3.7
Advantages of the approach proposed
Previous results have shown that the proposed constitutive model is capable of predicting material failure after applying several cycles to the material. However, this prediction capability do not present a major advantage compared to other approaches such as the Coffin-Manson rule, or any other analytical expression capable of defining the maximum number of cycles that can be applied for a given plastic strain.
The main advantage of the proposed approach is that the prediction of ULCF failure does not depend on the applied plastic strain, but on the energy dissipated during the cyclic process. Therefore, it is possible to vary the plastic strain in the cycles applied to the structure and the constitutive model will be still capable of predicting the material failure.
This is proved in the following example, where an irregular load, in frequency and amplitude, is applied to the SP sample defined in section 3.3 and used to validate the formulation. The load defined is depicted in figure 3.53. This load
-5e+008 -4e+008 -3e+008 -2e+008 -1e+008 0 1e+008 2e+008 3e+008 4e+008 5e+008 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Stress [Pa] Strain ’1 cycle’
Figure 3.53: Seismic-type load applied
is applied as a fixed displacement following the same procedure used for the SP sample. The stress strain graph obtained from the numerical model is plotted in
-0.0006 -0.0004 -0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 0 10 20 30 40 50 60 70 80 90 100 Displacement [m] Step increment ’Seismic Load’
Figure 3.54: Material stress-strain response
figure 3.54. As can be seen, the applied load produces several hysteresis loops, each one with a different plastic strain.
The model is capable of capturing the energy dissipated in each one of these loops and, therefore, to evaluate the energy available in the material after having applied the load, which is equivalent to the residual strength of the material. It is also possible to repeat several times the irregular load, as shown in figure 3.53, to study the number of repetitions that are required to reach material failure. Figure 3.56 shows the stress-strain response of the material after 10 cycles (figure 3.55). At this point there are some points in the model that have lost most of their fracture strength and specimen failure occurs. As occurred with the SP model, this simulation also shows some lateral displacement on the equivalent strains due to the plastic strains suffered by the whole specimen.
Concluding, in this chapter a new plasticity constitutive law has been developed for the ULCF case that takes into account isotropic and kinematic hardening. The constitutive law has been validated on small scale samples and on large scale simulations made on pipelines. From an industrial point of view the constitutive model has proved to be a valid tool for the simulation of pipelines subjected to complex load conditions, both for the monotonic case and for the cyclic one.
3.7. Advantages of the approach proposed 89 -0.0006 -0.0004 -0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 0 100 200 300 400 500 600 700 800 900 1000 Displacement [m] Step Increment ’Seismic Load’
Figure 3.55: Seismic-type load applied: ten seismic-type cycles
-5e+008 -4e+008 -3e+008 -2e+008 -1e+008 0 1e+008 2e+008 3e+008 4e+008 5e+008 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06
Equivalent Stress [MPa]
Equivalent Strain
’Cycle 01’ ’Cycle 10’
Chapter 4
Constitutive modelling of
High Cycle Fatigue
4.1
Introduction
A stepwise load-advancing strategy for cyclic loading will be presented in this chapter that yields convergence in reasonable computational time for highly non- linear behaviour occurring past the S-N curve. The algorithm is also effective when dealing with combinations of cyclical loads. The strategy is coupled to a continuum damage model for mechanical fatigue analysis. An overview of the constitutive model is also presented. The capabilities of the proposed procedure are shown in several numerical examples. The model is validated by comparison to experimental results.
The basis of the HCF constitutive model used was initially developed by Oller et al. [100]. The model establishes a relationship between the residual material strength and the damage threshold evolution, controlled by the material internal variables and by a new state variable of fatigue that incorporates the influence of
the cyclic load. A brief overview of the constitutive formulation for the HCF case is provided in order to clarify the material behaviour exhibited in the numerical examples. Several model assumptions are to be made. Defect concentration on the microscale occurs during the whole period of cyclic loading. This is reflected in the model in a continuous reduction of the material strength, occurring even in the elastic stage. Stiffness degradation occurs only in the post critical stage, once the S− N curve has been passed and, therefore, only in the final stage before failure. The damage parameter has a phenomenological significance indicating the irreversibility of the fatigue process.
Depending on the size of the domain chosen for a fatigue numerical simulation, computational time for a numerical analysis can vary considerably. Nowadays, running simulations at macroscale level (mechanical part, structural element) continues to be a challenge, especially if the high level of structural complexity attained at the microscale needs to be taken into account to some extent at other scales. This work offers a stepwise load-advancing strategy that allows a saving of computational time and can help push the barrier of what if possible in terms of numerical simulation one step further.
The strategy can be especially effective when dealing with HCF where material lives are in the range of 106 – 107 cycles. If a single loading cycle is described by n loading steps, then the number of loading steps required to complete a HCF analysis would be in the order of 107 x n. Furthermore, if the mechanical piece has a complex geometry and a high level of discretization is required at finite element level, then at each of the 107x n load steps a large number of constitutive operations need to be computed for each integration point. The above serve as a clear example of why load-advancing strategies are of the utmost importance in HCF simulations.
Furthermore, increasingly more attention has been given to material behaviour in the very high cycle regime from an experimental point of view. The general belief that steel experiences no alteration in its properties after reaching its fatigue limit at 107cycles has been invalidated [14], [15], [69]. In this context, this work provides a tool for rapid automatized time-advance that allows taking numerical simulations beyond the limit of 107 cycles in reasonable computational time.