8 other angles of 21.351, 158.651, 201.351, and 338.651. Beyond 1.2a,
T¼ þ4 6 þ þ ð15Þ the initial yielding locations occur at the two special angles of
1þυ b b ð1 þ υÞ2 b 1þυ b
θ ¼01 and 1801.
Note that when b c a, a=b  0, thus, the stress solutions under Furthermore, the radial distributions of the normalized effec-
Condition (II) can be approximated to the following form by tive stresses can be presented here as per normalized radial
158 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166

distance r/a at different angles. The radial change in the initial

yielding with respect to the normalized radial distance r/a at
several selected angles is plotted in Fig. 6, in which only a quarter
of the spot weld nugget is shown as per the symmetrical condi-
r a
tions. It can be observed that the locations of the initial yielding

ra
ra
r a change gradually from the normalized radial distance r/a¼ 1.34 to
o the edges of the spot weld nugget, and finally to the infinity as the
angle further increases.
The contours of the normalized stresses are also plotted in the
polar coordinate system as in Fig. 7, in which the distributions of
the normalized radial stress s^ rr (Fig. 7(a)), hoop stress s^ θθ (Fig. 7(b))
Fig. 5. The trend of the locations of the initial yielding failure along the different and shear stress s^ rθ (Fig. 7(c)) are quite similar. For the normalized
normalized radial distances r/a (note that the figure illustrates the location change
radial stress s^ rr (Fig. 7(a)), the maximum and minimum values
and is not scaled).
locate at the angles of θ ¼01, 1801 and θ ¼ 901, 2701, respectively, as
the normalized radial distance r/a goes to infinity. For the normal-
ized hoop stress s^ θθ (Fig. 7(b)), the locations of the maximum and
minimum values are opposite to the normalized counterparts,
occurring at the angles of θ ¼ 901, 2701 and θ ¼01, 1801, respec-
tively, as the normalized radial distance r/a goes to infinity. For the
normalized shear stress s^ rθ (Fig. 7(c)), the locations of the max-
imum and minimum values are different from those of s^ rr and s^ θθ ,
appearing at the angles of θ ¼ 1351, 3151 and θ ¼451, 2251,
respectively, as the normalized radial distance r/a goes to infinity.
The stress distribution of the normalized effective stress s^ e (Fig. 7(d))
is somewhat different from those three stress components, which
appears considerably complicated near the weld nugget but
ra
approaches to a constant (around 1.8) as the normalized radial
distance r/a goes to infinity regardless of the angles.
o r a
r a
3. Elasto-plastic analysis by finite element method
Fig. 6. The location change of the initial yielding near the rigid spot weld
nugget along the different normalized radial distance r/a at several special angles The analytical stress solutions can be used to outline the pre-
(only a quarter of the spot weld nugget is shown herein). liminary stress distributions and initial failure locations. However, the

Fig. 7. Schematic plots of the normalized stresses in the polar coordinate system; (a) the normalized radial stress s^ rr , (b) the normalized hoop stress s^ θθ , (c) the normalized
shear stress s^ rθ , (d) the normalized effective stress s^ e .
 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166 159

elastic solutions may not provide sufficient information to under- solutions as presented above. Then, the elasto-plastic FEA follows
stand the failure process of lap-shear under ramped loading [11]. to explore the development of plastic flow near the spot weld
For this reason, a plane stress elastic finite element analysis (FEA) is nugget in the lap-shear specimens.
conducted to validate the numerical solution to the analytical elastic
3.1. Determination of the mechanical properties of base materials

In order to obtain the stress distributions and failure onsets of

lap-shear specimen under ramped loading condition, it is neces-
sary to determine the mechanical properties of the base materials
[43,44]. The base material samples considered herein were made
of the advanced high strength biphase steels (DP600 and DP980).
In order to obtain their mechanical parameters, the standard
tensile tests were conducted first.
The schematic of the geometry and dimensions is presented in
Fig. 8. Schematic showing the geometry and dimensions of the DP steel specimens Fig. 8. Each base material has three specimens. The relationship
for uni-axial tensile tests (unit: mm).
between the engineering stress and engineering strain of speci-
mens is plotted in Fig. 9. In the proportional range, the Young
module E and the Poisson's ratio υ of the both base materials were
obtained as 210 GPa and 0.3, respectively. As seen from an inset of
the figure, there is very little difference in the engineering stress–
strain curves of different specimens for the same material, espe-
cially for DP600. It can be seen that the yield stress and strength of
the DP980 specimens were higher than those of the DP600
specimens.
The rupture or failure locations are displayed in Fig. 10. From
Fig. 10(b) and (d), the failure locations of these three specimens
seem to be identical for the same material group. Based on the
tensile tests, the mean yield stresses sy of DP600 and DP980 are
376.4 MPa and 663.6 MPa, and the yield strengths sb are 611.9 MPa
and 953.1 MPa, respectively, as summarized in Table 1. Note that
the macro-plastic behavior of quantities of engineering metal
materials can be expressed in the form of a power-law formula
(s ¼ Kεn ), the strength coefficient K and the strain hardening
exponent n obtained from each tensile test are also presented in
Table 1. Note that the strength coefficient K of DP980 is much
Fig. 9. Relationship between the engineering stress and engineering strain of higher than that of DP600 material, but the strain hardening
DP600 and DP980 materials. exponent n is lower.

Fig. 10. Comparing of failure locations of the base materials; (a) DP980 before tensile test, (b) Deformed DP980 after tensile test, (c) DP600 before tensile test, (d) Deformed
DP600 after tensile test.
160 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166

Table 1 In the analytical elastic solution, the normalized effective stress

The material properties of DP600 and DP980 from the uniaxial tensile tests. along the spot weld nugget must be smaller than the yield stresses
of DP600 (376.4 MPa) and DP980 (663.625 MPa). Based on the
Specimen numbers E/MPa υ sy /MPa sb /MPa K/MPa n
yield stresses and Eq. (20), Table 2 summarizes the critically
DP600 specimen—1 – – 376.237 611.479 1035.1 0.2038 distributed load qc, under which the yielding could be initiated.
DP600 specimen—2 – – 375.441 610.668 1023.3 0.1930 Specifically, as the distributed load q reaches 272.06 N/mm2 and
DP600 specimen—3 – – 377.523 613.692 1025.7 0.1907 479.66 N/mm2, the material near the spot weld nugget will yield
Mean value (DP600) 2.1e5 0.3 376.400 611.946 1028.0 0.1958
DP980 specimen—1 – – 649.604 939.101 1412.5 0.1308
and undergo plastic deformation.
DP980 specimen—2 – – 674.407 954.304 1345.9 0.1247 Fig. 12 compares the elastic results of FEA and analytical
DP980 specimen—3 – – 666.865 965.891 1380.4 0.1258 solutions under distributed load q¼ 100 N/mm2. Overall, the
Mean value (DP980) 2.1e5 0.3 663.625 953.099 1379.6 0.1271 numerical results are in fairly good agreement with the analytical
solutions. From the figure, four peak points are seen at 301, 1501,
2101 and 3301, respectively. These four peak stresses indicated that
the infinite plate the initial yielding locations would likely occur at these four
corresponding angles of the spot weld nugget. Note that the FEA
stress results are somewhat different from those of the analytical
solutions at some regions and the highest error occurs at angles 0o
and 180o. The main reason is that the mathematical approximation
was introduced when transforming from Eqs. (15) to (16). Further-
more, both the results match well at the angle intervals of
[451, 1351] and [2251, 3151], but differ at other angle intervals of
[01, 451], [1351, 2251] and [3151, 3601].
q Based on the numerical simulation, the contour of von Mises
q 2a stress is plotted in Fig. 13. Comparing the analytical solution in
Fig. 7(d) with numerical solution in Fig. 13(a), the stress distribu-
tions are rather similar. As in Fig. 13(b), the locations of the peak
von Mises stresses (i.e. 141.145 MPa, 146.598 MPa, 146.522 MPa
and 140.676 MPa at the corresponding four points A, B, C, and D)
along the spot weld nugget from FEA are similar to those from the
analytical solution at four specific angular locations of θ ¼ 38.021,
141.981, 218.021 and 321.981.

Fig. 11. A plane stress finite element model subjected to a uniformly distributed
Table 2
tensile load q along the left and right edges.
The critically distributed loads of the yielding and necking for these two material
specimens based on the analytical elastic solutions.

3.2. Elastic analysis near the spot weld nugget by finite element Base material specimens sy /MPa sb /MPa qya/(N/mm2) qnb/(N/mm2)
method
DP600 base material 376.400 611.946 272.06 442.30
DP980 base material 663.625 953.099 479.66 688.88
To simulate the mechanical problem shown in Fig. 4(a),
commercial FEA program ABAQUS 6.10-1 was used. As pointed a
Represents the uniformly distributed loads of the yielding failure.
out by Lin et al. [11], the ratio of the plate width to the spot weld b
Represents the uniformly distributed loads of the necking failure.
nugget diameter should be at least more than 20 as per the St
Venant principle. In this study, the plate width and nugget radius
are taken as 128 mm and 3.2 mm, respectively. In order to explore
the elastic stress distributions near the spot weld nugget by FEA,
eight-node quadrilateral elements (plane stress elements: CPS8R)
which is a quadratic 2D element with 2 DOFs per node are
adopted. The finite element model for a spot weld nugget in a
plate under the uniformly distributed load q is thus shown in
Fig. 11. There are 169 elements in the region of spot weld nugget
after a convergence test. The spot weld nugget is tightly connected
to the plate along the nugget circumference and the uniformly
distributed tensile load q is applied in the left and right edges for
the plates and stretched in the both directions. The upper and
lower edges of the lap-shear specimen are set as a traction-free.
Based on the analytical solutions shown in Fig. 5 and Fig. 7(d),
the initial yielding along the rigid spot weld nugget (i.e. at the
normalized radial distance r/a ¼1) locates at the four special
angles (θ ¼38.021, 141.981, 218.021, 321.981), respectively. So if
the normalized radial distance r/a and the angle are taken as 1 and
38.021, for instance, the normalized effective stress se can be
calculated from Eqs. (17) and (18) as follows.
Fig. 12. Comparisons of the elastic FEA and the elastic analytical solutions under
se ¼ 1:3835q ð20Þ the uniformly distributed load of q ¼100 N/mm2.
 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166 161

Fig. 13. (a) Distributions of the elastic von Mises stress of the plate subjected to the uniformly distributed loading along the left and right edges, (b) Enlarged distributions of
the elastic von Mises stress near the spot weld nugget.

where ε_ is the equivalent plastic strain rate defined as

pl
3.3. Elasto-plastic finite element analysis near the spot weld nugget
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
From Fig. 9, the homogenized plastic behavior of the base _εpl ¼ 2ε_ pl : ε_ pl ð25Þ
materials can be expressed in the power-law formula. In order to 3
investigate the effect of elasto-plastic material behavior on the
plastic flow near the spot weld nugget, the “elastic perfectly plastic" As such, the equivalent plastic strain εpl is obtained from the
constitutive model considered by Lin et al. [11] does not seem integration of ε_ pl over the deformation history as
sufficiently accurate and may characterize the behavior qualitatively Z
ε_ dt
pl
only. But for the power-law constitutive model, the necking failure εpl ¼ ð26Þ
occurs in the locations where the stress reaches the ultimate tensile
strength (UTS), compared with the yield strength used in the elastic
perfectly plastic model. As such, the constitutive model adopted in Note that equivalent plastic strain εpl can be used to indicate
this study allows describing the plastic behavior near the spot weld the extent of the plastic deformation.
nugget more accurately. From Table 2, when the stresses reach Based on Table 2, it is reasonable to define critical loads of
611.946 MPa and 953.099 MPa for DP600 and DP980, respectively, qc ¼272.06 and qc ¼ 479.66 N/mm2 for the DP600 and DP980,
the material undergoes necking or thinning. Based on Eq. (20), respectively, at which the plastic deformation is initiated. Thus,
the critical uniformly distributed loads qc (reach to initial necking we can present the FEA results under the critical distributed load
stress) are estimated at 442.3 N/mm2 and 688.88 N/mm2 for these qc. In the 2D elasto-plastic FEA, the uniformly distributed tensile
two biphase high strength steels, respectively. load q is applied along the left and right edges each at an
Despite the mechanical implication, the elastic analyses do increment of Δq ¼50 N/mm2 until the peak stresses reach or
not provide adequate insights into the failure mechanism. For exceed the UTS (i.e. 611.946 MPa and 953.099 MPa) in order to
this reason, the power law based elasto-plastic FEA is conducted explore the location change of the maximum plastic strain and
here to further explore the plastic strain distributions and initial necking failure (specifically, 272.06, 322.06, 372.06, 422.06
deformation patterns. and 472.06 N/mm2 for DP600 and 479.66, 529.66, 579.66, 629.66,
A fundamental assumption of the elasto-plastic model is that 679.66, 729.66 N/mm2 for DP980), respectively.
the strain rate ε_ can be divided into an elastic part ε_ el and a plastic Fig. 14 plots the contours of von Mises stress of biphase steel
(inelastic) ε_ pl part as, DP600 at different load levels q. Due to symmetrical feature, only
half of contour plot (right part) is presented herein. The minimum
ε_ ¼ ε_ el þ ε_ pl ð21Þ stresses locate at the middle (901 and 2701) and do not change
with the increase in loading q. Fig. 15 plots the angular distribu-
tions of the equivalent von Mises stress spl near the spot weld
The elastic behavior can be formulated as, nugget as increase in load q. Due to the similarity of von Mises
stress between the first and the third (or the second and the
s ¼ Del : ε ð22Þ fourth) Gaussian integration points, there are only two integration
points discussed here. From Fig. 15(a) and (b), the locations of the
where Del represents the fourth-order elasticity tensor and s and ε peak von Mises stresses remain unchanged with increase in
are the second-order stress and strain tensors, respectively. distributed load q. For the biphase steel DP980, the angular
For the plastic behavior, the equivalent von Mises stress can be distributions of the von Mises stress have a rather similar trend
defined as. to those of DP600 and are not presented here.
In order to determine the locations of the initial necking and
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 indicate the extent of the plastic deformation, the equivalent
f ðs αÞ ¼ ðS  αdev Þ : ðS αdev Þ ð23Þ plastic strain εpl and the principal strains along the spot weld
2
nugget are discussed here, which can be used to predict the
where S is the deviatoric stress tensor and αdev is the deviatoric location change of the plastic deformation [11]. The contours of
part of the back stress. The associated plastic flow is calculated as the equivalent plastic strain along the spot weld nugget with the
increasing in the distributed load q are exhibited in Fig. 16, in
∂f ðs  αÞ _ pl which the change in the peak equivalent plastic strain can be
ε_ pl ¼ ε ð24Þ
∂s observed.
162 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166

Fig. 14. Contours of the equivalent von Mises stress under different distributed loads q (for DP600 base material); (a) q ¼272.06 N/mm2, (b) q ¼322.06 N/mm2, (c)
q¼ 372.06 N/mm2, (d) q¼ 422.06 N/mm2, (e) q¼ 472.06 N/mm2.
 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166 163

Fig. 15. The angular distributions of the equivalent von Mises stress along the spot weld nugget with respect to the distributed load q (DP600); (a) and (b) at the first and
second integration point respectively.

Fig. 16. Contours of the equivalent plastic strain with increase of distributed load q with intervals of 50 N/mm2 (DP600).
164 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166

Fig. 17 plots the distributions of the strains (radial εrr , hoop εθθ , special angular intervals as the uniformly distributed load q increases.
and shear εrθ ), the major principal strain ε1 and the minor The schematic of the change zones of the maximum equivalent plastic
principal strain ε2 for DP600 under distributed load q ¼472.06 N/ strain points around the spot weld nugget of DP600 and DP980
mm2. In Fig. 17(a), the maximum radial strain εrr is about 0.025 materials are illustrated in Fig. 20(a) and (b), respectively. It can be
and located at the four special angles of θ ¼301, 1501, 3101, 3301. seen that the four necking or thinning failure zones are determined in
The maximum shear strain (εrθ ¼0.06) appears at other four each quadrant (Fig. 18) and the locations are fairly similar for both the
special angles fairly close to the values derived from the analytical base materials.
elastic solutions (θ ¼38.021, 141.981, 218.021, 321.981). In Fig. 17(b), In fact, the results generated from this study are of general
the major principal strain ε1 is presented and the curve has four implication to the different high strength steels. So it can be
peak strains at the middle of each quarter of the inclusion (i.e. speculated that the angular locations of the maximum equivalent
θ ¼451, 1351, 2251, 3151), where the minor principal strain ε2 plastic strain points or initial necking failure points locate at the
reaches its minimum value ( 0.025). Note that along the spot four angular intervals i.e. [19.591, 45.001], [131.341, 158.571],
weld nugget, the major principal strain ε1 is either positive or zero [199.291, 225.001], [318.861, 338.571], respectively. Such derived
whereas the minor principal strain ε2 is either negative or zero. In angular intervals could provide an effective reference to the analysis of
addition, the major principal strain ε1 is zero at the intervals of some other engineering structures connected in spot welds.
θ A [751, 1051] and [2551, 3051], in which the minor principal strain
ε2 reaches its maximum (zero too).
Based on the stress and strain distributions obtained, some 4. Conclusion
evident patterns can be observed around the spot weld. In order to
explore the angular locations of the maximum equivalent plastic In this paper, the failure onsets of resistance spot welding in
strain, the circle representing the spot weld nugget can be divided lap-shear specimens for advanced high strength steels (AHSS)
into four distinguishing zones for elasto-plastic analysis as shown (specifically, DP600 and DP980 base materials respectively) were
in Fig. 18. Fig. 19 plots the angular location against the distributed studied based on analytical elastic and elasto-plastic finite element
load q for the maximum equivalent plastic strain in these different analysis. The analytical solutions to a lap-shear specimen with a
zones. When load q is relatively low (e.g. 272.06 N/mm2 rqr spot weld nugget subjected to the uniformly distributed loading
372.06 N/mm2), the angular locations of the maximum equivalent condition were derived. The angular distributions of the normal-
plastic strains vary. However, when load q reaches or exceeds ized effective stress indicate that the yielding can initiate from four
372.06 N/mm2, the angular locations of the maximum equivalent special angles of 38.021, 141.981, 218.021 and 321.981. The contours
plastic strain remain unchanged as predicted from the analytical elastic of the normalized stress were plotted in the polar coordinate
solutions (i.e. θ ¼38.021, 141.981, 218.021, 321.981). Tables 3 and 4 system, which indicate the locations of the maximum and mini-
summarize all these results. mum values of the normalized radial, hoop, and shear stresses at
The bold characters in Tables 3 and 4 represent the minimum
and maximum angular locations in different zones. The shift of
y
angular locations of the maximum equivalent plastic strain εpl θ = 90°
appears different from the literature [11]. The main reason leading
zone2 zone1
to the discrepancy is that (1) the uniform tensile-shear load q was
applied to both the edges of the lap-shear specimen and (2) the
elasto-plastic power-law behavior was adopted here. Based on the
θ = 180° θ = 0°
results from [11], the maximum equivalent plastic strain εpl occurs o x
initially on the two side edges of the spot weld nugget (θ ¼ 01,
1801), and the angular locations shift to θ ¼451, 1351, 2251, 3151
zone3 zone4
when the imposed load increases, and at last, the locations occur
on the top and bottom of the spot weld nugget (θ ¼901, 2701). It is θ = 270°
noted that there is no such obvious shift in our study and the angular
locations of the maximum equivalent plastic strain εpl occur in some Fig. 18. Schematic of different zones for elastoplastic finite element analysis.

Fig. 17. Strains distributions vs the angles (DP600 and q ¼472.06 N/mm2); (a) the radial strain εrr , the hoop strain εθθ and the shear strain εrθ , (b) the major principal strain ε1
and the minor principal strain ε2 .
 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166 165

Fig. 19. The maximum equivalent plastic strain vs angular location as per distributed load q at two different integration points (DP600 material).

Table 3
The maximum equivalent plastic strain εpl around the spot weld nugget and their corresponding angular locations θ for DP600 base
material.

q/(N/mm2) IPa εpl

max
b Anglec θ/1

272.06 1 0.00078/0.00027/0.00078/0.00027 30.00/158.57/210.00/338.57

2 0.00027/0.00078/0.00027/0.00078 19.29/147.86/199.29/327.86
322.06 1 0.00434/0.00375/0.00434/0.00375 38.57/132.86/218.57/312.86
2 0.00252/0.00434/0.00252/0.00434 27.86/141.43/207.86/321.43
372.06 1 0.00648/0.00603/0.00648/0.00603 38.57/132.86/218.57/312.86
2 0.00603/0.00648/0.00603/0.00648 45.00/139.29/225.00/319.29
422.06 1 0.01923/0.01677/0.01922/0.01677 36.43/132.86/216.43/312.86
2 0.01677/0.01922/0.01677/0.01923 45.00/141.43/225.00/321.43
472.06 1 0.04243/0.03800/0.04243/0.03801 36.43/132.86/216.43/312.86
2 0.03800/0.04243/0.03801/0.04243 45.00/141.43/225.00/321.43

a
IP represents the different Gaussian integration points.
b
The maximum equivalent plastic strains in four individual different zones which are shown in Fig. 17.
c
Angular locations of the maximum equivalent plastic strains in corresponding different zones.

Table 4
The maximum equivalent plastic strain εpl around the spot weld nugget and their corresponding angular locations θ for DP980 base
material.

q/(N/mm2) IP εpl
max Angle θ/o

1 0.00083/0.00027/0.00083/0.00027 32.14/156.43/212.14/336.43
479.66
2 0.00027/0.00083/0.00027/0.00083 21.42/145.71/201.43/325.71
1 0.00300/0.00231/0.00300/0.00231 36.43/132.86/216.43/312.86
529.66
2 0.00187/0.00300/0.00187/0.00300 27.86/141.43/207.86/321.43
1 0.00484/0.00426/0.00484/0.00426 38.57/132.86/218.57/312.86
579.66
2 0.00426/0.00484/0.00426/0.00484 45.00/141.43/225.00/321.43
1 0.00634/0.00587/0.00634/0.00587 38.57/132.86/218.57/312.86
629.66
2 0.00587/0.00634/0.00587/0.00634 45.00/139.29/225.00/319.29
1 0.00750/0.00696/0.00750/0.00696 38.57/132.86/218.57/312.86
679.66
2 0.00696/0.00750/0.00696/0.00750 45.00/139.29/225.00/319.29
1 0.01332/0.01181/0.01332/0.01181 36.43/132.86/216.43/312.86
729.66
2 0.01181/0.01332/0.01181/0.01332 45.00/141.43/225.00/321.43

the angles of 01/1801 and 901/2701, 901/2701 and 01/1801, 1351/3151 match to the analytical results fairly well. From the maximum
and 451/2251, respectively, as the normalized radial distance r/a equivalent plastic strain εpl , it was found that the maximum equivalent
goes to infinity. The detailed stress distributions along the spot plastic strain points or initial necking failure points locate at the four
weld nugget can be useful for determining the stress intensity angular intervals. The stress and strain distributions can help under-
factor and analyzing the fatigue failure. stand the failure behavior and damage evolution near spot weld
The elasto-plastic finite element analyses were also conducted to nugget for fatigue and fracture failure analyses. The obtained four
explore the necking or thinning initiation. It was found that the failure intervals provide effective references for the analyses of
angular locations of simulated peak von Mises stress and plastic strain some other AHSS engineering structures joined in spot welds.
166 F. Xu et al. / International Journal of Mechanical Sciences 78 (2014) 154–166

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