A phase field formulation for hydrogen assisted cracking

We present a phase field modeling framework for hydrogen assisted cracking. The model builds upon a coupled mechanical and hydrogen diffusion response, driven by chemical potential gradients, and a hydrogen-dependent fracture energy degradation law grounded on first principles calculations. The coupled problem is solved in an implicit time integration scheme, where displacements, phase field order parameter and hydrogen concentration are the primary variables. We show that phase field formulations for fracture are particularly suitable to capture material degradation due to hydrogen. Specifically, we model (i) unstable crack growth in the presence of hydrogen, (ii) failure stress sensitivity to hydrogen content in notched specimens, (iii) cracking thresholds under constant load, (iv) internal hydrogen assisted fracture in cracked specimens, and (v) complex crack paths arising from corrosion pits. Computations reveal a good agreement with experiments, highlighting the predictive capabilities of the present scheme. The work could have important implications for the prediction and prevention of catastrophic failures in corrosive environments. The finite element code developed can be downloaded from www.empaneda.com/code

A phase field model for elastic-gradient-plastic solids undergoing hydrogen embrittlement

We present a gradient-based theoretical framework for predicting hydrogen assisted fracture in elastic-plastic solids. The novelty of the model lies in the combination of: (i) stress-assisted diffusion of solute species, (ii) strain gradient plasticity, and (iii) a hydrogen-sensitive phase field fracture formulation, inspired by first principles calculations. The theoretical model is numerically implemented using a mixed finite element formulation and several boundary value problems are addressed to gain physical insight and showcase model predictions. The results reveal the critical role of plastic strain gradients in rationalising decohesion-based arguments and capturing the transition to brittle fracture observed in hydrogen-rich environments. Large crack tip stresses are predicted, which in turn raise the hydrogen concentration and reduce the fracture energy. The computation of the steady state fracture toughness as a function of the cohesive strength shows that cleavage fracture can be predicted in otherwise ductile metals using sensible values for the material parameters and the hydrogen concentration. In addition, we compute crack growth resistance curves in a wide variety of scenarios and demonstrate that the model can appropriately capture the sensitivity to: the plastic length scales, the fracture length scale, the loading rate and the hydrogen concentration. Model predictions are also compared with fracture experiments on a modern ultra-high strength steel, AerMet100. A promising agreement is observed with experimental measurements of threshold stress intensity factor $K_{th}$ over a wide range of applied potentials

The role of plastic strain gradients in the crack growth resistance of metals

Crack advance from short or long pre-cracks is predicted by the progressive failure of a cohesive zone in a strain gradient, elasto-plastic solid. The presence of strain gradients leads to the existence of an elastic zone at the tip of a stationary crack, for both the long crack and the short crack cases. This is in sharp contrast with previous asymptotic analyses of gradient solids, where elastic strains were neglected. The presence of an elastic singularity at the crack tip generates stresses which are sufficiently high to activate quasi-cleavage. For the long crack case, crack growth resistance curves are predicted for a wide range of ratios of cohesive zone strength to yield strength. Remarkably, this feature of an elastic singularity is preserved for short cracks, leading to a severe reduction in tensile ductility. In qualitative terms, these predictions resemble those of discrete dislocation calculations, including the concept of a dislocation-free zone at the crack tip

Interaction of Void Spacing and Material Size Effect on Inter-Void Flow Localization

The ductile fracture process in porous metals due to growth and coalescence of micron scale voids is not only affected by the imposed stress state but also by the distribution of the voids and the material size effect. The objective of this work is to understand the interaction of the inter-void spacing (or ligaments) and the resultant gradient induced material size effect on void coalescence for a range of imposed stress states. To this end, three dimensional finite element calculations of unit cell models with a discrete void embedded in a strain gradient enhanced material matrix are performed. The calculations are carried out for a range of initial inter-void ligament sizes and imposed stress states characterised by fixed values of the stress triaxiality and the Lode parameter. Our results show that in the absence of strain gradient effects on the material response, decreasing the inter-void ligament size results in an increase in the propensity for void coalescence. However, in a strain gradient enhanced material matrix, the strain gradients harden the material in the inter-void ligament and decrease the effect of inter-void ligament size on the propensity for void coalescence

Interaction of Void Spacing and Material Size Effect on Inter-Void Flow Localization

The ductile fracture process in porous metals due to growth and coalescence of micron scale voids is not only affected by the imposed stress state but also by the distribution of the voids and the material size effect. The objective of this work is to understand the interaction of the inter-void spacing (or ligaments) and the resultant gradient induced material size effect on void coalescence for a range of imposed stress states. To this end, three dimensional finite element calculations of unit cell models with a discrete void embedded in a strain gradient enhanced material matrix are performed. The calculations are carried out for a range of initial inter-void ligament sizes and imposed stress states characterised by fixed values of the stress triaxiality and the Lode parameter. Our results show that in the absence of strain gradient effects on the material response, decreasing the inter-void ligament size results in an increase in the propensity for void coalescence. However, in a strain gradient enhanced material matrix, the strain gradients harden the material in the inter-void ligament and decrease the effect of inter-void ligament size on the propensity for void coalescence

A finite element framework for distortion gradient plasticity with applications to bending of thin foils

© 2016 Elsevier Ltd A novel general purpose Finite Element framework is presented to study small-scale metal plasticity. A distinct feature of the adopted distortion gradient plasticity formulation, with respect to strain gradient plasticity theories, is the constitutive inclusion of the plastic spin, as proposed by Gurtin (2004) through the prescription of a free energy dependent on Nye's dislocation density tensor. The proposed numerical scheme is developed by following and extending the mathematical principles established by Fleck and Willis (2009). The modeling of thin metallic foils under bending reveals a significant influence of the plastic shear strain and spin due to a mechanism associated with the higher-order boundary conditions allowing dislocations to exit the body. This mechanism leads to an unexpected mechanical response in terms of bending moment versus curvature, dependent on the foil length, if either viscoplasticity or isotropic hardening are included in the model. In order to study the effect of dissipative higher-order stresses, the mechanical response under non-proportional loading is also investigated.Dr. Andrea Panteghini and Prof. Samuel Forest are acknowledged for helpful discussions. The authors gratefully acknowledge financial support from the Danish Council for Independent Research under the research career programme Sapere Aude in the project “Higher Order Theories in Solid Mechanics”. E. Martínez-Pañeda also acknowledges financial support from the Ministry of Science and Innovation of Spain through grant MAT2011-28796-CO3-03, and the University of Oviedo through grant UNOV-13-PF and an excellence mobility grant within the International Campus of Excellence programme. L. Bardella additionally acknowledges financial support from the Italian Ministry of Education, University, and Research (MIUR)

A micro-mechanics based extension of the GTN continuum model accounting for random void distributions

Randomness in the void distribution within a ductile metal complicates quantitative modeling of damage following the void growth to coalescence failure process. Though the sequence of micro-mechanisms leading to ductile failure is known from unit cell models, often based on assumptions of a regular distribution of voids, the effect of randomness remains a challenge. In the present work, mesoscale unit cell models, each containing an ensemble of four voids of equal size that are randomly distributed, are used to find statistical effects on the yield surface of the homogenized material. A yield locus is found based on a mean yield surface and a standard deviation of yield points obtained from 15 realizations of the four-void unit cells. It is found that the classical GTN model very closely agrees with the mean of the yield points extracted from the unit cell calculations with random void distributions, while the standard deviation $\textbf{S}$ varies with the imposed stress state. It is shown that the standard deviation is nearly zero for stress triaxialities $T\leq1/3$, while it rapidly increases %in the interval $4/3\lesssim T \lesssim 5$ for triaxialities above $T\approx 1$, reaching maximum values of about $\textbf{S}/\sigma_0\approx0.1$ at $T \approx 4$. At even higher triaxialities it decreases slightly. The results indicate that the dependence of the standard deviation on the stress state follows from variations in the deformation mechanism since a well-correlated variation is found for the volume fraction of the unit cell that deforms plastically at yield. Thus, the random void distribution activates different complex localization mechanisms at high stress triaxialities that differ from the ligament thinning mechanism forming the basis for the classical GTN model. A method for introducing the effect of randomness into the GTN continuum model is presented, and an excellent comparison to the unit cell yield locus is achieved

A phase field formulation for hydrogen assisted cracking

We present a phase field modeling framework for hydrogen assisted cracking. The model builds upon a coupled mechanical and hydrogen diffusion response, driven by chemical potential gradients, and a hydrogen-dependent fracture energy degradation law grounded on first principles calculations. The coupled problem is solved in an implicit time integration scheme, where displacements, phase field order parameter and hydrogen concentration are the primary variables. We show that phase field formulations for fracture are particularly suitable to capture material degradation due to hydrogen. Specifically, we model (i) unstable crack growth in the presence of hydrogen, (ii) failure stress sensitivity to hydrogen content in notched specimens, (iii) cracking thresholds under constant load, (iv) internal hydrogen assisted fracture in cracked specimens, and (v) complex crack paths arising from corrosion pits. Computations reveal a good agreement with experiments, highlighting the predictive capabilities of the present scheme. The work could have important implications for the prediction and prevention of catastrophic failures in corrosive environments. The finite element code developed can be downloaded from www.empaneda.com/codes

A phase field model for elastic-gradient-plastic solids undergoing hydrogen embrittlement

We present a gradient-based theoretical framework for predicting hydrogen assisted fracture in elastic-plastic solids. The novelty of the model lies in the combination of: (i) stress-assisted diffusion of solute species, (ii) strain gradient plasticity, and (iii) a hydrogen-sensitive phase field fracture formulation, inspired by first principles calculations. The theoretical model is numerically implemented using a mixed finite element formulation and several boundary value problems are addressed to gain physical insight and showcase model predictions. The results reveal the critical role of plastic strain gradients in rationalising decohesion-based arguments and capturing the transition to brittle fracture observed in hydrogen-rich environments. Large crack tip stresses are predicted, which in turn raise the hydrogen concentration and reduce the fracture energy. The computation of the steady state fracture toughness as a function of the cohesive strength shows that cleavage fracture can be predicted in otherwise ductile metals using sensible values for the material parameters and the hydrogen concentration. In addition, we compute crack growth resistance curves in a wide variety of scenarios and demonstrate that the model can appropriately capture the sensitivity to: the plastic length scales, the fracture length scale, the loading rate and the hydrogen concentration. Model predictions are also compared with fracture experiments on a modern ultra-high strength steel, AerMet100. A promising agreement is observed with experimental measurements of threshold stress intensity factor Kth over a wide range of applied potentials

On fracture in finite strain gradient plasticity

In this work a general framework for damage and fracture assessment including the effect of strain gradients is provided. Both mechanism-based and phenomenological strain gradient plasticity (SGP) theories are implemented numerically using finite deformation theory and crack tip fields are investigated. Differences and similarities between the two approaches within continuum SGP modeling are highlighted and discussed. Local strain hardening promoted by geometrically necessary dislocations (GNDs) in the vicinity of the crack leads to much higher stresses, relative to classical plasticity predictions. These differences increase significantly when large strains are taken into account, as a consequence of the contribution of strain gradients to the work hardening of the material. The magnitude of stress elevation at the crack tip and the distance ahead of the crack where GNDs significantly alter the stress distributions are quantified. The SGP dominated zone extends over meaningful physical lengths that could embrace the critical distance of several damage mechanisms, being particularly relevant for hydrogen assisted cracking models. A major role of a certain length parameter is observed in the multiple parameter version of the phenomenological SGP theory. Since this also dominates the mechanics of indentation testing, results suggest that length parameters characteristic of mode I fracture should be inferred from nanoindentation