Approaches to Low-Cycle Fatigue Analysis
The traditional approach for determining the fatigue limit for a structure is to establish the S-N curves (load versus number of cycles to failure) for the materials in the structure. Such an approach is still used as a design tool in many cases to predict fatigue resistance of engineering structures. However, this technique is generally conservative, and it does not define a relationship between the cycle number and the degree of damage or crack length.
One alternative approach is to predict the fatigue life by using a crack/damage evolution law based on the inelastic strain/energy when the structure's response is stabilized after many cycles. Because the computational cost to simulate the slow progressive damage in a material over many load cycles is prohibitively expensive for all but the simplest models, numerical fatigue life studies usually involve modeling the response of the structure subjected to a small fraction of the actual loading history. This response is then extrapolated over many load cycles using empirical formulas such as the Coffin-Manson relationship (see Coffin, 1954, and Manson, 1953) to predict the likelihood of crack initiation and propagation. Since this approach is based on a constant crack/damage growth rate, it might not realistically predict the evolution of the crack or damage.
Low-Cycle Fatigue Analysis in Abaqus/Standard
The direct cyclic analysis capability in Abaqus/Standard provides a computationally effective modeling technique to obtain the stabilized response of a structure subjected to periodic loading and is ideally suited to perform low-cycle fatigue calculations on a large structure. The capability uses a combination of Fourier series and time integration of the nonlinear material behavior to obtain the stabilized response of the structure directly. The theory and algorithm to obtain a stabilized response using the direct cyclic approach are described in detail in Direct cyclic algorithm.
The direct cyclic low-cycle fatigue procedure models the progressive damage and failure both in bulk materials (such as in solder joints in an electronic chip packaging or intra-laminar crack growth in laminated composites) and at material interfaces (such as delamination in laminated composites). The former can be based on either a continuum damage mechanics approach or the principles of linear elastic fracture mechanics with the extended finite element method. The response is obtained by evaluating the behavior of the structure at discrete points along the loading history (see Figure 1). The solution at each of these points is used to predict the degradation and evolution of material properties that will take place during the next increment, which spans a number of load cycles, ΔN . The degraded material properties are then used to compute the solution at the next increment in the load history. Therefore, the crack/damage growth rate is updated continually throughout the analysis.

The elastic material stiffness at a material point remains constant and contact conditions remain unchanged when the stabilized solution is computed at a given point in the loading history. Each of the solutions along the loading history represents the stabilized response of the structure subjected to the applied period loads, with a level of material damage at each point in the structure computed from the previous solution. This process is repeated up to a point in the loading history at which a fatigue life assessment can be made.
In bulk material, there are two approaches to modeling the progressive damage and failure. One approach is based on continuum damage mechanics. This approach is more appropriate for ductile material, in which the cyclic loading leads to stress reversals and the accumulation of plastic strains, which in turn cause the initiation and propagation of cracks. The damage initiation and evolution are characterized by the stabilized accumulated inelastic hysteresis strain energy per cycle as illustrated in Figure 2. The other approach is based on the principles of linear elastic fracture mechanics with the extended finite element method. This approach is more appropriate for brittle material or material with small scale yielding, in which the cyclic loading leads to material strength degradation causing fatigue crack growth along an arbitrary path. The onset and growth of the crack are characterized by the relative fracture energy release rate at the crack tip based on the Paris law (Paris, 1961).

At interfaces of laminated composites, the cyclic loading leads to interface strength degradation causing fatigue delamination growth. The onset and growth of delamination are also characterized by the relative fracture energy release rate at the crack tip based on the Paris law (Paris, 1961).
Both the progressive damage mechanism in the bulk material and the progressive delamination growth mechanism at interfaces can be considered simultaneously, with the failure occurring first at the weakest link in a model.
Defining a low-cycle fatigue analysis using the direct cyclic approach is similar to defining a direct cyclic analysis. See Direct Cyclic Analysis for details on how to specify the number of Fourier terms, number of iterations, and the increment sizes. You specify the maximum numbers of cycles, Nmax , when you define the low-cycle fatigue analysis step.
Input File Usage
DIRECT CYCLIC, FATIGUE first data line , , Nmax
Abaqus/CAE Usage
Step module: Create Step: General: Direct cyclic; Fatigue: Include low-cycle fatigue analysis, Maximum number of cycles: Value: Nmax
Determining Whether to Use the Fourier Coefficients from the Previous Step
A low-cycle fatigue step using the direct cyclic approach can be the only step in an analysis, can follow a general or linear perturbation step, or can be followed by a general or linear perturbation step. Multiple low-cycle fatigue analysis steps can be included in a single analysis. In such a case the Fourier series coefficients obtained in the previous step can be used as starting values in the current step. By default, the Fourier coefficients are reset to zero, thus allowing application of cyclic loading conditions that are very different from those defined in the previous low-cycle fatigue step.
As in a direct cyclic analysis, you can specify that a low-cycle fatigue step in a restart analysis should use the Fourier coefficients from the previous step, thus allowing continuation of an analysis to simulate more loading cycles. In a low-cycle fatigue analysis, a restart file is written at the end of the stabilized cycle. Consequently, a restart analysis that is a continuation of a previous low-cycle fatigue analysis will start with a new loading cycle at t=0 (see Restarting an Analysis).
Input File Usage
Use the following option to specify that the current step is a continuation of the previous low-cycle fatigue step using the direct cyclic approach:
DIRECT CYCLIC, FATIGUE, CONTINUE=YES
Use the following option to reset the Fourier series coefficients to zero:
DIRECT CYCLIC, FATIGUE, CONTINUE=NO (default)
Abaqus/CAE Usage
Use the following option to specify that the current step is a continuation of the previous low-cycle fatigue step using the direct cyclic approach:
Step module: Create Step: General: Direct cyclic; Basic: Use displacement Fourier coefficients from previous direct cyclic step; Fatigue: Include low-cycle fatigue analysis
Use the following option to reset the Fourier series coefficients to zero:
Step module: Create Step: General: Direct cyclic; Fatigue: Include low-cycle fatigue analysis