Tip Fields Research Articles

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.

Strain gauge experimental study on mode I rock fracture characteristics under impact loading

Based on the analysis of the strain field of the dynamic crack tip under impact loading, a specific acute (obtuse) angle multi-point strain gauge method is proposed to study the mode I rock crack dynamic fracture. Compared with the dynamic caustics method, the feasibility of the strain gauge method for calculating the crack propagation velocity and dynamic stress intensity factor (SIF) is verified. The rock specimen mode I crack dynamic fracture behaviour under impact loading was studied by the strain gauge method and assisted by high-speed photography and scanning electron microscopy. The results showed that the mode I crack propagation behaviour calculated by two strain gauge methods under impact loading was close to the caustics experimental results. Compared with the dynamic caustics method, the multi-point strain gauge method could roughly record the crack propagation behaviour of rock under impact loading, which provides an experimental basis for studying non-transparent impact fracture. The rock section mainly experienced tensile stress failure under impact loading, manifested as mode I cracking, and the granite specimen fracture damage occurred in white spot micro-crack propagation. Under impact loading, cracks in both polymethyl methacrylate (PMMA) and granite specimens propagated stably in the middle stage of propagation, the average granite specimen crack propagation velocity was 647.9 m/s, and the average SIF value of was 5.37 MN/m3/2, which were higher than the PMMA specimen experimental results.

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.

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.

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.

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.

Nanoscale Vector Electric Field Imaging Using a Single Electron Spin.

The ability to perform nanoscale electric field imaging of elementary charges at ambient temperatures will have diverse interdisciplinary applications. While the nitrogen-vacancy (NV) center in diamond is capable of high-sensitivity electrometry, demonstrations have so far been limited to macroscopic field features or detection of single charges internal to the diamond itself. In this work, we greatly extend these capabilities by using a shallow NV center to image the electric field of a charged atomic force microscope tip with nanoscale resolution. This is achieved by measuring Stark shifts in the NV spin-resonance due to AC electric fields. We demonstrate a near single-charge sensitivity of ηe = 5.3 charges/√Hz and subelementary charge detection (0.68e). This proof-of-concept experiment provides the motivation for further sensing and imaging of electric fields using NV centers in diamond.

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.

Meshless Numerical Simulation of Singular Fields at Crack Tips of Branched Crack

One of the most important tasks in the numerical analysis of the fracture problem is to achieve a high-precision approximation of the singular stress fields at crack tips. Partially or fully enriched basis functions are often used as enhancements on the whole problem domain or just near the crack tips when constructing the meshless shape functions to simulate the singularity fields at crack tips, namely: global enhancement with fully enriched basis, global enhancement with partially enriched basis, local enhancement with fully enriched basis and local enhancement with partially enriched basis. The scheme of the enriched basis has been extensively and successfully employed in handling problems with a single crack, but its applicability in the branched crack problem remains to be studied. In this paper, the stress intensity factors, the displacement field and the stress field near crack tip are calculated by these four schemes, respectively, and the singular field at the crack tip of a branching crack is also considered. The numerical results show that: for the problems with a single crack, the accuracy of the stress intensity factor obtained by the meshless method with different kinds of enriched basis is basically the same, but the accuracy of the crack tip field variables is obviously different; for the problems with branched crack, the accuracy of the stress intensity factors differs depending on the problems.

New algorithm for optimised fitting of DIC data to crack tip plastic zone using the CJP model

The work presented in this paper provides guidance in optimally determining the appropriate fitting region used in digital image correlation (DIC) when fitting a crack tip field model to DIC data. In this work, the technique is applied to the CJP crack tip field model and uses the Levenberg-Marguardt (L-M) iterative method to solve for the crack tip position. This is combined with an error analysis of the main term of the Williams series expansion for crack tip stresses and the extrapolation of Saint-Venant’s principle to determine the optimum fitting region for use in DIC fit with the multi-parameter solution for the CJP crack tip field model. A comparison is then made between the size and shape of the crack tip plastic zone obtained using the Williams and the CJP models. This demonstrated that as the plastic zone size increased, there was a deterioration in accuracy of the description of the crack tip plastic zone using the Williams series expansion, while the CJP description remains more accurate. In addition, KF, the stress-intensity factor that drives crack growth in the CJP model has a strong linear relationship with the plastic zone area, which indicates that KF is a direct driving force for crack growth and there is a clear relationship between ΔKCJP and ΔKI.

Elastic-plastic solutions for corner and surface cracks emanating from stress concentrators

Three-dimensional (3D) constraint effects for part-through cracks, such as corner and surface cracks, emanating from stress concentrators play important role in damage tolerance design of structures, but have hardly been investigated in the frame of 3D elastic–plastic fracture theory. Here the ability of current elastic–plastic solutions to describe 3D constraints and their effects on crack tip fields of corner and surface cracks emanating from stress concentrators are systematically analyzed using finite element (FE) method. Distribution of the out-of-plane constraint factor Tz and corresponding in-plane constraint parameters QT and AT can quantify the effect of stress concentration on crack tip fields. Good agreement between 3D elastic–plastic solutions (J-Tz, J-Tz-QT, J-Tz-AT) and FE results is obtained for four typical specimens at relatively high and low loading levels, showing that these solutions can serve as the theory foundation for advanced damage tolerant analyses of engineering structures.

On the second order term asymptotic solution for sharp V-notch tip field in elasto-viscoplastic solids

A two-order asymptotic solution for sharp V-notch tip mechanics field in elasto-viscoplastic solids with power-law form under mode I loading is developed. The stress exponents and angular distributions for various notch angles and viscoplastic exponents for the second terms are reported. The stress exponent of the first order is singular and is non-singular except for low creep exponent with small notch opening angle. Compared with finite element results from several V-notched specimens, it is found that the two-order solution can have a better estimate for those cases with small notch opening angles, e.g., α = 30° under n = 5, than that of one-order solution regardless of viscoplastic extent, specimen type and loading level. One-order term solution is demonstrated to be sufficient to characterize the full field of sharp V-notch specimen for those conditions that elastic term is considered into higher order term, e.g., α = 60° under n = 5, in all specimens under various viscoplasticity extents.

Effect of tip injection on the flow characteristics of an aeroengine compressor rotor

Existing experiments, simulations, and analysis have shown that the flow field of the blade tip has a significant effect on rotating stall and acoustic resonance in aeroengine compressors. Based on the numerical simulation of NASA rotor 37, the steady influence of air injection at blade tip was investigated in this investigation. The results show that the injected flow could obviously extend the stability margin of the axial compressor to varying degrees. The tip injection reduces the blockage caused by the tip leakage, and reduces the backflow in the trailing edge area of the blade tip, thereby delaying the occurrence of stall. Furthermore, in this paper, different types of rotors are constructed to verify the effect of injection parameters on the stability extension. The results show that the improvement of rotor stall margin is positively correlated with the injector inlet total pressure, and when the injector inlet flow conditions are of the same magnitude, the improvement of the rotor stall margin is negatively related to the distance between the injector and the leading edge of the blade.

Addressing a lattice of rotatable molecular dipoles with the electric field of an STM tip.

Functional molecular groups mounted on specific foot structures are ideal model systems to study intermolecular interactions, due to the possibility to separate the functionality and the adsorption mechanism. Here, we report on the rotational switching of a thioacetate group mounted on a tripodal tetraphenylmethane (TPM) derivative adsorbed in ordered islands on a Au(111) surface. Using low temperature scanning tunnelling microscopy, individual freestanding molecular groups of the lattice can be switched between two bistable orientations. The functional dependence of this rotational switching on the sample bias and tip-sample distance allows us to model the energy landscape of this molecular group as an electric dipole in the electric field of the tunnelling junction. As expected for the interaction of two dipoles, we found states of neighbouring molecules to be correlated.

Effect of crack curvature radius on the stress fields under nonlinear deformation

The influence of crack curvature radius and plastic properties of material on the crack tip fields is evaluated by conventional mechanism-based strain gradient (CMSG) plasticity theory, and classical plasticity theory. Stress fields in single edge tension specimen are investigated numerically for a wide range of the crack curvature radius variations. The strain hardening exponent varied from N=0.075 to N=0.4. The local areas of the influence of the crack tip curvature radius are established for the strain gradient plasticity and classical plasticity. The couple effects of material length scale parameter and plastic properties of material on crack tip fields are established for strain gradient plasticity. The boundaries of CMSG plasticity dominated zone are determined.

Phase field modelling of fracture and fatigue in Shape Memory Alloys

We present a new phase field framework for modelling fracture and fatigue in Shape Memory Alloys (SMAs). The constitutive model captures the superelastic behaviour of SMAs and damage is driven by the elastic and transformation strain energy densities. We consider both the assumption of a constant fracture energy and the case of a fracture energy dependent on the martensitic volume fraction. The framework is implemented in an implicit time integration scheme, with both monolithic and staggered solution strategies. The potential of this formulation is showcased by modelling a number of paradigmatic problems. First, a boundary layer model is used to examine crack tip fields and compute crack growth resistance curves (R-curves). We show that the model is able to capture the main fracture features associated with SMAs, including the toughening effect associated with stress-induced phase transformation. Insight is gained into the role of temperature, material strength, crack density function and fracture energy homogenisation. Secondly, several 2D and 3D boundary value problems are addressed, demonstrating the capabilities of the model in capturing complex cracking phenomena in SMAs, such as unstable crack growth, mixed-mode fracture or the interaction between several cracks. Finally, the model is extended to fatigue and used to capture crack nucleation and propagation in biomedical stents, a paradigmatic application of nitinol SMAs.

Crack tip fields and fracture resistance parameters based on strain gradient plasticity

The crack tip mechanics of strain gradient plasticity solids is investigated analytically and numerically. A first-order mechanism-based strain gradient (MSG) plasticity theory based on Taylor’s dislocation model is adopted and implemented in the commercial finite element package ANSYS by means of a user subroutine. Two boundary value problems are considered, a single edge tension specimen and a biaxially loaded plate. First, crack tip fields are characterized. Strain gradient effects associated with dislocation hardening mechanisms elevate crack tip stresses relative to conventional plasticity. A parametric study is conducted and differences with conventional plasticity predictions are quantified. Moreover, the asymptotic nature of the crack tip solution is investigated. The numerical results reveal that the singularity order predicted by the first-order MSG theory is equal or higher to that of linear elastic solids. Also, the crack tip field appears not to have a separable solution. Moreover, contrarily to what has been shown in the higher order version of MSG plasticity, the singularity order exhibits sensitivity to the plastic material properties. Secondly, analytical and numerical approaches are employed to formulate novel amplitude factors for strain gradient plasticity. A generalized J-integral is derived and used to characterize a nonlinear amplitude factor. A closed-form equation for the analytical stress intensity factor is obtained. Amplitude factors are also derived by decomposing the numerical solution for the crack tip stress field. Nonlinear amplitude factor solutions are determined across a wide range of values for the material length scale l and the strain hardening exponent N. The domains of strain gradient relevance are identified, setting the basis for the application of first-order MSG plasticity for fracture and damage assessment.

An analytical velocity field of spiral tips in reaction–diffusion systems

Spiral waves are ubiquitous in diverse physical, chemical, and biological systems. The tip (phase singularity) of a spiral wave is considered to represent its organizing center. Here, we derive an analytical velocity field of spiral tips based on the variables of a general two-variable reaction–diffusion (RD) equation. From this velocity field, we can predict the velocities of spiral tips at time t as long as the values of the variables are given at that time. Numerical simulations with two-variable RD models are in quantitative agreement with the analytical results. Furthermore, we also demonstrate the velocity field of spiral tips in the Luo–Rudy model for cardiac excitation.

Crack Tip Field of a Double-Network Gel: Visualization of Covalent Bond Scission through Mechanoradical Polymerization

Quantitative characterization of the energy dissipation zone around a crack tip is the focal point in the fracture mechanics of soft materials. In this report, we present a mechanochemical techniqu...