NUMERICAL MODELING OF HYDROGEN EMBRITTLEMENT
A Dissertation
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Chuanshi Huang
May, 2020
NUMERICAL MODELING OF HYDROGEN EMBRITTLEMENT
Chuanshi Huang
Dissertation
Approved: Accepted:
Advisor Department Chair Dr. Xiaosheng Gao Dr. Sergio Felicelli
Committee Member Dean of the College Dr. Gregory Morscher Dr. Craig Menzemer
Committee Member Dean of the Graduate School Dr. Yalin Dong Dr. Marnie Saunders
Committee Member Date Dr. Gary L. Doll
Committee Member Dr. Chien-Chung Chan
ii
ABSTRACT
This dissertation developed a comprehensive numerical framework for the prediction of hydrogen embrittlement. The framework contains a numerical implementation of the hydrogen diffusion model, a study of hydrogen’s effect on ductile fracture under various stress states, a development of the phase field method for ductile and brittle fracture and the fracture mechanism transition from ductile to brittle caused by hydrogen, and a coupled numerical solution strategy that combines the phase field model with hydrogen diffusion model.
The first part of this dissertation uses a unit cell model to study the effect of hydrogen enhanced localized plasticity (HELP) on ductile fracture. The void growth and coalescence of the unit cell is influenced by the initial uniformly hydrogen distribution. The hydrogen redistribution caused by the stress field and plastic strain is observed. The results show that hydrogen reduces the ductility of the material by accelerating void growth and coalescence, and the effect of hydrogen on ductile fracture is strongly influenced by the stress state experienced by the material, as characterized by the stress triaxiality and the Lode parameter.
This dissertation presents a phase field model for simulating brittle and ductile fracture. The new developments include the introduction of a degradation function to the yield surface and the modification of the crack driving force function by including the plastic contribution. As the phase field value increases, the yield surface is degraded at the
iii same rate as the elastic modulus is, which maintains the integrity of the elasto-plastic constitutive equations. Parameters in the modified crack driving force function include the critical energy release rate and a plastic adjustment factor, which is an exponential function of the plastic strain. The conjoint effect of the plastic adjustment function and the value of the critical energy release rate on the crack driving force reflects the competition between the brittle and ductile fracture mechanisms. A numerical algorithm is proposed to implement the plasticity model with phase field and to solve the coupled system equations monolithically. A strategy using the crack driving force increment to control the size of solution increment is shown to be computationally efficient to assure solution accuracy.
Various numerical examples are carried out to demonstrate the capability of the proposed model and to illustrate the influences of model parameters on the simulation results.
To modify the phase field model for the simulation of hydrogen embrittlement, a
HEDE model is proposed account for hydrogen’s effect on the critical energy release rate, the hydrogen diffusion coefficient and the hydrogen trapping density function are modified as functions of the phase field value, the HELP model is considered in the material constitutive equations.
iv
ACKNOWLEDGEMENTS
First, I would like to express my sincere appreciation to my advisor Dr. Xiaosheng
Gao for his continuous support of my study and research. He always has been patient, motivative and inspiring. His guidance helped me throughout the years of my academic and industrial works at the University of Akron. His mentoring trained me to be a better engineer and researcher which with no doubt will continue benefit me in the future.
Beside my advisor, I would like to thank the rest of my committee members: Dr.
Chien-Chung Chan, Dr. Gary L. Doll, Dr. Gregory Morscher and Dr. Yalin Dong for their time to review my dissertation, and their insightful comments and questions. Additional thanks to Dr. Chang Ye who served as my committee member for my proposal.
I also want to thank my colleagues and friends in the group: Dr. Jinyuan Zhai, Dr.
Tuo Luo, Clayton Reakes, Chuan Zeng and Guanyue Rao for their helps and advises. they helped me a lot in my research and daily life during this period of time.
v
TABLE OF CONTENTS
Page
LIST OF TABLES ...... ix
LIST OF FIGURES ...... x
CHAPTER
I. INTRODUCTION ...... 1
1.1 Hydrogen Embrittlement Understanding ...... 1
1.2 Hydrogen transport and trapping ...... 2
1.3 Hydrogen enhanced localized plasticity...... 3
1.4 Hydrogen enhanced decohesion ...... 5
1.5 Phase Field Method ...... 7
1.6 Fracture mechanisms ...... 8
1.7 Research objective ...... 9
II. THE EFFECT OF HYDROGEN ON DUCTILE FRACTURE ...... 12
2.1 Hydrogen Diffusion Formulation ...... 12
2.2 Unit cell model ...... 15
2.3 HELP effect under different stress triaxialities ...... 19
vi
2.4 HELP effect under different Lode parameters ...... 25
2.5 Summary and conclusions ...... 30
III. PHASE FIELD MODELING FOR BRITTLE AND DUCTILE FRACTURE ...... 33
3.1 Geometrical phase field method for ductile fracture...... 34
3.1.1 Phase field modeling ...... 34
3.1.2 Degradation function ...... 36
3.1.3 Free energy function ...... 37
3.2 Plasticity Model and Numerical Algorithm ...... 41
3.3 Finite Element Implementation ...... 46
3.4 Parameter Studies ...... 49
3.4.1 Effect of the Increment Size ...... 49
3.4.2 Effect of the Critical Energy Release Rate ...... 55
3.4.3 Effects of α ...... 58
3.4.4 Effect of the Yield Stress ...... 61
3.5 Additional Examples ...... 62
3.5.1 Flat specimen without notch ...... 63
3.5.2 Compact tension specimens ...... 66
3.6 Summary and Conclusions ...... 73
IV. PHASE FIELD MODELING OF HYDROGEN EMBRITTLEMENT ...... 75
4.1 Hydrogen Transport Coupled with Displacement and Phase Field ...... 75
vii
4.2 Hydrogen trapping ...... 78
4.3 Hydrogen embrittlement modeling ...... 80
4.4 Yield function...... 81
4.5 Numerical implementation ...... 81
4.6 Compact Tension Specimen ...... 84
4.6.1 Lattice Hydrogen Diffusion and Hydrogen Trapping ...... 84
4.6.2 Hydrogen Embrittlement Mechanisms ...... 90
4.7 Double Notched Flat Specimen ...... 95
4.7.1 Lattice Hydrogen Diffusion and Hydrogen Trapping ...... 96
4.7.2 Hydrogen Embrittlement Mechanisms ...... 97
4.8 Conclusions ...... 101
V. LOADING SPEED DEPENDENCE ...... 103
5.1 Introduction ...... 103
5.2 Numerical simulations procedure and results ...... 104
5.3 Conclusions ...... 107
VI. CONCLUSION AND FUTURE WORK ...... 109
6.1 Conclusions ...... 109
6.2 Future works ...... 111
BIBLIOGRAPHY ...... 113
viii
LIST OF TABLES
Table Page
3.1 Material properties ...... 50
4.1 material parameters for the study of hydrogen embrittlement ...... 83
ix
LIST OF FIGURES
Figure Page
2.1 A typical 1/2 -symmetric finite element mesh of the unit cell ...... 17
2.2 The stress state imposed on the unit cell ...... 17
2.3 Curves of normalized void volume fraction vs. macroscopic effective strain of the RMV as the stress triaxiality varies from T = 0.8 to T = 1.8 while the Lode parameter is kept as L = 0.15...... 20
2.4 (a) Variation of the failure strain with stress triaxiality, (b) the HELP effect on the failure strain under different stress triaxialities...... 20
2.5 Contours of von Mises stress, plastic strain (SDV13) and trapping hydrogen concentration (SDV26) when the macroscopic effective strain of the RMV is equal to 0.038 for the case of high stress triaxiality (T = 1.8) and Lode parameter L = 0.15: (a) von Mises stress without HELP effect, (b) von Mises stress with HELP effect, (c) plastic strain without HELP effect, (d) plastic strain with HELP effect, (e) trapping hydrogen concentration without HELP effect, (f) trapping hydrogen concentration with HELP effect...... 22
2.6 Contours of von Mises stress, plastic strain and trapping hydrogen concentration when the macroscopic effective strain of the RMV is equal to 0.038 for the case of low stress triaxiality (T = 0.8) and Lode parameter L = 0.15: (a) von Mises stress without HELP effect, (b) von Mises stress with HELP effect, (c) plastic strain without HELP effect, (d) plastic strain with HELP effect, (e) trapping hydrogen concentration without HELP effect, (f) trapping hydrogen concentration with HELP effect...... 23
x
2.7 Contours of the von Mises stress for the case of stress triaxiality T = 0.8 and Lode
parameter L = 0.15 at different macroscopic effective strain levels (Ee)...... 24
2.8 Comparison of the void growth rate for cases of stress triaxiality T = 0.8 and T = 1.8, where the Lode parameter is fixed at L = 0.15. Letter “h” in the legends corresponds to the results with the HELP effect included...... 25
2.9 Curves of normalized void volume fraction vs. macroscopic effective strain of the RMV as the Lode parameter varies from L = -0.3 to L = 0.3 while the stress triaxiality is kept at T = 1.0...... 26
2.10 (a) Variation of failure strain with Lode parameter, (b) the HELP effect on failure strain under different Lode parameters...... 26
2.11 Contours of von Mises stress, plastic strain and trapping hydrogen concentration when the macroscopic effective strain of the RMV is equal to 0.038 for the case of stress triaxiality T = 1.0 and Lode parameter L = -0.15: (a) von Mises stress without HELP effect, (b) von Mises stress with HELP effect, (c) plastic strain without HELP effect, (d) plastic strain with HELP effect, (e) trapping hydrogen concentration without HELP effect, (f) trapping hydrogen concentration with HELP effect...... 27
2.12 Contours of von Mises stress, plastic strain and trapping hydrogen concentration when the macroscopic effective strain of the RMV is equal to 0.038 for the case of stress triaxiality T = 1.0 and Lode parameter L = 0.3: (a) von Mises stress without HELP effect, (b) von Mises stress with HELP effect, (c) plastic strain without HELP effect, (d) plastic strain with HELP effect, (e) trapping hydrogen concentration without HELP effect, (f) trapping hydrogen concentration with HELP effect...... 28
2.13 Comparison of the void growth rate for cases of L = -0.15 and L = 0.3, where the stress triaxiality is kept at 1.0. Letter “h” in the legends corresponds to the results with the HELP effect included...... 29
xi
2.14 (a) Variation of failure strain with stress triaxiality and Lode parameter without HELP effect, (b) variation of failure strain with stress triaxiality and Lode parameter with HELP effect, (c) reduction of failure strain due to HELP effect...... 31
3.1 Effects of the plastic adjustment function and the critical energy release rate on the evolution of the phase field value...... 40
3.2 Flow chart of a return-mapping algorithm using the Newton-Raphson method to integrate the plastic rate equations...... 42
3.3 Schematics of stress update and strain energy calculation ...... 44
3.4 A double notched, dog bone specimen: (a) geometry and boundary conditions (dimensions in mm); (b) finite element mesh...... 49
3.5 Computed load-displacement curves with different increment sizes ...... 51
3.6 Crack phase field contours at different loading stages, where 2δ and 4δ represent the increment sizes used in the simulations, and a1, a2 and a3 correspond to the loading stages indicated in Figure 3.5...... 52
3.7 Final crack phase field obtained with different increment sizes ...... 53
3.8 Variation of the maximum increment of crack driving force with different increment sizes of applied displacement ...... 54
3.9 Comparison of the computed load-displacement curve with the increment size controlled using the proposed strategy with those obtained using fixed increment sizes ...... 55
3.10 Load-displacement curve and variation of the maximum plastic strain during the loading history ...... 56
3.11 Load-displacement curves and variations of the maximum plastic strain for different
values of Gc ...... 57
xii
3.12 Final crack phase field resulted from different values of Gc ...... 58
3.13 Effect of α on the load-displacement response and maximum plastic evolution: (a) 2 2 Gc= 1.2 mJ/mm , (b) Gc= 1 mJ/mm ...... 59
3.14 Final crack phase field resulted from different values of α and Gc...... 60
3.15 Effect of yield stress on the load-displacement response and maximum plastic strain evolution...... 61
3.16 Final crack phase field resulted from different values of yield stress...... 62
3.17 A smooth dog bone specimen: (a) geometry and boundary conditions (dimensions in mm); (b) finite element mesh...... 63
3.18 Load-displacement curves and variations of the maximum plastic strain for different values of critical energy release rate ...... 64
3.19 Final crack phase field resulted from different Gc values ...... 65
3.20 Load-displacement curve and evolution of maximum phase field value for the Gc = 1.2 mJ/mm2 case...... 65
3.21 Contours of crack phase field at two load levels indicated in Figure 20 ...... 66
3.22 (a) A modified CT specimen, (b) a CT specimen (dimensions in mm) ...... 67
3.23 Load-displacement curve and variation of the maximum plastic strain for the modified CT specimen with two holes...... 68
3.24 Crack phase field contours at different loading stages indicated in Figure 3.23 for the modified CT specimen with two holes...... 69
3.25 Load-displacement curve and variation of the maximum plastic strain for the CT 2 specimen. Here Gc = 0.2 mJ/mm ...... 69
xiii
3.26 Crack phase field contours at different loading stages indicated in Figure 3.25 for the 2 CT specimen. Here Gc = 0.2 mJ/mm ...... 70
3.27 Load-displacement curve and variation of the maximum plastic strain for the CT 2 specimen. Here Gc = 1.2 mJ/mm ...... 71
3.28 Crack phase field contours at different loading stages for the CT specimen. Here Gc = 1.2 mJ/mm2...... 71
3.29 Crack phase field and stress contours the applied displacement level a2 as indicated in Figure 3.27. (a) Crack phase field contour, (b) stress component normal to the crack 2 plane, (c) hydrostatic stress. Here Gc = 1.2 mJ/mm is used in the computation...... 72
4.1 (a) Dimensions (in mm) of a CT specimen. (b) Finite element mesh of the CT specimen...... 84
4.2 Distributions of crack phase field value (a) and lattice hydrogen concentration (b) on the mid-surface of the specimen before the onset of crack initiation...... 85
4.3 Distributions of rack phase field values (a) and lattice hydrogen concentration (b) on the mid-surface of the specimen after some amount of crack propagation...... 86
4.4 Distributions of crack phase field value (a), trapped hydrogen concentration (b), and total hydrogen concentration (c) on the mid-surface of the specimen prior to fracture initiation...... 88
4.5 Distributions of crack phase field value (a), trapped hydrogen concentration (b), and total hydrogen concentration (c) on the mid-surface of the specimen after some amount of crack propagation...... 89
4.6 Load-displacement curves (solid lines) and maximum plastic strain-displacement curves (dashed lines) for different cases, where the HELP parameters are ϑ=0.9 and η=0.5, and the HEDE parameters are ζ=0.9 and ξ=0.5...... 90
xiv
4.7 Load-displacement curves (solid lines) and maximum plastic strain-displacement curves (dashed lines) obtained with different HELP parameters...... 91
4.8 Load-displacement curves (solid lines) and maximum plastic strain-displacement curves (dashed lines) obtained with different HEDE parameters...... 92
4.9 Distributions of crack phase field value (a1 – a4), total hydrogen concentration (b1 – b4), and hydrogen embrittlement effects (c1 – c4) on the mid-surface of the specimen at different loading stages (number 1 – 4 in each row)...... 93
4.10 (a) Dimensions of a flat specimen with double notches (in mm); (b) finite element mesh...... 95
4.11 Distributions of crack phase field value (a1, a2), the lattice hydrogen(b1, b2), the trapped hydrogen (c1, c2), and the total hydrogen (d1, d2) on the mid-surface of the specimen before (a1, b1, c1, d1) and after (a2, b2, c2, d2) the onset of crack initiation...... 96
4.12 Load-displacement curves (solid lines) and maximum plastic strain-displacement curves (dashed lines) for different cases, where the HELP parameters are ϑ=0.9 and η=0.5, and the HEDE parameters are ζ=0.9 and ξ=0.5...... 98
4.13 Distributions of the effective plastic strain (a1, a2), the trapped hydrogen(b1, b2), the total hydrogen(c1, c2) and the crack phase field value (d1, d2) on the mid-surface of the specimen for case 1 (a1, b1, c1, d1) and case 2 (a2, b3, c2, d2) at the loading stage 1 indicated in Figure 4.12...... 99
4.14 Distributions of the effective plastic strain (a1, a2), the trapped hydrogen(b1, b2), the total hydrogen (c1, c2) and the crack phase field value (d1, d2) on the mid-surface of the specimen for case 1 (a1, b1, c1, d1) and case 3 (a2, b3, c2, d2) at the loading stage 2 indicated in Figure 4.12...... 100
5.1 Load-displacement curves for different cases, where the HELP parameters are ϑ=0.9 and η=0.5, and the HEDE parameters are ζ=0.9 and ξ=0.5...... 105
xv
5.2 Distributions of the hydrostatic stress (a1 to a4 representing loading speeds v to v/1000), the lattice hydrogen concentration (b1 to b4 representing loading speeds v to v/1000)...... 106
5.3 Distributions of the plastic strain (a1 to a4 representing loading speeds v to v/1000), the trapped hydrogen concentration (b1 to b4 representing loading speeds v to v/1000)...... 106
5.4 Distributions of the total hydrogen concentration (a1 to a4 representing loading speeds v to v/1000), ...... 107
xvi
CHAPTER I
INTRODUCTION
1.1 Hydrogen Embrittlement Understanding
Hydrogen embrittlement refers to the phenomenon that the dissolved hydrogen in metals will degrade the mechanical properties of the material. The affected metals will lose its strength and ductility and fail at a lower load level than the material can normally bear.
Most hydrogen attacked materials fail without any sign of embrittlement until catastrophic failure occurs. Hydrogen embrittlement is one of the most common failure causes of the high strength steel components in military aircraft, it is also one of the most critical corrosion failures in oil and gas industry, some high strength steel components used in construction are often affected by hydrogen as well. Thus, the predicting of hydrogen embrittlement is very important especially for high strength steels because it is more vulnerable to hydrogen.
Hydrogen may be introduced into metals during the material forming, machining and finishing process or environmental corrosion process. Once hydrogen entered the metal, it can transport between interstitial lattice sites and being trapped at microstructural sites, e.g., grain boundaries, dislocations, and matrix-particle interfaces. There are interactions between the dissolved hydrogen and the mechanical field variables such as hydrostatic stress and plastic strain etc.
1
Johnson first observed the phenomenon of hydrogen embrittlement in 1875, he immerged a piece of iron wire in strong hydrochloric or dilute sulphuric acids for a few minutes and find a significant reduction of the toughness and break strain of that immerged iron. Then he found this phenomenon is only temporary due to the observed fact that the iron wire regains its original toughness after exposed to a certain temperature for certain amount of time[1]. His experiment then initiated the study of the hydrogen embrittlement of metals. Many mechanisms have been proposed try to interpret hydrogen embrittlement such as hydrogen enhances localized plasticity (HELP), hydrogen enhanced decohesion
(HEDE) and hydride formation etc.
1.2 Hydrogen transport and trapping
The understanding of how hydrogen resides and diffuses in the metal is essential for the predicting of hydrogen embrittlement. Hydrogen diffusion is via the drifting of hydrogen atoms between normal interstitial lattice sites (NILS) in metals, it is dominated by the gradients of chemical potential which is controlled by the gradients of hydrogen concentration and the gradients of hydrostatic stress. Frohmberg et al.[2] observed that the positive hydrostatic stress causes an accumulation of dissolved hydrogen in this area.
Johnson et al.[3] suggested that the hydrogen tend to diffuses to the crack tip due to the high hydrostatic stress at the crack tip. Besides the NILS, there is another group of sites which hydrogen resides is called trapping sites. The Darken and Smith[4] first observed that hydrogen is trapped by the imperfections of a cold worked steel. McNabb and Faster[5] proposed a new formulation for the hydrogen diffusion modeling with the consideration of the hydrogen trapping effect. Oriani [6] developed their formulation based on the assumption of a local equilibrium between the diffusive hydrogen (lattice hydrogen) and
2 the trapped hydrogen. Sofronis and McMeeking[7] adopted Oriani’s equilibrium theory derived non-linear hydrogen diffusion equations and implemented the proposed diffusion model with numerical method, then studied the coupled transient hydrogen diffusion and plastic straining around the blunting crack tip. They also proposed that the trap density is a function of the equivalent plastic strain based on Kumnick and Johnson’s [8] experimental observation. Lufrano and Sofronis[9] further modified Sofronis and
McMeeking’s hydrogen diffusion model by adding the hydrogen induced dilatation to the material constitutive equations.
1.3 Hydrogen enhanced localized plasticity
Hydrogen enhanced localized plasticity mechanism is based upon the experimental result that the presence of hydrogen in ductile material will decrease the local flow stress of the material thus promote ductile fracture occurs at a lower load level. However, this mechanism only valid in a certain range of temperature. Beachem[10] observed that hydrogen will decrease the local flow stress by comparing torsion tests between hydrogen charged and hydrogen free steel pipes, and proposed a unified hydrogen embrittlement model suggests that sufficiently concentrated hydrogen at the crack tip will promote whatever deformation processes that the microstructure allowed. Matsui et al.[11] and
Moriya et al.[12] systematically studied the effect of hydrogen on mechanical properties of purity iron under different temperatures and hydrogen concentrations. They concluded that the trapped hydrogen at the screw dislocation core will promote the screw dislocations and hydrogen will impede the edge dislocations when the temperature is under a sufficiently low level. Sofronis and Birnbaum[13] used a numerical method to study the effect of a hydrogen on the interaction of dislocations, and proposed a hydrogen shielding
3 effect which can account for the hydrogen promoted dislocation mobility. Ferreira et al.[14] observed that the elastic interactions between dislocations is reduced by hydrogen in stainless steel. Robertson[15] performed in-situ TEM deformation experiments on different metal materials. Results showed persuasive evidence that hydrogen can enhance the mobility of dislocations. Besides the experimental observations which provide the direct evidence for the HELP mechanism, atomistic simulations provide a microscopic point of view for a more comprehensive understanding of the HELP mechanism[16].
The HELP effect can reduce the ductility of metals, i.e., embrittle the metals by promoting the plastic localization processes in ductile fracture. Theoretical studies have been done try to develop a continuum model to simulate the phenomenon of hydrogen enhanced localized plasticity. Tabata and Birnbaum[17] observed that the flow stress decreases with the increasing hydrogen concentration and proposed that there is a softening ratio between the flow stress and hydrogen concentration. Sofronis et al.[18] and Liang et al.[19] proposed a HELP model for numerical simulations, their study suggest that the
HELP effect will lead to a macroscopic shear band localization and cause plastic instability.
Ahn et al.[20] used a unit cell model to study the HELP effect on void growth and coalescence, numerical results show that hydrogen promotes the void growth and coalescence, and the effect increases with higher triaxiality. Huang et al.[21] studied the effect of HELP on ductile fracture under different stress triaxiality and Lode parameter.
Tuo et al.[22] considered the loading speed as an influence of the steady state hydrogen distribution, and studied the effect of HELP on ductile fracture under different loading speeds and stress states.
4
Numerical studies suggest that, the void growth and coalescence and shear bands formation is promoted by hydrogen, causes easier ductile fracture, leads to a reduction of the material ductility. However, the HELP mechanism only affects the yield condition of the material, in order to fully describe the hydrogen embrittlement, it needs to be accompanied with other mechanisms such as HEDE.
1.4 Hydrogen enhanced decohesion
Hydrogen enhanced decohesion postulates that the presence of hydrogen in metals will reduce the bonding energy between metal atoms. HEDE Phenomenon including the reduction of cohesive strength along the grain boundary[23], and the fracture mechanism transition from ductile to brittle[24]. A related phenomenon is the reduction of the surface energy caused by dissolved hydrogen[24][25].
Troiano[27] introduced the concept that the cohesive strength of the iron lattice is lowered by the dissolved hydrogen. Oriani[28] further discussed the decohesion theory and pointed out that for high-strength steel, hydrogen embrittlement can be avoided either by preventing hydrogen from being absorbed into the steel or by increasing the ductility of the steel. Beachem[10] observed that the hydrogen concentration, along with the stress intensity at the crack tip will influence the fracture modes changing between intergranular, quasi-cleavage and micro-void coalescence. Oriani and Josephic[23] experimentally demonstrated that in a high-strength steel, the local maximum tensile stress and the maximum cohesive resistive force of the lattice is reduced by the high concentration hydrogen in the area adjacent to the crack tip. Gerberich and Chen[29] proposed a quantitative correlation between the applied stress intensity and the hydrogen concentration at the crack tip and suggested that this correlation is governed by the yield strength, initial
5 hydrogen concentration, stress state and temperature. Vehoff and Rothe[30] use in situ
SEM observed the microscopic process which occurred during hydrogen embrittlement in
FeSi- and Ni-single crystals, they found that there is a combination of plastic shearing off processes and cleavage-like cleavage processes within 100 nm from the crack tip.
Gerberich et al.[31] suggested that the local Griffith value linearly reduced by the increasing hydrogen concentration. Jiang et al.[32] use Born-Haber thermodynamic cycle to calculate the fracture energy of hydrogen attacked metals (Fe and Al), they find that the fracture energy decreases with increasing of hydrogen concentration. Based on this calculation, Serebrinsky[33] proposed a quadratic relationship between the cohesive strength and the hydrogen concentration.
Wang et al.[34] conducted atomic simulations to analyze the hydrogen induced decohesion of the grain boundary under different hydrogen charging conditions, and observed a intergranular fracture when the reduction of grain boundary cohesive energy caused by hydrogen is 37%. They suggested that the intergranular fracture is result from the combination of HEDE and HELP. Depover et al.[24] observed the fracture surface of the pre-charged specimen and find that there is brittle zones on the fracture surface of the ductile material where hydrogen resides.
Another mechanism related to and can be modeled together with HEDE mechanism is the surface energy reduction caused by hydrogen. Petch[25] suggested that the adsorbed hydrogen on the Griffith crack surface will lower the surface energy thus reduce the fracture stress of the metal. Tromans[35] concluded that the highly localized tensile stresses are necessary to produce a sufficient reduction of surface energy to promote cracking.
6
1.5 Phase Field Method
The basic understanding of hydrogen embrittlement has been briefly reviewed in previous sections. To comprehensively model hydrogen embrittlement of metals for finite element simulation purpose, it is obvious that a numerical method for the fracture prediction is needed. This numerical method needs to be capable of simulating the ductile fracture, brittle fracture and the transition from ductile fracture to brittle fracture. From the energetic point of view, fracture occurs when the total strain energy equals to the energy needed for creating new crack surfaces plus any dissipated energy. Based on different fracture criteria, various numerical methods have been developed in recent years to predict the crack initiation and growth in materials. Among which the phase field method has shown the advantage of flexible implementation and ability to simulate complex crack initiation, propagation and branching. A large amount of research works have shown the suitability of the phase field method for the modeling of brittle fracture [36]–[45]. Francfort and Marigo[46] proposed a variational model of quasi-static crack evolution that does not require a preexisting crack or a well-defined crack path, overcoming the limitations of the approaches based on the Griffith’s theory[47]. The follow-up work by Bourdin et al.[36][48] demonstrated the numerical feasibility of this method. Their numerical framework regularizes the sharp crack surface topology in the material by diffusive crack zones governed by a scalar auxiliary variable, i.e., the phase field, which interpolates between the unbroken and the broken states of the material. Miehe et al.[40][41] proposed a thermodynamically consistent approach of phase field models of crack propagation in elastic solids, and implemented the proposed model using an operator split algorithm that incrementally updates the fracture phase field and the displacement field. Borden et al.[42]
7
[43] extended the quasi-static model presented by Miehe et al.[41] to the dynamic case, and proposed a fourth-order model for the phase-field approximation. Miehe et al.[45] generalized the continuum phase field models for brittle fracture to include fully coupled thermo-mechanical and multi-physics cases at large strains. However, so far, only a limited number of studies have attempted to model ductile fracture using the phase field method[49]–[54]. Miehe et al.[49] extended the their model to ductile fracture by adding the non-elastic contribution to the crack driving force. The crack driving force is a function of elastic and plastic work densities and barrier functions related to critical values of the inelastic state variables. Ambati et al.[50] [51] introduced a coupling between the degradation function applied to the tensile portion of the elastic energy and the plastic strain state to enable the initiation of fracture in the regions of plastic strain localization, and compared model predictions with experimental results. More recently, Miehe et al.[52][54] included the gradient plasticity and gradient damage into the phase field model for ductile fracture, and Borden et al.[53] presented a model that includes a yield surface degradation and a measure of stress triaxiality as a driving force for crack initiation and propagation.
Huang and Gao[55] proposed development of the phase field method which is able to simulate ductile and brittle fracture and the transition between ductile and brittle fracture.
1.6 Fracture mechanisms
Ductile fracture is a common failure mode of structures made of metals. It is caused by localized plastic deformation, which leads to void nucleation, growth and coalescence or micro shear bands. Extensive research has been conducted resulting in various models for establishing the criterion for ductile fracture. These models can be classified into two categories: uncoupled models[56][57] and coupled models[58][59]. The uncoupled model
8 neglects the effect of damage on the yield surface of the material while the coupled model incorporates damage accumulation into the constitutive equation. Li et al.[60] provided a comprehensive review of the ductile fracture models. Comparing to the uncoupled models, which are easy for numerical implementation in the finite element programs, the coupled models represent a sounder physical background as they account for the deterioration of material due to damage evolution. Gurson[58] developed a homogenized yield surface for void containing materials based on the maximum plastic work principle, where the void volume fraction is used as the damage parameter. Lemaitre’s continuum damage mechanics model[59] is based on the thermodynamic and effective stress concept and uses an internal variable to describe the damage evolution and progressive degradation of the material. More recent research has resulted in various modifications and extensions of these models[61][62][63].
1.7 Research objective
The research objective of this dissertation is to develop a numerical framework for the simulation of hydrogen embrittlement. Which is capable of the simulation of hydrogen diffusion and trapping, HELP and HEDE effect on the fracture behavior of the material and the fracture mechanism transition between ductile and brittle. The detailed research objectives are:
• Use a unit cell model to study the effect of HELP mechanism on ductile fracture under
different stress states. Analyze how hydrogen distribute is affected by different stress
states and local plastic strain, and how the redistributed hydrogen influences the fracture
behavior in return.
9
• Develop a phase field model as a numerical method for the simulation of brittle and
ductile fracture, and the transition between the ductile and brittle fracture.
• Combine the HELP, HEDE and hydrogen diffusion models with the developed phase
field model for the comprehensively simulation of hydrogen embrittlement.
The outline of the dissertation is showed in the following:
Chapter I briefly introduces the concept of hydrogen embrittlement, reviews development of the hydrogen diffusion, hydrogen enhanced localized plasticity and hydrogen enhanced decohesion. The research of phase field method and ductile fracture models are also reviewed in Chapter I.
Chapter II develops a numerical method to simulate the hydrogen diffusion and trapping under different stress states and plastic strains. A unit cell model is used to investigate the hydrogen effect on the void growth and coalescence under different stress triaxiality and Lode parameter.
Chapter III proposes a development of a phase field model for ductile fracture, and a numerical solution strategy for the weak form equations of the phase field coupled with the displacement field. Studies the effect of different phase field model parameters on the fracture mechanisms.
Chapter IV modifies the proposed phase field model by taking hydrogen embrittlement mechanisms HELP and HEDE into consideration, adjusts the hydrogen diffusion model by considering the effect of crack surface described by the phase field value. And combines those two models for the simulation of hydrogen embrittlement.
10
Chapter V discusses the loading speed sensitivity of the hydrogen embrittlement developed in Chapter IV.
Chapter VI concludes the main achievement of this dissertation and pointing out several directions for the future research.
11
CHAPTER II
THE EFFECT OF HYDROGEN ON DUCTILE FRACTURE
Hydrogen enhanced localized plasticity (HELP) is a major mechanism for material ductility reduction cause by hydrogen embrittlement. In this chapter, the effect of hydrogen on ductile fracture is demonstrated by its influence on the processes of void growth and coalescence. The hydrogen diffusion model developed by Sofronis and McMeeking[7] is adopted to simulate the hydrogen trapping and transportation in metals. Hydrogen induced dilatation is considered in the constitutive equations. Assuming an initially uniform hydrogen distribution and a periodic array of spherical voids present in the material, a series of finite element analyses of a representative material volume (RMV) subjected to various stress states were carried out. The evolution of local stress and deformation states results in hydrogen redistribution in the material, which in turn changes the material’s flow property due to the HELP effect.
2.1 Hydrogen Diffusion Formulation
Dissolved hydrogen in metals can be found at the normal interstitial lattice sites
(NILS) as well as the reversible trapping sites, therefore the total hydrogen concentration can be expressed as
퐶 = 퐶L + 퐶T = 훽L휃L푁L + 훽T휃T푁T, (2. 1)
12 where 퐶L is the hydrogen concentration at NILS, 퐶T is the hydrogen concentration at the trapping site, 훽Lis the number of NILS per solvent atom, 훽T is the number of sites per trap,
휃L is the occupancy of NILS, 휃T is the occupancy of the trapping sites, 푁L is the number of solvent atom per unit lattice volume, 푁T is the trapping density.
The two populations are assumed to be in equilibrium and the relationship between
θL and θT can be described as[6]
휃T 휃L 푊B = exp ( ) , (2. 2) 1 − 휃T 1 − 휃L 푅훩
where WB is the binding energy of the trapping sites, 푅 is the gas constant, 훩 is the absolute temperature.
The governing equation for hydrogen transport proposed by Sofronis and
McMeeking[7] is written as
p 퐶L + 퐶T(1 − 휃T) 휕퐶L 퐷L퐶L푉H d푁T 휕휀̅ − ∇(퐷L∇퐶L) + ∇ ( ∇휎h) + 휃T p = 0 (2. 3) 퐶L 휕푡 푅훩 d휀̅ 휕푡 where 퐷L is the hydrogen diffusion constant through NILS, 푉H is the partial molar volume
휕휀̅p of hydrogen, 휎 is the hydrostatic stress, 휀p̅ is the plastic strain, is plastic strain rate. h 휕푡
Taha and Sofronis[64] provided a relation between trap density and plastic strain based on the experimental result of Kumnick and Johnson[8], the expression of the relation is
p log(푁T) = 23.26 − 2.33 exp(−5.5휀̅ ) (2. 4)
13
The total strain rate consists of an elastic part, 휀̇e, a plastic part, 휀̇p and a part due to hydrogen-induced lattice deformation, 휀̇h. The hydrogen-induced deformation rate can be expressed as [65]
h d (푐 − 푐0)푉H 휀푖푗̇ = {ln [1 + ]} 훿푖푗 (2. 5) d푡 3휔푁A
where 푐 is the current total hydrogen concentration 푐0 is the initial hydrogen concentration,
휔 is the mean atomic volume of the host metal atom, 푉H is the partial molar volume of hydrogen in solution, NA is the Avogadro’s constant.
Previous studies show that the dissolved hydrogen in iron increases the mobility of dislocation, causing the local yield stress to decrease with the increase in hydrogen concentration[13][17]. Sofronis and Liang proposed a phenomenological model to describe the hydrogen effect on the local yield stress[19][18]
푁 휀푝 휎Y(휀푝, 푐) = 휎0(푐) (1 + ) , (2. 6) 휀0 with
푐 [(휗 − 1) ] 휎0 + 1 휎0(푐) > 휂휎0 휎0(푐) = { 푐0 , (2. 7) 휂휎0 휎0(푐) ≤ 휂휎0
where 휎0 is the initial yield stress in the presence of hydrogen, and 휗, 휂 are the softening parameters. In the above equations, ε0 = σ0/E with σ0= σ0(0) representing the initial yield in the absence of hydrogen, E is the Young’s modulus, and ησ0 is the lowest possible value of the yield stress with η varying between 0 and 1, considering that in reality hydrogen cannot cause the yield stress to vanish.
14
The above models were implemented in ABAQUS via user subroutines UMATHT and UMAT. Although ABAQUS does not provide a user interface for solving the coupled hydrogen transport equation, it does provide a built-in program for heat transfer analysis and allows the user to define the thermal behavior of the material for transient heat transfer analysis via a user subroutine UMATHT. The analogous structure of the Fourier’s equation of thermal conduction and the hydrogen transport equation makes it possible to implement the hydrogen diffusion model in ABAQUS, where a UMATHT subroutine is used to match variables of the governing equations for heat transfer analysis with those for hydrogen diffusion.
2.2 Unit cell model
It is generally accepted that ductile fracture in metallic alloys is a process of void nucleation, growth and coalescence (van Stone et al[66]; Garrison and Moody[67]) and the material’s ductility, often measured by the strain to failure, is strongly affected by the stress state (Gao and Kim[68]; Bai and Wierzbicki[69]). Assuming a periodic distribution of voids, the material can be considered as comprised of a number of voids containing RMVs.
A straight-forward approach to study the ductile fracture mechanism as well as the effects of material properties and stress state on the fracture process is to conduct a series of unit cell analyses. As the unit cell deforms, the void volume increases and the void shape/orientation changes, thus the ligament between voids in adjacent unit cells decreases.
When the localization criterion is reached, rapid internal necking of the ligament occurs, leading to void coalescence.
Here a uniform distribution of spherical voids is considered and the ratio between the void diameter and the void spacing is taken as 0.2, resulting in an initial void volume
15 fraction of f0 = 0.004189. To simulate flow localization into a band, Barsoum and
Faleskog[70] proposed a unit cell model neglecting voids outside the central layer. This unit cell model was subsequently adopted by many other researchers[71][72], and is employed in this study. Figure 2.1 shows a typical ½-symmetric finite element mesh of a
unit cell with an initial size of X 0 2X 0 X 0 containing a centered, spherical void. To study the behavior of the unit cell under various stress states, the stresses imposed on the unit cell should include both normal and shear components. Here the applied stress consists of three normal stress components and a shear stress component, the same as in Barsoum and Faleskog[70], Dunand and Mohr[71], and Wong and Guo[72]. Figure 2.2 illustrates the stress state imposed on the unit cell.
Let Σij be the macroscopic Cauchy stress tensor and Sij be the stress deviator, Sij = Σij
- Σmδij, where δij denotes the Kronecker delta, Σm represents the mean stress, Σm = Σii/3, and the summation convention is adopted for repeated indices. The von Mises equivalent stress is defined as
푁 휀푝 휎푌(휀푝, 푐) = 휎0(푐) (1 + ) . (2. 8) 휀0
The stress triaxiality is defined as the ratio between the mean stress and the von Mises equivalent stress, T = Σm / Σe, and the Lode parameter is defined as
휋 퐿 = √3 tan(휑 − ) , (2. 9) 6
3 where 휑 denotes the Lode angle, with cos(3휑) = 27퐽3/(2휎푒 ) and 퐽3 = 푆푖푗푆푗푘푆푘푖/3[73].
The material properties used in this study are the same as those in[74][65], with the yield stress in air σ0 = 400 MPa, the Young’s modulus E = 200 GPa, the Poisson’s ratio υ 16
= 0.3, and the power-law straining hardening exponent n = 0.1. The hydrogen lattice
-2 2 diffusion coefficient DL = 1.27×10 mm /s [7]. The molar volume of iron VM =
3 3 19 7.116×10 mm /mol and the lattice site density is NL = NA / VM = 8.46×10 solvent lattice
3 atoms/mm . The number of interstitial lattice sites per solvent atom, 훽L, is equal to 6 for
BCC materials and number of sites per trap, 훽T, is assumed to be 1 [75]. The partial molar
3 3 volume of hydrogen in solid solution VH = 2.0×10 mm /mol. The initial hydrogen
12 3 distribution in the material is assumed to be uniform with CL = C0 = 2.084×10 atoms/mm .
Figure 2.1 A typical 1⁄2 -symmetric finite element mesh of the unit cell
Figure 2.2 The stress state imposed on the unit cell
17
The displacement boundary conditions on the outer surfaces of the unit cell are prescribed such that the macroscopic stress triaxiality and Lode angle of the unit cell are kept constant during the entire deformation history. This is realized by using the method described in[76] by Barsoum and Faleskog. By varying the boundary conditions, different stress state can be imposed on the unit cell and the material failure criterion in terms of failure strain as a function of stress triaxiality and Lode parameter can be obtained.
Localization of plastic deformation within a narrow band is an important precursor to ductile fracture. Barsoum and Faleskog[76] presented a numerical approach for determining the material failure criterion based on the theoretical framework of plastic localization into a band by Rice[77]. Ductile fracture (void coalescence) occurs when the deformation gradient rate inside the band becomes much larger than the deformation gradient rate outside the band. The macroscopic deformation gradient of the RMV, 푭̅, can be calculated by taking a volume average of the local deformation gradient
1 1 퐹̄ = ∫ 퐹 푑 푉 = 훿 + ∫ 푢 푛0푑 푆 , (2. 10) 푖푘 푉 푖푘 0 푖푘 푉 푖 푘 0 0 푉0 0 푆0
where 푉0 is the volume of the RMV in the undeformed configuration, 푆0 is the outer
0 surface with the outward normal 풏 , 푢푖 is the displacement component. Similarly, the average deformation gradient outside the localization band, 퐅0 , can be calculated by considering a subregion away from the localization band. Consequently, the localization criterion can be expressed as
‖푭̄̇ ‖⁄‖푭̇ 0‖ → ∞, (2. 11) where ( ̇ ) denotes taking the time derivative, ‖ ‖ denotes taking the norm.
18
In literature, many researchers have used critical values between 2 and 10
[70][71][72][76]. Dunand and Mohr[71] showed that changes in the predicted failure strain is insignificant when the critical value for localization increases from 2 to 5. In this study, a critical value of ‖퐹̅̇ ‖/‖퐹̇ 0‖ is set to be 10, and when ‖퐹̅̇ ‖/‖퐹̇ 0‖ = 10 , the macroscopic effective strain of the RMV is defined as the failure strain of the material.
Previous study[76] suggests that the effect of Lode parameter on void growth rate increases as the stress triaxiality decreases, and for the same stress triaxiality, the effect of
Lode parameter becomes more significant as L is close to zero. To manifest the Lode parameter effect, the range of stress triaxiality, from 0.8 to 1.8, and the range of Lode parameter, from -0.3 to 0.3, are chosen in this study.
2.3 HELP effect under different stress triaxialities
In the following, the HELP effect is systematically analyzed under various stress states. It is well-known that the void growth and coalescence speed is strongly influenced by the stress triaxialiy[78]. There for, in this section, the effect of stress triaxiality is analyzed. First, the Lode parameter imposed on the unit cell is kept constant and different stress triaxialities are considered. The results of the void growth rete and the effective strain at which the void coalescence occurred are plotted out.
Figure 2.3 compares the void growth rates as the stress triaxiality varies from T = 0.8 to T = 1.8 while the Lode parameter is kept as L = 0.15. Here f0 represents the initial void volume fraction, f represents the current void volume fraction, and Ee represents the macroscopic effective strain of the RMV. In these calculations the HELP effect is not
19 included. As expected, higher stress triaxiality results in higher void growth rate, and void growth accelerates as deformation increases.
Figure 2.3 Curves of normalized void volume fraction vs. macroscopic effective strain of the RMV as the stress triaxiality varies from T = 0.8 to T = 1.8 while the Lode parameter is kept as L = 0.15.
Figure 2.4 (a) Variation of the failure strain with stress triaxiality, (b) the HELP effect on the failure strain under different stress triaxialities.
Figure 2.4 (a) shows an exponential decay of the failure strain as the stress triaxiality increases, and the reduction of the failure strain due to the HELP effect. Figure 2.4 (b) shows the variation of 휀fh/휀f0 with the stress triaxiality, where 휀fh and 휀f0 represent the failure strain with and without the HELP effect, respectively. Figure 2.4 (b) suggests that
20
HELP has little influence on the failure strain when the stress triaxiality is low but has a significant effect on the failure strain when the stress triaxiality becomes high. For L = 0.15 and in the range of stress triaxiality between T = 0.8 and T = 1.8, the 휀fh/휀f0vs. stress triaxiality relation can be approximated as a linear function. Comparing Figure 2.4 and
Figure 2.3, it is found that the higher the void growth rate is, the more significant the HELP effect is on the ductility of the material.
To explain the influence of stress triaxiality on the HELP effect, two cases with the same Lode parameter, L = 0.15, but different stress triaxialities, T = 0.8 and T = 1.8, are compared. Figure 2.5 and Figure 2.6 show the contours of von Mises stress, plastic strain and trapping hydrogen concentration when the macroscopic effective strain of the RMV is equal to 0.038. Figures. (a), (c) and (e) are the results not considering the HELP effect, while Figures. (b), (d) and (f) are the results with the HELP effect included. Figure 2.5 shows the case of stress triaxiality T = 1.8 and Figure 2.6 shows the case of stress triaxiality
T = 0.8. Considering the stress triaxiality T = 1.8 case, without the HELP effect, the plastic strain is observed localizing in an area near the void surface, Figure 2.5 (c). This high plastic strain causes more hydrogen trapped in this area, Figure 2.5 (e). With the HELP effect the plastic strain localization and hydrogen trapping become more significant, Figure
2.5 (d) and (f). The phenomenon of trapped hydrogen accumulating in the area of high plastic strain is also reported in the work by Taha and Sofronis[64]. The high concentration of trapped hydrogen in the area near the void surface, Figure 2.5 (f), in turn works with the
HELP effect resulting in significant material softening (low Mises stress) in this region as shown in Figure 2.5 (b). Consequently, the void growth is accelerated and the material failure strain is reduced.
21
Figure 2.5 Contours of von Mises stress, plastic strain (SDV13) and trapping hydrogen concentration (SDV26) when the macroscopic effective strain of the RMV is equal to 0.038 for the case of high stress triaxiality (T = 1.8) and Lode parameter L = 0.15: (a) von Mises stress without HELP effect, (b) von Mises stress with HELP effect, (c) plastic strain without HELP effect, (d) plastic strain with HELP effect, (e) trapping hydrogen concentration without HELP effect, (f) trapping hydrogen concentration with HELP effect. 22
Figure 2.6 Contours of von Mises stress, plastic strain and trapping hydrogen concentration when the macroscopic effective strain of the RMV is equal to 0.038 for the case of low stress triaxiality (T = 0.8) and Lode parameter L = 0.15: (a) von Mises stress without HELP effect, (b) von Mises stress with HELP effect, (c) plastic strain without HELP effect, (d) plastic strain with HELP effect, (e) trapping hydrogen concentration without HELP effect, (f) trapping hydrogen concentration with HELP effect.
23
The results of plastic strain localization and hydrogen trapping near the void surface are similar but less significant for the stress triaxiality T = 0.8 case, Figure 2.6 (c-f).
Contrary to Figure 2.5 (b), there is no softening region near void surface in Figure 2.6 (b).
However, it is worth mentioning that the HELP effect does lead to a small softening area appearing on the void surface at a relatively large deformation level in Figure 2.7 (a). But this softening area quickly disappears as deformation continues, Figure 2.7 (b) and (c). The less significant and shorter lasting softening region shows that HELP has less effect on failure strain when the stress triaxiality is low.
Figure 2.7 Contours of the von Mises stress for the case of stress triaxiality T = 0.8 and
Lode parameter L = 0.15 at different macroscopic effective strain levels (Ee).
Figure 2.8 compares the void growth rates with and without the HELP effect under different stress triaxialities. Here letter “h” in the legends corresponds to the results with the HELP effect included. Again, two cases with the same Lode parameter, L = 0.15, but different stress triaxialities, low (T = 0.8) and high (T = 1.8), are considered. The void growth rate is much higher for the high stress triaxiality case than for the low stress triaxiality case. As shown in Figure 2.8, void growth is significantly accelerated by the
24
HELP effect for the high stress triaxiality case while the HELP effect on the void growth rate is less pronounced for the low stress triaxiality case.
Figure 2.8 Comparison of the void growth rate for cases of stress triaxiality T = 0.8 and T = 1.8, where the Lode parameter is fixed at L = 0.15. Letter “h” in the legends corresponds to the results with the HELP effect included.
2.4 HELP effect under different Lode parameters
In this section, the stress triaxiality is kept at T = 1.0 and different Lode parameters are considered. Figure 2.9 shows the void growth rate varying with different Lode parameters, from L = -0.3 to L = 0.3. In these calculations the HELP effect is not included.
The effect of the Lode is more complex. The L = -0.15 case displays the highest void growth rate when Ee < 0.3, but as the deformation continues, the void growth rate for the
L = -0.3 case becomes faster. For all the L > 0 cases, the void growth rate decreases with the increase of the Lode parameter.
Figure 2.10 (a) shows the effect of the Lode parameter on the failure strain. The failure strain is minimum when the Lode parameter is about -0.15 and increases as the Lode parameter moves away from this value. The HELP effect reduces the failure strain as
25 demonstrated by the variation of 휀fh/휀f0vs. Lode parameter in Figure 2.10 (b). The HELP effect reduces the failure strain by about 25-30% when the Lode parameter is between L =
-0.1 and L = -0.25, with the most effect occurs when the Lode parameter is about L = -0.15.
The HELP effect steadily decreases to less than 5% as the Lode parameter increases to L
=0.3.
Figure 2.9 Curves of normalized void volume fraction vs. macroscopic effective strain of the RMV as the Lode parameter varies from L = -0.3 to L = 0.3 while the stress triaxiality is kept at T = 1.0.
Figure 2.10 (a) Variation of failure strain with Lode parameter, (b) the HELP effect on failure strain under different Lode parameters.
26
Figure 2.11 Contours of von Mises stress, plastic strain and trapping hydrogen concentration when the macroscopic effective strain of the RMV is equal to 0.038 for the case of stress triaxiality T = 1.0 and Lode parameter L = -0.15: (a) von Mises stress without HELP effect, (b) von Mises stress with HELP effect, (c) plastic strain without HELP effect, (d) plastic strain with HELP effect, (e) trapping hydrogen concentration without HELP effect, (f) trapping hydrogen concentration with HELP effect.
27
Figure 2.12 Contours of von Mises stress, plastic strain and trapping hydrogen concentration when the macroscopic effective strain of the RMV is equal to 0.038 for the case of stress triaxiality T = 1.0 and Lode parameter L = 0.3: (a) von Mises stress without HELP effect, (b) von Mises stress with HELP effect, (c) plastic strain without HELP effect, (d) plastic strain with HELP effect, (e) trapping hydrogen concentration without HELP effect, (f) trapping hydrogen concentration with HELP effect.
28
To interpret the influence of the Lode parameter on the HELP effect, two cases with the same stress triaxiality, T = 1.0, but different Lode parameters, L = -0.15 and L = 0.3, are considered here. Figure 2.11 and Figure 2.12 show the contours of von Mises stress, plastic strain and trapping hydrogen concentration when the macroscopic effective strain of the RMV is equal to 0.038. Figure 2.11 shows the results of the L = -0.15 case and Figure
2.12 shows the results of the L = 0.3 case. Figures. (a), (c) and (e) are the results not considering the HELP effect while Figures. (b), (d) and (f) are the results with the HELP effect included. Plastic strain localization near the void surface leads to trapped hydrogen accumulating, and thanks to the HELP effect, material becomes softer in this area.
Consequently, void grows faster with the HELP effect, and the failure strain is reduced.
The plastic strain localization and the resulted hydrogen trapping and accumulation near the void surface are less significant for the L = 0.3 case than for the L = -0.15 case, and so is the HELP effect on the failure strain.
Figure 2.13 Comparison of the void growth rate for cases of L = -0.15 and L = 0.3, where the stress triaxiality is kept at 1.0. Letter “h” in the legends corresponds to the results with the HELP effect included.
29
Figure 2.13 compares the void growth rates with and without the HELP effect for the
L = -0.15 case and the L = 0.3 case. Without the HELP effect, the void growth rate is higher for the case of L = -0.15, but the difference is not very significant. However, with the HELP effect, the void growth rate is significantly accelerated for the L = -0.15 case but only slightly increased for the L = 0.3 case.
2.5 Summary and conclusions
Hydrogen effect on material ductility is studied by incorporating the hydrogen diffusion process and the induced HELP effect into a finite element program. A series of finite element analyses of a representative material volume subjected to various stress states were carried out. The evolution of local stress and deformation states result in hydrogen redistribution in the material, which in turn changes the material’s flow property.
In general, the HELP effect accelerates void growth and promotes material failure.
Furthermore, the HELP effect on ductile fracture is influenced by the stress state, as characterized by the triaxiality and Lode parameter. The following briefly summarizes the findings of this study:
• Under plastic deformation, plastic strain localizes near the void surfaces, leading to
more trapped hydrogen accumulating in this area. Due to the HELP effect, material in
this area softens, causing void growth to accelerate and failure strain to decrease.
• The effect of hydrogen on failure strain varies with the stress triaxiality and the Lode
parameter. Hydrogen causes a reduction in ductility, and this effect increases as the
stress triaxiality increases. The reduction in failure strain is more pronounced when the
30
Lode parameter is around -0.15 and gradually diminishes as the Lode parameter
increases or becomes more negative.
• Despite the complex effect of the stress triaxiality and the Lode parameter, hydrogen
tends to have a stronger effect on failure strain when the void growth rate is higher.
Figure 2.14 (a) Variation of failure strain with stress triaxiality and Lode parameter without HELP effect, (b) variation of failure strain with stress triaxiality and Lode parameter with HELP effect, (c) reduction of failure strain due to HELP effect.
• For the range of stress triaxiality and Lode parameter considered in this study, 0.8 ≤ T
≤ 1.8 and -0.3 ≤ L ≤ 0.3, the predicted variations of failure strain with stress triaxiality
31 and Lode parameter with and without HELP effect as well as the reduction of failure strain due to HELP effect are shown in Figure 2.14.
32
CHAPTER III
PHASE FIELD MODELING FOR BRITTLE AND DUCTILE FRACTURE
In this chapter, attempts are made to develop a phase field model for the simulation of ductile and brittle fracture. The new model is based on the frame work proposed by
Miehe et al.[79]. A degradation function is added to the yield function so that the yield surface shrinks as the phase field value of the diffusive crack increases, which ensures the consistency of the stored elastic energy calculation. A new crack driving force function is proposed by accounting for the plastic contribution, which accelerates the crack propagation after the plastic deformation reaches a critical value. The conjoint effect of the plastic adjustment function and the value of the critical energy release rate on the crack driving force reflects the competition between the ductile and brittle fracture mechanisms.
A coupled monolithic solution strategy is used by implementing the phase field model into the finite element software ABAQUS via a user defined element (UEL) subroutine. The
Newton-Raphson method is used to solve the coupled phase field and displacement field, where the crack driving force is kept constant during each increment and takes the value computed at the end of the previous increment. For this reason, small increment size needs to be used to ensure the accuracy of the result. An 8-node hexahedral element with the phase field value as one of the nodal degrees of freedom is developed. The B-bar method is used to avoid volumetric locking after plastic deformation occur. FEM simulations are conducted to validate the developed phase field model.
33
3.1 Geometrical phase field method for ductile fracture
3.1.1 Phase field modeling
Miehe and co-workers[40] introduced a phase field modeling of the diffusive crack.
Consider an infinite one-dimensional bar 훺, and use a field variable 푑 ∈ (0,1) to describe the crack topology, with 푑 = 0 representing the unbroken state and 푑 = 1 representing the fully broken state. Assuming the crack is at location 푥 = 0, an exponential function of the phase field variable is expressed as
|푥| 푑(푥) = exp ( ) , (3. 1) 푙 where 푙 is a length scale parameter controlling the smoothness of the crack topology and
푑(푥) is the phase field value of the diffusive crack surface. Note that Equation 3.1 is the solution of the homogeneous differential equation 푑(푥) − 푙2푑′′(푥) = 0 with boundary conditions: 푑(0) = 1 and 푑(±∞) = 0. This differential equation is the Euler equation of the variational problem 푑 = Arg inf{퐼(푑)} with 푑 ∈ {푑|푑(0) = 1, 푑(±∞) = 0 } in terms of the function
1 푙2 퐼(푑) = ∫ ( 푑2 + 푑′2) d푉 . (3. 2) 훺 2 2
Assuming that the cross-section of the bar is 훤 and integrating the right-hand side of
Equation 3.2 result in 퐼(푑) = 푙훤. Therefore Equation 3.2 can be rewritten as
1 푙 훤(푑) = ∫ ( 푑2 + 푑′2) 푑푉 = ∫ 훾(푑, 푑′)d푉 ， (3. 3) 훺 2푙 2 훺
34 here 훤(푑) can be regarded as the crack surface itself, and 훾(푑, 푑′) as the crack surface density function in one dimension. To extend it to multiple dimensions, the crack surface density function is given as
1 푙 훾(푑, |∇푑|) = 푑2 + |∇푑|2， (3. 4) 2푙 2 where ∇푑 represents the spatial gradient of 푑.
Miehe and co-workers introduced the crack driving force[41] and later generalized the formulation to make it suitable for various constitutive models[49]. For rate- independent case, it is assumed that the time derivative of the crack surface follows
d 1 훤(푑) = ∫ 2(1 − 푑)H푑̇d푉 ， (3. 5) d푡 푙 훺 where H represents the crack driving force. To satisfy the irreversibility of the crack evolution, H must satisfy the following constraints:
d H = 0, H = ∞, H ≥ 0. (3. 6) unbroken broken d푡
By taking the time integration of Equation 3.5, the crack surface can be updated according to the equation
1 2 2 훤(푑) = 훤(푑푛) + ∫ [(2푑 − 푑 ) − (2푑푛 − 푑푛)]Hd푉 ， (3. 7) 푙 훺
where 푑 denotes the current phase field value and 푑푛 denotes the value of phase field at the beginning of the increment, while H is held constant throughout the increment.
Multiplying Equation 3.7 by 푙 and rewriting the equation result in
35
2 2 ∫ {푙훾(푑푛, |∇푑푛|) − 푙훾(푑, |∇푑|) + [(2푑 − 푑 ) − (2푑푛 − 푑푛)]H}d푉 = 0 . (3. 8) 훺
By taking the variation of Equation 3.8, the weak form of the equilibrium equation can be obtained as
∫ [−푑 + 푙2∆푑 + 2(1 − 푑)H]d푉 = 0 ， (3. 9) 훺 where ∆푑 is the Laplacian of the phase field.
3.1.2 Degradation function
In order to describe the transition from the unbroken state to the broken state of material, Miehe et al.[40] proposed a degradation function
푔(푑) = (1 − 푑)2. (3. 10)
In the previous studies[49]–[51], this degradation function was applied only to the
Young’s modulus to reduce the stiffness of the material as the phase field value increases while the yield stress is assumed invariant with the phase field value. Thus, the yield condition is expressed as
p p 푓(휎̅, 푑, 휀̅ ) = 푔(푑)휎̅ − 휎y(휀̅ )， (3. 11)
p where 휎̅ represents the equivalent stress, 휀̅ represents the equivalent plastic strain, and 휎y represents the current yield stress. However, this can cause an unreasonable consequence, i.e., the material will undergo elastic unloading as the phase field value increases leading to no plastic strain accumulation at the final stage of the crack initiation. This contradicts the plastic damage theory, and thus in Borden et al.[53], a degradation function is applied to the yield stress. This is analogous to concept adopted in the coupled ductile damage
36 models, such as the Gurson model[58], where the yield surface shrinks as damage progresses. In this study, we assume both Young’s modulus and yield surface deteriorate with the increase of the phase field value, thus the yield condition is expressed as
p p 푓(휎̅, 푑, 휀̅ ) = 푔(푑)휎̅ − 푔(푑)휎푦(휀̅ ). (3. 12)
3.1.3 Free energy function
For an elasto-plastic material, the total free energy function can be expressed as
퐸(풖, 푑) = ∫ [Ψe(풖, 푑) + Ψp(풖, 푑) + Ψc(풖, 푑)]d푉 ， (3. 13) 훺 where 풖 denotes the displacement field, Ψe denotes the density of stored elastic energy,
Ψp denotes the density of plastic dissipation, and Ψc denotes the density of fracture energy.
It is convenient to split the stress and strain tensors into volumetric and deviatoric components. Since crack closure does not cause material deterioration, degradation should only be applied to the tensile part of the volumetric stress and the deviatoric part of the total stress. To evaluate Ψe(풖, 푑), the total elastic energy density is split into two parts
Ψe+ = 퐾⟨tr(훆)⟩2 + 2휇(훆e : 훆e ) + dev dev e− 2 ， (3. 14) Ψ = 퐾⟨tr(훆)⟩−
where the bracket ⟨푥⟩± is defined as (푥 ± |푥|)⁄2, 퐾 is the bulk modulus, 휇 is the shear
e modulus, tr(훆) is the trace of the strain tensor, and 훆dev is elastic part of the deviatoric
e 풆 strain tensor defined as 훆dev = 훆 − tr(훆)퐈⁄3 with 퐈 being the second order identity tensor.
Based on the assumption that both the dissipated plastic energy and the stored elastic energy are degraded by the crack phase field, we define the total artificial energy density as
37
퐺 푊 = (1 − 푑)2Ψe+ + Ψe− + Ψp + c (푑2 + 푙2|∇푑|2)， (3. 15) 2푙
where 퐺c is the critical energy release rate, the first two terms in the right-hand side represent the density of stored elastic energy, i.e.,
Ψe(풖, 푑) = (1 − 푑)2Ψe+ + Ψe−， (3. 16) and the density of plastic dissipation Ψp can be evaluated by integrating over the time history
푡 p 2 p p Ψ = ∫ [(1 − 푑) 휎푦(휀̅ )휀̇̅ ]d푡 . (3. 17) 0
Taking the variational derivative of Equation 3.15 leads to
퐺 훿 푊 = −2(1 − 푑)Ψe+훿푑 + 훿 Ψp + c (푑 − 푙2Δ푑)훿푑. (3. 18) 푑 푑 푙
Based on the Euler equation of the variational principle for the evolution of the gradient damage derived by Miehe[80], Equation 3.18 can be rewritten as
퐺 훿 Ψp 2(1 − 푑)Ψe+ = c (푑 − 푙2Δ푑) + 푑 . (3. 19) 푙 훿푑
Note that because of the complicated form of the plastic dissipation as defined by
Equation 3.17 it is very difficult to express its variational derivative explicitly. Thus, exact evaluation of the last term in Equation 3.19 becomes not possible. To simplify the calculation while approximately accounting for the plastic contribution, we introduce a modified crack surface energy function
퐺 푊̃ = c (푑 − 푙2Δ푑)， (3. 20) f 푙퐴
38 here 퐴 is a plastic adjustment function, and its form can be written as
α휀p̅ 퐴 = exp ( ) ， (3. 21) 휀f
where coefficient α is a material parameter and 휀f is the critical failure strain. Using 푊̃f to approximate the right-hand side of Equation 3.19 results in
퐺 2(1 − 푑)훹푒+ = 푐 (푑 − 푙2훥푑). (3. 22) 푙퐴
Compare Equation 3.22 and Equation 3.9, the crack driving force function H can be defined as
Ψe+ H̃ = . (3. 23) 퐺c⁄푙 퐴
The value of H̃ (풙, 푠) depends on the location 풙 and the current step time 푠. The constraint given by Equation 3.6 leads to H = max H̃ (푥, 푠) within the time history 푠 ∈ [0, 푡].
The plastic adjustment function 퐴 in Equation 3.23 promotes the evolution of the phase field value upon the occurrence of plastic deformation as shown in Figure 3.1. In
Figure 3.1 (a), the yellow curve represents the result when the plastic adjustment function
퐴 is not included in the model, and the other three curves show the results computed using the model that includes the plastic adjustment function, in which α takes different values while 휀f and 퐺c are kept the same for the three cases. In the elastic region, the four curves coincide. Once the yield strain (휀y) is reached, the plastic adjustment function results in an acceleration in the evolution of the phase field value. As α increases, the effect of the plastic adjustment function increases. On the contrary, the effect of the plastic adjustment
39 function is reduced as 휀f increases, Figure 3.1 (b). The effect of 퐺c is shown in Figure 3.1
(c). A Larger 퐺c value results in a smaller phase field value at the same strain level.
Consequently, a sufficiently high 퐺c value may suppress crack initiation under elastic deformation. The competing effects of the plastic adjustment function and the critical energy release rate make it possible to capture the transition of fracture mechanism from ductile to brittle. Furthermore, it is worth noting from Figure 3.1 that without the plastic adjustment function, d increases very slowly with strain, making fracture difficult to occur.
The introduction of the plastic adjustment function facilitates the increase of d, and with the proper choices of parameters α, 휀f and 퐺c, the model has the flexibility of describing the fracture process in various materials.
Figure 3.1 Effects of the plastic adjustment function and the critical energy release rate on the evolution of the phase field value.
40
3.2 Plasticity Model and Numerical Algorithm
The material is assumed to follow the von Mises plasticity theory. Noting that degradation only occurs to the tensile part of the volumetric stress and the deviatoric part of the total stress, thus the stress-strain relation can be expressed as
2 2 e 훔 = [1 − 퐻(휀푘푘)푑] 퐾휀푘푘퐈 + 2(1 − 푑) 휇훆dev， (3. 24)
where 퐻(휀푘푘) is the Heaviside function. At each time increment [푡푛, 푡푛+1], the trial elastic strain is given as
e trial e 훆푛+1 = 훆푛 + ∆훆， (3. 25)
e where 훆푛 represents the elastic strain at the end of the previous increment and ∆훆 represents the total strain increment. Therefore, the volumetric component and the deviatoric component of the elastic trial stress are obtained as
2 퐩e trial = [1 − 퐻(휀e trial )푑] 퐾휀e trial 퐈; 푛+1 푘푘 푛+1 푘푘 푛+1 (3. 26) e trial 2 e trial 퐬푛+1 = 2(1 − 푑) 휇훆dev 푛+1,
e trial e trial where 퐩푛+1 is volumetric elastic trial stress and 퐬푛+1 deviatoric elastic trial stress.
Figure 3.2 shows the flow chart of a return-mapping algorithm using the Newton-Raphson method to integrate the plastic rate equations. Here the degradation function is applied to the elastic moduli as well as the yield surface.
To explain why it is necessary to apply the degradation function to the yield surface, consider the definitions of the total elastic energy density, Equation 3.14, which is used for the calculation of the crack driving force as shown in Equation 3.23. There is no problem
41 if the deformation is purely elastic. However, when there is plastic deformation, a contradiction will arise if the yield surface is not degraded.
Figure 3.2 Flow chart of a return-mapping algorithm using the Newton-Raphson method to integrate the plastic rate equations.
For illustration purpose, consider the stress update process for a one-dimensional problem. As shown in Figure 3.3 (a), from increment 푛 to increment 푛 + 1, the total strain increment is ∆ε, and the degraded Young’s modulus 푔(푑푛+1)퐸 is used to calculate the elastic trial stress. Point 5 represents the resulting elastic trial stress. By returning to the
42 yield surface, the updated stress is given by point 2, thus the total elastic strain after increment 푛 + 1 is given by the line segment 1-3. Note that the slope of line 1-2 is
푔(푑푛+1)퐸 . However, the total elastic energy density defined by Equation 3.14 is calculated using the undegraded Young’s modulus 퐸, and is equal to the area of triangle
1-4-3. Since the slope of line 1-4 is 퐸, point 4 lies outside the yield surface. As a result, the total elastic energy Ψe+ calculated according to Equation 3.14 is not purely elastic now, which presents a contradiction to the definition of Ψe+.
On the contrary, as shown in Figure 3.3 (b), the upper curve is the original stress- strain curve, the lower curve is the degraded stress-strain curve, and in the stress updating process, the degraded yield surface is used. Same as the previous case, the total elastic strain is given by the line segment 1-3 and the total elastic energy density calculated according to Equation 3.14 is given by the area of triangle 1-4-3. As shown in Figure 3.3(b), different from the previous case, this time point 4 lands on the original yield surface. The above discussion demonstrates that degrading the yield surface by the same ratio as what is used to degrade the Young’s modulus ensures the consistency of the elastic energy density definition, Equation 3.14, and the stress-strain relation coupled with phase field.
Moreover, as illustrated in Figure 3.3 (c), if the elastic modulus is degraded but the yield stress is not, the material may undergo elastic unloading as the phase field value increases. Let point 6 on the yield surface denote the stress at the end of increment 푚. Since the phase field value is monotonically increasing, the Young’s modulus to be used to compute the elastic trial stress is going to be smaller than the value used in the previous increment. As a result, for a small strain increment ∆휀, the computed elastic trial stress may become less than the yield stress for increment 푚 + 1, as indicated by point 5 in Figure 3.3
43
(c). This means that the total strain increment ∆휀 is purely elastic, which is not reasonable.
In order for the computed result to have a continuous increasing in plastic strain, the yield stress needs to be degraded with the same ratio as the Young’s modulus so that point 5 lies outside the yield surface as shown in Figure 3.3 (d).
Figure 3.3 Schematics of stress update and strain energy calculation
44
Therefore, the degraded yield surface is expressed as
3 풇(퐬, 푑, 휀p̅ ) = √ 퐬: 퐬 − 푔(푑)휎 (휀p̅ )， (3. 27) 2 푦 where 퐬 is the deviatoric stress tensor, and the return-mapping procedure described in
Figure 3.2 results in
6Δ훆̅p[(1 − 푑)2휇]2 푒 e trial ( ) 흈푛+1 = 풑푛+1 + 풔푛+1 = [푫 − trial 핀d] : 휺푛+1 ， 3. 28 휎̅푛+1
1 where 핀 is the deviatoric projection tensor defined as 핀 = 핀 − 퐈⨂퐈, 핀 is the fourth order d d 3
1 symmetric tensor defined as 핀 = (훿 훿 + 훿 훿 ), and the elasticity tensor 푫푒 is 푖푗푘푙 2 푖푘 푗푙 푖푙 푗푘 expressed as
푒 2 2 푫 = [1 − 퐻(휀푘푘)푑] 퐾퐈⨂퐈 + 2(1 − 푑) 휇핀d. (3. 29)
The elastoplastic consistent tangent modulus is obtained by differentiating Equation
3.29
p[( )2 ]2 휕훔푛+1 푒 6Δ훆̅ 1 − 푑 휇 = 푫 − trial 핀d 휕훆푛+1 휎̅ 푛+1 ， (3. 30) Δ훆̅p 1 [( )2 ]2 +6 1 − 푑 휇 ( trial − 2 ) 풏̅푛+1⨂풏̅푛+1 휎̅푛+1 3(1 − 푑) 휇 + 퐻
trial trial where 풏̅푛+1represent the unit flow normal, i.e., 풏̅푛+1 = 풔푛+1 /‖풔푛+1‖. By differentiating
Equation 3.23, 휕흈⁄휕푑 is obtained as
휕훔 = −2[1 − 퐻(휀 )푑]퐾휀 퐈 − 4(1 − 푑)퐺훆 . (3. 31) 휕푑 푘푘 푘푘 dev
45
3.3 Finite Element Implementation
In this section, the phase field model is implemented in the commercial finite element code ABAQUS via a user defined subroutine UEL to solve the coupled displacement field and phase field. An 8-node hexahedral element with the crack phase field value treated as one of the nodal degrees of freedom is developed and the body forces are neglected. Miehe et al.[41] proposed an operator split scheme, which uses the staggered strategy to solve the weak form of equilibrium equations of the displacement field and the phase field separately.
However, in this study, both equations are coded in the UEL subroutine and solved monolithically using the ABAQUS built in Newton-Raphson solver.
The complexity of the crack driving force H makes the partial derivatives of the phase field residual with respect to the nodal displacements hard to obtain. Thus, an approximate scheme is adopted here, where for each time interval [푡푛, 푡푛+1], the crack driving force is kept constant and takes the value H푛 computed at the end of the previous time interval. To maintain the accuracy of the solution, the increment size is limited to be relatively small.
Rewriting Equation 3.8 in spatial version results in
2 1 2 푙 2 2 2 (퐍퐝푛) + [(퐍,푥퐝푛) + (퐍,푦퐝푛) + (퐍,푧퐝푛) ] 2 2 2 ( ) ∫ 1 2 푙 2 2 2 d푉 = 0 ， 3. 32 훺 − (퐍퐝) − [(퐍,푥퐝) + (퐍,푦퐝) + (퐍,푧퐝) ] 2 2 2 2 { +[2퐍퐝 − 2퐍퐝푛 − (퐍퐝) + (퐍퐝푛) ]H }
where 퐍 is a vector containing the eight shape functions, 퐍,푥 , 퐍,푦 , and 퐍,푧 are the derivatives of shape functions with respect to coordinates, 퐝 is a vector containing the
46 current nodal phase field values, and 퐝푛 contains the nodal phase field values at the beginning of the increment,
퐍 = [푁1 푁2 ⋯ 푁8] 퐍,푥 = [푁1,푥 푁2,푥 ⋯ 푁8,푥] 퐍,푦 = [푁1,푦 푁2,푦 ⋯ 푁8,푦]， (3. 33) 퐍,푧 = [푁1,푧 푁2,푧 ⋯ 푁8,푧] T 퐝 = [푑1 푑2 ⋯ 푑8] and the B matrix is given as
퐁 = [퐁1 퐁2 ⋯ 퐁8]， (3. 34) with
푁 0 0 푖,푥 0 푁푖,푦 0 0 0 푁 퐁 = 푖,푧 . (3. 35) 푖 푁 푁 0 푖,푦 푖,푥 푁푖,푧 0 푁푖,푥 [ 0 푁푖,푧 푁푖,푦]
To avoid volumetric locking as stresses reach the yield surface, the B-bar method is adopted in finite element formulation [81].
Taking variation of Equation 3.32 with respect to the crack phase field value, the weak form of the equilibrium function for the phase field can be expressed as
2 ∫ {[1 − (퐍퐝)]퐍H − (퐍퐝)퐍 + 푙 [(퐍,푥퐝)퐍,푥 + (퐍,푦퐝)퐍,푦 + (퐍,푧퐝)퐍,푧]}d푉 = 0. (3. 36) 훺
At each increment, the Newton-Raphson method is used to iteratively reduce the residuals of the nodal forces and the phase field. The residual of the nodal force is defined as
47
퐑u = ∫ 퐁̅T훔푑푉 − ∫ 푵퐮퐛푑푉 − ∫ 푵퐮퐭d푆 ， (3. 37) 훺 훺 휕훺 where 퐛 represents the body force vector and 퐭 represents the surface traction on 휕훺. The mapping matrix 푵퐮 is defined as
퐮 퐮 퐮 퐮 푵 = [푵1 푵2 ⋯ 푵8]， (3. 38) with
푁푖 0 0 퐮 푵푖 = [ 0 푁푖 0 ] . (3. 39) 0 0 푁푖
The residual for phase field is defined as
d 2 퐑 = ∫ {[1 − (퐍퐝)]퐍H − (퐍퐝)퐍 + 푙 [(퐍,푥퐝)퐍,푥 + (퐍,푦퐝)퐍,푦 + (퐍,푧퐝)퐍,푧]}d푉 ，(3. 40) 훺
Therefore, the tangent stiffness matrix can be partitioned into 퐊uu, 퐊ud, 퐊du, and 퐊dd, where
∂퐑u ∂훔 퐊uu = − = − ∫ 퐁̅T 퐁̅d푉 ， (3. 41) ∂퐮 훺 ∂훆
∂퐑u ∂훔 퐊ud = − = − ∫ 퐁̅T d푉 ， (3. 42) ∂퐝 훺 ∂퐝
d dd ∂퐑 2 퐊 = − = ∫ {퐍⨂퐍H + 퐍⨂퐍 − l [퐍,푥⨂퐍,푥 + 퐍,푦⨂퐍,푦 + 퐍,푧⨂퐍,푧]}d푉 ，(3. 43) ∂퐝 훺
∂퐑d 퐊du = − = ퟎ， (3. 44) ∂퐮 where 퐮 denotes the nodal displacement vector, and the spatial version of the stress derivative with respect to the crack phase field is expressed as 48
∂훔 −2[1 − 퐍퐝]퐾휀 퐈⨂퐍 − 4(1 − 퐝)휇휺 ⨂퐍 ε ≥ 0 = { 푘푘 dev 푘푘 . (3. 45) ∂퐝 −4(1 − 퐝)휇휺dev⨂퐍 ε푘푘 < 0 Note that 퐊du= 0 is due to the assumption of constant crack driving force value during the time interval.
3.4 Parameter Studies
3.4.1 Effect of the Increment Size
Figure 3.4 A double notched, dog bone specimen: (a) geometry and boundary conditions (dimensions in mm); (b) finite element mesh.
The algorithm described in the previous section to implement the phase field model requires the increment size to be sufficiently small to ensure the accuracy of the solution.
To demonstrate the effect of the increment size, consider a dog bone specimen having two
49 asymmetrical, semicircular notches. The geometry (in mm) and boundary conditions of the specimen are shown in Figure 3.4 (a), and the thickness of the specimen is 3mm. Figure
3.4 (b) shows the finite element model with refined mesh in the middle of the specimen where the fracture is expected to occur. The element size of the refined mesh is 0.4 mm.
The uniaxial stress-strain behavior of the material is assumed to follow a bilinear model so that the flow properties are represented by a yield stress and a constant hardening modulus.
Table 3.1 lists the material properties.
Table 3.1 Material properties
Properties Values Units
Young’s modulus 퐸 = 200 GPa
Possion’s ratio 휐 = 0.3 -
Yield stress 휎푦 = 250 MPa
Hardening modulus ℎ = 100 MPa
2 Critical energy release rate 퐺c = 0.2 mJ/mm
Fracture strain 휀f = 0.1 -
Coefficient 훂 α = 1.0 -
The applied displacement is imposed incrementally on the top surface of the specimen. Let δ = 6.25 × 10−6 mm. Analyses are performed with the increment sizes of
δ, 2δ, 4δ, ⋯, and 128δ respectively, and the computed load vs. displacement results are shown in Figure 3.5. It can be seen that the load-displacement results tend to converge as the increment size reduces. At point a1, corresponding to displacement 푢 = 0.0138 mm and load 퐹 = 1706 N , crack initiated and starts to propagate. The computed load-
50 displacement results before point a1 show indistinguishable difference as the increment size becomes smaller than 4δ.
Figure 3.5 Computed load-displacement curves with different increment sizes
Figure 3.6 compares the contour plots of the phase field value computed using the increment sizes of 2δ and 4δ at three applied load levels corresponding to points a1, a2 and a3 indicated in Figure 3.5 respectively. At point a1, cracks start to propagate from the notch tips, where the phase field value reaches 1. The contour plots corresponding to point a3 using increment sizes 2δ and 4δ show the same levels of the phase field values and the same amount of crack propagation. But as indicated in Figure 3.5, it requires a slightly larger amount of applied displacement to reach point a3 with the increment size of 4δ.
Figure 3.5 and Figure 3.6 suggest that larger increment size results in slower degradation of material stiffness and slower evolution of crack phase field. However, as the increment
51 size becomes sufficiently small, the numerical results, including the computed load- displacement curve and the prediction of crack initiation and propagation, do converge.
Figure 3.6 Crack phase field contours at different loading stages, where 2δ and 4δ represent the increment sizes used in the simulations, and a1, a2 and a3 correspond to the loading stages indicated in Figure 3.5.
The final crack topology of the eight different cases with different increment sizes show similar features as demonstrated in Figure 3.7. The cracks propagate separately from each notch to the other side of the specimen and do not merge with each other. However, this pattern is slightly changing as the increment size increases. For larger increments, as the two cracks propagate towards the middle of the specimen, they tend to curve toward each other. The crack path with the increment size of 128δ is more curved than other cases.
52
Figure 3.7 Final crack phase field obtained with different increment sizes
The maximum increment of the crack driving force ∆H is plotted in Figure 3.8. At each displacement increment, ∆H takes the highest value among the computed values at all the Gauss points. Four curves are plotted here, corresponding to the increment sizes of
δ, 2δ, 4δ and 8δ. In general, ∆H stays at a very low level at the early loading stage and dramatically increases as the crack is to start propagating. From Figure 3.8 is seen that ∆H becomes sensitive to increment size only when it starts to dramatically increase with the applied displacement. Therefore, this study proposed a strategy to control the increment size by setting a limit value for ∆H. When the computed value of ∆H becomes larger than the limit value, the current increment will be abandoned and the increment size will be
53 reduced. This allows for larger increment size being used in the early loading stage when
∆H stays low and the increment size being reduced when ∆H increases. Comparing to the strategy of using small increment size from the beginning, this new strategy significantly reduces the computational time while maintaining the solution accuracy.
Figure 3.8 Variation of the maximum increment of crack driving force with different increment sizes of applied displacement
Figure 3.9 shows an example in which the strategy described above is used to control the increment size. The initial increment size is 16δ and the limit value of ∆H is set to be
0.002. The purple curve (indicated by “CDF Limit” in the graph legend) shows the computed load-displacement response. For comparison, the results using fixed increment size of 훿 and 2훿 are also shown in Figure 3.9. The three curves coincide before cracks start to propagate while the curve with the increment size controlled by ∆H and the curve
54 computed using a fixed increment size of 훿 show negligible difference during the crack propagation process. However, the curve with the increment size controlled by ∆H is computed with only 372 increments while the curve computed with fixed increment size of 훿 requires 2250 increments.
Figure 3.9 Comparison of the computed load-displacement curve with the increment size controlled using the proposed strategy with those obtained using fixed increment sizes
3.4.2 Effect of the Critical Energy Release Rate
Consider again the example studied in the previous section. Figure 3.10 shows the computed load-displacement curve (solid line) and variation of the maximum plastic strain over the loading history (dashed line). In each increment, the maximum plastic strain is obtained by comparing the values computed at all Gauss points in the mesh. As is shown in Figure 3.10, the maximum plastic strain remains zero until the crack initiates. Note that
55 based on the proposed phase field model, the plastic strain will keep increasing after fracture is fully developed, which explains the rapid increase of the plastic strain after the crack starts to propagate. Nonetheless, there is no plastic strain before the crack initiation, which indicates that the failure mode is brittle.
It is well known that ductile fracture is associated with a higher energy than brittle fracture, thus the critical energy release rate is higher for the ductile fracture than that for brittle fracture[82]–[84]. In their phase field modeling of fracture, Ulmer et al.[85] employed different values of critical energy release rate for the ductile fracture and the brittle fracture. In this section, the effect of the critical energy release rate on the phase field model prediction is studied. The material properties used here are the same as in those presented in Table 3.1 except for the value of 퐺c. Four different 퐺c values are considered,
0.8, 1.0, 1.2, and 1.4 mJ/mm2, and the model predictions are compared.
Figure 3.10 Load-displacement curve and variation of the maximum plastic strain during the loading history
56
Figure 3.11 compares the computed load-displacement curves and variations of the maximum plastic strain for the four cases with different 퐺c values. The red dots indicate the maximum plastic strain value reached when the crack starts to propagate. As is seen in
Figure 3.11, crack starts to propagate at a higher applied displacement as the value of 퐺c increases, leading to a higher load carrying capacity of the specimen. It is worth noting that the value of 휀f in the plastic adjustment function for crack driving force is taken as 0.1 here.
When the crack starts to propagate, the maximum plastic strain has exceeded 휀f for the
2 cases with 퐺c = 1.0, 1.2, 1.4 mJ/mm , but has not reached 휀f for the case with 퐺c =
0.8 mJ/mm2.
Figure 3.11 Load-displacement curves and variations of the maximum plastic strain for
different values of 퐺c
57
Figure 3.12 shows the crack paths resulted from the four 퐺c values. When 퐺c =
0.8 mJ/mm2, the crack initiates from one of the notches and propagates to the other side
2 of the specimen. When 퐺c = 1 mJ/mm , cracks initiate from both notches, but show
2 different propagation speeds and paths. For the cases of 퐺c = 1.2 and 1.4 mJ/mm , cracks initiate from both notches, curve towards each other while propagating, and finally merge together.
The above numerical examples show that small 퐺c values lead to brittle fracture while large 퐺c values result in ductile fracture. The 퐺c value also influence the crack path and the load carrying capacity of the specimen. The proposed phase field model is capable of simulating both brittle fracture and ductile fracture as well as the ductile-brittle transition if 퐺c is set to be a function of field variables such as strain rate and temperature.
Figure 3.12 Final crack phase field resulted from different values of 퐺c
3.4.3 Effects of α
In this section, the effect of the parameter 훼 in the plastic adjustment function for
2 crack driving force is discussed. Here two 퐺c values,1.0 and 1.2 mJ/mm , together with
58 four different 훼 values, 0.5, 1.0, 1.5 and 2.0 are used in eight simulations of the double notched, dog bone specimen shown in Figure 3.4. Other material properties are the same as those listed in Table 3.1.
Figure 3.13 Effect of 훼 on the load-displacement response and maximum plastic 2 2 evolution: (a) 퐺c = 1.2 mJ/mm , (b) 퐺c = 1 mJ/mm .
59
Figure 3.13 compares the computed load-displacement curves and variations of the
2 maximum plastic strain using different 훼 values, where 퐺c = 1.2 mJ/mm in Figure 3.13
2 (a) and 퐺c = 1 mJ/mm in Figure 3.13 (b). The results show that a larger 훼 leads to earlier crack initiation and lower ductility. However, for a fixed 퐺c value, the maximum load the specimen can carry does not change significantly with different 훼 values. Figure 3.14 show the crack paths for different combinations of 퐺c and 훼. For the cases with low plastic strain resulted from lower 퐺c and larger 훼, cracks tend to propagate unevenly from the two notches and do not merge. For the cases with high plastic strain resulted from higher 퐺c and smaller 훼, cracks initiated from the two notches tend to curve toward each other and merge together.
Figure 3.14 Final crack phase field resulted from different values of 훼 and 퐺c.
60
It is important to point out that the parameter 훼 plays a role only in the cases that have plastic strain. Its effect diminishes when 퐺c is very small and the failure mode is brittle fracture.
3.4.4 Effect of the Yield Stress
In this section, two values of yield stress are considered, 휎푦 = 250 MPa and
2 500 MPa. The critical energy release rate is taken as 퐺c = 1.2 mJ/mm , and other material properties are the same as those listed in Table 3.1.
Figure 3.15 Effect of yield stress on the load-displacement response and maximum plastic strain evolution.
Figure 3.15 compares the computed load-displacement curves and variations of the maximum plastic strain for the two cases. For 휎푦 = 500 MPa, there is no plastic strain before the crack initiation. The maximum plastic strain stays at zero until the load-
61 displacement curve starts to drop. For 휎푦 = 250 MPa, there is plastic strain accumulation before crack initiation. At the instant when the load-displacement curve starts to drop rapidly, the maximum plastic strain has exceeded the value of 휀f.
Figure 3.16 compares the crack path for the two cases. For 휎푦 = 500 MPa, cracks initiate at the two notches and propagate separately toward the opposite side of the specimen. The two cracks do not merge. For 휎푦 = 250 MPa, cracks initiate at the two notches and merge together as they propagate toward the middle of the specimen.
The numerical results shown in Figure 3.15 and Figure 3.16 agree with the common knowledge about material’s ductility and strength, i.e., stronger materials tend to be more brittle.
Figure 3.16 Final crack phase field resulted from different values of yield stress
3.5 Additional Examples
In the previous section, various cases of crack initiation and propagation in a double notched, dog bone specimen are discussed, where the notch tip region provides a crack initiation site. To demonstrate the versatility of the numerical model, additional specimens,
62 a flat specimen without notch, a compact tension (CT) specimen and a modified CT specimen are considered in this section.
3.5.1 Flat specimen without notch
Figure 3.17 (a) shows the geometry (in mm) and boundary conditions of a smooth dog bone specimen. The thickness of the specimen is 3 mm. Figure 3.17 (b) shows the finite element model with refined mesh in the middle of the specimen where the fracture is expected to occur. The element size of the refined mesh is 0.4 mm. The bottom surface of the specimen is fixed, and displacement is applied on the top surface. Three different
2 critical energy release rates, 퐺c = 1.2, 1.3, and 1.4 mJ/mm , are used in the simulations presented in this section. All other material properties are the same as those listed in Table
3.1.
Figure 3.17 A smooth dog bone specimen: (a) geometry and boundary conditions (dimensions in mm); (b) finite element mesh.
63
Figure 3.18 compares the computed load-displacement curves and variations of the maximum plastic strain for the three cases. Same trend is observed as in the previous example, i.e., crack initiates at a higher applied displacement as the value of 퐺c increases, leading to a higher load carrying capacity of the specimen. Interestingly, for the three different 퐺c values, the maximum plastic strain values in the specimen at which crack starts to propagate are virtually the same, around the critical value 휀f = 0.1.
Figure 3.18 Load-displacement curves and variations of the maximum plastic strain for different values of critical energy release rate
Figure 3.19 shows the final crack phase field resulted from different critical energy
2 release rates. For 퐺c = 1.2 mJ/mm , crack initiates at the center, propagates horizontally,
2 and tilts slightly as it approaches the side surfaces. For 퐺c = 1.3 mJ/mm , the crack path is inclined except for a small segment in the center where it is horizontal. For 퐺c =
64
1.4 mJ/mm2, the whole crack path is inclined. The inclined crack path is similar to the ones described in the studies of Xue [86] and Besson et al [87].
Figure 3.19 Final crack phase field resulted from different 퐺c values
Figure 3.20 Load-displacement curve and evolution of maximum phase field value for 2 the 퐺c = 1.2 mJ/mm case.
65
In Figure 3.18, it is observed that for all three cases, there is a sudden drop of load near the peak of the load-displacement curve (marked by red arrows). To explain this
2 phenomenon, let’s consider the 퐺c = 1.2 mJ/mm case. The load-displacement curve and evolution of maximum phase field value are plotted in Figure 3.20. From point a1 to point a2, there is a sudden drop in load and increase in maximum phase field value. Figure 3.21 plots the contours of crack phase field corresponding to load levels a1 and a2 respectively.
At load level a1, the crack phase field is uniform in a large region of the gage section of the specimen. But at load level a2, the crack phase field is getting localized into the middle of the specimen. The sudden drop of the load-displacement curve from a1 to a2 is due to the onset crack phase field localization.
Figure 3.21 Contours of crack phase field at two load levels indicated in Figure 20
3.5.2 Compact tension specimens
Figure 3.22 shows the dimensions and boundary conditions of the compact tension specimens considered in this section. The thickness of both specimens is 2.5 mm. The
66 modified CT specimen shown in Figure 3.22 (a) contains two holes with a diameter of 2 mm.
Figure 3.22 (a) A modified CT specimen, (b) a CT specimen (dimensions in mm)
67
2 First, consider the case of brittle fracture with 퐺c = 0.2 mJ/mm . All other material properties used in the simulations are the same as those listed in Table 3.1. Figure 3.23 shows the computed load-displacement curve of the modified CT specimen, and Figure
3.24 shows the crack profiles at different loading stages. At the applied displacement level a1, the crack initiates at the notch tip and then quickly propagates ahead. In the meantime, the specimen experiences a quick load drop.
Figure 3.23 Load-displacement curve and variation of the maximum plastic strain for the modified CT specimen with two holes.
Figure 3.24 (a5) shows the crack phase field contour corresponding to point a5 in
Figure 3.23. It can be seen that the crack initiation site is at the midplane. The crack is soon deviated towards the top hole and stops after reaching the hole as the applied displacement is increased to level a2. During crack arrest, the load gradually increases as the applied displacement increases. At the applied displacement level a3, the crack initiates again at the surface of the top hole and starts to propagate to the left side, and consequently, the load drops rapidly thereafter. Figure 3.24 (a4) shows the final crack path.
68
Figure 3.24 Crack phase field contours at different loading stages indicated in Figure 3.23 for the modified CT specimen with two holes.
Figure 3.25 Load-displacement curve and variation of the maximum plastic strain for the 2 CT specimen. Here 퐺c = 0.2 mJ/mm .
69
Using the same material parameters, the CT specimen shown in Figure 3.22 (b) is simulated and the results are shown in Figure 3.25 and Figure 3.26. In this case, the crack initiates at the notch tip and propagates straightly to the left side.
Figure 3.26 Crack phase field contours at different loading stages indicated in Figure 3.25 2 for the CT specimen. Here 퐺c = 0.2 mJ/mm .
From Figure 3.23 and Figure 3.25 it is seen that the maximum plastic strain stays at almost zero until crack initiation, representing the brittle fracture mechanism. To simulate ductile fracture of the CT specimen, a higher value of critical energy release rate, 퐺c =
1.2 mJ/mm2, is adopted while all other material properties remain the same as those listed in Table 3.1. Figure 3.27 shows the computed load-displacement curve and variation of the maximum plastic strain for the CT specimen. Figure 3.28 shows the crack phase field contours at different loading stages for the CT specimen. It is seen from Figure 3.27 that the maximum plastic strain has exceeded the critical value of 휀f = 0.1 when crack initiates. 70
Figure 3.28 (a3) shows the tunneling effect of the crack front, which is similar to previous ductile fracture simulation results such as [88] and experimental observations such as [89].
Figure 3.27 Load-displacement curve and variation of the maximum plastic strain for the 2 CT specimen. Here 퐺c = 1.2 mJ/mm .
Figure 3.28 Crack phase field contours at different loading stages for the CT specimen. 2 Here 퐺c = 1.2 mJ/mm .
71
Figure 3.29 Crack phase field and stress contours the applied displacement level a2 as indicated in Figure 3.27. (a) Crack phase field contour, (b) stress component normal to 2 the crack plane, (c) hydrostatic stress. Here 퐺c = 1.2 mJ/mm is used in the computation.
72
It is very important to make sure that the phase field method correctly predicts the stress distribution around the crack tip. Figure 3.29 (b) and (c) show the opening stress and hydrostatic stress distributions on the mid-plane of the specimen corresponding to the applied displacement level a2 as indicated in Figure 3.27, while Figure 3.29(a) show the crack tip location at this load level. As shown in Figure 3.29, the stress contours at the crack tip predicted by the phase field method display the same features as those presented in previous papers and fracture mechanics textbooks.
3.6 Summary and Conclusions
A phase field model is proposed that can simulate both brittle and ductile fracture.
The new developments include the introduction of a degradation function to the yield surface and the modification of the crack driving force function by including the plastic contribution. As the phase field value increases, the yield surface is degraded at the same rate as the elastic modulus is, which maintains the integrity of the elasto-plastic constitutive equations. Parameters in the modified crack driving force function include the critical energy release rate and a plastic adjustment factor, which is an exponential function of the plastic strain. The conjoint effect of the plastic adjustment function and the value of the critical energy release rate on the crack driving force reflects the competition between the brittle and ductile fracture mechanisms. A numerical algorithm is proposed to implement the plasticity model with phase field and solve the fully coupled system equations monolithically. A strategy using the crack driving force increment to control the size of solution increment is shown to be computationally efficient to assure solution accuracy.
Various numerical examples are carried out to demonstrate the capability of the proposed
73 phase field model and to illustrate the influences of model parameters on the simulation results. The following summarizes the main findings:
• The numerical algorithm requires sufficiently small increment size to ensure solution
accuracy. The increment size can be controlled by setting a limit on the maximum
increment of the crack driving force.
• The critical energy release rate, 퐺c, is an important model parameter. Small 퐺c values
lead to brittle fracture while large 퐺c values result in more plastic deformation and
ductile fracture. The 퐺c value also influences the crack path and the load carrying
capacity of the specimen.
• The plastic adjustment function is controlled by two parameters 훼 and 휀f, and plays an
important role when 퐺c is large and the material experience plastic deformation before
fracture. A larger 훼 leads to earlier crack initiation and lower ductility.
• With other parameters kept the same, the simulations show that an increase in yield
stress results in a change of fracture mode from ductile to brittle.
• Numerical examples of the smooth dog bone specimen, notched dog bone specimen,
compact tension (CT) specimen and modified CT specimen show that the proposed
phase field model is capable of predicting crack initiation and propagation and capturing
detailed features of crack path and crack front profile
.
74
CHAPTER IV
PHASE FIELD MODELING OF HYDROGEN EMBRITTLEMENT
In this chapter, the phase field mode developed in chapter III is used to model hydrogen embrittlement. The modeling including: considering the effect of the phase field variable, modifying the governing equation of hydrogen diffusion, proposing a new relationship between the hydrogen trapping density and the plastic strain. Hydrogen embrittlement modeling considers two mechanisms. The HELP mechanism is modeled by decreasing the local flow stress with hydrogen concentration, and the HEDE mechanism is modeled by reducing the critical energy release rate with hydrogen concentration. A compact tension specimen and a flat specimen with a double notch are used to demonstrate the numerical model, and the simulation results are presented and discussed.
4.1 Hydrogen Transport Coupled with Displacement and Phase Field
The hydrogen diffusion and trapping in metals has been briefly explained in chapter
II. The hydrogen transport equation coupled with phase field model is derived step by step in this section. Consider a domain Ω with surface ∂Ω, the local balance equation of total hydrogen is
∫ (퐶L̇ + 퐶Ṫ )d푉 + ∫ J ∙ 퐧d푆 = 0 (4. 1) 훺 휕훺 where 퐧 is the outward unit normal of the domain surface and J is the hydrogen diffusion flux
75
Equation 2.1 and 2.2 described the two different groups of dissolved hydrogen, lattice hydrogen and trapping hydrogen, and the equilibrium relationship between the two groups. Due to the fact that the solubility of hydrogen atoms in steel is very low, so that the occupancy of NILS 휃L ≪ 1, thus Equation 2.2 can be rewrite as
휃T = 휃L퐾T, (4. 2) 1−휃T where 퐾T = exp(푊B⁄푅훩) . From Equation 2.1 and Equation 4.2 , we can derive the trapping hydrogen concentration expression as
훽 푁 퐶 = T T . (4. 3) T 훽 푁 1 + L L 퐾T퐶L
Taking the partial derivative of trapping hydrogen with respect to the lattice hydrogen get
휕퐶 퐶 (1 − 휃 ) T = T T . (4. 4) 휕퐶L 퐶L
And the partial derivative of trapping hydrogen with respect to the effective plastic strain get
휕퐶T 휕퐶T d푁T d푁T = = 휃T . (4. 5) 휕휀p 휕푁T d휀p d휀p
Taking the time derivative of trapped hydrogen is
휕퐶T 휕퐶T 퐶Ṫ = 퐶L̇ + 휀ṗ . (4. 6) 휕퐶L 휕휀p
Substitute Equation 4.4 and 4.5 into Equation 4.6 . The time derivative of trapping hydrogen in Equation 4.6 can be rewritten as
퐶T(1 − 휃T) d푁T 퐶Ṫ = 퐶L̇ + 휃T 휀ṗ . (4. 7) 퐶L d휀p
76
The total hydrogen flux J is defined as
J = −푀 퐶 ∇휇 − 푀 퐶 ∇휇 , (4. 8) L L L T T T where 푀L and 푀T are the mobility of lattice and trapped hydrogen respectively, 휇L and 휇T respectively indicate the chemical potential of the lattice and trapped hydrogen. It is assumed that trapping sites are isolated and transport between traps is by lattice diffusion
[90]. Thus, the mobility of trapped hydrogen MT is assumed to be zero. Moreover, the mobility of the hydrogen in lattice sites ML is assumed to degrade with the occurrence of facture. With these assumptions, the total hydrogen diffusion flux J becomes
2 J = −(1 − 푑) 푀L퐶L∇휇L . (4. 9)
Here the degradation function is incorporated into the hydrogen diffusion flux function, which means that when the material is fully broken (푑 = 1), there is no hydrogen diffusion along or cross the newly developed crack surfaces.
The chemical potential of the lattice hydrogen is
0 퐶L 휇L=휇L + 푅훩ln ( ) − 푉H휎h , (4. 10) 푁L
0 where 휎h = (휎11 + 휎22 + 휎33)⁄3, 휇L is the initial chemical potential, substitute Equation
4.10 into Equation 4.9, then we can derive the hydrogen flux
퐷̅ 퐶 푉 J= − 퐷̅ ∇퐶 + L L H ∇휎 . (4. 11) L L 푅훩 h
2 with the effective lattice diffusion coefficient 퐷̅L = (1 − 푑) 푀L푅훩 . Note that the hydrogen diffusion coefficient in Equation 2.3 is 퐷L=푀L푅훩, so the relation between two
2 diffusion coefficients is 퐷̅L = (1 − 푑) 퐷L. Now the Equation 4.1 can be rewrite as
77
퐶 + 퐶 (1 − 휃 ) d푁 ∫ [ L T T 퐶̇ + 휃 T 휀̇ ] d푉 L T p 훺 퐶L d휀p = 0 . (4. 12) 퐷̅L퐶L푉H + ∫ [−퐷̅L∇퐶L + ∇휎h] ∙ 퐧푑푆 { 휕훺 푅훩 }
Apply the Gauss’s divergence theorem get
퐶 + 퐶 (1 − 휃 ) L T T ̇ ̅ 퐶L − ∇ ∙ (퐷L∇퐶L) 퐶L ∫ ̅ d푉 = 0 . (4. 13) 훺 퐷L퐶L푉H d푁T +∇ ∙ ( ∇휎H) + 휃T 휀ṗ [ 푅훩 d휀p ]
Equation 4.13 is the weak form of the governing equation of the hydrogen diffusion coupled with stress field and phase field. The strong form is
퐶L + 퐶T(1 − 휃T) 퐷̅L퐶L푉H d푁T 퐶L̇ − ∇ ∙ (퐷̅L∇퐶L) + ∇ ∙ ( ∇휎H) + 휃T 휀ṗ = 0 . (4. 14) 퐶L 푅훩 d휀p
The difference between Equation 4.14 and Equation 2.3 is that the hydrogen diffusion coefficient is a function of the hydrogen concentration. The governing Equation 4.14 for hydrogen transport that takes into account the effect of the crack phase field. From
Equation 4.14 we can see that the diffusion of lattice hydrogen is not only affected by the gradient of lattice hydrogen concentration, but also affected by the trapped hydrogen concentration, the gradient of hydrostatic stress, the plastic deformation, and the phase field value.
4.2 Hydrogen trapping
In Equation 2.1, the trapped hydrogen concentration is a function of the number of sites per trap 훽T, the occupancy of the trapping sites 휃T and the trapping density 푁T. Here,
훽T is the material constant, 휃T is a factor for the equilibrium relations between trapping
78 hydrogen and lattice hydrogen. As for the trapping density 푁T, Kumnick and Johnson [91] showed that in iron, it is associated with dislocations developed during plastic deformation.
Sofronis and McMeeking [7] suggested that in BCC iron, the trapping density 푁T increases monotonically with the increased plastic strain and it is independent of temperature. They also assumed that the trapping density saturates as the plastic strain becomes larger than
0.8. According to the experimental observations presented by Kumnick and Johnson [91],
Taha and Sofronis [64] proposed that for iron and steels, the trapping density is a function of the plastic strain, and the relation is described in Equation 2.4.
The experiment study to observe the hydrogen trapping behavior alone the fracture surface during the crack initiation and propagation process has not been done yet. However, atomic simulation [92] suggests that as a crack propagates through a hydrogen-rich region, the hydrogen atoms are trapped along the newly created crack surfaces, and behind the new crack tip hydrogen concentration does not increase. However, in the phase field model, the crack topology is represented in a diffusive manner using the phase field value, d, and the plastic strain in the area of the diffusive crack still increases monotonically after fracture occurred, resulting in a monotonic increase in trapping density there according to Equation
4.15. To overcome this problem, we propose a new trapping density function in an incremental form as follows
p log(푁T) = 23.26 − 2.33 exp(−5.5휀̃ ) , (4. 15) different from Equation 2.4, here the plastic factor 휀̃p is not the effective plastic strain, but a function of the effective plastic strain and the crack phase field value. At time increment
p p 2 p 푛 + 1，the plastic factor is defined as 휀푛̃ +1 = 휀푛̃ + (1 − 푑푛+1) Δ휀̅ .
79
4.3 Hydrogen embrittlement modeling
Various mechanisms contribute to hydrogen induced cracking, among which the
HELP and HEDE mechanisms are incorporated into the phase filed model in this study.
The numerical model of HELP and its effect on ductile fracture has been discussed in
Chapter 2. This section is focusing on the modeling of the HEDE mechanism.
The basic explanation of hydrogen enhanced decohesion is that the hydrogen in metals will reduce the bonding energy between atoms. In this study, phenomenon including the reduction of cohesive strength along the grain boundary [23], the fracture mechanism transition from ductile to brittle [24], and the reduction of the surface energy caused by dissolved hydrogen [25], [26] are all considered as HEDE. Here a phenomenological model is used to account for the HEDE effect, in which the critical energy release rate is assumed to be a decreasing function of the hydrogen concentration
퐺c̅ = 퐺c(푐) , (4. 16)
where 푐 is the total hydrogen concentration. And the embrittlement function 퐺c(푐) is assumed to take the form of
푐 [(휁 − 1) ] 퐺c 퐺c(푐) > 휉퐺c 퐺c(푐) = { 푐0 , (4. 17) 휉퐺c 퐺c(푐) ≤ 휉퐺c where 휁 and 휉 are the parameters controlling the initial and maximum reduction of the critical energy release rate. It is worth pointing out that Equation 4.16 and Equation 2.7 have the same structure but different parameters. Equation 4.16 is not derived from experimental data nor based on the the results from atomic simulation, it is a function phenomenologically describe the fact that the critical energy release rate 퐺c been reduced
80 by hydrogen. With the definition of 퐺c̅ , the crack driving force function as shown in
Equation 3.23 now is
Ψe+ H̃ = . (4. 18) 퐺c̅ ⁄푙 퐴
4.4 Yield function
When only the HELP is considered in chapter II, the yield condition is a function of the hydrogen concentration and effective plastic strain. To incorporate the HELP effect in the phase field model, Equation 3.12 should be combined with Equation 2.7. Therefore, the yield condition becomes a function of the hydrogen concentration, the phase field value and the effective plastic strain
p p 푓(휎̅, 푐, 푑, 휀̅ ) = 휎̅(푑) − 휎y(푐, 푑, 휀̅ ) . (4. 19)
The proposed form of the yield function in this study is
p p 푓(휎̅, 푐, 푑, 휀̅ ) = 푔(푑)휎̅ − 푠(푐)푔(푑)휎y(휀̅ )， (4. 20) where 푔(푑) is the degradation function as defined in Equation 3.12. And 푠(푐) is the softening function which is similar to the Equation 2.7
푐 [(휗 − 1) ] + 1 s(푐) > 휂 푠(푐) = { 푐0 , (4. 21) 휂 푠(푐) ≤ 휂
4.5 Numerical implementation
The user subroutine UEL for a commercial finite element software ABAQUS developed in Chapter III is adopted to implement the modified phase field model, with the phase field value treated as the degrees of freedom #11. The hydrogen diffusion governing
81
Equation is solved using an ABAQUS user subroutine UMATHT as described in Chapter
II. A common block is used to pass variables between UEL and UMATHT.
The HELP effect is included in the phase field model by taking into account the hydrogen effect on the yield stress as described in Equations 4.20 and 4.21. The HEDE effect is included in the phase filed model by varying the critical energy release rate according to Equations 4.16 and 4.17 . The results in Chapter II suggest that HELP embrittles ductile materials by accelerating the growth of micro-voids, and results in
Chapter III suggest that lowering the critical energy release rate reduces material’s strength as well as ductility. In the present hydrogen embrittlement model, both HELP and HEDE are accounted for. To demonstrate the proposed model, numerical simulations of a compact tension (CT) specimen and a double notched flat specimen are conducted for four cases: 1) no HELP or HEDE effect, 2) only HELP effect, 3) only HEDE effect, and 4) with both
HELP and HEDE effects. Both specimens are under displacement-control. The loading speeds are set to be slow enough to ensure sufficient time for lattice hydrogen diffusion so that the hydrogen concentration field remains steady state during the loading process.
Assume a uniformly initial hydrogen distribution in both specimens with the initial value
12 3 CL = 2.084×10 atoms/mm , and the trapped hydrogen concentration CT is calculated according to the equilibrium relation. The “no flux boundary condition” is imposed on the exterior surfaces of both specimens. Hydrogen redistribution in the specimen is driven by the varying stress and deformation fields. The material’s mechanical properties and parameters for hydrogen diffusion are listed in Table 4.1 [55] [21]. The numerical results of the CT specimen are presented and discussed first in the next section, followed by the results of the double notched specimen.
82
Table 4.1 Material parameters for the study of hydrogen embrittlement
Properties Values Units
Young’s modulus 퐸 = 200 GPa
Possion’s ratio 휐 = 0.3 -
Yield stress 휎푦 = 500 MPa
Hardening modulus ℎ = 100 MPa
2 Critical energy release rate 퐺c = 20 mJ/mm
Fracture strain 휀f = 0.05 -
Coefficient 훂 α = 8 -
2 Diffusion coefficient 퐷L = 0.0127 mm /s
3 Molar volume of iron 푉M = 7160 mm /mol
19 3 Lattice site density 푁L = 8.46 × 10 atoms/mm
3 Molar volume of hydrogen in solid solution 푉H = 2000 mm /mol
Binding energy 푊B = 60 kJ/mol
Number of NILS per solvent atom 훽L = 6 -
Number of sites per trap 훽T = 1 -
83
4.6 Compact Tension Specimen
Figure 4.1 (a) Dimensions (in mm) of a CT specimen. (b) Finite element mesh of the CT specimen.
In this section, simulations of a CT specimen are conducted and the numerical results are discussed. Figure 4.1 (a) shows the dimensions (in mm) of the CT specimen having a thickness of 2.5 mm. Figure 4.1 (b) displays the finite element mesh showing refined mesh in the region where fracture is expected to occur. Results of hydrogen diffusion and hydrogen trapping of the case with no HELP or HEDE effect are presented and discussed first, followed by the discussion of the effects of HELP and HEDE on hydrogen embrittlement.
4.6.1 Lattice Hydrogen Diffusion and Hydrogen Trapping
Figure 4.2 shows the distributions of crack phase field value and lattice hydrogen concentration on the mid-surface of the specimen just before the onset of crack initiation.
The lattice hydrogen concentration is high in the area ahead of the crack tip as a result of high positive hydrostatic stress in this area. As the phase field values reaches one, crack
84 starts to propagate. During crack propagation, the area of high positive hydrostatic stress moves with the crack tip [55]. Figure 4.3 shows the distributions of crack phase field value and lattice hydrogen concentration on the mid-surface of the specimen after some amount of crack propagation. The red color in Figure 4.3 (a) represents the new crack surfaces.
Figure 4.3 (b) shows the area of high lattice hydrogen distribution moves to the new crack tip where positive hydrostatic stress exists. These results are in accord with experimental observations by previous researchers [2], [93].
Figure 4.2 Distributions of crack phase field value (a) and lattice hydrogen concentration (b) on the mid-surface of the specimen before the onset of crack initiation.
85
Figure 4.3 Distributions of rack phase field values (a) and lattice hydrogen concentration (b) on the mid-surface of the specimen after some amount of crack propagation.
Figure 4.4 and Figure 4.5 show the distributions of crack phase field value, trapped hydrogen concentration and total hydrogen concentration prior to and after fracture initiation respectively. Here Figures. 4.4 (a) and 4.5 (a), showing the crack phase field distributions, are the same as Figures. 4.2 (a) and 4.3 (a) respectively, whose purpose is to indicate the crack tip location. Prior to the onset of crack initiation, trapped hydrogen is concentrated in a small region ahead of the crack tip where plastic deformation takes place,
Figure 4.4 (b). After crack starts propagating, more hydrogen is trapped along the newly
86 created crack surfaces as shown in Figure 4.5 (b). This agrees with the atomistic simulation result [92] that trapped hydrogen atoms tend to stay on the newly created crack surfaces.
Figure 4.4 (c) shows the distribution of total hydrogen concentration (lattice hydrogen plus trapped hydrogen) prior to crack initiation, where the hydrogen concentration is the highest at the crack tip region. Figure 4.5 (c) shows the distribution of total hydrogen concentration ahead of the crack tip after some amount of crack propagation.
Note that in Figure 4.5 (c), fractured elements with phase field value equal to one are removed from the picture, which is for the purpose of a better illustration of the total hydrogen concentration in the material. Figure 4.5 (c) indicates that hydrogen keeps transporting to and being trapped in the area ahead of the growing crack tip.
It is worth mentioning that by comparing the crack tip hydrogen concentrations shown in Figures. 4.4 (c) and 4.5 (c), it is noticed that the total hydrogen concentration at the crack tip region decreases as crack propagates. This is because the total amount of hydrogen is fixed in the simulation (no flux boundary condition on the exterior surfaces) and more hydrogen is trapped along the newly created crack surfaces as crack propagates.
This result agrees with the atomistic simulation result by Song and Curtin [37].
The simulation results presented above confirms that the numerical model is capable of capturing the hydrogen diffusion and hydrogen trapping mechanisms and correctly simulating the hydrogen transport phenomenon. Coupling the hydrogen concentration with the HELP and HEDE mechanisms, parameters affecting hydrogen embrittlement can be analyzed and discussed.
87
Figure 4.4 Distributions of crack phase field value (a), trapped hydrogen concentration (b), and total hydrogen concentration (c) on the mid-surface of the specimen prior to fracture initiation.
88
Figure 4.5 Distributions of crack phase field value (a), trapped hydrogen concentration (b), and total hydrogen concentration (c) on the mid-surface of the specimen after some amount of crack propagation.
89
4.6.2 Hydrogen Embrittlement Mechanisms
To demonstrate the influence of HELP and HEDE on hydrogen embrittlement, four cases, 1) no HELP or HEDE effect, 2) only HELP effect, 3) only HEDE effect, and 4) with both HELP and HEDE effects, are analyzed and compared. Case 1 serves as a baseline for comparison. The HELP effect on the yield stress is defined by Equation 4.17, and the values of 휗 and 휂 are set to be 0.9 and 0.5 respectively. The HEDE effect on the critical energy release rate is defined by Equation 4.21, and the values 휁 and 휉 are also set to be
0.9 and 0.5 respectively. Figure 4.6 compares the load-displacement curves (solid lines) for the four cases. The variations of the maximum plastic strain during the loading history
(dashed lines) for the four cases are also shown in Figure 4.6.
Figure 4.6 Load-displacement curves (solid lines) and maximum plastic strain- displacement curves (dashed lines) for different cases, where the HELP parameters are 휗 = 0.9 and 휂 = 0.5, and the HEDE parameters are 휁 = 0.9 and 휉 = 0.5.
90
Comparing the result of case 2, which includes only the HELP effect, to the baseline case, where neither HELP nor HEDE effect is considered, the results show that HELP reduces the load carrying capacity of the specimen, i.e., it reduces the strength and fracture toughness of the material. Moreover, HELP leads to more plastic deformation in the specimen as reflected by a higher value of plastic strain at the same applied displacement in the maximum plastic strain-displacement curve. These results agree with previous findings of the HELP effect on ductile fracture [13][21]. Comparing the result of case 3, which includes only the HEDE effect, with the baseline case, it is seen that the HEDE effect also lowers the strength and ductility of the material, however there is no noticeable difference between the maximum plastic strain-displacement curves of the two cases before crack initiation. With both HELP and HEDE mechanisms included in the simulation, case
4 exhibits the lowest load carrying capacity.
Figure 4.7 Load-displacement curves (solid lines) and maximum plastic strain- displacement curves (dashed lines) obtained with different HELP parameters.
91
To further discuss the HELP effect, simulations are conducted with different values of parameter 휗. Figure 4.7 compares the results obtained from three cases: (휗 = 0.9; 휂 =
0.5), (휗 = 0.8; 휂 = 0.5), and the baseline case. The sudden drop of the load-displacement curve indicates the onset of crack propagation. The red dots on the dashed lines represent the maximum plastic strain values when the crack start to propagate. Figure 4.7 shows that with only HELP effect, the load carrying capacity decreases as the value of 휗 decreases. In addition, the maximum value of plastic strain at fracture initiation is higher when the HELP effect is considered, and it increases as the value of 휗 decreases.
Figure 4.8 Load-displacement curves (solid lines) and maximum plastic strain- displacement curves (dashed lines) obtained with different HEDE parameters.
Similarly, to further discuss the HEDE effect, simulations are conducted with different values of parameter 휁. Figure 4.8 compares the results obtained from three cases:
(휁 = 0.9; 휉 = 0.5), (휁 = 0.8; 휉 = 0.5), and the baseline case. With only HEDE effect, the
92 load carrying capacity decreases as the value of 휁 decreases. In addition, the maximum value of plastic strain at fracture initiation is lower when the HEDE effect is considered, and it decreases as the value of 휁 decreases.
Figure 4.9 Distributions of crack phase field value (a1 – a4), total hydrogen concentration (b1 – b4), and hydrogen embrittlement effects (c1 – c4) on the mid-surface of the specimen at different loading stages (number 1 – 4 in each row).
Figure 4.9 compares the hydrogen embrittlement effects (HELP and HEDE mechanisms) at different loading stages. The HELP and HEDE effects are evaluated by the values of 휎0(푐)/휎0 and 퐺c(푐)/퐺c respectively. The values of 휎0(푐)/휎0 and 퐺c(푐)/퐺c are the same because the values of the parameters used to describe the HELP and HEDE effects are the same (휗 = 휁 = 0.9 and 휂 = 휉 = 0.5). Figures. 4.9 (a1 – a4) show the distribution
93 of the crack phase field value at four different loading stages, with the red color represents the newly created crack. Figures. 4.9 (b1 – b4) show the distribution of the total hydrogen concentration at the four loading stages. The peak hydrogen concentration is at the crack tip and its value is decreasing from loading stage 1 to loading stage 4 because of the no hydrogen diffusion flux boundary condition and more hydrogen being trapped along the newly created crack surfaces. As a result of the decrease of the total hydrogen, the embrittlement effect at the crack tip is weakening as the loading process continues, Figures.
4.9 (c1 – c4).
The strongest embrittlement effect takes place at the crack tip before crack starts to initiate, Figure 4.9 (c1), where the values of 휎0(푐)/휎0 and 퐺c(푐)/퐺c are the lowest (~ 0.86).
At this stage, the embrittlement effects are controlled by the accumulated hydrogen (lattice hydrogen and trapped hydrogen) at the crack tip as well as the embrittlement parameters 휗 and 휁. As loading continues and crack propagates, the values of 휎0(푐)/휎0 and 퐺c(푐)/퐺c gradually increase to close to the initial value of 0.9 due to diminished hydrogen accumulation at the crack tip. From this stage on, the embrittlement effects are controlled only by the embrittlement parameters 휗 and 휁.
Results presented above suggest that the proposed phase field model has the ability to predict hydrogen embrittlement resulted from the HELP and HEDE mechanisms. HELP promotes the localization of plastic strain and accelerates material failure. HEDE reduces the critical energy release rate and facilitates crack propagation. The simulation results are in qualitative agreement with the previous investigations [13][21][55].
94
4.7 Double Notched Flat Specimen
In this section, simulations of a flat specimen with double notches are conducted, and the results are presented to further validate the proposed numerical model. Figure 4.10 (a) shows the geometry of the double notched specimen having a thickness of 3 mm, and
Figure 4.10 (b) shows the finite element mesh. The material properties and hydrogen diffusion parameters are the same as those listed in Table 4.1, expect for a higher critical
3 energy release rate, 퐺c = 60 mJ/mm , being used in simulations conducted in this section.
Four cases, 1) no HELP or HEDE effect, 2) only HELP effect, 3) only HEDE effect, and
4) with both HELP and HEDE effects, are analyzed and compared.
Figure 4.10 (a) Dimensions of a flat specimen with double notches (in mm); (b) finite element mesh.
95
4.7.1 Lattice Hydrogen Diffusion and Hydrogen Trapping
Figure 4.11 Distributions of crack phase field value (a1, a2), the lattice hydrogen(b1, b2), the trapped hydrogen (c1, c2), and the total hydrogen (d1, d2) on the mid-surface of the specimen before (a1, b1, c1, d1) and after (a2, b2, c2, d2) the onset of crack initiation.
96
The results of hydrogen diffusion and hydrogen trapping for the baseline the case1
(no HELP or HEDE effect) are displayed in Figure 4.11. Pictures a1, b1, c1 and d1 show the distributions of the crack phase field value, the lattice hydrogen, the trapped hydrogen and the total hydrogen prior to crack propagation respectively. Pictures a2, b2, c2 and d2 show the corresponding contours after some amount of crack propagation. Similar to the results of presented in the previous section for the CT specimen, the lattice hydrogen accumulates at the notch tips prior to crack propagation, Figure 4.11 (b1), and the new crack tips after crack propagation, Figure 4.11 (b2). More hydrogen is trapped at the notch tips before crack propagation, Figure 4.11 (c1), and the trapped hydrogen remains on the newly created crack surfaces after crack propagation, Figure 4.11 (c2). The total hydrogen concentration is highest at the notch tips before fracture, Figure 4.11 (d1), and at the new crack tips after crack propagation, Figure 4.11 (d2). The peak value of total hydrogen concentration reduces as the crack propagates.
4.7.2 Hydrogen Embrittlement Mechanisms
The load-displacement curves (solid lines) and the variations of the maximum plastic strain curves (dashed lines) during the loading process of the four cases are plotted in
Figure 4.12, where reductions in strength and ductility caused by the HELP and HEDE effects can be observed. One difference that worth mentioning is that, in Figure 4.12, the load-displacement curve of case 2 is lower and the sudden drop occurs earlier than case 3, while Figure 4.6 shows an opposite trend for the CT specimen. In Figure 4.12, the displacements at which the load curves drop rapidly are marked by two vertical dashed lines labeled as “loading stage 1” and “loading stage 2” for case 2 (with HELP effect) and case 3 (with HEDE effect) respectively. This result suggests that, for the double notched
97 specimen, the HELP effect on the strength and ductility is stronger than the HEDE effect, while it is opposite for the CT specimen. This difference may result from not only the geometrical disparity of the notch tips in two specimens but also the higher critical energy release rate used for the simulations of a flat specimen with a double notch.
Figure 4.12 Load-displacement curves (solid lines) and maximum plastic strain- displacement curves (dashed lines) for different cases, where the HELP parameters are 휗 = 0.9 and 휂 = 0.5, and the HEDE parameters are 휁 = 0.9 and 휉 = 0.5.
To analyze how the HELP mechanism embrittles the material, Figure 4.13 compares the simulation results for case 1 (no HELP or HEDE) and case 2 (with HELP) at the loading stage 1 indicated in Figure 4.12. The HELP effect results in higher plastic strain at the notch tip, Figure 4.13 (a1, a2), and higher plastic strain results in more trapped hydrogen,
Figure 4.13 (b1, b2), therefore, leading to higher total hydrogen concentration at the notch tip area, Figure 4.13 (c1, c2). The accumulated hydrogen at the notch tip further promotes
98 local plastic straining in this area, which in turn facilitates more hydrogen accumulation.
As a consequence of this effect, crack initiation occurs faster for case 2 than case 1, Figure
4.13 (d1, d2).
Figure 4.13 Distributions of the effective plastic strain (a1, a2), the trapped hydrogen(b1, b2), the total hydrogen(c1, c2) and the crack phase field value (d1, d2) on the mid-surface of the specimen for case 1 (a1, b1, c1, d1) and case 2 (a2, b3, c2, d2) at the loading stage 1 indicated in Figure 4.12.
99
Figure 4.14 Distributions of the effective plastic strain (a1, a2), the trapped hydrogen(b1, b2), the total hydrogen (c1, c2) and the crack phase field value (d1, d2) on the mid- surface of the specimen for case 1 (a1, b1, c1, d1) and case 3 (a2, b3, c2, d2) at the loading stage 2 indicated in Figure 4.12.
Figure 4.14 examines the HEDE effect by comparing the simulation results for case
1 (no HELP or HEDE) and case 3 (with HEDE) at the loading stage 2 indicated in Figure
4.12. As shown in Figure 4.14 (a1, a2, b1, b2), the HEDE mechanism also facilitates plastic straining and promotes hydrogen trapping at the notch tip. But comparing to Figure 4.13
100
(a1, a2, b1, b2), the effect of HEDE on plastic straining and hydrogen trapping is not as significant as the effect of HELP. Thus, the difference of the total hydrogen concentration in Figure 4.14 (c1) and Figure 4.14 (c2) is hardly noticeable. However, as a result of the reduction of the critical energy release rate caused by the HEDE effect, the crack initiation speed is still faster than the baseline case, Figure 4.14 (d1, d2).
Results in Figure 4.13 and Figure 4.14 suggest that the HELP mechanism embrittles the material by promoting local plastic deformation, and it has a strong effect on the hydrogen distribution at the notch tip. On the other hand, the HEDE mechanism does not show a strong influence on the hydrogen redistribution prior to crack initiation, but it embrittles the material by reducing the critical energy release rate.
4.8 Conclusions
This study presents a numerical model for hydrogen embrittlement at the macroscopic scale. A previously proposed phase field method is extended by incorporating hydrogen transport and the resulting HELP and HEDE effects into the governing equations.
The equation controlling hydrogen diffusion through lattice sites is modified to include the effect of the crack phase field. A new trapping density function is proposed so that the trapped hydrogen remains on the newly created crack surfaces but the concentration does not increase during crack propagation. The HELP effect is modeled by reducing the yield stress with the presence of hydrogen and the HEDE effect is modeled by reducing the critical energy release rate with hydrogen concentration. A compact tension specimen and a flat specimen with a double notch are used to demonstrate the numerical model. Results of a series of numerical simulations suggest that:
101
• Hydrogen atoms tend to accumulate at the crack/notch tip region since the positive
hydrostatic stress in this area promotes lattice hydrogen cumulation and the plastic
deformation in this area generates more traps.
• HELP promotes the localization of plastic strain and accelerates material failure. The
load carrying capacity of the specimen decreases as the value of the HELP parameter 휗
decreases.
• HEDE reduces the critical energy release rate and facilitates crack propagation. The
load carrying capacity of the specimen decreases as the value of the HEDE parameter 휁
decreases.
• The proposed numerical model, which combines the HELP and HEDE effects, can
comprehensively simulate hydrogen embrittlement and predict the transition from
ductile to brittle caused by the presence of hydrogen.
102
CHAPTER V
LOADING SPEED DEPENDENCE
5.1 Introduction
A comprehensive hydrogen embrittlement numerical model was developed and implemented into finite element method in Chapter IV. Simulations were conducted to demonstrate the capability of the proposed model. However, in all the simulation cases in
Chapter IV, the loading speed is limited at a low level to ensure enough diffusion time of the dissolved hydrogen. The purpose of the slow loading speed is to eliminate the model’s loading speed dependence. Therefore, in this chapter, more simulation results under different loading speeds are presented and the loading speed dependence of the proposed hydrogen embrittlement model is discussed.
The conclusions in Chapter IV suggested that the accumulated hydrogen at the notch tip plays a critical role in hydrogen embrittlement processes. Thus, weather the simulation duration is sufficient to allow hydrogen diffuses to the notch tip will obviously influence the hydrogen embrittlement effect. Krom et al. [94] demonstrated that hydrogen diffusion and accumulation is depending on the strain rate. Wang et al. [95] studied the crosshead speed dependence of the notch tensile strength with the presence of hydrogen. Results show that more hydrogen accumulates to the notch tip and the notch tensile strength decreases with decreasing crosshead speed. Experiment study [24] on a tensile test found that the lower test speed led to more significant hydrogen embrittlement effect.
103
This chapter first demonstrated the loading speed dependence of the proposed hydrogen embrittlement model by comparing the load-displacement curves at various loading speeds. Then, detailed analysis of hydrogen redistribution and fracture processes influenced by different loading speeds was presented. Finally, the discussion and conclusion are outlined to describe how the material strength and ductility is influenced by the loading speed via hydrogen redistribution.
5.2 Numerical simulations procedure and results
To demonstrate the loading speed sensitivity of the hydrogen embrittlement model developed in Chapter IV, simulations of the compact tension specimen as show in Figure
4.1 with different loading speeds were conducted. Both the HELP and the HEDE effect are considered in the simulations. The material properties are the same as shown in Table 4.1, and the parameters for the HELP effect are 휗 = 0.9 and 휂 = 0.5, paramters for the HEDE effect are 휁 = 0.9 and 휉 = 0.5. Same uniformly distributed initial hydrogen concentration as applied in Chapter IV is imposed on the specimen. With the fastest loading speed 푣 =
0.00125 mm⁄s, 4 cases with different loading speeds, 푣, 푣/10, 푣/100 and 푣/1000 are conducted.
Figure 5.1 shows the load-displacement curves of the cases with 4 different loading- speeds. The vertical dash line indicates the loading stage at which the fracture of the case with slowest loading stage starts to initiate. The results show that specimens with faster loading speed showed higher strength and ductility. And both the strength and ductility are gradually decreasing with slower loading speeds. At loading stage 1 as indicated in Figure
5.1, the distributions of lattice hydrogen accumulated at the notch tip are shown in Figure
5.2. Figures 5.2 (a1 – a4) show the hydrostatic stresses at the notch tip from loading speed
104
푣 to 푣/1000 respectively. The hydrostatic stress value is slightly lower as the loading speed decreases, but the difference is not significant. Figures 5.2 (b1 – b4) show the lattice hydrogen concentrations from loading speed 푣 to 푣/1000 respectively. Contrary to the decreasing trend of hydrostatic stress with slower loading speed, the lattice hydrogen concentration accumulated at the notch tip is increasing. The distributions of the plastic strain and the trapped hydrogen concentration at loading stage 1 are shown in Figure 5.3, where pictures a1 – a4 are the plastic strains from loading speed 푣 to 푣/1000 respectively, pictures b1 – b4 are the trapped hydrogen concentrations from loading speed 푣 to 푣/1000 respectively. Both the plastic strain and the trapped hydrogen are slightly increasing as the loading speed decreases. Similar to the hydrostatic stresses, the various of the plastic strains and the trapped hydrogens with different loading speeds are barely noticeable. However, the total hydrogen concentrations as shown in Figure 5.4 are increasing as the loading speed decreases.
Figure 5.1 Load-displacement curves for different cases, where the HELP parameters are 휗 = 0.9 and 휂 = 0.5, and the HEDE parameters are 휁 = 0.9 and 휉 = 0.5.
105
Figure 5.2 Distributions of the hydrostatic stress (a1 to a4 representing loading speeds 푣 to 푣/1000), the lattice hydrogen concentration (b1 to b4 representing loading speeds 푣 to 푣/1000).
Figure 5.3 Distributions of the plastic strain (a1 to a4 representing loading speeds 푣 to 푣/1000), the trapped hydrogen concentration (b1 to b4 representing loading speeds 푣 to 푣/1000).
106
Figure 5.4 Distributions of the total hydrogen concentration (a1 to a4 representing loading speeds 푣 to 푣/1000),
Above contour results indicate that at loading stage 1, there is almost no noticeable difference of the hydrostatic stresses and the plastic strains at different applied loading speeds, so as the hydrogen trapped by the plastic strain. However, the lattice hydrogen concentration at the notch tip shows significant incremental trend as loading speed slows.
One thing that worth mentioning is that when the loading speed is 푣, the lattice hydrogen concentration at the notch tip is lower than the other area where is not influenced by the stresses and plastic strain. This is because the lattice hydrogen in this area is trapped by the plastic strain created new trapping sites and the fast loading speed allows no sufficient time for lattice hydrogen diffuses into this area. The total hydrogen concentration at the notch tip increases as the loading speed decreases. This increasing trend of the total hydrogen concentration caused more reduction of the material’s ductility and strength with slower loading speed as shown in Figure 5.1.
5.3 Conclusions
This chapter demonstrated loading speed sensitivity of the hydrogen embrittlement model proposed in Chapter IV. The model showed the ability to capture the hydrogen diffusion effected by the loading speed. And the mechanical properties that effected by the
107 loading speed via hydrogen diffusion is also captured by the model. Simulation results indicates that:
• Faster loading speed allows less time for hydrogen diffusion which leads to less
hydrogen accumulates to the notch/crack tip.
• The difference of total hydrogen concentration is mainly cause by the lattice hydrogen
due to the lattice diffusion is strongly influenced by loading speed.
• Prior to crack, the trapped hydrogen showed no significant difference with difference
because the trapping sites are created by plastic strain, and the plastic strain is not
strongly affected by the loading speed before fracture.
• Faster loading speed diminishes the effect of hydrogen embrittlement, and increases the
material strength and ductility.
108
CHAPTER VI
CONCLUSION AND FUTURE WORK
6.1 Conclusions
This dissertation first presented a study of the effect of HELP on material ductility.
The hydrogen diffusion process and the HELP effect are incorporated into a finite element program. A series of finite element analyses of a representative material volume subjected to various stress states were carried out. The simulation results indicate that, the HELP effect accelerates void growth and coalesce by promoting plastic strain localization. And the localized plastic strain traps more hydrogen in this area which will further soften the material. Furthermore, the HELP effect on ductile fracture is influenced by the stress state, as characterized by the triaxiality and Lode parameter. Despite the complex effect of the stress triaxiality and the Lode parameter, the HELP mechanism tends to have a stronger effect on ductile failure strain when the void growth rate is higher.
To comprehensively model hydrogen embrittlement, a phase field model is developed for the simulation of both brittle and ductile fracture. A numerical algorithm uses the finite element method to solve the coupled displacement field and phase field is proposed. The new developed phase field model incorporated plastic contribution into the function of crack driving force, assumed a higher critical energy release rate for ductile fracture and assumed yield surface is degraded by the increasing phase field value. Various
109 numerical examples ware carried out to demonstrate the capability of the proposed phase field model and to illustrate the influences of model parameters on the simulation results.
The following summarizes the main findings: The proposed phase field model is capable of predicting crack initiation and propagation and capturing detailed features of crack path and crack front profile; The proposed numerical algorithm requires sufficiently small increment size to ensure solution accuracy. The increment size can be controlled by setting a limit on the maximum increment of the crack driving force; The small critical energy release rate values lead to brittle fracture while large values result in more plastic deformation and ductile fracture.
A comprehensive hydrogen embrittlement model at the macroscopic scale is proposed by modified the phase field model developed in Chapter IV. The phase field method is extended by incorporating hydrogen transport and the HELP mechanism as discussed in Chapter II, and the HEDE effects by the reduction of critical energy release rate caused by the presence of hydrogen. The equation controlling hydrogen diffusion through lattice sites is modified to include the effect of the crack phase field. A new trapping density function is proposed. Results of a series of numerical simulations suggest that the proposed hydrogen embrittlement model is capable of predict the hydrogen diffusion and embrittlement mechanisms which including: Hydrogen atoms tend to accumulate at the crack/notch tip region which is similar to previous studies; The HELP promotes the localization of plastic strain and accelerates material failure; The HEDE reduces the critical energy release rate and facilitates crack propagation. The proposed numerical model, which combines the HELP and HEDE effects, can comprehensively simulate hydrogen embrittlement and predict the transition from ductile to brittle caused
110 by the presence of hydrogen. Simulations also showed the loading speed dependence of the proposed hydrogen embrittlement model.
6.2 Future works
The phase field model for simulations of ductile and brittle fracture was developed in Chapter III, but there are several improvements that needed further study in the future.
First, the finite element implementation of the developed phase field model solves the displacement field implicitly and solves the phase field explicitly. This is due to the tangent modulus of the phase field residual with respect of displacements is hard to derive.
Therefore, future study could focus on the derivation of the tangent modulus of the phase field residual with respect to the displacement. and develop a fully implicit finite element procedure. Furthermore, a fully explicit finite element procedure should also be considered to overcome the stress instability during the fracture process. Another advantage of a fully explicit implementation is that, it does not need tangent modulus to solve the system matrix which can bypass the complex derivation of the tangent modulus.
The trapping density as discussed in Chapter IV is assumed to be a function of plastic strain which neglect the other trapping sites at grain boundaries and precipitates etc.
Therefore, future research is needed to develop a universal trapping density function considers the grain boundaries, precipitates and fracture etc. as trapping sites. The implementation of the hydrogen embrittlement model is via user subroutine UEL and
UMATHT. This method requires a common block to pass variables between UEL and
UMATHT and makes the displacement field, phase field, hydrogen field are not fully coupled. Thus, to eliminate this effect, a fully coupled implicit implementation via UEL or an explicit implementation via VUEL is needed in the future.
111
The general capability and the loading speed sensitivity of the hydrogen embrittlement model was demonstrated in Chapter IV and V respectively. Future parameter study such as traps binding energy, initial yield stress, hardening modulus and the embrittlement parameters etc. should be considered to further validate the proposed model.
112
BIBLIOGRAPHY
[1] W. H. Johnson, “On some remarkable changes produced in iron and steel by the action of hydrogen and acids,” Nature, vol. 11, no. 281, p. 393, Jan. 1875.
[2] R. P. Frohmberg, W. J. Barnett, and A. R. Troiano, “Delayed Failure and Hydrogen Embrittlement in Steel WADC Technical Report 54-320,” 1954.
[3] H. H. Johnson, J. G. Morlet, and A. R. Troiano, “Hydrogen, crack initiation, and delayed failure in steel,” Trans. Met. Soc. AIME, vol. 212, 1958.
[4] L. S. DARKEN and R. P. SMITH, “Behavior of Hydrogen in Steel During and After Immersion in Acid,” CORROSION, vol. 5, no. 1, pp. 1–16, Jan. 1949.
[5] A. Mcnabb and P. K. Foster, “A new analysis of the diffusion of hydrogen in iron and ferritic steels,” Trans. Metall. Soc. AIME, vol. 227, pp. 618–627, 1963.
[6] R. . Oriani, “The diffusion and trapping of hydrogen in steel,” Acta Metall., vol. 18, no. 1, pp. 147–157, Jan. 1970.
[7] P. Sofronis and R. M. McMeeking, “Numerical analysis of hydrogen transport near a blunting crack tip,” J. Mech. Phys. Solids, vol. 37, no. 3, pp. 317–350, Jan. 1989.
[8] A. J. Kumnick and H. H. Johnson, “Deep trapping states for hydrogen in deformed iron,” Acta Metall., vol. 28, no. 1, pp. 33–39, Jan. 1980.
[9] J. Lufrano and P. Sofronis, “Enhanced hydrogen concentrations ahead of rounded notches and cracks-competition between plastic strain and hydrostatic stress,” Acta Mater., vol. 46, no. 5, pp. 1519–1526, Mar. 1998.
[10] C. D. Beachem, “A new model for hydrogen-assisted cracking (hydrogen ‘embrittlement’),” Metall. Trans., vol. 3, no. 2, pp. 441–455, Feb. 1972.
[11] H. Matsui, H. Kimura, and S. Moriya, “The effect of hydrogen on the mechanical properties of high purity iron I. Softening and hardening of high purity iron by hydrogen charging during tensile deformation,” Mater. Sci. Eng., vol. 40, no. 2, pp. 207–216, Oct. 1979.
113
[12] S. Moriya, H. Matsui, and H. Kimura, “The effect of hydrogen on the mechanical properties of high purity iron II. Effect of quenched-in hydrogen below room temperature,” Mater. Sci. Eng., vol. 40, no. 2, pp. 217–225, Oct. 1979.
[13] H. K. Birnbaum and P. Sofronis, “Hydrogen-enhanced localized plasticity-a mechanism for hydrogen-related fracture,” Mater. Sci. Eng. A, vol. 176, no. 1–2, pp. 191–202, Mar. 1994.
[14] P. J. Ferreira, I. M. Robertson, and H. K. Birnbaum, “Hydrogen effects on the interaction between dislocations,” Acta Mater., vol. 46, no. 5, pp. 1749–1757, Mar. 1998.
[15] I. M. Robertson, “The effect of hydrogen on dislocation dynamics,” Eng. Fract. Mech., vol. 64, no. 5, pp. 649–673, Nov. 1999.
[16] O. Barrera et al., “Understanding and mitigating hydrogen embrittlement of steels: a review of experimental, modelling and design progress from atomistic to continuum,” Journal of Materials Science, vol. 53, no. 9. Springer US, pp. 6251– 6290, 06-May-2018.
[17] T. Tabata and H. K. Birnbaum, “Direct observations of the effect of hydrogen on the behavior of dislocations in iron,” Scr. Metall., vol. 17, no. 7, pp. 947–950, Jul. 1983.
[18] P. Sofronis, Y. Liang, and N. Aravas, “Hydrogen induced shear localization of the plastic flow in metals and alloys,” Eur. J. Mech. A/Solids, vol. 20, no. 6, pp. 857– 872, Nov. 2001.
[19] Y. Liang, P. Sofronis, and N. Aravas, “On the effect of hydrogen on plastic instabilities in metals,” Acta Mater., vol. 51, no. 9, pp. 2717–2730, May 2003.
[20] D. C. Ahn, P. Sofronis, and R. H. Dodds, “On hydrogen-induced plastic flow localization during void growth and coalescence,” Int. J. Hydrogen Energy, vol. 32, no. 16, pp. 3734–3742, Nov. 2007.
[21] C. Huang, T. Luo, X. Gao, and S. M. Graham, “Modeling the Effect of Hydrogen on Ductile Fracture,” Mater. Perform. Charact., vol. 7, no. 2, pp. 101–118, 2018.
[22] T. Luo, C. Huang, and X. Gao, “An investigation of the effect of hydrogen on ductile fracture using a unit cell model,” Int. J. Hydrogen Energy, Mar. 2019.
[23] R. A. Oriani and P. H. Josephic, “Equilibrium aspects of hydrogen-induced cracking of steels,” Acta Metall., vol. 22, no. 9, pp. 1065–1074, Sep. 1974.
114
[24] T. Depover, E. Wallaert, and K. Verbeken, “Fractographic analysis of the role of hydrogen diffusion on the hydrogen embrittlement susceptibility of DP steel,” Mater. Sci. Eng. A, vol. 649, pp. 201–208, Jan. 2016.
[25] N. J. Petch, “The lowering of fracture-stress due to surface adsorption,” Philos. Mag., vol. 1, no. 4, pp. 331–337, Apr. 1956.
[26] D. Tromans, “On surface energy and the hydrogen embrittlement of iron and steels,” Acta Metall. Mater., vol. 42, no. 6, pp. 2043–2049, Jun. 1994.
[27] TROIANO and A. R., “The Role of Hydrogen and Other Interstitials in the Mechanical Behavior of Metals,” Trans. ASM, vol. 52, pp. 54–80, 1960.
[28] R. A. Oriani, “A mechanistic theory of hydrogen embrittlement of steels,” Berichte der Bunsengesellschaft für Phys. Chemie, vol. 76, no. 8, pp. 848–857, Aug. 1972.
[29] W. W. Gerberich and Y. T. Chen, “Hydrogen-controlled cracking-An approach to threshold stress intensity,” Metall. Trans. A, vol. 6, no. 2, pp. 271–278, Feb. 1975.
[30] H. Vehoff and W. Rothe, “Gaseous hydrogen embrittlement in FeSi- and Ni-single crystals,” Acta Metall., vol. 31, no. 11, pp. 1781–1793, Nov. 1983.
[31] W. W. Gerberich, P. G. Marsh, and J. W. Hoehn, “Hydrogen Induced Cracking Mechanisms - Are There Critical Experiments?,” in Hydrogen Effects in Materials, Hoboken, NJ, USA: John Wiley & Sons, Inc., 2013, pp. 539–554.
[32] D. E. Jiang and E. A. Carter, “First principles assessment of ideal fracture energies of materials with mobile impurities: implications for hydrogen embrittlement of metals,” Acta Mater., vol. 52, no. 16, pp. 4801–4807, Sep. 2004.
[33] S. Serebrinsky, E. A. Carter, and M. Ortiz, “A quantum-mechanically informed continuum model of hydrogen embrittlement,” J. Mech. Phys. Solids, vol. 52, no. 10, pp. 2403–2430, Oct. 2004.
[34] S. Wang, M. L. Martin, I. M. Robertson, and P. Sofronis, “Effect of hydrogen environment on the separation of Fe grain boundaries,” Acta Mater., vol. 107, pp. 279–288, Apr. 2016.
[35] D. Tromans, “On surface energy and the hydrogen embrittlement of iron and steels,” Acta Metall. Mater., vol. 42, no. 6, pp. 2043–2049, Jun. 1994.
[36] B. Bourdin, G. A. Francfort, and J. J. Marigo, “Numerical experiments in revisited brittle fracture,” J. Mech. Phys. Solids, vol. 48, no. 4, pp. 797–826, 2000.
115
[37] A. Karma, D. A. Kessler, and H. Levine, “Phase-field model of mode III dynamic fracture,” Phys. Rev. Lett., vol. 87, no. 4, pp. 45501-1-45501–4, 2001.
[38] H. Henry and H. Levine, “Dynamic instabilities of fracture under biaxial strain using a phase field model,” Phys. Rev. Lett., vol. 93, no. 10, 2004.
[39] V. Hakim and A. Karma, “Laws of crack motion and phase-field models of fracture,” J. Mech. Phys. Solids, vol. 57, no. 2, pp. 342–368, 2009.
[40] C. Miehe, F. Welschinger, and M. Hofacker, “Thermodynamically consistent phase- field models of fracture: Variational principles and multi-field FE implementations,” Int. J. Numer. Methods Eng., vol. 83, no. 10, pp. 1273–1311, Sep. 2010.
[41] C. Miehe, M. Hofacker, and F. Welschinger, “A phase field model for rate- independent crack propagation: Robust algorithmic implementation based on operator splits,” Comput. Methods Appl. Mech. Eng., vol. 199, no. 45–48, pp. 2765– 2778, 2010.
[42] M. J. Borden, C. V. Verhoosel, M. A. Scott, T. J. R. Hughes, and C. M. Landis, “A phase-field description of dynamic brittle fracture,” Comput. Methods Appl. Mech. Eng., vol. 217–220, pp. 77–95, Apr. 2012.
[43] M. J. Borden, T. J. R. Hughes, C. M. Landis, and C. V. Verhoosel, “A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework,” Comput. Methods Appl. Mech. Eng., vol. 273, pp. 100–118, 2014.
[44] M. Ambati, T. Gerasimov, and L. De Lorenzis, “A review on phase-field models of brittle fracture and a new fast hybrid formulation,” Comput. Mech., vol. 55, no. 2, pp. 383–405, 2014.
[45] C. Miehe, L. M. Schänzel, and H. Ulmer, “Phase field modeling of fracture in multi- physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids,” Comput. Methods Appl. Mech. Eng., vol. 294, pp. 449–485, Sep. 2015.
[46] G. A. Francfort and J. J. Marigo, “Revisiting brittle fracture as an energy minimization problem,” J. Mech. Phys. Solids, vol. 46, no. 8, pp. 1319–1342, Aug. 1998.
[47] A. A. Griffith, “The Phenomena of Rupture and Flow in Solids,” Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., vol. 221, no. 582–593, pp. 163–198, Jan. 1921.
116
[48] B. Bourdin, G. A. Francfort, and J.-J. Marigo, “The Variational Approach to Fracture,” J. Elast., vol. 91, no. 1–3, pp. 5–148, 2008.
[49] C. Miehe, M. Hofacker, L. M. Schänzel, and F. Aldakheel, “Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic-plastic solids,” Comput. Methods Appl. Mech. Eng., vol. 294, pp. 486–522, Sep. 2015.
[50] M. Ambati, T. Gerasimov, and L. De Lorenzis, “Phase-field modeling of ductile fracture,” Comput. Mech., vol. 55, no. 5, pp. 1017–1040, May 2015.
[51] M. Ambati, R. Kruse, and L. De Lorenzis, “A phase-field model for ductile fracture at finite strains and its experimental verification,” Comput. Mech., vol. 57, no. 1, pp. 149–167, Jan. 2016.
[52] C. Miehe, F. Aldakheel, and A. Raina, “Phase field modeling of ductile fracture at finite strains: A variational gradient-extended plasticity-damage theory,” Int. J. Plast., vol. 84, pp. 1–32, 2016.
[53] M. J. Borden, T. J. R. Hughes, C. M. Landis, A. Anvari, and I. J. Lee, “A phase- field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects,” Comput. Methods Appl. Mech. Eng., vol. 312, pp. 130–166, 2016.
[54] C. Miehe, F. Aldakheel, and S. Teichtmeister, “Phase-field modeling of ductile fracture at finite strains: A robust variational-based numerical implementation of a gradient-extended theory by micromorphic regularization,” Int. J. Numer. Methods Eng., vol. 111, no. 9, pp. 816–863, Aug. 2017.
[55] C. Huang and X. Gao, “Development of a phase field method for modeling brittle and ductile fracture,” Comput. Mater. Sci., vol. 169, p. 109089, Nov. 2019.
[56] J. R. Rice and D. M. Tracey, “On the ductile enlargement of voids in triaxial stress fields,” J. Mech. Phys. Solids, vol. 17, no. 3, pp. 201–217, 1969.
[57] J. Zhou, X. Gao, M. Hayden, and J. A. Joyce, “Modeling the ductile fracture behavior of an aluminum alloy 5083-H116 including the residual stress effect,” Eng. Fract. Mech., vol. 85, pp. 103–116, May 2012.
[58] A. L. Gurson, “Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I—Yield Criteria and Flow Rules for Porous Ductile Media,” J. Eng. Mater. Technol., vol. 99, no. 1, p. 2, Jan. 1977.
117
[59] J. Lemaitre, “A continuum damage mechanics model for ductile fracture,” J. Eng. Mater. Technol., vol. 107, no. 1, pp. 83–89, 1985.
[60] H. Li, M. W. Fu, J. Lu, and H. Yang, “Ductile fracture: Experiments and computations,” Int. J. Plast., vol. 27, no. 2, pp. 147–180, Feb. 2011.
[61] V. Tvergaard and A. Needleman, “Analysis of the cup-cone fracture in a round tensile bar,” Acta Metall., vol. 32, no. 1, pp. 157–169, Jan. 1984.
[62] L. Xue, “Damage accumulation and fracture initiation in uncracked ductile solids subject to triaxial loading,” Int. J. Solids Struct., vol. 44, no. 16, pp. 5163–5181, Aug. 2007.
[63] J. Zhou, X. Gao, J. C. Sobotka, B. A. Webler, and B. V. Cockeram, “On the extension of the Gurson-type porous plasticity models for prediction of ductile fracture under shear-dominated conditions,” Int. J. Solids Struct., vol. 51, no. 18, pp. 3273–3291, Sep. 2014.
[64] A. Taha and P. Sofronis, “A micromechanics approach to the study of hydrogen transport and embrittlement,” Eng. Fract. Mech., vol. 68, no. 6, pp. 803–837, Apr. 2001.
[65] H. Peisl, “Lattice strains due to hydrogen in metals,” in Hydrogen in metals I, Springer, Berlin, Heidelberg, 1978, pp. 53–74.
[66] R. H. Van Stone, T. B. Cox, J. R. Low, and J. A. Psioda, “Microstructural aspects of fracture by dimpled rupture,” Int. Met. Rev., vol. 30, no. 1, pp. 157–180, Jan. 1985.
[67] W. M. Garrison and N. R. Moody, “Ductile fracture,” J. Phys. Chem. Solids, vol. 48, no. 11, pp. 1035–1074, Jan. 1987.
[68] X. Gao and J. Kim, “Modeling of ductile fracture: Significance of void coalescence,” Int. J. Solids Struct., vol. 43, no. 20, pp. 6277–6293, Oct. 2006.
[69] Y. Bai and T. Wierzbicki, “A new model of metal plasticity and fracture with pressure and Lode dependence,” Int. J. Plast., vol. 24, no. 6, pp. 1071–1096, Jun. 2008.
[70] I. Barsoum and J. Faleskog, “Rupture mechanisms in combined tension and shear— Micromechanics,” Int. J. Solids Struct., vol. 44, no. 17, pp. 5481–5498, Aug. 2007.
[71] M. Dunand and D. Mohr, “Effect of Lode parameter on plastic flow localization after proportional loading at low stress triaxialities,” J. Mech. Phys. Solids, vol. 66,
118
pp. 133–153, May 2014.
[72] W. H. Wong and T. F. Guo, “On the energetics of tensile and shear void coalescences,” J. Mech. Phys. Solids, vol. 82, pp. 259–286, Sep. 2015.
[73] S. M. Graham, T. Zhang, X. Gao, and M. Hayden, “Development of a combined tension–torsion experiment for calibration of ductile fracture models under conditions of low triaxiality,” Int. J. Mech. Sci., vol. 54, no. 1, pp. 172–181, Jan. 2012.
[74] J. Kim, X. Gao, and T. S. Srivatsan, “Modeling of void growth in ductile solids: effects of stress triaxiality and initial porosity,” Eng. Fract. Mech., vol. 71, no. 3, pp. 379–400, Feb. 2004.
[75] J. P. Hirth, “Effects of hydrogen on the properties of iron and steel,” Metall. Trans. A, vol. 11, no. 6, pp. 861–890, Jun. 1980.
[76] I. Barsoum and J. Faleskog, “Micromechanical analysis on the influence of the Lode parameter on void growth and coalescence,” Int. J. Solids Struct., vol. 48, no. 6, pp. 925–938, Mar. 2011.
[77] J. R. Rice and J. R. Rice, “The localization of plastic deformation,” W.T. KOITER (ED.), Theor. Appl. Mech., pp. 207--220, 1976.
[78] T. Pardoen and J. . Hutchinson, “An extended model for void growth and coalescence,” J. Mech. Phys. Solids, vol. 48, no. 12, pp. 2467–2512, Dec. 2000.
[79] C. Miehe, M. Hofacker, L.-M. Schänzel, and F. Aldakheel, “Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic–plastic solids,” Comput. Methods Appl. Mech. Eng., vol. 294, pp. 486–522, Sep. 2015.
[80] C. Miehe, “A multi-field incremental variational framework for gradient-extended standard dissipative solids,” J. Mech. Phys. Solids, vol. 59, no. 4, pp. 898–923, Apr. 2011.
[81] T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. 1987.
[82] E. Orowan, “Energy Criteria of Fracture,” Weld. J. Res. Suppl., vol. 34, no. Supplement, pp. 157–160, 1954.
[83] J. W. Kysar, “Energy dissipation mechanisms in ductile fracture,” J. Mech. Phys.
119
Solids, vol. 51, no. 5, pp. 795–824, May 2003.
[84] C. T. Sun and C. Y. Wang, “A new look at energy release rate in fracture mechanics,” Int. J. Fract., vol. 113, no. 4, pp. 295–307, 2002.
[85] H. Ulmer, M. Hofacker, and C. Miehe, “Phase Field Modeling of Brittle and Ductile Fracture,” PAMM, vol. 13, no. 1, pp. 533–536, Dec. 2013.
[86] L. Xue, “Ductile fracture modeling: theory, experimental investigation and numerical verification,” PhD Thesis, p. 251, 2007.
[87] J. Besson, D. Steglich, and W. Brocks, “Modeling of plane strain ductile rupture,” Int. J. Plast., vol. 19, no. 10, pp. 1517–1541, Oct. 2003.
[88] X. Gao and J. Kim, “Modeling of ductile fracture: Significance of void coalescence,” Int. J. Solids Struct., vol. 43, no. 20, pp. 6277–6293, Oct. 2006.
[89] D. Dawicke, M. Sutton, J. Newman, and C. Bigelow, “Measurement and Analysis of Critical CTOA for an Aluminum Alloy Sheet,” Fract. Mech. 25th Vol., pp. 358– 379, 1995.
[90] A. H. M. Krom, “Numerical Modelling of Hydrogen Transport in Steel,” 1998.
[91] A. J. Kumnick and H. H. Johnson, “Deep trapping states for hydrogen in deformed iron,” Acta Metall., vol. 28, no. 1, pp. 33–39, Jan. 1980.
[92] J. Song and W. A. Curtin, “Atomic mechanism and prediction of hydrogen embrittlement in iron,” Nat. Mater., vol. 12, no. 2, pp. 145–151, 2013.
[93] H. Johnson and F. Army, “Hydrogen , Crack Initiation , and Delayed Failure in Steel,” osti.gov, vol. 3, 1956.
[94] A. H. M. Krom, H. J. Maier, R. W. J. Koers, and A. Bakker, “The effect of strain rate on hydrogen distribution in round tensile specimens,” Mater. Sci. Eng. A, vol. 271, no. 1–2, pp. 22–30, Nov. 1999.
[95] M. Wang, E. Akiyama, and K. Tsuzaki, “Crosshead speed dependence of the notch tensile strength of a high strength steel in the presence of hydrogen,” Scr. Mater., vol. 53, no. 6, pp. 713–718, Sep. 2005.
120