{ ∆λt ∆dt } k = { ∆λt ∆dt } k−1 − ( ∂HP ∂∆λ ) t ( ∂HP ∂∆d ) t ( ∂HD ∂∆λ ) t ( ∂HD ∂∆d ) t −1 k−1 { HP(∆λ t, ∆dt) HD(∆λ t, ∆dt) } k−1 (5.20) Despite having the analytical expression of the tangent constitutive tensor, equa- tion 5.13, the calculation of this tensor is extremely costly and, depending on the yield and damage functions used, its approximation does not provide correct results. To overcome this drawback this tensor is calculated numerically by a perturbation method. This is obtained as
Cijabep = δ ˙Sij
δ ˙Eab
(5.21)
with δ ˙Eab an infinitesimal perturbation applied to the mechanical strain tensor,
and δ ˙Sij the stress variation produced by the strain perturbation. With this pro-
cedure, it is necessary to apply twice a×b perturbations to obtain the complete tangent tensor. However, despite the computational cost, it provides an accurate approximation that improves the global convergence of the problem [74].
5.3
Plastic damage model oriented to fatigue ana-
lysis
The effects of a cyclic load on the constitutive behaviour of a material range from the accumulation of plastic strain in the case of ULCF to the reduction of material stiffness when dealing with high cycle fatigue. LCF induces changes in the material that are a combination of the aforementioned phenomena. In the following the methodology and motivation for taking into account the effects generated by the cyclical load will be presented.
5.3.1
Ultra-low cycle fatigue
The model is able to account for ULCF effects by incorporating a new law, especially developed for steel structures, that has been designed to reproduce their hardening and softening performance under monotonic and cyclic loading conditions (figure 5.1). This law depends on the fracture energy of the material.
Figure 5.1: Evolution of the equivalent stress
The equivalent stress state shown in figure 5.1 has been defined to match the uniaxial stress evolution shown by most metallic materials. This curve is divided in two different regions. The first region is defined by fitting a curve to a given set of equivalent stress-equivalent strain points. This curve is a polynomial of any given order and is fitted by using the least squares method. The data given to define this region is expected to provide an increasing function, in order to obtain a good performance of the formulation for cyclic analysis.
The second region is defined with an exponential function to simulate softening. The function starts with a null slope that becomes negative as the equivalent plastic strains increase. The exact geometry of this last region depends on the fracture energy of the material. The exact formulation of the constitutive law can be found in Martinez et al. [73],[11] and has been presented in detail in chapter 3.
Characteristic of this type of fatigue is the Bauschinger effect that is taken into ac- count in the constitutive model by combining isotropic hardening with kinematic hardening. The energy dissipated in each hysteresis loop is monitored and failure under cyclical loads is reached when the total available fracture energy of the material is spent.
5.3. Plastic damage model oriented to fatigue analysis 149
The plastic damage formulation presented in this paper can be used to improve result accuracy when simulating the softening behaviour under ULCF loads.
5.3.2
Low cycle fatigue
The behaviour of a material subjected to cyclical loads that induce LCF exhibits both accumulation of plastic strain and a reduction of stiffness (Figure 5.2). While ULCF can be described exclusively by plastic models and HCF by damage models, LCF should be modelled with coupled plastic damage models. It is often difficult to predict at which moment in the material life the stiffness reduction begins, since the boundaries between these types of fatigue are rather arbitrary. Stiffness reduction is understood as the change in slope when unloading occurs in a cycle of loading, and is generally present only when softening has begun. The model this work proposes aims at making a contribution in correctly assessing the fatigue life for materials subjected to ULCF and LCF and is particularly effective for the transition zone between these two phenomena.
-6e+008 -4e+008 -2e+008 0 2e+008 4e+008 6e+008 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 Stress(Pa) Strain (-) Experimental curve
Figure 5.2: Experimental stress-strain curve for X52 steel [95]
In the context of the hardening law proposed for ULCF, the plastic damage model presented in this work activates itself in the softening region. This is justified by the physical implications behind the damage phenomenon, as damage induces
porosity that leads to stress relaxation. This implies that regions 1 and 2 in figure 5.1 are governed by plasticity ensuring that only the cyclical loads that last a long enough number of cycles get to experience damage effects. This is important as the formulation is meant to guarantee that, for a material life clearly in the ULCF range (dozens of cycles or less), the constitutive equations governing are those of plasticity. By regulating the extension of regions 1+2 with respect to region 3 discrimination is made between materials that exhibit more sensitivity to ULCF with respect to LCF or the opposite.
5.3.3
Energy distribution law
The hardening law proposed in figure 5.1 marks as the onset of softening, the level of equivalent plastic strain inputted by user [73]. This is the triggering point for the plastic damage model to activate itself. At this point the total energy dissipated by the plasticity model has been quantified and, by subtracting it from the total fracture energy of the material, the energy available for the plastic damage model is obtained.
The next issue to be addressed is then: how much energy goes to the plasticity model and how much to the damage model? The following law is proposed in order to assess this issue, where N is the number of cycles the material has been subjected to, up until the first increment when softening begins, limU LCFis
the limit between the ULCF domain and the LCF one and limHCF is the limit
between the LCF domain and the HCF one (figure 5.3):
p%dam= (N− limU LCF)
(limHCF − limU LCF)× 100
(5.22)
The percentage of energy allocated to plasticity is the complementary part,
p%plast = 100− p%dam. By multiplying these percentages to the energy avail-
able for the softening process, Gsof tf = Gf− Ghardf , the nominal energy for each process is obtained: Gsof t,plastf = Gsof tf ×p%plastand Gsof t,damf = Gsof tf ×p%dam.
figure 5.3: Schematic representation of the energy distribution law in softening
over the entire fatigue domain (X axis not scaled).
5.3. Plastic damage model oriented to fatigue analysis 151
Figure 5.3: Schematic representation of the energy distribution law in softening over the entire fatigue domain (X axis not scaled)
limU LCF, then the p%dam= 0 and p%plast = 1, thus marking the behaviour as
completely governed by plasticity. When the number of cycles is greater than the limit between LCF and HCF then the entire energy available for the softening part goes to damage.
Although the energy distribution law is formulated in a straightforward and simple manner, the main difficulty lies in correctly assessing the number of cycles considered as a limit in between ULCF and LCF, and LCF and HCF. These lim- its can be derived statistically if an experimental program is available for small scale specimens. The statistical analysis has to be made taking into account the loading cycle when softening begins.
These limits are material dependent, as each material exhibits a different be- haviour in terms of the vulnerability to ULCF or HCF conditions. Consider for instance two materials that exhibit the same total fatigue life (hardening + soften- ing) for a certain straining amplitude, but have different onsets for the softening process. The stress-strain hysteresis loop for the entire load history is different and a more accurate monitoring of the exact onset of softening leads to a finer tuning in terms of strain based design.