**5. Crystal plasticity model**

Even though multiple models of crystal elasto-plasticity and elasto-visco elastoplasticity have been added in several kinds of research to analyze the plastic distortion and fatigue fracture of various structures; there is an absence of in-depth and systematic documentation of the process mechanism involved and essential outcomes.

Nearly in the mid- 1980s, based on extensive experimentation and analysis, Panin et al. [31, 32] added a hypothesis to analyze a deformed material as a complex arrangement or multi-scale system. It proposed to originate a novel theme based on solid-state physics, also known as physical meso-mechanics. In other studies, Panin et al. [33, 34] documented the concepts involved and typical approaches for physical meso-mechanics. These studies were based on a classical assumption of considering a deformed solid as a multi-scale system where the plastic flow takes place because of a loss in shear stability in stress regions on variable atomic levels. These discussions (based on experimentation and theoretical hypothesis) resulted in a new approach to consider a deformed solid as a multi-scale self-organizing system. It permitted configuring a multi-scale model of a deformed object that may have a complete know-how of structural scales of deformation.

Firstly, Pierce et al. [35] proposed the concept of crystal plasticity finite element method (CPFEM) to study the tensile properties of a single crystal. They analyzed only two symmetric slip systems in their model at the initial stage because of the high computational cost. At later stage, Harren et al. [36] implemented the CPFEM to a polycrystalline material by applying a 2D model. They analyzed the mechanical characteristics for a polycrystalline Cu under tensile, compressive, and shear loading. The modeling of channel die compression in an FCC Al material under twelve slip systems is taken by Becker [37]. At a later stage, several other researchers used the CPFEM tool for various purposes due to its increased computational power. For nickel-base super-alloy, Manonukul and Dunne [38] added the CPFEM concept to analyze the crack formation characteristic in LCF and HCF. For the same material, Guan et al. [39] incorporated CPFEM and high-resolution digital image correlation technique to analyze the strain localization under fatigue. It is reported that for a wide range of parameters, the CPFEM can predict the material behavior very accurately.

Kysar [40, 41] analyzed the crack formation behavior across the copper/sapphire interface developed by diffusion welding by multiple experimentation and theoretical studies. They noticed that the propagation velocity of brittle crack was much higher compared to that of ductile one. The behavior of both the cracks was different, which may be due to the variation in slip system orientation in the case of single crystalline copper with respect to the direction of crack growth. In simple words, it can be said that variation in ductility may be due to the differences in dislocation substructure evolution that may promote variable stress properties at the fracture or crack tip. Van der Giessen et al. [42] reported an analysis of discrete dislocation simulation for material response at the crack tip. In their work, they took into account only the edge dislocations having dislocation lines that were perpendicular to the modeled plane. They observed that the stress values across the crack zone were significantly higher compared to those obtained in a direct elastoplastic model. Flouriot et al. [43] proposed theoretical cum experimental analysis of stress rate in the close region of fatigue crack for FCC crystal. Their theoretical investigations were based on the crystal elasto-viscoplastic model [44] with powerlaw for shear rate. Across the crack region, strain localization bands, kink bands, and lattice rotation, the theoretical and experimental outcomes showed good agreement. To analyze the damage and fracture, Clayton [45] proposed two-level

### *Advances in Fatigue Prediction Techniques DOI: http://dx.doi.org/10.5772/intechopen.99361*

direct elasto-viscoplastic model. He detailed about the kinematic, dynamic, and thermodynamic properties as well as their relations at each level. In his analysis, an assumption to have internal discontinuities and displacement field discontinuities were made for kinematic equations. He incorporated crystal plasticity model added with anisotropic hardening law to take into account the effect of temperature field. Boudifa et al. [46] incorporated a self-consistent crystal plasticity model to investigate the fracture behavior and strain localization. To determine the damage in mesoscale, averaging of the required parameters over a representative macro volume was done.

For friction stir welding (FSW) [47–49] process, mechanical properties of joints are widely governed by microstructure patterns [50–52] and crystal structure characteristics. Across the stirred zone (SZ) in Al alloys, the strain field is not continuous for every texture band joint and contains high angle grain boundaries at most [53]. Some models have been employed to determine the influence of microstructure on the mechanical performance for the FSW joint. Dhondt et al. [53] used CPFEM to document the impact of texture on strain rate across the SZ of FSW joints. Romanova et al. [54] employed 3D microstructure-based model of FSW steel joints using a mesoscale deformation process. The effects of polycrystalline microstructure on the material flow process and fracture failure of FSW joints for Al alloy were studied by Balokhonov et al. [55]. In the case of FSW joints of Mg alloys, the texture analysis on deformation characteristics was modeled using a plastic finite element method by He et al. [56].

For laser-weld joints, Tu et al. [57] employed the Rousselier model for analyzing the fracture properties of Al alloys. Gaur et al. [58] added CPFEM technique to analyze the role of mean stress and weld defects to the fatigue life for Al alloys. They employed a two-dimensional model to simulate the fatigue loading, which replicated the microstructure pattern of the respective metal also. They added an anisotropic tessellation algorithm, as used by Briffod et al. [59] in their work for analyzing the grain shape and size taken by EBSD data. The orientation of crystals for each grain is governed by the algorithm proposed by Melchior and Delannay [60] and was relied on the orientation distribution function (ODF) of EBSD.

The R-ratio (,min∕,max) distribution as an aggregate for the last loading cycle at the lowest applied stress ranges at the respective R-ratios is shown in **Figure 5**.

#### **Figure 5.**

*Inhomogeneities in R- ratio distribution and macroscopic stress strain hysteresis curve at (a)* Δσ *= 97 MPa, R = 0.5 (b)* Δσ *= 125 MPa, R = 0.1 (c)* Δσ *= 150 MPa, R = -0.5 (d)* Δσ *= 150 MPa, R = -1 [58].*

As depicted, a significant amount of heterogeneity for R-ratio distribution with respect to its increasing value is foreseen. When R-ratios are low, macroscopic and local R-ratios are more or less the same in most of the regions. Similarly, when R-ratios are low, their difference is high. A plot depicting the R-ratio distribution at the steady-state condition when *R* = 0.5, i.e., during the last load cycle, and maximum and minimum stress values are shown in **Figure 6**. For a fair interpretation, a plot of stress vs. strain is made across the four randomly chosen elements. In the case of far-field, R-ratio comes out 0.5; however, R-ratio across the four elements are 0.54, −0.61, 5.37 and −4.32. The results emphasized that precise prediction of crack initiation could not be possible by macroscopic parameters. It is because some inhomogeneities may be developed at the microstructural level. This outcome backs the results reported in the previously published literature [61]. While using the MIG technique to join Al-Mg alloy with different filler-wire, Gaur et al. [62, 63] analyzed the fatigue properties of weld joints. It was observed that the damaging action of mean stress was because of a decrease in crack-nucleation time and crack closure effects. Also, at low R-ratios (<0.1), maximum fatigue failures were surface-initiated. However, at high R-ratios ( ≥ **0 1**. ), mostly the defect-induced failures were predominant. A phenomenon of shift in crack development mechanism can be understood because of local cyclic plasticity under stress-concentration factor.

Several researchers [58, 64, 65] have implemented the CPFEM model to predict the material properties and found that the predicted results were in good agreement with the experimental data, of course within some acceptable scatter. For example, Ye et al. [65] analyzed the fatigue crack initiation behavior in an Al beam with a hole under 4-point bending. They employed both in-situ experiments (EBSD and digital image correlation) and CPFEM simulations to investigate the slip bands and crack initiation sites at the microstructure scale. Based on EBSD maps, a realistic microstructure model was developed. They noted that the simulation results had a good agreement with the experimental outcomes in several aspects. Gaur et al. [58] predicted fatigue lives and its comparison with experimental data at different R-ratios without considering any defect (**Figure 7**). The anticipated results are observed to have a good agreement with the experimental outcomes for all the R-ratios. It is also

*Advances in Fatigue Prediction Techniques DOI: http://dx.doi.org/10.5772/intechopen.99361*

#### **Figure 7.**

*Prediction of fatigue lives at different R-ratios without considering any defect [58].*

important to note that the scatter in fatigue lives has increased significantly upon considering defects in FE simulations.
