Cliff Butcher

Void evolution and coalescence in porous ductile materials in simple shear

The fracture of porous ductile materials subjected to simple shear loading is numerically investigated using three-dimensional unit cells containing voids of various shapes and lengths of the inter-void ligament (void spacing). In shear loading, the porosity reduction is minimal while the void rotates and elongates within the shear band. The strain at coalescence was revealed to be strongly related to the initial void spacing and void shape. It is observed that a transitional spacing ratio for shear coalescence exists with coalescence being unlikely at spacing ratios lower than 0.35. Initially prolate voids are particularly prone to shear coalescence while initially oblate (flat) voids are most resistant to shear failure. The cell geometry becomes sensitive to shear coalescence for increasing void aspect and spacing ratios. In addition, the macroscopic shear stress response becomes independent of the void shape at high spacing ratios while showing a weak dependence on the void shape when the voids are far apart.

Application of the Complete Percolation Model

The performance of the complete percolation model is evaluated by applying the model to a notched tensile test specimen of AA5182 sheet. All of the fundamentals established in the previous chapters are used in this application. The calibrated void evolution models in Chap. 4 are used to model the individual voids in the particle field. The homogenization scheme of Chap. 9 is used to obtain the stress state in the particles as a function of their composition, size and shape to predict nucleation using the particle cracking-based model of Chap. 9. The particle field generator of Chap. 8 is used to generate five representative particle fields of AA5182 to capture the experimentally observed variation in the fracture strains. The percolation elements developed in Chap. 10 are placed at the notch root of the sample to capture the initiation of a macro-crack as observed experimentally. Finally, the predicted fracture strains, porosity and nucleation trends are compared and validated with the experiment data and as well as the data available in the literature.

Introduction to Ductile Fracture Modelling

Plastic deformation is widely employed in metal shaping for various applications. Ideally, metals will be formed to a permanent shape as the final product or for subsequent processes. However, due to material heterogeneity, metals sometimes will fail prematurely through ductile fracture. During metal forming processes, the existence of initial second-phase particles and inclusions in metal alloys offers sites where damage can nucleate in the form of microvoids (Fig. 1.1b). With continued deformation, nucleation of damage will continue, accompanied by the growth of the existing voids, as shown in Fig. 1.1c. At a certain stage of deformation, the interaction of neighbouring voids triggers void coalescence which eventually leads to the formation of macro-cracking and failure through ductile fracture. While easily categorized into independent regimes, the reality is that these mechanisms are tightly interwoven and are related to many additional factors such as the second-phase particle/void distribution, void geometry, stress state, strain rate, material hardening and temperature (Horstemeyer et al. 2003). Furthermore, the highly localized and dynamic nature of ductile fracture makes observing void initiation and evolution extremely difficult. Due to the complexity of ductile fracture, damage-based constitutive models must employ many approximations and simplifications to reach a tractable analytical or numerical solution.

Estimation of the Stress State Within Particles and Inclusions and a Nucleation Model for Particle Cracking

Despite great strides in developing physically motivated models for void growth, shape evolution and coalescence, a suitable treatment for void nucleation remains an open question. Accurate modeling of void nucleation is difficult within a Gurson-based framework due to the intrinsic assumption that the material does not contain any second-phase particles. Consequently, the nucleation models employed in these constitutive models are overly simplistic as the particle shape, composition, stress state and load-sharing are neglected, lumped into a single calibration parameter (Beremin 1981) or indirectly accounted for in a phenomenological manner (Chu and Needlman 1980). The lack of progress in developing physically sound nucleation models has not been for lack of effort but a result of the inherently complex nature of the nucleation process. Void nucleation is very difficult to capture experimentally since it is a relatively random and instantaneous event that cannot be captured in-situ without the aid of high resolution x-ray tomography. Additionally, the local stress state near a particle of interest is typically unknown, as well as the particle composition and mechanical properties. The nucleation mechanism can occur by debonding or particle cracking and is influenced by the particle size, shape, composition, distribution, strain rate and temperature. From an engineering perspective, one can clearly see the attraction in adopting a phenomenological nucleation model whose parameters can be adjusted to give good agreement with the experiment data. Nevertheless, there is ample opportunity to improve the physical foundation of the current nucleation models, especially in regards to percolation modeling.

On Simulation of Edge Stretchability of an 800MPa Advanced High Strength Steel

In the present work, the edge stretchability of advanced high strength steel (AHSS) was investigated experimentally and numerically using both a hole expansion test and a tensile specimen with a central hole. The experimental fracture strains obtained using the hole expansion and hole tension test in both reamed and sheared edge conditions were in very good agreement, suggesting the tests are equivalent for fracture characterization. Isotropic finite-element simulations of both tests were performed to compare the stress-state near the hole edge.

Three-Dimensional Particle Fields

There are two distinct challenges in the development of a multi-scale damage percolation model: (i) the development and application of the micromechanical models used to predict damage initiation and evolution and (ii) obtaining the experimental particle distributions for the model to use. While the development of the micromechanical models and implementation of the percolation model is a significant endeavour, obtaining the experimental particle distributions is equally as challenging, and arguably more tedious.

Void Growth to Coalescence: Unit Cell and Analytical Modelling

The voids in a ductile material subjected to plastic deformation change shape according to the local plastic flow of the material since the voids are not internally pressurized. As a result, the void growth rate and shape evolution are intrinsically linked because the void shape (and orientation) induce anisotropy, altering the stress state and the growth rate in a non-linear fashion. The standard Gurson-Tvergaard model maintains its isotropic formulation by enforcing the void to remain spherical. The influence of the void shape on the stress response of the material is shown in Fig. 4.1 as well as the variation in the growth rate in Fig. 4.2 for a practical stress triaxiality of 2/3 (equal-biaxial stretching).

Averaging Methods for Computational Micromechanics

The averaging or homogenization process is the foundation of all unit cell models and any yield criterion derived from them. In porous materials, the presence of a void gives rise to an overall response of the bulk or aggregate material that is different than that of a damage-free material. The averaging process is employed to transition from the micro-scale (unit-cell) to the macro-scale to quantify the overall response of the material and these average quantities are frequently referred to as “macroscopic” quantities. The study of homogenization techniques is a very rich field and a proper treatment is outside the scope of this book and the interested reader is referred to basic textbooks on plasticity as well as the work of Eshelby (1957), Mori and Tanaka (1973), Nemat-Nasser (1993a, b) as well as Ponte Casteneda and Suquet (1998). Only a brief explanation of the extremum theory of plasticity is provided here since a great deal of attention will be paid to the application of upper and lower bound-based yield criteria for porous ductile materials.

Modelling Void Growth to Coalescence in a 3-D Particle Field

Modeling of the growth and shape evolution of the voids and cracks in the percolation model requires certain assumptions since the evolution models are designed for spheroids and not for general 3-D ellipsoids. The main challenge in adapting these models to the general case is their implicit dependence upon the loading direction. By assuming a periodic distribution of axisymmetric voids, the void aspect ratio can be defined as a state variable with a definitive initial value that can evolve during deformation. However, in the general case, the loading direction is not constrained to a specific direction and the aspect ratio is not an independent variable, but a function of the loading direction. This point is best illustrated if we consider a penny-shaped void that has just nucleated from a cracked particle. If the principal loading direction happens to be aligned with the opening direction of the void, it will appear as a penny-shaped void as viewed from the loading direction and there is no issue. If the loading direction happens to be transverse to the void opening direction, the penny-shaped void appears as an extremely prolate or needle-shaped void that will experience negligible growth and shape evolution. Certainly the void growth and shape evolution rules for a penny-shaped void do not apply in this case.

Two-Dimensional (2D) Damage Percolation/Finite Element Modeling of Sheet Metal Forming

Experimental evidence and numerical simulation have established that ductile damage critically limits the formability of sheet metals (Gelin 1998; Hu et al. 2000; Tang et al. 1999). To accurately predict formability and to optimize material processing to achieve enhanced formability, it is important to understand how heterogeneously distributed micro-defects affect the macromechanical behaviour of sheet metal. Therefore, it is of both theoretical and practical interest to investigate damage evolution during sheet metal forming.