# Analysis of the Crack Initiation and Growth in Crystalline Materials Using Discrete Dislocations and the Modified Kitagawa–Takahashi Diagram

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## Abstract

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## 1. Introduction

## 2. Crack Growth Analysis

**σ**, work done by the applied stress, and work expended in creating two new surfaces. The total energy reaches a peak, where crack length longer than the critical size

_{y}**a**will expand continuously with the reduction in the total energy, Figure 1b. The energy gradient provides the crack-tip driving force. If the energy to nucleate crystal dislocations from the crack tip is lower than the energy needed for the crack to expand further as an elastic crack, then the crack undergoes plastic relaxation, causing a reduction in the total energy of the system.

_{c}**μ**.

#### 2.1. Discrete Dislocation Models

**μ**). If the frictional stress reaches zero, the emitted dislocations can go to infinity. On the other hand, if the stress is infinite, the crack reduces to an elastic crack. Thus, a range of material behavior can be obtained by changing this relative ratio,

**μ**.

#### 2.2. Continuous Elastic-Plastic Crack

**μ**, the emission of the second dislocation is prevented due to the back stress from the previously emitted dislocation. Hence, the calculations are somewhat tedious and become intensive as the crack grows with changing glide to cleavage components depending on

**μ**and the surface energy of the material. However, the case represents a more realistic situation with the plastic zone accumulating in the wake of the growing crack. The continuous elastic-plastic crack also captures crack growth history. Thus, depending on the

**μ**and

**γ**(surface energy) values, the relative components of glide vs. cleavage components change. The total energy of the incipient crack increases with an increase in the length of the crack until it reaches a peak value. Further increase in crack size only reduces the total energy, resulting in the acceleration of the crack, contributing to the total failure, Figure 2b. The material can harden as the crack grows (thus changing the

**μ**value), thereby altering the energy-crack length curve or contributing to crack growth toughness. Figure 1c shows the log of stress vs. the log of critical crack size (at the peck energy value). Calculations show that in the log–log coordinates, the stress vs. crack length for the continuous elastic-plastic crack follows a straight line but with the slope less that of the elastic Griffith crack, which is 0.5. The slope decreases with a decrease in

**μ**. This is similar to the effect of the decrease of the yield stress of the material. Conversely, as the yield stress increases, the crack growth behavior approaches that of the elastic crack.

**a**value. One can think of an infinitesimal change in the applied stress to keep the energy at the same peak level, without it increasing or decreasing. This process can be continued with a continuous decrease in the applied stress as the crack length slowly increases to maintain the growth in equilibrium. Since the total energy remains constant, such a crack grows at a quasi-steady state. For an elastic crack, the applied stress has to be reduced, maintaining the Griffith stress with the increasing crack length. Hence in the log–log plot, the stress vs. crack length line represents the quasi-steady crack growth condition for continuously decreasing stress. If the stress is higher than the Griffith line, then the growing crack accelerates. On the other hand, if the stress falls below the line, the growing crack is arrested. This forms the condition for the crack arrest of an incipient growing crack due to a sharp decrease of applied or internal stresses that are contributing to the growth of a crack. It also leads to the Kitagawa–Takahashi type of diagram [20], as will be discussed below.

_{c}#### 2.3. Crack Initiation at Pre-Existing Stress Concentrations

_{t}and the notch tip radius, ρ. Figure 3a shows an incipient crack initiated at the notch tip. The stress at the notch tip corresponds to K

_{t}σ but decreases with distance depending on the notch tip radius, ρ, approaching the remote stress, σ. For sharp notches, the stress gradient is sharp, while for blunt notches, the rate of decrease is slower. We have analyzed the growth of a short crack at the notch tip using elastic-plastic fracture mechanics [16]. The results are shown schematically in Figure 3c. The stress intensity factor for the short crack increases sharply from zero, decreases to some minimum, and then increases slowly with a further increase in the crack length. When the short crack length is zero, K for the short crack is also zero. The sharp increase is due to the very high notch tip stresses. Hence the initial sharp increase can be considered as within the process zone or from the point of dislocations within the core region of the notch. The decrease of K as the short crack grows is due to the gradient in the notch tip stress field. Further increase in the K value arises as the crack grows due to the remote applied stress since K increases with the crack length for a given stress. Hence, the depicted behavior of K

_{sc}is expected due to the notch tip stress gradient. It may be noted that for just purely elastic calculation, K

_{sc}monotonically increases and does not show the observed minimum [22].

_{th}for crack growth. Otherwise, the incipient crack that is growing in the high-stress field of the notch is arrested when K

_{sc}drops below K

_{th}. The minimum of the K

_{sc}value is related to the internal stress (notch tip stress) magnitude and its gradient. For very sharp notches (ρ ~ 0), the stress gradient can be sharp, leading to arrest of the growing short crack leading to non-propagating cracks at sharp notches. This is observed, particularly under fatigue, leading to fatigue stress concentration factor, K

_{FC}, differing from the elastic stress concentration factor, K

_{t}. The magnitude of the stress at the notch tip also depends on the applied stress, σ

_{apl}. We have shown [16] that the minimum applied stress needed for the continuous growth of incipient crack near the stress concentration can be expressed as:

_{th}corresponds to the threshold for crack growth. It can be a threshold for any subcritical crack growth (thresholds for fatigue, stress corrosion, corrosion-fatigue, sustained load, or even for a fracture, such as K

_{1C}). K

_{t}and ρ are elastic stress concentration factor and notch-tip root radius, respectively. The equation has been successfully applied to the extensive notch-fatigue data available in the open literature. Recently, the equation has been applied to determine the pit to crack transition under corrosion fatigue [23].

_{t}values but for a fixed ρ, showing how minimums in K

_{sc}values become sharper with increasing K

_{t}value.

#### 2.4. Crack Initiation at the In Situ Generated Stress-Concentrations

#### 2.5. Role of Internal Stresses and the Modified Kitagawa-Takahashi Diagram

^{7}cycles are needed for the crack to initiate and grow. These cycles are required for the development of the needed internal stresses and their gradients for an incipient crack to form and grow. The initiation and growth of the short crack in the endurance have been accounted for by the fracture mechanics community by invoking the similitude break down and proposing that the short crack threshold is different from that of long crack thresholds due to crack closure. We have shown using the dislocation theory that the crack-closure concept is inherently faulty in the plane strain regim e, and no similitude break down is needed to account for the short crack growth behavior. The short crack grows due to the presence of both applied and in situ generated internal stresses arising from inhomogeneous deformations in the polycrystalline materials. The thresholds do not depend on the crack size, and one has to properly account for the local build-up of internal stresses and their gradients due to dislocation pile-ups. A detailed review of short crack growth was provided recently [30].

#### 2.6. Role of Chemical Forces

_{max}and Δσ for Stress vs. number of cycles for failure (S-N fatigue) fatigue or two stress intensity factors (K

_{max}, ΔK) for fatigue crack growth [30]. However, for stress corrosion or sustained load crack growth, there is no subcritical crack growth in an inert environment for a reference state. Only the fracture toughness value in an inert medium provides the reference.

## 3. Experimental Support for the Above Concepts

_{2}(SO)

_{4}. The data are plotted in the form of the Kitagawa–Takahashi diagram. The endurance stress is similar to the minimum failure stress, σ

_{th}, of a smooth specimen loaded in the corrosive environment. The mechanical equivalent of chemical internal stress is defined in the figure. The extent of experimental data on smooth and fracture mechanics specimens in corrosive media available in the open literature is limited. Nevertheless, the analysis shows that transition from short cracks to long cracks and the role of internal stresses in accentuating the crack initiation and growth process are general for all subcritical crack growth processes in materials.

#### 3.1. Application to Fracture Toughness

_{1C}lines. The extension of the K

_{1C}lines defines the internal stresses needed to initiate a crack in a smooth specimen. The internal stress triangle is large for the low yield stress 2024-T3 alloy in comparison to the high yield stress 7075-T6 alloy, as expected.

#### 3.2. Discrete Dislocation Models

**μ**, the ratio of friction stress to applied stress on the growing elastic-plastic cracks is shown. With the decrease of

**μ**, the glide component increases with the cleavage component. Figure 11 shows the slope of the log(applied stress) vs. the log(crack length) decreases with the decrease of

**μ**. Nevertheless, the exponential relation remains with the exponent of σ vs. a decrease from 0.5. Figure 12 shows the total energy of the growing crack as a function of crack size for two

**μ**values, 0.6 and 0.8, based on the calculations reported in ref. [14]. The crack grows along the path that has lower energy. The figure shows that glide and cleavage components fluctuate until the crack becomes unstable. The relative proportions of the two components vary depending on the

**μ**value. Experiments undertaken with varying H concentrations support these results.

#### 3.3. Effect of Hydrogen Pressure

**μ**, the ratio of lattice friction to applied stress, the slope of log(stress) vs. log(crack length) increases as the hydrogen pressure increases, due to the increased cleavage component, and the curve moves closer to the Griffith crack.

#### 3.4. Crack Initiation Ahead of the Main Crack

## 4. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## Symbols Used

ASTM—American Society of Testing and Materials |

KT diagram—Kitagawa Takahashi |

μ—Ratio of frictional stress to applied stress. |

BCS model—Bilby, Cottrell, and Swindon model |

a—Crack Length |

a_{c}—Critical Crack length |

σ_{y}—Normal Stress |

E_{T}—Total Energy of the System |

E_{S}—Self Energy of all dislocations |

E_{I}—Interaction energy of all dislocation |

E_{γ}—Surface energy |

E_{σ}—Work done by applied stress |

τ_{xy}—Lattice frictional stress |

α_{c}—Crack mouth angle |

K_{t}—Elastic Stress Concentration Factor |

ρ—Notch tip radius |

K_{SC}—Stress intensity factor for short crack |

K_{th}—Threshold stress intensity factor for crack growth |

K_{IC}—Fracture Toughness |

K_{ISCC}—Stress Corrosion Crack Growth Threshold |

K_{pl}—Stress intensity factor for short crack in the elastic plastic notch tip field. |

ΔK_{th}—Fatigue crack growth threshold Stress intensity range. |

σ_{apl}—Applied Stress |

σ_{ys}—Yield Stress |

σ_{e}—Endurance Stress |

σ_{FS}—Fracture Stress |

E—Elastic Modulus |

γ—Surface Energy |

ν—Poisson ratio |

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**Figure 1.**Analysis of crack initiation and growth for elastic to elastic-plastic cracks: (

**a**) crack configurations (

**b**) variation of total energy with crack length (

**c**) log of stress vs. log of crack length for elastic and elastic-plastic cracks.

**Figure 2.**Discrete dislocation modeling of a continuously growing elastic-plastic crack. (

**a**) Crack with crack and crystal dislocations with applied and lattice frictional forces. (

**b**) Continuously expanding elastic-plastic crack with inbuilt history. Based on the total energy of the system, the crack can expand elastically or emit crystal dislocations. The crack mouth angle depends on the relative ratio of glide and cleavage planes. (

**c**) Quasi-static growth of the crack under continuously decreasing stress holding the total energy constant after its first nucleation.

**Figure 3.**Crack initiation at a notch tip. (

**a**) The stress field ahead of the notch tip. (

**b**) Crack initiation at grain boundaries ahead of the notch tip due to hydrostatic stresses. (

**c**) The variation of the stress intensity factor K

_{sc}of the incipient crack growing nearing a notch tip.

**Figure 4.**Typical results of actual calculations of K

_{sc}(called K

_{pl}due to elastic-plastic stress fields) for different K

_{t}values but for a fixed ρ, showing the minimum in K

_{sc}.

**Figure 5.**Stress field in the next grain due to dislocation pile-up, from Eshelby, and X-ray data from Gao et al. (2014).

**Figure 6.**(

**a**) Griffith crack representation with yield stress defining the required minimum internal stress magnitude and gradient. (

**b**) Parallel representation of the modified Kitagawa–Takahashi diagram for subcritical crack growth.

**Figure 7.**Role of chemical forces and their quantification. (

**a**) Total energy as a function of crack length and (

**b**) log(stress) vs. log(crack length) plot.

**Figure 8.**Corrosion-fatigue crack growth in alloy steels with varying yield stress, from Usami, 1981.

**Figure 9.**Modified Kitagawa–Takahashi diagram for stress corrosion crack growth in 4340 steel in H2SO4 solution, extracted from Hiroshi–Mura data.

**Figure 10.**Application to K

_{1C}, fracture data. (

**a**) Experimental data from Bucci, 1996. (

**b**) Representation of the data in terms of the modified Kitagawa–Takahashi diagram showing the internal stress triangle for the two cases, 7075-T6 and 2024-T3 Al-alloys.

**Figure 12.**Changes in the relative glide and cleavage components with the change in

**μ**values for continuous elastic-plastic cracks.

**Figure 13.**Effect of hydrogen pressure on crack growth for a given

**μ**ratio. With the increase in H pressure, the lines move closer to the elastic Griffith crack behavior.

**Figure 14.**Experimental results from Vehoff and Roth, 1983, on Fe-3%Si showing the effect of hydrogen pressure on crack growth. (

**a**) With the increase in the hydrogen pressure cot(α) increases (α decreases) due to the increase in the cleavage component. (

**b**) The associated micrograph indicates the crack mouth angle decreases when hydrogen pressure increases.

**Figure 15.**Results of discrete dislocation analysis showing that the stress-intensity factor of an incipient crack shows a saturation effect with H coverage.

**Figure 16.**Experimental results showing the saturation effect with the embrittling species on the crack growth threshold.

**Figure 17.**Discrete dislocation models of growth of an incipient crack from ahead of the main crack for various hydrogen coverages.

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**MDPI and ACS Style**

Sadananda, K.; Adlakha, I.; Solanki, K.N.; Vasudevan, A.K.
Analysis of the Crack Initiation and Growth in Crystalline Materials Using Discrete Dislocations and the Modified Kitagawa–Takahashi Diagram. *Crystals* **2020**, *10*, 358.
https://doi.org/10.3390/cryst10050358

**AMA Style**

Sadananda K, Adlakha I, Solanki KN, Vasudevan AK.
Analysis of the Crack Initiation and Growth in Crystalline Materials Using Discrete Dislocations and the Modified Kitagawa–Takahashi Diagram. *Crystals*. 2020; 10(5):358.
https://doi.org/10.3390/cryst10050358

**Chicago/Turabian Style**

Sadananda, Kuntimaddi, Ilaksh Adlakha, Kiran N. Solanki, and A.K. Vasudevan.
2020. "Analysis of the Crack Initiation and Growth in Crystalline Materials Using Discrete Dislocations and the Modified Kitagawa–Takahashi Diagram" *Crystals* 10, no. 5: 358.
https://doi.org/10.3390/cryst10050358