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The strengthening and hardening of metals increase with diminishing size due to plastic strain gradient accompanied by Geometrically Necessary Dislocations (GNDs). In contrast, the inertia forces under severe dynamic loading conditions decrease. To shed the light on the interaction between the two mechanisms, the present work focuses on the localization in notched bars subject to high strain-rate tensile loading and first lays out a new computational model where the Fleck and Willis strain gradient plasticity theory is implemented into a dynamic finite element framework. The new framework allows accounting for strain gradient strengthening through the non-local plasticity material model and models the “smaller is stronger” mechanism (controlled by a built-in length parameter), while material inertia governs the “larger is more dynamic” effect. The work by Needleman (2018) [Effect of size on necking of dynamically loaded notched bars, Mech. Materials, 116:180-188] is taken as a starting point, and the present study first conveys the message that the scale-dependent dynamic effect observed in visco-plastic tensile bars also exists for rate-independent materials across scales (questioned by Needleman, 2018). Subsequently, the notched bar's size and corresponding loading are scaled to the micrometer size range where plastic strain gradients influence the material response. Focusing attention on the shift between plastic localization in the notch and away from the notch (in the region subject to uni-axial tension) allows cementing this shift at the micron scale. The study shows that gradient strengthening owing the GNDs stabilizes the early localization in the notch and delays the shift between the localization mechanisms when the size of the notched bar diminishes while keeping the material length parameter constant. Moreover, severe gradient strengthening gives rise to a slow-growing neck-like (micron scale) localization mechanism where a neck-type deformation evolv

The two distinct tearing mechanisms observed in ductile metal plates are the void-by-void advance of the crack tip, and the simultaneous interaction of multiple voids on the plane ahead of the crack tip. Void-by-void crack advance, which leads to a cup-cup crack surface morphology, is the dominant mechanism if the plate contains a low number of small void nucleation sites (i.e., second phase particles). Conversely, a large number and/or size of nucleation sites trigger the simultaneous interaction of multiple voids resulting in a slanted crack. The present work aims to provide further insight into the parameters controlling the mechanisms and energy dissipation of plate tearing by focusing on the shape and, thereby, the orientation of the nucleation sites. The study uses a two-dimensional plane strain finite element domain to model the cross section of a plate, subject to mode I tearing, with discretely modeled, randomly distributed, finite-sized elliptic void nucleation sites. The developed finite element setup can capture the dependence of the crack surface morphology on the microstructure of the plate. The simulation results confirm that cup-cup crack propagation develops by intense plastic straining throughout the thinning region of the plate. Conversely, slanted and cup-cone cracks propagate in thin localized shear deformation bands. The energy dissipation is, therefore, greater for cup-cup cracks. The study demonstrates that the damage-related microstructure has a significant role in determining the overall hardening capacity of a plate, which in turn dictates the tearing mode and energy., TuBTAKTurkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [315M133]; Department of Mechanical Engineering at the Technical University of Denmark; Independent Research Fund Denmark [DFF-7017-00121, 0136-00194B], The authors gratefully acknowledge the financial support by TuBTAK (Project No: 315M133). RGA is financially supported by the Department of Mechanical Engineering at the Technical University of Denmark in the project "Advancing Numerical Analysis of Large Scale Crack Propagation in Plate Structures". KLN is financially supported by Independent Research Fund Denmark, partly, in the project "Advanced Damage Models with InTrinsic Size Effects" (Grant no: DFF-7017-00121), and, partly, in the project "Why, Where and When metals fail" (Grant no: 0136-00194B).

The two distinct tearing mechanisms observed in ductile metal plates are the void-by-void advance of the crack tip, and the simultaneous interaction of multiple voids on the plane ahead of the crack tip. Void-by-void crack advance, which leads to a cup-cup crack surface morphology, is the dominant mechanism if the plate contains a low number of small void nucleation sites (i.e., second phase particles). Conversely, a large number and/or size of nucleation sites trigger the simultaneous interaction of multiple voids resulting in a slanted crack. The present work aims to provide further insight into the parameters controlling the mechanisms and energy dissipation of plate tearing by focusing on the shape and, thereby, the orientation of the nucleation sites. The study uses a two-dimensional plane strain finite element domain to model the cross section of a plate, subject to mode I tearing, with discretely modeled, randomly distributed, finite-sized elliptic void nucleation sites. The developed finite element setup can capture the dependence of the crack surface morphology on the microstructure of the plate. The simulation results confirm that cup-cup crack propagation develops by intense plastic straining throughout the thinning region of the plate. Conversely, slanted and cup-cone cracks propagate in thin localized shear deformation bands. The energy dissipation is, therefore, greater for cup-cup cracks. The study demonstrates that the damage-related microstructure has a significant role in determining the overall hardening capacity of a plate, which in turn dictates the tearing mode and energy.

Cohesive zone traction-separation relations, and the related phenomenological parameters, for steady-state ductile plate tearing, are strongly tied to the micro-mechanics governing the void nucleation and growth process leading to localized deformation and micro-crack formation. The effects of such local variations on the damage evolution and cohesive zone parameters, respectively, are brought out in this study. A 2D plane strain model setup, first considered in Nielsen and Hutchinson (Int J Impact Eng 48:15-23 (2012)], is adopted, but here by discretely modeling a finite number of finite-size void nucleation sites distributed randomly in the plate material. It is found that the heterogeneous material conditions, resulting from the nucleation process, strongly affect the localization of damage and fracture, which influence the cohesive energy. By considering a number of realizations of the random distribution for each material configuration, it is concluded that: (i) the peak force in the cohesive traction-separation relation is, essentially, unaffected by the heterogeneity coming into play through the damage-related microstructure, while (ii) the cohesive energy decreases when either increasing the number or the size of the nucleation sites. The cohesive energy is found to be in the range of those previously reported for homogeneous materials, but a direct comparison should be made with caution. The results imply that care should be taken if the actual material configuration diverges from a homogeneous microstructure such as when considering very thin plates and for plates with a few void nucleation sites.

The objective of this paper is to investigate the ductile fracture behavior of thin metal plates under nominally plane strain conditions, with an emphasis on the differences between far-field plane strain tension and plane strain bending. Recent experimental results show that the fracture strain obtained using far-field plane strain tension tests is significantly lower than that obtained under plane strain bending conditions. Investigating the source of these differences is of critical importance for multiple industries since it affects formability, crashworthiness and ductile failure of thin metal plates. The present work provides critical insights into this issue based on the results of a detailed numerical investigation of both tests to failure, using the micro-mechanics based Gurson material model. The numerical simulations reveal many similarities between plane strain bending and far-field plane strain tension until the onset of localization. In the far-field plane strain tension case, after Considère thinning, a through-thickness shear localization occurs, followed by a micro-crack formation in the center of the plate. This is accompanied by an increase of the stress triaxiality well above the nominal plane strain tension level (i.e., ∼ 0.57 for thin plates). In contrast, the absence of thinning in plane strain bending yields a stress triaxiality that equals nominal plane strain tension, which results in a significantly lower void growth rate (in reference to far-field plane strain tension). A surface instability (undulations) eventually forms on the tension side of the plate in plane strain bending, followed by multiple shear bands. The crack initiates inside the dominant shear band and propagates from the surface into the plate. These differences in the localization mechanisms and local stress states are responsible for the difference in the macroscopic fracture strain obtained under far-field plane strain tension and plane strain bending.

Ductile crack initiation, governed by void growth to coalescence, is controlled by two distinct mechanisms, namely: (i) void-by-void growth, where substantial growth of the void closest to the crack tip precedes growth of other voids, and (ii) multiple void interaction, where the voids grow simultaneously and interact in the vicinity of the crack tip prior to crack advance. Tvergaard and Hutchinson (2002) [Int. J. Solids and Struct, 39: 3581-3597] studied the two mechanisms in detail under far-field mode I loading conditions and demonstrated their interference by changing the size and spacing of discretely modeled voids. The present numerical work focuses on mixed mode loading conditions rather than limiting the study to mode I and takes up the question; how will a change in the far-field loading conditions affect the shift between the two mechanisms? A pre-existing straight crack with nearby discretely modeled voids in an elastic-plastic material is considered by a specialized 2D plane strain setting capable of out-of-plane displacements to allow for combinations of mode I, mode II, and mode III. Details of the void-by-void growth versus the multiple void interaction mechanisms are laid out for a range of load cases and geometrical void configurations, and increasing the mode mixity is found to favor void-by-void growth accompanied by a higher load intensity required on the far-field boundary to initiate the crack. The change between the two mechanisms is tied to a change in the deformation field that surrounds the voids, and the study reveals significant deformation that stretches above (and below) the row of discrete voids for increasing shear mode contributions yielding an increase in the void rotation.

The present study investigates a sequence of failure events related to steady-state tearing of large-scale ductile plates by employing the micro-mechanics based Gurson-Tvergaard-Needleman (GTN) model. The fracture process in front of an advancing crack is approximated by a series of 2D plane strain finite element models to facilitate a comprehensive study of mixed mode fracture behavior as well as a parameter study of the cohesive energy and tractions involved in the process. The results from the conducted GTN model simulations are used to define cohesive zone models suitable for plate tearing simulations at large scale. It is found that mixed mode loading conditions can have a significant effect on the cohesive energy as well as relative displacement (in reference to pure mode I loading), while peak traction is practically unaffected. Specifically, increasing mode II contribution leads to monotonic increase of the cohesive energy. In contrast, the effect of mode III is more complicated as it leads to reduction of the mixed mode cohesive energy (in reference to pure mode I loading) at low to medium levels of mode mixity ratios (0–0.3). However, increasing mode III contribution beyond the mode mixity ratio of 0.3, reverses this trend with cohesive energy potentially exceeding the pure mode I level when at mode mixity ratio of 0.6 or higher. This behavior cannot be captured by the interactive cohesive zone models that rely on a simple rotational sweep of mode I traction-separation relation. Depending on the shear mode contribution, i.e., mode II or mode III, these models can lead to overly conservative (mode II) or unconservative (mode III) prediction of the crack growth resistance.

In a cohesive zone model, neighboring elements are connected by nonlinearsprings (instead of directly sharing a node) which are designed to i) absorb the fracture energy, ii) release the connection between elements in the fracture zone. Unfortunately ,it is often necessary to conduct lab tests combined with curve fitting in order to define the potential energy and the peak traction of the nonlinear springs.