2.1. SCF of Welds
The formation of welded joints is the way in which two or more pieces meet together by the welding process. They are generally made of base metal (BM), weld metal (WM), fusion zone (FZ) and heat affected zone (HAZ). Welded joints experience a series of complicated processes such as local heating, metal melting and solidification, etc., which result in a region with heterogeneous ingredients, organization and mechanical properties.
According to the geometry of the weld, different welded joints have distinct SCF degrees. If the nonlinear plasticity is considered, stress and strain have different SCF, which are described as follows:
where
is the stress concentration factor,
is the strain concentration factor,
is the maximum stress,
is the nominal stress,
is the maximum strain and
is the nominal strain.
In the research of fatigue fields, stress range, or strain range is the much more important controlling parameter. As such, this paper defines corresponding SCF parameters as follows:
where
is the stress range concentration factor,
is the strain range concentration factor,
is the maximum stress range,
is the nominal stress range,
is the maximum strain range and
is the nominal strain range.
There are two kinds of typical welded joints commonly used in engineering, which are fillet joints (as shown in
Figure 1) and butt joint (as shown in
Figure 2). Their stress SCF formulas are shown as Equations (5) and (6) [
7].
where
is the weld inclination angle,
is the reinforcement angle,
is the plate thickness,
is the fillet leg length,
is the weld toe radius and
is the weld un-penetration size.
It can be seen from the above formulas that for fillet joints, mainly depends on the weld inclination angle, the ratio of the weld un-penetration size to the fillet leg length and the ratio of the plate thickness to the weld toe radius. Smaller weld inclination angle, larger weld toe radius and larger fillet leg length will result in smaller SCF at the weld toe. For butt joints, mainly depends on the reinforcement angle, , and the ratio of the plate thickness to the weld toe radius. With constant plate thickness and weld width, smaller reinforcement angle and larger weld toe radius will be helpful to decrease the SCF. Compared with butt joints, fillet joints usually have larger weld inclination angle, and the fillet leg length cannot be large without limitation due to the plate thickness. Therefore, the weld shape transition of fillet joints is much sharper, which is prone to produce severer SCF in the weld toe. On the contrary, the transition geometry of butt joints changes mildly and the SCF is small. Therefore, fillet joints are generally inferior to butt joints in their fatigue performances.
In fact, the transitional area of the weld toe is not an ideal rounded shape, as shown in
Figure 3. It can be seen that the local weld toe of the real weld is often composed by a series of small arcs with different radii, in which the smaller radius is close to the dimension of surface roughness. Although the smaller arcs do not affect the overall geometry of the weld section, it has a key influence on SCF at the weld toe, unquestionably. In Equations (5) and (6), only an ideal arc radius is used to approximately characterize the weld toe radius, so both the stress state analysis of the weld toe and the fatigue life prediction of the welded joint are undoubtedly inaccurate.
In addition, due to the complicated process of solidification of weld metal, not only is the actual weld toe profile on the weld section is complex, but also the profiles on different weld sections are randomly variable along the longitudinal direction of the weld, which is more arresting to the weld toe curve. Initial weld defects such as undercuts, blow holes and slag inclusions usually appear on weld toes, which makes SCF along with weld toes more complicated and severe. Therefore, it is difficult to obtain sufficient accuracy by using constant weld geometry parameters to analyze and investigate the mechanical properties of actual welded structures in engineering application.
2.2. Cyclic Stress-Strain Response Behavior
Since the weld toe of welded joints has strong stress concentration effect, even if the nominal stress is in linear elastic phase, the actual stress and strain at the weld toe tend to be in plastic phase. Furthermore, compared with the elastic-plastic behavior under monotonic loading, the stress-strain response under cyclic loading is far more complicated.
Lots of fatigue tests concerning different metals have shown that the strain amplitude decreases gradually as the number of cycles increases, even if under the symmetrical loading with constant amplitude, which is called cyclic hardening. Conversely, some metals have the property of cyclic softening, which means strain amplitude increases with cycles increasing. Cyclic hardening and cyclic softening phenomena occur in not only strain-controlled tests but also stress-controlled tests. It is generally believed that the cyclic loading causes the metal to repeatedly undergo plastic deformation, resulting in some changes to the plastic flow characteristics of the material. Then the ability to resist deformation is enhanced or reduced, which reflects cyclic hardening or softening behavior. Commonly, cyclic hardening or softening happen in the early stage of cyclic loading, then it will disappear, and the stress amplitude or strain amplitude will stabilize gradually. Such behaviors can generally be described by cyclic stress-strain curve (
−
curve) and hysteresis loop curve (
−
curve) [
36].
The cyclic stress-strain curve can be expressed by Equation (7), which does not reflect the loading path. The hysteresis loop curve can cover this shortcoming and can be expressed by Equation (8).
where
is elastic modulus,
is cyclic strength coefficient,
is cyclic strain hardening exponent,
is strain amplitude,
is strain range,
is elastic strain,
is plastic strain,
is stress amplitude and
is stress range.
For welded joints, it is difficult to determine the actual stress at the weld toe by the force signals recorded by the testing machine. Therefore, strain signals at the weld toe are often obtained and used to calculate the corresponding stresses by combining Equations (7) and (8). Furthermore, the stress-strain response curve can be drawn. The computing procedure is as follows:
- (1)
For the first loading cycle, σa is calculated according to εa by cyclic stress-strain curve (Equation (7)), in which εa is obtained from the strain testing data.
- (2)
For other cycles, the hysteresis loop curve (Equation (8)) is used to calculate ∆σ according to ∆ε, in which ∆ε is obtained by rain flow counting for the strain testing data.
It is worth noting that when the strain reaches the same reversal point for the second time, a closed-loop will be formed, and the material memory effect should be introduced. Since the closed-loop does not affect the subsequent response process, stress and strain here should be calculated according to the previous loading path.
2.3. FBG Strain Sensor
FBG is a periodic modulation of the index of refraction, which is inscribed into the core of a single-mode optical fiber [
37]. Essentially, it works as a wavelength filter [
38]. When a broadband light source is injected into the fiber, the FBG reflects a narrowband portion of the incident light with a specific wavelength, while the rest portion of the broadband light passes through, as shown in
Figure 4.
Generally, the center wavelength of the reflection spectrum is called the Bragg wavelength (λ
B), which is correlated with the grating period (the distance between the consecutive grooves).
where
is the effective refractive index of the optical fiber, and
is the periodicity of the grating.
When the FBG is subjected to the external environment (e.g., strain and temperature), the grating period increases or decreases, and then the Bragg wavelength increases or decreases, respectively. Thus, the peak shift of the Bragg wavelength introduced by the strain variation (
) and the temperature variation (
) can be expressed as:
where Δλ
B is the shift of the Bragg wavelength,
is the photo-elastic coefficient of the optical fiber,
αf is the thermal expansion coefficient,
ξ is the thermal-optic coefficient.
Typically, the fractional wavelength changes of λ
B are on the order of 10 pm/℃ and 1.2 pm/με for FBG with a center wavelength of 1550 nm [
37]. So, the Bragg wavelength shift can be considered linear to the strain changes under constant temperature condition.
It should be noted that the optical fiber is a cylindrical filament composed of a cladding layer with an outer diameter of 125 μm and a core with a diameter of 9 μm. In order to enhance the mechanical properties of the optical fiber, a protective layer with a certain thickness may be coated outside the optical fiber. Therefore, the fiber (including grating) sectional dimension is about 0.25 mm, which is much smaller than the base size of the strain gauge. Therefore, it is possible to arrange multiple FBG sensors along the longitudinal direction of the weld toe. In addition, the optical fiber with secondary coating protection has good flexural deformation resistance and can be bonded along the weld toe transition arc even across the weld toe characteristic point. Since the wavelength signal collected by the FBG sensor is the maximum value of the Bragg wavelength in the grating, for the strain testing for weld joints, it can be considered that the obtained strain data is just the exact value at the most severe position ( which refers to weld toe local position) on the certain weld section. So, the FBG is a kind of ideal strain sensors which can be used for local strain measurement at the weld toe.