bainitic phases Ni K, et al, 2004. The region around the nugget, the so-called heat- affected zone HAZ, has a mixed microstructure consisting of martensite, bainite, ferrite
and pearlite. The nugget is much harder than the base material due to the quenching effect, while the HAZ has a gradient mechanical property and a mixed microstructure with the
strength decreasing from the nugget to the base. In many cases, failures of spot welded joints tend to occur around this region, specifically around the heat-affected zone HAZ
Mukhopadhyay M., 2009.
Many research has been conducted to improve the understanding on spot welded joint as the interactions between electrical, thermal, metallurgical and mechanical phenomena.
Hou Z., et al, 2006. One active research field is on the prediction of the dimension of spot welded joints by simulating the welding process with the finite element modelling
Emmanuel H., et al, 2007; Rahman M. M., et al, 2008. Another active research field is
on the study of microstructure development Sun D.Q., et al, 2007; Bakavos D., et al,
2010. The microstructure models have to consider the thermo-physical properties of the
materials in order to describe the phase transformations during heating and cooling stages Jou M.,2003, Tan J.C., et al, 2007, Chigurupati P., 2010. These works have resulted in
several models to describe the simultaneous formation that has made it possible to predict the microstructure development and transformations during spot welding process, and also
to investigate the characteristics and behavior of materials, relating with the applied load
conditions on the spot weld joint 2.5. Mechanical integrity of welded joints
Spot welded joints are widely used in many loads bearing situations and their mechanical strength has a strong influence on the integrity of the whole structure. A large amount of
research works have been conducted to study the deformation of spot welded joints, experimentally or numerically, under different loading, such as tensile, bending, impact
etc. Darwish, 2003, Cavalli et al, 2003, Yang et al, 2005. Figure 2.5. shows typical mechanical testing methods of spot welded joint, including peel test, lap shear test, cross-
tension tests, Impact test, corrosion fatique test, and stress corrosion cracking tests Alenius, 2006. Each test has its own objectives and the selection of tests hsould be
dependent on the service condiiton of the structure being concerned.
2.6 The materials behaviours of metallic materials and the properties of different welded zones
The plastic behaviour is normally described by the constitutive material equations. In many cases, the three parameter power law hardening rule Eq. 2.1 is used for steels:
= + 2.1
where the parameter ‘
σ
’ is the yield stress, ‘
K
’ is the strength coefficient and ‘n’ is the strain hardening exponent. These material parameters influence both the yielding strength
and work hardening behaviour of the spot welded joint. The engineering measures of stress and strain, denoted as
and respectively, are determined from the measured the load and deflection using the original specimen cross-sectional area
and length as
=
;
= 2.2
In the elastic portion of the curve, many materials obey Hooke’s law, so that stress is proportional to strain with the constant of proportionality being the modulus of elasticity
or Young’s modulus, denoted E: = 2.3
Using the true stress = PA rather than the engineering stress = ⁄ can give a
more direct measure of the material’s response in the plastic flow range. A measure of strain often used in conjunction with the true stress takes the increment of strain to be the
incremental increase in displacement dL divided by the current length L: = =
= ln
2.4
This is called the “true” or “logarithmic” strain. During yield and the plastic-flow regime following yield, the material flows with negligible change in volume; increases in length
are offset by decreases in cross-sectional area. Prior to necking, when the strain is still uniform along the specimen length, this volume constraint can be written:
= 0 =
= 2.5
The ratio is the extension ratio, denoted as λ. Using these relations, it is easy to develop relations between true and engineering measures of tensile stress and strain
= 1 + = . λ = ln 1 + = λ 2.6
These equations can be used to derive the true stress-strain curve from the engineering curve.
The failure of spot welded joints can be overload failure and fracture Lee H., et al, 2005. Fracture is separation, or fragmentation of a solid body into two or more parts under the
action of stress. Fractures of metals are classified two sorts as the fracture behaviour: brittle fracture and ductile fracture. The ductile fracture process in metals usually follows
a sequence of three stages: a Formation of a free surface at an inclusion or second phase particle by either interface decohesion or particle cracking, b Growth of the void around
the particle, by means of plastic strain and hydrostatic stress, c Coalescence of the growing void with adjacent voids. The linking of voids may occur on planes which are
perpendicular to the applied stress normal rupture, or are parallel to it delimitation, or along shear bands at angle to it shear fracture. Most attempts at understanding ductile
fracture have considered only the first type. Once a microvoid has been nucleated in a plastically deformed matrix, by either the bending or cracking of a second-phase particles
or inclusions, the resulting stress-free surface of the void causes localized stress and strain concentration of the adjacent plastic field. The fracture of spot welded joints is a ductile
fracture. Gurson model is widely used in ductile fracture mechanics, in which, the fracture of
material is considered as the result of void growths in the material volume. The Homogenous material surrounding the void is called matrix material. The Gurson model
can realistically represent failure, provided the loading state in the coupon used to determine the Gurson parameters, is similar to that in the rupture zone of the structure.
The most commonly used model based on the Gurson was called Gurson-Tvergaard-
Needleman GTN model ABAQUS Theory Manual 6.9, which is briefly described below. The original model developed by Gurson, assumed plastic yielding of a porous
ductile material, where the yield surface was a function of a spherical void as follows =
+,
+ - .01
2 +,
− 4 + - = 5 2.7
Where
6
is the yield stress of the material,
7
is the mean stress, 8 is the void volume
fraction. 8 = 0 implies that the material is fully dense, and the Gurson yield condition
reduces to that of von Mises; 8 = 1 implies that the material is fully voided and has no
stress carrying capacity. 9
:;
is the components of stress deviator , = 1, 2,3 , defined
as =
−
2
A
2.8
And B
:;
is the Kronecher delta
A = C4 =
5 ≠ E
Theoretical micromechanical studies for materials containing periodic distribution of cylindrical or spherical voids have been carried out by Tvergaard 2000. By considering
the influence of neighbouring voids on each pair of voids, three parameters, q1, q2, q3,
have been added to equation 2.9
=
+,
+ F
4
- .01
F
2 +,
− 4 + F -
= 5 2.9
G1, G2, G3 are material constants and it was found that matching of test results can be achieved for most alloys, by taking the following material values ABAQUS Theory
Manual 6.4: G = 1.5 G
I
= 1 G
J
= G
I
= 2.25
Tvergaard and Needleman have further modified the Gurson model by replacing 8 with an
effective void volume fraction, 8
∗
8 :
-
∗
= L - - - ≤ -
N
-
N
+
-
O
PPPPQ-
N
-
O
Q-
N
- − -
N
-
O
PPP - - ≥ -
O
E - -
N
- -
O
2.10
Where :
-
O
PPP = F
4
+ TF
4
− F F
The parameter 8
U
and 8
V
model the material failure due to mechanisms such as micro fracture and void coalescence.
8
U
is a critical value of the void volume fraction, and 8
V
is the value of void volume fraction at which there is a complete loss stress carrying capacity
in the material The total charge in void volume fraction is given as
-W = -W
XY
+ -W
Z[N\
2.11
Where 8W
]
is charge due to growth of existing voids and 8W
_`U
is change due to nucleation of new voids. Growth of the existing voids is based on the law of conservation of mass
and is expressed in terms of the void volume fraction -
XY
W = 4 − - aW.
b\
∶ d
e
:;
= B
:;
the unit second order tensor The nucleation of voids is given by a strain-controlled relationship:
-W
Z[N\
= f aPW
2 b\
Where
f =
-
Z g
√ i
jkl m−
4
n
aP
2 b\
Q a
g g
o p
2.12
The normal distribution of the nucleation strain has a mean value
q
and standard deviation
9
q
, 8
_
is the volume fraction of the nucleated voids. Gurson model has been implemented in several FE modelling software, like ABAGUS, ADINA, SYSTUS, etc,
and is widely used by researchers ABAQUS Theory Manual 6.9 Fatigue failure is another problem associated with spot welded joints. Fatigue is the
progressive and localized structural damage that occurs when a material is subjected to cyclic loading. The maximum stress values are less than the ultimate tensile stress limit,
and may be below the yield stress limit of the material. Some common fatigue mechanisms on metallic material include Time-varying Loading fatigue, Thermal fatigue,
Corrosion Fatigue, SurfaceContact Fatigue, combined creep and Fatigue Matos A., 2010 Time-varying loading fatigue can be defined as a process caused by time-varying
load which never reach a high enough level to cause failure in a single application, and yet results in progressive localized permanent damages on the material. The damages, usually
cracks, initiate and propagate in regions where the strain is most severe. When the local damages grow out of control, a sudden fracturerupture ends the service life of the
structure. Common categories and approaching methods, include: The S-N curve Stress Life Method, is the basic method presenting fatigue failure in high
cycles N 10
5
which implies the stress level is relatively low and the deformation is in elastic range. Users may check the maximum and minimum stress directly. Define R is the
ratio of minimum stress to the maximum stress. Alternatively, define A is the ratio of alternating stress to mean stress.
r =
s
tuv
s
twx
2.13
=
s
w
s
t
=
Qy zy
2.14 An approximation based on the zero-mean stress S-N curve proposed by Goodman and
Gerber is written as
{
=
|
}1 − }
s
t
s
~
•
6
• 2.15
Where
|
is the stress at fatigue fracture when the material under zero mean stress cycled loading
7
is the mean stress of the actual loading.
`
is the tensile strength of the material R
= 1 is called Goodman line which is close to the results of notched specimens. r
= 2 is the Gerber parabola which better represents ductile metals. Individual contributions, known as Palmgren-Miners linear damage hypothesis or Miners
rule.
∑
_
u
•
u
q
u
•
u
‚ :ƒ
= „
2.16
Where k is the total number of different stress magnitudes in a spectrum
9
:
1 ≤ ≤ … is the magnitudes of each different stress in sprectrum
:
9
:
is the actual number of cycle under the specific stress 9
:
†
:
9
:
is the total number of cycle to failure under the specific stress 9
:
0.7 c 2.2 is an material dependent constant obtained by experiments. Set c = 1, if there is no further information available
Derived relationship for the stage II crack growth with cycle N, in term of the cyclical component ∆K of the stress intensity factor K
{ q
= ‡ ∆
7
2.17
Where ‰ †
⁄ is the crack length and m is typically in the range 3 to 5 for metal The impact strength of spot welds is a very important quality index in the automotive
industry, impact test conducted to determine the impact energy Impact performance of weld through impact process characteristics such as impact force and displacement
profiles Zhang H., et al, 2001. Impact tests were also performed to evaluate of resistance to dynamic failure in the characteristics and behaviour of thin welds of different grades of
steels in dynamic loading conditions Bayraktar E., 2004. The absorbed energy is
determined by integration of the force vs. deformation curve. The mean crush load for a given deformation is defined as the absorbed energy divided by the deformation
The energy absorbed by the specimen during crushing, EA, is equal to the area beneath the load vs. displacement curve and can be defined as follows Joosten M.W., 2010:
EA = 2.18
where is the stroke or displacement of the crosshead and is the applied axial force. The
specific energy absorption, SEA, is defined as the energy absorbed per unit mass of destroyed structure and can be defined as follows:
SEA =
Š‹Œ•ŠŽ Ž••• 9‘•’„‘’•Ž “Ž•ℎ‘
SEA =
•– —
2.19 which is a function of the density,
˜, the cross sectional area of the structure, A, and the absorbed energy, EA, over a crushing stroke,
. Note that the cross sectional area in the steeple trigger zone is not constant, and is calculated as a function of the axial
displacement, .
On axial crushing test with high speed crushing, the value of impact energy similar with that given in eq.2.20
EA =
I
› œ
I
2.20
Where m is the mass of the crosshead and v the crush velocity 2.7. Mixed experimental and numerical methods in characterising material
constitutive parameters The elastic-plastic material parameters and the fracture parameters of materials can be
readily determined when standard specimens are available. However, for a spot welded joint, standard testing is not applicable to characterise the HAZ and nugget due to their
complex structure and small size. One potential approach to be developed in this work is
to use a mixed experimental-numerical method based on the indentation test to inversely characterise the parameters of the constitutive material laws for the nugget, HAZ and the
base. In a mixed numerical-experimental approach, the load-deformation data of the material is used as input data to a finite element FE model that simulate the geometry
and boundary conditions of the experiment. This approach has been used on some non- standard specimens or surface loading situations, such as in vivo within a living
organism tension, indentation, etc. Meuwissen et al, 1998, Shan et al, 2003; Gu et al, 2003; Bolzon et al, 2004, Ren et al, 2006. With indentation tests, the local plastic
properties can be calculated by solving the inverse problem via finite element analysis by incrementally varying properties in 3D modeling to find a similar simulated load–
displacement curve as compared with experimental one.
Fig. 2.1 The setup of a spot welding system and the resistances occurring in the welding
Aslanlar S., 2006.
Fig. 2.2 Basic welding cycles for spot welding Aslanlar S., 2006.
Figure.2.3 Typical spot welding lobe curve Aravinthan A., 2003.
a Different structure zone of steel weldment Bayraktar E., 2004
b Typical structure of welded joint of dissimilar materials, A- Asymmetric penetration in the dissimilar metal joint of EN 1.4318 2B and ZStE260BH steels; B - dissimilar metal
joint of EN 1.4318 2H and DX54DZ steels Alenius, 2006 Fig. 2.4 Different structure zones of welded joints.
Base Nugget
Heat affected zone HAZ
Grain growth zone
Recrysatllised zone
Partial transformed zone
Tempered zone
Testing Methods Specimen
Peel test.
Lap shear test.
Cross tension test.
Impact test.
fatigue test
Figure. 2.5 Different testing methods of spot welded joints.
aDifferent types of indenters
b
Figure. 2.6 a Different indenter types; b Schematic illustration of a typical P-h
response of an elasto plastic material to sharp indentation
P = C P
Load
h
Depth
Fig. 2.7 Parametric modelling approach to extract the material properties Ren et al 2006.
3.
3. Inverse Material Properties Prediction Base On
