Crack Problems Research Articles

Realization of adaptive mesh refinement for phase-field model of thermal fracture within the FEniCS framework

This paper presents an adaptive phase field model for simulating fracture due to coupled interactions (such as thermal quenching and hot cracking in additive manufacturing). The proposed model is implemented in an open-source finite element framework, FEniCS. The model considers spatial variations of the fracture toughness and differential coefficients of thermal shrinkage. Several paradigmatic case studies are addressed to demonstrate the potential of the proposed modeling framework. Specifically, we (a) benchmark our crack predictions for mechanical and thermal boundary condition interactions with the results from alternative numerical methods, (b) accurately reproduce experimentally observed complex crack trajectories due to thermal quenching and hot cracking in additive manufacturing, and (c) demonstrate the ease of extending of the proposed framework to thermal cracking problems in three dimensions. The current implementation provides the basic for an efficient framework for fracture problems due to multi-physical interactions for practical engineers with less programming expertise.

Analytical solutions for the plane thermoelastic problem of a nano-open crack in one-dimensional hexagonal quasicrystal non-periodic plane

As a stable material at high temperatures, the presence of cracks and other defects greatly reduces the performance of quasicrystals (QCs). The plane thermoelastic problem of one-dimensional hexagonal quasicrystals (1DHQ) non-periodic plane containing a nano-open crack is analyzed using surface elasticity theory. The closed form solutions are obtained to describe the thermoelastic field corresponding to the nano-crack by the Fourier integral transformation technique. The analytical solutions of the thermal stress intensity factors (TSIFs) and the strain energy density factor (SEDF) at the crack tip are derived explicitly. The influence of the applied loads, the surface effects and the multi-field coupling effects on the TSIFs and SEDF are discussed. The numerical results reveal that the surface effects have an obvious impact on TSIFs with small applied loads, and the surface effects will inhibit the propagation of nano-open cracks. The research results of this study provide some reliable theoretical references for the development of nano-QCs.

An application of Lobatto–Chebyshev method to a nonhomogeneous plane problem with two cracks

In this manuscript, Lobatto–Chebyshev method, which is an effective collocation method, is applied to a system of singular integral equations, which leads from a nonhomogeneous plane problem. It is assumed that there are two cracks in a nonhomogeneous medium. The problem is formulated in the view of the basics of the elasticity theory and boundary conditions of the problem. By using the method of singular integral equation, the problem is converted to a system of first kind Cauchy type singular integral equations. It is aimed to determine the stress intensity factors (SIFs) of the crack problem. It is seen that Lobatto–Chebyshev quadrature has many advantages in determining SIFs. To verify the validity of the method, the obtained results corresponding to the one crack case are compared with the results in literature.

The Multi-Virtual Crack Closure Technique for three-dimensional interface crack problems

The Virtual Crack Closure Technique (VCCT) is a simple method that was found in the past to produce good results for determining stress intensity factors of cracks in homogeneous bodies and poor results for interface cracks. In recent papers, the VCCT was found to produce excellent results for two-dimensional interface cracks where many elements were used in the virtual crack extension (VCE). This approach is called here the Multi-VCCT. A criterion for the optimal number of elements used in the VCE was also found. Here, the M-VCCT is extended to three-dimensional interface crack problems. Results for three-dimensional problems containing a straight through finite length interface crack and a penny-shaped interface crack are presented. The interfaces include that between two homogeneous and isotropic dissimilar materials, as well as two transversely isotropic materials.

Investigation of stress states around a finite-height crack

This paper is concerned with the analysis of a crack that has a finite-height at its crack tip, loaded under Mode I and II loading. The stress field around the “crack-height” region has been analysed and local Williams’ eigensolutions collocated using the H-integral. Results of the analysis help us to predict which one of the three possible places that fracture failure could occur is most likely, i.e., (1) at the middle plane of the crack tip (collinear with the sharp crack problem), (2) at the “notch corner” or (3) at the intermediate plane between the notch corner and the middle plane (intermediate state). A practical example of an infinite plane with a rectangular shaped crack has also been shown to demonstrate the practical use of the results of this paper.

Dynamic fracture behavior for hole-initiated cracks in functionally graded magneto-electro-elastic bi-materials

In this article, the dynamic fracture behavior of hole-initiated cracks in functionally graded magneto-electro-elastic (FGMEE) bi-materials with exponential variation is studied by Green’s function method, and SH-wave is considered as an external load acting on the bi-materials. The mechanical model of the cracks is constructed through interface-conjunction and crack-deviation techniques, thereby simplifying the crack problem to solving a series of the first kind of Fredholm’s integral equations, from which the dynamic stress intensity factor (DSIF) at the crack tips is expressed. Numerical results clarified the factors that affect the DSIF, including wave number, incident angle, the geometry of the defect, and the gradient of the bi-materials. The accuracy of the present methods is examined by comparing the obtained results with the reference solutions. Compared with previous traditional numerical and analytical methods, the methods proposed in this paper are relatively more effective and applicable, and provide a new perspective for studying the fracture problems of FGMEE materials with more complex defects in practical engineering.

A Boundary-Element Analysis of Crack Problems in Multilayered Elastic Media: A Review

Crack problems in multilayered elastic media have attracted extensive attention for years due to their wide applications in both a theoretical analysis and practical industry. The boundary element method (BEM) is widely chosen among various numerical methods to solve the crack problems. Compared to other numerical methods, such as the phase field method (PFM) or the finite element method (FEM), the BEM ensures satisfying accuracy, broad applicability, and satisfactory efficiency. Therefore, this paper reviews the state-of-the-art progress in a boundary-element analysis of the crack problems in multilayered elastic media by concentrating on implementations of the two branches of the BEM: the displacement discontinuity method (DDM) and the direct method (DM). The review shows limitation of the DDM in applicability at first and subsequently reveals the inapplicability of the conventional DM for the crack problems. After that, the review outlines a pre-treatment that makes the DM applicable for the crack problems and presents a DM-based method that solves the crack problems more efficiently than the conventional DM but still more slowly than the DDM. Then, the review highlights a method that combines the DDM and the DM so that it shares both the efficiency of the DDM and broad applicability of the DM after the pre-treatment, making it a promising candidate for an analysis of the crack problems. In addition, the paper presents numerical examples to demonstrate an even faster approximation with the combined method for a thin layer, which is one of the challenges for hydraulic-fracturing simulation. Finally, the review concludes with a comprehensive summary and an outlook for future study.

Nanosecond laser grooving of water-immersed silicon carbide assisted by high intensity focused ultrasound (HIFU)

Silicon carbide (SiC) is a very important semiconductor and ceramic material with many applications. However, its micromachining is difficult and it is still challenging to achieve a good combination of high quality, productivity and low cost. The corresponding author previously proposed the “ultrasound-assisted water-confined laser micromachining” (UWLM) process, a new machining technology utilizing in-situ ultrasound to improve laser machining of a water-covered workpiece. The author's previous studies show that UWLM based upon a high intensity focused ultrasound (HIFU) transducer is able to produce a relatively high machining rate and quality for metals. Thus, it is natural to guess the HIFU-based UWLM process might also possibly help provide a good solution for SiC micromachining. However, SiC have properties very different from metals. An experimental study is critically needed to confirm whether or not HIFU-based UWLM can still produce a high machining rate and quality for SiC, and overcome additional challenge(s) arising (if any). However, such a study is seldom reported to the authors' knowledge. This paper reports a study on the micro grooving of SiC via the HIFU-based UWLM process. With the conditions in this study, it has been found that UWLM can produce a much better apparent groove quality (i.e., much less recast material and/or debris re-deposition) than the laser machining process in the ambient air, and a much higher machining depth per pulse than the laser machining process in water with no ultrasound. The likely fundamental mechanism is the in-situ cleaning effect due to ultrasound-induced cavitation flow and pressure waves in water. The laser machining process in water with or without ultrasound can produce cracks at relatively low pulse repetition rates (PRRs). Time-resolved pressure measurements in water have been performed to help analyze stress sources causing the cracks. The crack problem appears to be effectively solved by using a relatively high PRR of 5 kHz. One possible major reason is expected to be the inter-pulse thermal accumulation effect at the high PRR, improving workpiece fracture toughness and/or reducing temperature gradients.

Efficient healing of existed cracks in cement via synergistic effects of cement matrix activation and monomer polymerization

The effective repair of existing cracks can prolong the service life of infrastructures, which is of great significance for energy saving and emission reduction. A great deal of research on self-healing cement has been undertaken to efficiently prevent possible future cracking problems in buildings, but these methods do not deal with existing cracks. Herein, the efficient healing of existing cracks is achieved by the synergistic reaction of cement matrix activation and organic polymerization of sodium acrylate (ANa). ANa polymerization was systematically investigated for its effect on stimulated calcium silicate hydrate (C-S-H) and true healing systems in this study. The interconnected porous structure of sodium polyacrylate (PANa) observed during study suggested that ANa had been polymerized successfully under room temperature and the denser short rod-like C-S-H was also formed during the polymerization process. The formation of this organic-inorganic hybrid structure facilitates the rapid healing of existing cracks. When applied to cement, results showed that the compressive strength of specimens with the healing agent SS/PANa recovered 119.99% of its original strength and the micro-cracks of approximately 50 μm were completely sealed after 7-day curing. The ANa in the cracks helps enhance the reactivity of matrix and the polymerization improves the binding ability, forming an organic-inorganic composite network of the PANa and C-S-H. The composition and microstructure of the healed area revealed that the healing products were mainly composed of calcite (CaCO3), calcium hydroxide (CH), C-S-H and PANa.

On an annular crack near an arbitrarily graded interface in FGMs

Annular cracks can be generated during the fabrication of materials and around the pre-existing defects. Previous works on the annular crack problems were most limited to homogenous materials. This paper extends the analysis and examines the axisymmetric problem of a flat mixed-mode annular crack near and parallel to an arbitrarily graded interface in functionally graded materials (FGMs). The crack is modelled as plane circular dislocation loop and an efficient solution for dislocation in FGMs is used to calculate the stress field at the crack plane. The three-parts mixed boundary value problem is solved by a series expansion method. Analytical solutions of the stress intensity factors are obtained. Compared with many other existing analytical treatments to the crack problems in FGMs with the assumption of special gradation, the present method can consider arbitrarily graded shear modulus and Poisson's ratio. It is shown that the present solution can reduce to existing solutions in literatures for annular crack in homogenous materials. Numerical study is conducted to investigate the fracture mechanics of annular crack in FGMs. A practical example of an annular crack in a real Epoxy-Glass FGM with measured data of material properties is also considered.

Scalable manufacturing of the Al-based master composites containing TiB2 and TiC particles and their modification effect on the hot cracking of rapidly solidified Al alloys

This work proposed a scalable manufacturing method for preparing the Al-based master composites containing micrometer-TiB2 and nanometer-TiC particles through molten salt reaction combined with vibration assisted dispersion. The modification effect of the Al-based master composites on the hot cracking and microstructure of 6000 and 7000 series Al alloys with high crack sensitivity during rapid solidification is initially investigated. The results show that the addition of the mixed particles could significantly refine the grains, effectively suppress the hot cracks, and improve the mechanical properties of rapidly solidified Al alloys. Meanwhile, the modification effect of the mixed particles is better than those of the single-phased micrometer-TiB2 particles or nanometer-TiC particles. This scalable and low-cost manufacturing process provides a practical way for preparation of metal matrix composites with low concentration of particles, and proposed a new path to solve the hot cracking problems in the rapid solidification areas such as additive manufacturing.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

A novel approach to compute the spatial gradients of enriching functions in the X-FEM with a hybrid representation of cracks

The eXtended Finite Element Method (X-FEM) is a versatile technique to model discontinuities by enriching the trial functions with a prior solution. In the X-FEM, a crack can be explicitly represented by a set of triangles or implicit signed distances, i.e., level set functions, of the points of interest from the crack surface and the crack front. In the explicit representations, it is crucial to accurately evaluate surface normal, conormal, and tangent vectors along crack fronts for computations of the gradients of the enriching functions. The solution is very sensitive to these directional vectors, especially for non-planar 3D cracks. We here propose a novel approach to compute these gradients without evaluating the directional vectors. Level set functions are first set up in a hexahedral grid independent of the background mesh in the X-FEM. We can thus implement the Weight Essentially Non-Oscillatory (WENO) scheme to compute the gradients of level sets. The gradients of the enriching functions at any integration point can therefore be computed by interpolations and chain rules. We compare the implementation procedures of the explicit representation and the proposed hybrid representation in detail. A three-dimensional lens crack problem is studied to demonstrate the accuracy of the proposed method, especially for coarse meshes.

Mode-I penny-shaped crack problem in an infinite space of one-dimensional hexagonal piezoelectric quasicrystal: exact solutions

Mode-I penny-shaped crack problem in an infinite space of one-dimensional hexagonal piezoelectric quasicrystal: exact solutions

Bone crack inspired pair of Griffith crack opened by forces at crack faces

The mathematical theory of elasticity helps in the study of physical quantities in the problem of structures. The structures face the problem of crack’s presence, which makes the problem difficult but not impossible to deal. Integral equations are useful in a variety of problems. Integral equations are used to solve problems like fracture mechanics or fracture design. The physical interest in the fracture design criterion is due to stress and crack opening displacement components. We have an accurate form of stress and displacement components for a pair of longitudinal crack propagations in the bone fracture of the human body at the interface of an isotropic and orthotropic half-space that are bounded together in the proposed study. The expression was calculated using the Fourier transform approach near the crack tips, but these components were evaluated using Fredholm integral equations and subsequently reduced to coupled Fredholm integral equations. We employ the Lowengrub and Sneddon problem in this research and reduce it to triple integral equations. The Srivastava and Lowengrub method reduces the solution of these equations to a coupled Fredholm integral equation. The problem is further reduced to a decoupled Fredholm integral equation of the second kind. Triple integral equations are solved, and the problem is reduced to a coupled Fredholm integral equation. The Fredholm integral equation is solved and reduced to a decoupled Fredholm integral equation of the second kind. Stress and crack opening displacement components drive physical interest in fracture design criteria. Finally, the stress and displacement components may be simply calculated in their exact form.

Failure Analysis and Countermeasures for Cement Sheath Interface Sealing Integrity in Shale Gas Wells

Summary Shale gas development usually uses large displacement horizontal well and staged fracturing technology to increase operation production. The complex environmental and construction conditions often lead to wellbore sealing integrity problems in the shale gas production process. This study shows a new method for evaluating the sealing integrity of shale gas cement sheath interfaces, which aims to understand the failure mechanism during shale gas production and to propose countermeasures that can effectively improve the sealing integrity of cement sheath interfaces in shale gas cementing. The study results showed that the oil contamination of cement sheath interface will greatly weaken its sealing performance. After repeated cyclic loading, the sealing performance of the conventional and expanded cement sheath assemblies is damaged, and a gas channel is formed, which is caused by the combination of microcracks and microgaps. Furthermore, oil contamination of the cement sheath interface will accelerate its sealing failure. The addition of an expansion agent is helpful to solve the problem of microgap destruction, and the fibers or whiskers can alleviate the problem of tensile cracking. The field application in the three wells proved that the toughened expanded cement slurry significantly improved the sealing integrity of the cement sheath interface in shale gas wells. The research results can evaluate and predict the sealing performance of the cement sheath interface in shale gas wells under the conditions of staged fracturing and have some directional significance for the cement slurry system optimization in the field.

Adaptive phantom node method: An efficient and robust approach towards complex engineering cracks

An efficient and robust approach in the framework of the finite element method for numerical analysis of crack problems is presented. In this method, the discontinuity due to the presence of cracks is described by the technique of phantom nodes. The idea of introducing adaptive mesh refinement along with crack extension into the original phantom node method is presented. The hanging nodes which present in the locally refined mesh is treated by the simple technique of constrained approximation. No additional enrichment functions (and the corresponding extra degrees of freedom) or special elements are involved. As a result, the proposed adaptive phantom node method is almost as simple (and its implementation is as convenient) as the standard finite element method. Numerical results show that the proposed method is able to pass discontinuous patch test accurately and to model crack extension efficiently. In particular, according to its comparison with the original phantom node method in the computation of stress intensity factors, the proposed method consumes less than one percent of the CPU time of the original phantom node method, but it still shows much better accuracy. The capability of the proposed method in modeling multiple cracks, even in complex engineering structures, is also demonstrated.

Theoretical and computational aspects of configurational forces in three-dimensional crack problems

Mechanics in material space yields configurational forces (CFs) as an appropriate means to describe driving forces acting at defects or inhomogeneities in magnitude and direction. A thermodynamically based approach to obtain a weak formulation of CFs with virtual defect displacements as test functions is introduced first, starting from a global variational energy balance instead of from the currently employed local configurational balance equation. A theoretical basis and issues of three-dimensional implementation are presented to calculate vectors of CFs along a crack front, providing information on local loading and deflection angles. Spurious nodal CFs are separated from physical ones, inter alia requiring surface integrals along the crack faces. Within a linear elastic framework, these are efficiently replaced by an algebraic expression, providing CFs at crack front nodes in a very good approximation, which is eventually demonstrated by means of analytical and numerical examples.

Study on mechanical properties of multi-fiber reinforced fly ash concrete

Aiming at the problems of low strength and easy cracking of concrete with large dosages of fly ash, SSF, ARF, and PPF were mixed into concrete respectively according to the same volume ratio. Various indexes were tested and the change rule was analyzed. Besides, the fiber spacing theory was adopted to explore the anti-cracking mechanism of fiber on concrete. The results indicate that the compressive strength and flexural strength of concrete were improved by three kinds of fiber in different dosage; the RFC of 7d was between 14% and 18%, and that of 28d was between 13% and 20%; SSF in concrete would produce strong bonding force and mechanical bite force. ARF has a small impact on the compressive strength of concrete, but a significant impact on the flexural strength of concrete. PPF can be more evenly dispersed in concrete to disperse and release a certain amount of stress and play a reinforcing role. Combined with SEM Microscopic Morphology Analysis of Concrete, a concrete mix ratio of three fibers 1:1:1 is recommended.

Effective elastic properties of three-dimensional multiple crack problems with the isogeometric boundary element parallel fast direct solver

We propose an isogeometric boundary element (IGABEM) parallel fast direct solver based on OpenMP to solve three-dimensional multiple crack problems. In this paper, the orderly and randomly distributed cracks are studied to observe the influence of the multiple crack interaction on the effective elastic properties. This paper designs the parallel implementation of the matrix low rank approximation and the matrix assembly for the hierarchical off-diagonal low rank (HODLR) matrix. In the matrix low rank approximation, we propose the m-n tree to improve the accelerated hybrid algorithm to balance the CPU performance and memory usage. The numerical results show that the effective elastic properties can respond to the strong interaction to a certain extent, but the maximum stress intensity factors (SIFs) of multiple cracks still need to be calculated to diagnose the possible crack initiation points. On the other hand, the numerical results also show that the IGABEM parallel fast direct solver can greatly improve the computational efficiency on the premise of ensuring the computational accuracy.