Crack Tip Fields Research Articles

Cascading crack bifurcations in soda-lime glass: Quantification of fracture mechanics-based precursors using Digital Gradient Sensing

Dynamic crack growth in high-stiffness and ultralow-toughness amorphous materials such as soda-lime glass (SLG) often involves seemingly unprovoked crack branching events which are yet to be fully explained. The absence of optical tools for performing mechanical field measurements at sufficiently high spatial and temporal resolutions to decipher highly localized deformations at the tip of a crack growing in excess of mile-a-second speed has perpetuated this knowledge gap. The full-field method of Digital Gradient Sensing (DGS) used in conjunction with ultrahigh-speed photography has overcome some of these limitations allowing direct quantification of fracture parameters associated with different phases of crack growth in SLG (Sundaram and Tippur, 2018; Dondeti and Tippur, 2020). In this work, time-resolved stress gradients are measured in SLG plates of two different geometries, first one producing a single crack bifurcation event and the second two-tier cascading crack bifurcations, during dynamic wedge-loading experiments. The measurements are then used along with the asymptotic crack-tip fields to extract fracture parameters and identify crack branching precursors based on crack velocity, stress intensity factors, and higher order coefficients. The fracture surface roughness and other features are also separately quantified via high-resolution post-mortem examination to corroborate them with the optically measured quantities.

Essential work of fracture of soft elastomers

Stress concentration occurs at the crack tip of a notched sample when loaded. The crack tip region has been recognized of consisting of two regions: the inner (or end) region, which is also called the fracture process zone, and the surrounding outer region where inelastic deformation occurs. The size of these regions is strongly related to the fatigue threshold and the toughness of the material, and has been extensively studied both experimentally and numerically. Here, we study the crack tip fields of highly stretchable elastomers by adopting the concept of essential work of fracture. In the experiments, double deep edge notched tension (DENT) specimens are loaded to rupture, during which the loading curves are recorded. By changing the ligament length, two length scales around the crack tip are identified: one is the size of the fracture process zone, and the other is that of the inelastic zone. Their experimental measurements are comparable with the fractocohesive length and fracto-dissipative length of the materials, respectively. Moreover, we find that the essential work of fracture weof a highly stretchable elastomer, obtained by extrapolating the specific work of fracture wf to zero ligament length, resembles the fatigue thresholdTth obtained by a separate fatigue test. Furthermore, we compare the fatigue threshold with those predicted by Lake-Thomas model and Wells model. The present work links the essential work of fracture to the fatigue threshold, and elucidates the physics embedded in the crack tip fields of highly stretchable materials.

Effect of creep-fatigue loading condition on crack tip fields of grade 91 steel at crack initiation and growth

In this paper, the effect of creep-fatigue loading condition on crack tip deformation and stress fields of Grade 91 steel at 600 °C at crack initiation and growth is studied. By using finite element debond analysis method, creep-fatigue crack growth tests using C(T) specimens under various load ratios and hold times are simulated, and the effect of creep-fatigue load ratio and hold time on the crack opening profile, equivalent stress, crack-opening stress and triaxiality at crack initiation and growth is presented. It is found that the crack tip deformation and stress fields under tension–compression creep-fatigue loading are overall different from those under tension-tension creep-fatigue and pure creep loading, due to re-sharpening of the crack tip.

Effect of plastic deformation on the elastic stress field near a crack tip under small-scale yielding conditions: An extended Irwin's model

The effective stress intensity factor derived from Irwin’s model is greater than the linear elastic counterpart. This is in contradiction to the shielding effect of crack-tip plastic zone. Besides, the classic Irwin’s model underestimates the size of the crack-tip plastic zone compared with Dugdale’s model and FE analysis. In this paper, an extended Irwin’s model for elastic perfectly-plastic material under small-scale yielding conditions is developed through imposing static equivalence between the plasticity-affected stress field and linear elastic stress field in the plasticity-affected zone. The extended Irwin’s model is capable of quantifying the effect of the crack-tip plastic zone on the crack-tip elastic stress field. It reduces to the classic Irwin’s model in the limiting case. This new version of Irwin’s model provides new insights in elucidating the effect of the crack-tip plastic zone on fracture behavior.

A unified perspective of incremental J-integrals by the generalized Griffith framework

The novel point of this paper is that the Griffith framework of elastic materials is explained from two different perspectives, which have produced two statements of the generalized Griffith framework suitable for crack growth in elastic–plastic materials. The two statements have encompassed all of these incremental J-integrals. Four aspects of the crack growth in elastic–plastic materials are explored: 1) an explanation is given on why the unique overall energy rate balance has resulted in two different incremental J-integrals. By generalizing Griffith’s idea into the elastic–plastic materials, this paper has proposed two perspectives of the unique power balance on crack propagation: one is based on the Helmholtz free energy, and the other is based on the internal energy. Two expressions of crack driving force are then derived, from which common incremental J-integrals can be directly obtained. 2) A new viewpoint is provided to explain the weaker singularity of growing crack-tip fields in elastic–plastic materials, starting from the physical meaning of the crack driving force. This weaker singularity is attributed to the discontinuous distribution of plastic strain from the plastic unloading behind the crack tip. 3) The physical meanings of finite-contour incremental J-integrals are presented. Interestingly, only the J-integral based on Helmholtz free energy represents the driving force on a finite contour. 4) To tackle the dilemma that the theoretical analyses of cracked bodies considering fracture process zone (FPZ) are very complex, we have proposed a conventional continuum model of such a cracked body, which has transferred this body to a traditional bi-material model.

Characterizations of material constraint effect for creep crack in center weldment under biaxial loading

Material mismatch effect on the cracking behavior is an important topic for those welding structures. Characterization of the material constraint effect based on rigorously asymptotic solution is studied in this paper. Based on decomposition of the second order term, the constraint effect characterization parameter is decomposed as material constraint parameter and geometry constraint parameter. In general, the total constraint level for crack tip under undermatch condition is higher than overmatch condition. The specimen with positive biaxiality could lead to a higher constraint level compared with that of negative biaxiality. Geometry constraint effect and material constraint effect could not be separated independently from rigorously asymptotic solution for those cases with positive biaxiality. For a crack tip field under non-positive biaxiality, the material constraint effect can be characterized independently although it is approximate. For these conditions, the proposed material constraint effect and geometry constraint effected characterized are approximately independent on material mismatch factor, crack depth ratio and stress biaxiality. An empirical formula has been presented to characterize the geometry constraint effect and material constraint effect for the crack tip in the weldment under biaxial loading, which has been verified with fine accuracy.

Effects of specimen type and thickness on elastic and elastic–plastic fracture parameters for I-II and I-III mixed mode cracks

This paper focuses on the effects of specimen type and thickness on the fracture parameters for I-II and I-III mixed mode cracks. Results show that mode III and mode II components emerge simultaneously near free surfaces of specimens under I-II and I-III mixed mode loadings, which is called coupling effect. For specimen under I-II mixed mode loading, the level of coupling effect is independent of specimen geometry. While for specimen under I-III mixed mode loading, the coupling effect is much affected by specimen geometry. Thicker middle cracked specimen has smaller coupling effect than thinner single edge notched specimen. Due to the non-negligible coupling effect on crack tip field and fracture behavior, a new mixity parameter taking the coupling effect into consideration is proposed to describe the mixed mode crack. Furthermore, mode I component is the main reason that constraint of I-II or I-III mixed mode crack exists, which is related with loading mode like tensile opening loading and bending opening loading. Specimen under tensile opening loading like middle cracked specimen is in a lower constraint than that under bending opening loading like single edge cracked specimen. Results also shows that the effect of specimen type on constraint is greater than that of thickness.

Characteristics and implications of material mismatch on mode II creep crack tip field: Theoretical analysis and numerical investigation

The mismatch effect in weldment is widely to be found in engineering practices. In this paper, the material mismatch effect on the mode II creep crack tip field is investigated and discussed. The mismatch effect on the C(t)-integral is presented. Both the local mismatch and the general mismatch are found to play important role in C(t)-integral under mode II condition. The mismatch effect on the stress field of mode II creep crack is also studied. The two-order term solutions are adopted to characterize the material mismatch effect on the mode II creep crack. The effect of creep coefficient, creep exponent as well as the HAZ thickness on the high order term solutions are also presented. Some typical cases by considering general mismatch and local mismatch effect are given to make a comparisons between the HRR field, FE solutions and the two-order term solutions. It can be seen that the two-order term solutions can coincide with the full field FE solutions quite reasonably regardless of creep extent, creep exponent, mismatch factor, and HAZ thickness. The material mismatch effect on the high order term solutions is significant under some conditions. A theoretical formula is presented to show the implication of material mismatch effect on creep crack growth under mode II condition based on stress damage model.

Fracture of 28 buckled two-dimensional hexagonal sheets

Mode-I stress intensity factors are estimated for 28 buckled two-dimensional hexagonal materials, including 6 mono-elemental (silicene, indiene, blue phosphorene, arsenene, antimonene and bismuthene) and 22 binary (CS, CSe, CTe, SiO, SiS, SiSe, SiTe, SiGe, GeO, GeS, GeSe, GeTe, SnO, SnS, SnSe, SnTe, SnGe, SnSi, InAs, InSb, GaAs and AlSb) two-dimensional materials. The crack-tip displacement field revealed from linear elastic fracture mechanics is adopted to find the stress intensity factor. Atomic-scale finite element method with Stillinger-Weber potentials is used to simulate the tensile tests. Mode-I stress intensity factors of these 28 two-dimensional materials appear ≤ 0.8 which are very small in comparison with boronitrene and graphene. Most of them exhibit their fracture toughness below 0.5 Our findings are helpful in the design of nano-devices with these two-dimensional materials.

Modeling Dislocation-Mediated Hydrogen Transport and Trapping in Face-Centered Cubic Metals

Abstract The diffusion of hydrogen in metals is of interest due to the deleterious influence of hydrogen on material ductility and fracture resistance. It is becoming increasingly clear that hydrogen transport couples significantly with dislocation activity. In this work, we use a coupled diffusion-crystal plasticity model to incorporate hydrogen transport associated with dislocation sweeping and pipe diffusion in addition to standard lattice diffusion. Moreover, we consider generation of vacancies via plastic deformation and stabilization of vacancies via trapping of hydrogen. The proposed hydrogen transport model is implemented in a physically based crystal viscoplasticity framework to model the interaction of dislocation substructure and hydrogen migration. In this study, focus is placed on hydrogen transport and trapping within the intense deformation field of a crack tip plastic zone. We discuss the implications of the model results in terms of constitutive relations that incorporate hydrogen effects on crack tip field behavior and enable exploration of hydrogen embrittlement mechanisms.

Effects of multiple families of nonlinear fibers on finite deformation near a crack tip in a neo-Hookean sheet

In this paper, we investigate the asymptotic crack tip fields in a neo-Hookean sheet reinforced by two families of nonlinear fibers, where the fibers are characterized by the standard reinforcing model. In the asymptotic analysis, the fibers with stronger stiffening compared to the matrix dominate the mechanical behavior at the crack tip, which simplifies the analysis. A hodograph transformation is used to solve for the leading and higher order eigenmodes for the nonlinear eigenvalue problem derived from the model. Asymptotic path-independent J-integrals are constructed to evaluate the leading order parameters separately. The asymptotic crack tip fields agree well with finite element results for different combinations of fiber orientation angles and modulus ratios. We find that the peak value of stress for the case with two families of fibers decreases significantly compared to the case for a single family of fibers.

Biaxial Loading Effects on Strain Energy Release Rate and Crack-Tip Strain Field in Elastic Hydrogels

We characterized the strain energy release rate and local crack-tip strain field for elastic hydrogels under several types of biaxial loading: planar, unequal biaxial, and equibiaxial extensions. The strain energy release rate (G) in each type of biaxial strain is characterized via two different approaches. One evaluates G from the change in strain energy with respect to the notch length (Xc) using single-edge notched specimens with various Xc values. The other estimates G from the difference in strain energy density between the two regions far behind and ahead of the crack tip. The excellent agreement between the G values obtained by the two methods demonstrated their reliability. We also revealed the features of the local crack-tip strain field in biaxial loading, including crack-tip opening displacement, strain field area, strain magnitude resulting from crack opening, and strain singularity exponent for strain growth near the crack tip. Importantly, these properties are governed exclusively by G, independent of the biaxial loading type. The results provide an important basis for a comprehensive understanding of the fracture behavior of elastic soft materials under complicated deformations.

Explicit edge-based smoothed numerical manifold method for transient dynamic modeling of two-dimensional stationary cracks

Modeling and prediction of dynamic behaviors of cracked bodies are vital to the safety evaluation of structures. The numerical manifold method (NMM) has achieved considerable success in crack analysis as it provides a powerful representation of complex discontinuities. Strain smoothing has been incorporated into enriched methods such as NMM to improve their accuracy, efficiency, and tolerance to mesh distortion. However, conventional strain smoothing cannot reproduce the large spatial variance in the strain surrounding crack tips. Consequently, the accuracy is reduced when it is applied to crack-tip enrichment bases. High-fidelity representation of crack-tip fields can be achieved by crack-tip enrichment. Therefore, it is desirable to combine compatible strain for the enriched areas with edge-based smoothed strain for the remaining areas. When explicit time integration is employed, it is beneficial to use a lumped mass matrix because it provides a larger critical time step and a straightforward solution of the linear algebraic equation. A local-approximation-based mass lumping strategy is used to obtain a block-diagonal mass matrix with small blocks. Finally, the proposed methodology is applied to various static and dynamic crack analyses, and highly accurate and stable results are obtained.

Hodograph transformation for crack-tip fields in hyperelastic sheets: higher order eigenmodes and asymptotic path-independent integrals

Hodograph transformations can be used to linearize a nonlinear partial differential equation by judicious use of physical quantities (e.g. velocities or displacement gradients) as coordinate variables in the hodograph plane. This approach has been found useful for obtaining the leading order terms of eigenproblems that govern asymptotic singular crack fields in nonlinear materials. There is little work on the use of the hodograph transformation for obtaining higher order terms in the asymptotic expansion of the crack tip fields. In this paper, we develop a framework to obtain such higher order terms using the hodograph transformation. The method relies heavily on the representation of physical quantities of interest in terms of hodograph plane variables. We demonstrate the method via application to a generalized neo-Hookean material. In addition, asymptotic path-independent J-integrals are expressed in terms of either physical or hodograph variables and are used to compute the leading-order amplitude coefficients. A relationship between the asymptotic J-integrals and the energy release rate is established for a mixed crack mode. The asymptotic results are compared with numerical results from finite element computation and excellent agreement is obtained.

Extracting fracture mechanics parameters (higher-order coefficients of Williams series expansion) from FEM analysis and digital photoelasticity method

The paper describes new algorithms and procedures proposed for determining fracture mechanics parameters from finite element analysis using the over deterministic method. The multi-parameter crack tip stress field description is used. The algorithms and procedures based on multi-parameter stress field representations in series form are shown to be a powerful tool for reliable and accurate parameter determination. The technique is aimed at the determination of coefficients of the Williams series expansion from finite element analysis and is based on the over deterministic approach. The methodology is illustrated and applied to several cases of cracked specimens. Examples are presented for crack-tip fields recorded using digital photoelasticity. The results of finite element analysis are compared with the digital photoelasticity experiments. The results are in good agreement. The principal stresses obtained from finite element method are in good agreement with the isochromatic fringe patterns obtained by the photoelasticity method. Explanation has been made for giving guidance to a user on how best to approach implementation of the method from a practical standpoint.

Atomistic aspects of the temperature effect on fracture toughness of a silicon single crystal

The temperature effect on the fracture toughness of a Si single crystal was explored through molecular dynamics simulations over the temperature range of 1–1000 K. A sharp crack model with the (010)[001] (plane)[front] crystallographic system was considered, in which the interatomic force was characterized using the modified embedded atom method potential. The computational experiment of a mode-I fracture was carried out employing the displacement based on the mode-I near tip field, and the fracture toughness was evaluated using the critical stress intensity factor for the onset of crack growth. The results showed that the fracture toughness in the temperature range of 1–750 K slightly decreased with increasing temperature, where the effect of lattice trapping was reduced at the elevated temperature. However, when the temperature was higher than 800 K, the fracture toughness increased resulting from the nucleation of nanovoids in front of the crack tip. Detailed atomistic analysis showed that, in the small-scale yielding limit, the temperature effect on the hydrostatic component of the local crack tip field was important to crack-tip plasticity.

On the crack-tip stress field due to the presence of isotropic dilatational inclusion: theoretical and numerical analysis

In this paper, the stress field at the crack-tip of a semi-infinite crack interacting with an isotropic dilatational inclusion is studied. The fundamental solution for a semi-infinite crack interacting with an eigenstrained point is presented first. By employing it as Green’s function, a crack enclosed by a homogenous inclusion is formulated and its Kolosov–Muskhelishvili complex potentials are obtained. Then the approximate solutions for the full-field stress in the vicinity of the crack-tip are derived. Finally, numerical method is performed, which not only reveals the distribution of the stress field, but also validates the theoretical analysis. As expected, theoretical and numerical results show that the presence of dilatational inclusion will produce a significant shielding effect on crack-tip. Importantly, it is found that a stress jump occurs at the interface between inclusion and matrix, which may unexpectedly trigger a micro-damage prior to the semi-infinite crack-tip. This mechanism is capable of leading to the crack growth fundamentally, when it is exposed to an oxidizing or corrosive environment.

Estimation of the creep crack-tip stress field considering the effect of threshold stress

Because of the presence of second-phase particles or nanoscale particles, a term of threshold stress incorporated into the power-law creep model of some materials, such as magnesium, aluminum, and chromium alloys. The estimation of the creep crack-tip stress field for these materials is notably important to facilitate their exploitation in high-temperature structures. In the present work, a two-term asymptotic expansion of the stress field is developed to predict the creep crack-tip stress field for power-law creep materials with threshold stress. The finite element method is used to verify the accuracy of the two-term asymptotic expansion of the stress field. The simulation results reveal that the two-term asymptotic expansion of the stress field is accurate enough as an engineering method in estimating the creep crack-tip stress field. In addition, the existing estimation method of the time-dependent C-integral of C(t) can be directly implemented in power-law creep materials with threshold stress.

On the Topology Update of the Numerical Manifold Method for Multiple Crack Propagation

Crack initiation and propagation are the two key issues of concern in the geotechnical engineering. In this study, the numerical manifold method (NMM) is applied to simulate crack propagation and the topology update of the NMM for multiple crack propagation is studied. The crack-tip asymptotic interpolation function is incorporated into the NMM to increase the accuracy of the crack-tip stress field. In addition, the Mohr-Coulomb criterion with tensile cut off is adopted to be the crack propagation criterion to judge the direction of crack initiation and propagation. Then a crack tip searching method is developed to automatically update the position of the crack tips. The inapplicability of the original loop search method in the NMM is also illustrated and a novel loop search method based on manifold elements is developed for physical loop updating. Moreover, methods for the manifold element updating and physical cover updating are provided. Based on the above study, the developed numerical method is capable to simulate multiple crack propagation. At last, typical rock rupture problems are numerically simulated to manifest the effectiveness of the developed numerical method.

Crack tip fields in a neo-Hookean sheet reinforced by nonlinear fibers

In this paper, we derive the asymptotic crack tip fields in a neo-Hookean material reinforced by nonlinear fibers. The fiber are characterized by a power law model which exhibits significant stiffening behavior at large stretch. A specific case of the model yields the so-called standard reinforcing model (Triantafyllidis and Abeyaratne, 1983). A series of transformations is used to set the stage for the asymptotic solution. We begin with a coordinate scaling (Liu and Moran, 2020b) to account for fiber orientation aspects. This is followed by a hodograph transformation to the strain plane to deal with nonlinearities. Finally, an additional nonlinear algebraic transformation is employed to render the equations suitable for a separable solution. The asymptotic solutions are compared with and agree well with finite element solutions for different fiber modulus ratios, fiber orientation angles, and loading modes. We find that, due to the stiffening effects of the fibers, the deformation fields can be divided into three regions, where the deformation in the two regions near to the crack faces is significantly larger than in the middle region. Stress peaks are observed at the interfaces between regions. The solutions in the paper may provide insight into damage modes and crack initiation at a crack tip in fiber-reinforced soft composites.