Crack Problems Research Articles

Interface crack analysis in 2D bounded dissimilar materials using an enriched physics-informed neural networks

This study explores the application of physics-informed neural networks (PINNs) to analyze interface crack problems within the context of elastic bimaterial fracture mechanics. Bimaterial interface cracks exhibit a distinct behavior compared to cracks in homogeneous materials, and this behavior often involves oscillatory phenomena that can pose challenges in numerical modeling. By employing neural networks for solution approximation, PINNs are meshless and are trained using batches of collocation points, which may be randomly or strategically sampled across the computational domain. To effectively capture the oscillatory singular behavior in the crack-tip regions, this paper introduces an enhanced PINNs formulation that enables the modeling of interface cracks without requiring any refinement near the crack-tip. The trainable parameters of the current PINNs are dynamically optimized throughout the training process to fulfill both the underlying differential equations and the associated initial/boundary conditions. One of the significant advantages of the present PINNs is that it uses enrichment functions to capture the behavior around the crack region, allowing for greater flexibility in handling irregular or complex crack paths. The method's accuracy and stability are validated across several benchmark examples.

Multi-domain wavelet boundary element method for calculating two-dimensional stress intensity factors

In order to improve the accuracy of stress intensity factors (SIFs) calculated by traditional boundary element methods (BEM), the multi-domain wavelet boundary element method (WBEM) is proposed. Firstly, by adjusting the nodes of the B-spline wavelet element on the interval, crack-tip elements are constructed. Since B-spline wavelet on the interval (BSWI) has excellent compact support characteristics and is particularly suitable for describing solution domains with large gradient changes, the constructed crack-tip can reduce the numerical oscillation effect near the crack tip. Secondly, the crack-tip elements are implemented into WBEM. And the combination of WBEM and multi-domain technology can effectively handle interface cracks. Thirdly, the crack problem solving strategy based on multi-domain WBEM can directly evaluate the SIFs of cracks. Finally, several numerical examples involving homogeneous media and bi-material models are given to verify that the proposed method is simple and highly accurate.

Bearing Capacity of a Concrete Grouting Pad on the Working Surface of a Highway Tunnel Shaft

A grouting pad is the key structure for the construction of water inrush grouting on the shaft working surface. Previous methods of calculating the bearing capacity have limitations due to a lack of understanding of the failure mode. To investigate the bearing capacity of a concrete grouting pad on the working surface of a shaft, this paper establishes a mechanical model for the punching shear failure of a grouting pad under symmetrical loading conditions. A unified solution for the bearing capacity is derived, and the influence of parameters is discussed. In addition, a new method for designing the plastic limit thickness is proposed based on this research. The results show that the reason for the grouting pads’ punching shear failure resulted from the formation of peripheral grouting holes “weak ring” caused by the reduction of the bearing capacity. When the thickness of B0 remains constant, the bearing capacity qu of the grouting pad is inversely proportional to the ratio of the diameter and the area of the bottom load. Therefore, following the method of “dividing, interval, and jumping holes” during grouting construction is recommended. The greater the thickness of the grouting pad, the greater the bearing capacity qu will be. When the grouting pad diameter is 2r2 and the thickness B0 is constant, the bearing capacity qu increases with the material tensile strength ft. When designing grouting pads, following the principles of “large thickness, uniform strength theory, high strength materials” will improve bearing performance. The findings have been implemented in the design of the grouting pad thickness for the Tianshan Shengli Tunnel shaft project, which can successfully solve the problem of frequent cracking caused by the weak bearing capacity of a grouting pad. The findings can provide a theoretical basis and reference for the design and construction of grouting pads in a highway tunnel shaft.

Highly reflective TiO2 layer on AlN substrate: Enabling high quality laser lighting

Phosphors in glass films (PiFs) is considered ideal conversion materials for laser lighting, but heat dissipation issue is the main challenge limiting its development. Therefore, it is urgent to find more excellent heat dissipation materials. Herein, we proposed to use AlN with high thermal conductivity as the substrate, and innovatively introduced TiO2 coating to improve the interface reflection ability and solve the possible problems of shedding and cracking. After a series of optimizations, the LSN PiG-TiO2-AlN (PTA) color converter obtained a high thermal conductivity of 29W m−1k−1 and a low surface temperature. At an input power of 5.2W, the luminous efficiency can reach 175 lm/W and the relative initial intensity can still be maintained by 95.3% at 200 °C. Benefiting from the extremely strong heat dissipation capacity of the AlN substrate, the LSN PTA color converter maintains its relative luminous intensity at 95 % of the initial intensity after 24 h of continuous illumination, which is the best value recorded so far. In addition, LSN PTA is combined with CaAlSiN3:Eu2+ phosphor to achieve high-quality white light with a CRI of 82.5. The above results indicate that LSN PiG-TiO2-AlN color converters have a promising research prospect in the field of laser lighting.

Fracture Analysis of Planar Cracks in 3D Thermal Piezoelectric Semiconductors

The study of piezoelectric semiconductor (PSC) materials and devices is experiencing rapid growth, giving rise to a burgeoning research domain known as piezotronics. Understanding the fracture behavior of PSC materials is essential for designing robust devices and ensuring their long-term functionality. A planar crack of arbitrary shape in the isotropic plane of a three-dimensional (3D) transversely isotropic thermal PSC body subjected to combined thermal-electrical-mechanical loadings is studied via the extended displacement discontinuity (EDD) boundary integral equation method. Under the framework of linearized basic equations and one-way thermal coupling, the hypersingular boundary integral equations with the volume integrals containing the temperature and carrier concentration are obtained and expressed by the EDDs which include displacement, electric potential, carrier concentration and temperature discontinuities across the crack surfaces and are the basic variables in analyzing crack problems. Based on the finite part of hypersingular integrals, the EDDs near the crack border are proved to have the classical behavior of O(r). Furthermore, the asymptotic solutions of extended stresses including the stress, electric displacement, current density and heat flux at the crack front are presented and display classical singular behavior of O(1/r). The extended stress intensity factors are defined and given in terms of EDDs. A 3D numerical model of a penny-shaped crack in the thermal PSC cylinder is established and used to verify the presented formulae.

A stable numerical framework for long-time dynamic crack analysis

This paper introduces a stable numerical framework designed to address dynamic crack problems over long-time intervals. The initial step involves the temporal discretization of the governing dynamic equilibrium equations using the arbitrary order Krylov deferred correction method. To ensure precise boundary condition matching, a novel numerical implementation is incorporated into the Krylov deferred correction technique. Subsequently, the resulting system of spatial partial differential equations at each time node is solved using the meshless generalized finite difference method with 4th-order expansions in regions close to the crack-tips and 2nd-order expansions in areas far from the crack-tips. This combined approach capitalizes on the strengths of both the Krylov deferred correction technique and the generalized finite difference method, enabling stable simulations of dynamic cracks with the large time step and without the need for mesh generation. Notably, we refine the collocation nodes near crack tips to attain accurate numerical results for displacement and stress field. Several numerical experiments involving diverse impact loadings are conducted to validate the developed framework. Furthermore, a comparison is made between the dynamic stress intensity factors obtained using our approach and those from existing methods.

The strain gradient viscoelasticity full field solutions for Mode-I and Mode-II crack problems

The impacts of size and viscosity become distinctly evident when considering micro- and nano-scale phenomena for advanced materials with micro- and nanostructures. In this study, the mechanical behavior of the advanced material is characterized by using the strain gradient viscoelasticity theory, and a novel solution is presented for the mode-I and mode-II cracks, which is formulated based on the strain gradient viscoelasticity theory, employing the Wiener-Hopf method. Besides, the gradient-dependent viscoelastic crack solutions are directly derived by applying the correspondence principle that aligns strain gradient viscoelasticity with strain gradient elasticity within the Maxwell's standard linear solid model. Relative to the influence of elastic strain gradient effects, the involvement of viscous gradient effects instigates a reinforcement to the stress field around the crack tip, thereby offering a more reasonable representation of advanced materials crack behavior. When the viscosity effect is omitted or as time tends to infinity, the solutions based on strain gradient viscoelasticity theory converges to those on the classical strain gradient elasticity theory.

Physically consistent nonlocal macro–meso-scale damage model for quasi-brittle materials: A unified multiscale perspective

Despite great efforts in capturing the damage evolution law in multiscale approaches in damage mechanics, much less attention has been paid to the conversion from geometric damage to energy dissipation. In the present paper, the newly proposed nonlocal macro–meso-scale damage (NMMD) model is physically consolidated and a multiscale point of view is consistently adopted in both the damage evolution and energy dissipation. In this model, for each macroscopic material point a mesoscopic structure is attached such that all the point pairs composed of this point and the points in its influence domain are connected to it when the material is intact. The deformation field under loading will result in deformation of point pairs, and thus mesoscopic damage related to a point pair will occur if the deformation index of this point pair exceeds the prescribed threshold. Such mesoscopic change among point pairs connected to a material point within its influence domain has two-fold consequences: geometrically the accumulation of mesoscopic damage will result in macroscopic discontinuity, and simultaneously the free energy will be dissipated, leading to degradation of mechanical behavior of quasi-brittle materials. In other words, the damage as a metric of the geometric discontinuity will lead to the damage in the sense of energy dissipation and degradation of mechanical behavior for quasi-brittle materials, which can be physically captured by the ratio of summation of all the dissipated mesoscopic free energy in point pairs to the free energy of intact material within the influence domain. This energy-based macroscopic damage could then be inserted into the framework of continuum damage mechanics. Therefore, the conversion from geometric damage to energetic damage is implemented on the mesoscale instead of the macroscale, and thus a macroscopic energetic degradation function is not needed. In addition, the structured strain which is suitable for quasi-brittle materials can be adopted to determine the deformation index of point pairs. Correspondingly, the macroscopic free energy can be split into dissipative and non-dissipative parts. In this way, several inconsistencies in the previous NMMD models are remedied. The influence of mesoscopic model parameters is investigated in depth, and several numerical examples including mode-I, mixed-mode and compressive splitting fracture problems of quasi-brittle materials are carried out. Numerical results indicate that the proposed model can not only well capture the cracking process with or without initial cracks but also quantitatively predict the load–deformation curves without mesh size sensitivity. Moreover, due to the inherent physical consistency, in mixed-mode cracking problems the proposed model performs better than the previous NMMD model. Problems to be further studied are also discussed.

Study on Hydration Heat Release Model and Its Influence Coefficient of Addition Concrete

The concrete of the main structure of subway stations is highly constrained, so the temperature effect and shrinkage effect generated during the construction process often cause large restraint stress, thus causing the problem of structural cracking. Therefore, low-heat-of-hydration cements are usually used in such construction work to reduce the temperature during the aging of concrete. Therefore, based on the hydration heat release subprogram of concrete, an effective numerical simulation was carried out, and compared with the semi-adiabatic temperature rise test, a method of numerical calculation and parameter fitting to solve the influence coefficient of the hydration heat model was proposed, and the sensitivity of the hydration heat release model and the influence coefficient for several concrete mixtures with additives was studied. The main conclusions are as follows: (1) Combined with the measured temperature curves of several types of mineral-admixed concrete, the composite exponential exothermic model of several types of mineral-admixed concrete can be obtained by adjusting the parameter fitting to solve the influence coefficient of the exothermic hydration model proposed in this paper. (2) One can adjust the influencing coefficient a to control the maximum temperature of concrete. Similarly, the temperature curve of concrete can be adjusted by modifying the influencing coefficient b.

Plane crack problems within strain gradient elasticity and mixed finite element implementation

An alternative approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method (FEM) based on the concept of the Galerkin method is used. To construct finite-dimensional subspaces separate approximations of displacements, deformations, stresses, and their gradients are implemented by choosing the different sets of piecewise polynomial basis functions, interrelated by the stability condition of the mixed FEM approximation. This significantly simplifies the pre-requirement for approximating functions to belong to class C1 and allows one to use the simplest triangular finite elements with a linear approximation of displacements under uniform or near-uniform triangulation conditions. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses and their gradients are treated as local unknowns. The conditions of existence, uniqueness, and continuous dependence of the solution on the problem’s initial data are formulated for discrete equations of mixed FEM. These are solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. This avoids the need to form a global stiffness matrix for the problem of strain gradient elasticity since it is enough to calculate only the residual vector in the current approximation. A set of modeling plane crack problems is solved. The obtained solutions agree with the results available in the relevant literature. Good convergence is achieved by refining the mesh for all scale parameters. All three problems under study exhibit specific qualitative features characterizing strain gradient solutions namely crack stiffness increase with length scale parameter and cusp-like closure effect.

Thermal contraction coordination behavior between unbound aggregate layer and asphalt mixture overlay based on the finite difference and discrete element coupling method

AbstractThe constraint action of the unbound aggregate layer underneath plays an important role in affecting the temperature strains in the top asphalt layer. The focus of the present paper is to investigate the interactive thermal contraction mechanisms between the asphalt mixture and granular base layers to offer a new perspective in promoting the understanding of the thermal cracking disease. In this paper, both experiments and simulations were conducted for the thermal contraction tests. A novel numerical modeling of virtual composite structures was proposed using wall‐zone coupling operation based on the finite difference and discrete element coupling method. The experimental composite structures containing three types of asphalt mixture overlays and five types of unbound aggregate base layers were developed for verification. The thermal contraction and restraint mechanism were revealed from both macro and microscopic scales. The proposed numerical modeling shows a 94% high accuracy. The particle displacement vector and contact force chain could explain the thermal contraction behavior of composite structures. Smaller particle motion displacement or stronger contact force chain result in a higher restraint strain of the asphalt overlay. The thermal contraction behavior can be coordinated through the compaction and loosening of unbound aggregates. The material parameters and cooling temperature differences in the asphalt overlay have slight effects on the constraint action of a base layer, while the gradation and mechanical parameters of the unbound aggregate layer show significant impaction. The parameters, cohesion C and friction angle φ, show a quadratic function with the restraint coefficient. This work has significant guidance on the selection of pavement structure and materials to improve the thermal cracking problem and lay the basis for the mechanical theoretical calculation to predict thermal cracking.

State-of-the-art review on meshless methods in the application of crack problems

Meshless or meshfree methods (MMs) are practical and excellent numerical techniques for solving crack problems. With the inherent limitations of mesh-based methods such as the finite element method (FEM) in modeling cracks, MMs have captivated the attention of researchers worldwide because of their conspicuous merits, including mesh independence and higher-order continuity. However, the most recent review literature on MMs in crack problems is only available until 2018, and no relevant analyses have been reported in detail since then. To fill this gap, this state-of-the-art review explores the novel potential and advancements of MMs in crack problems. Generally, eight approaches are implemented with MMs to reproduce crack behaviors. To improve efficiency and stability, the existing strategies are explored in terms of numerical integration, method coupling, and adaptive analysis. Based on these techniques, 82 highly cited and influential articles are analyzed to determine the specific MMs, simulation techniques, and the enhanced strategies that have been adopted. According to existing studies, various novel MMs have been proposed to analyze cracks. Furthermore, flexible coupling of multiple techniques to simulate cracks can be applied to different forms of cracks in various materials. However, adaptive analysis still needs to be widely performed in the MMs for crack problems due to the absence of a solid mathematical foundation for error estimators. This paper presents the research gaps and the future prospect of MMs in the application of crack problems. It should be stated that significant works on numerical methods for modeling cracks predating 1994 are not cited in this paper, a decision considered by the authors to be non-essential to its scope and objectives.

Application of a hierarchical quadrature element method to interface crack analysis in bi-material systems

This paper presents an extension of the hierarchical quadrature element method (HQEM) with p-convergence to evaluating fracture mechanical parameters of bi-material interface cracks. The numerical implementation of HQEM in combination with the virtual crack closure method (VCCM) is comprehensively illustrated, and the formulas for calculating the strain energy release rate (SERR) are rederived. The relationship between components of SERR and the complex stress intensity factors (SIFs) is established. In addition, the effectiveness of an auxiliary parameter related to the dual energy release rates in measuring the accuracy of the present method is examined.The feasibility of applying HQEM to the interface crack problems is demonstrated by investigating the benchmark problem of an infinite bi-material plate with a central interface crack. Two loading cases are considered, including pure tensile loading and mixed tensile-shear loading. The influence of the crack tip element size Δa on the convergence characteristics and computational accuracy of fracture parameters is quantitatively investigated to provide guidance for choosing the most reasonable value of Δa. Numerical results show that even with very coarse meshes, together with a virtual crack extension consisting of only one element, HQEM yields highly accurate results for the total SERR and the complex SIFs, making it suitable for complex, large-scale interface crack problems in dissimilar media.

Influence of material anisotropy on the mechanics of surface damage

The problem of multiple cracks originating at free surface (surface damage) in an anisotropic material is considered. We focus on the effect of material anisotropy on the mechanics of crack interactions – in particular, on the suppressing effect of interactions on crack nucleation and growth under the tensile loading (stress shielding). Importantly, this effect may change to the opposite one of enhancement, under the shear loading. These effects strongly depend on the material anisotropy. Calculations were performed for the orthotropic materials and the effect of anisotropy was examined by case studies of specific configurations that included comparison with the case of the isotropic material.

Singular integral based closed-form solutions for modified EMPS models in semipermeable magneto-electro-elastic materials

The singular integral equation method is applied to present analytical solutions for modified electric-magnetic-polarization saturation (EMPS) models subjected to semipermeable center cracked magneto-electro-elastic (MEE) materials. A generalized methodology is presented to explicitly solve the EMPS model under any arbitrary saturated electric and magnetic conditions using the distributed dislocation method (DDM) and finding the analytical solutions for developed singular integral equations. The explicit expressions are derived for distributed dislocation densities, crack opening displacement (COD), crack opening potential (COP), crack opening induction (COI), saturated zone lengths, and local stress intensity factor (LSIF) for particular cases of the generalized EMPS model such as constant, linear, quadratic and cubic polynomial varying saturated electric and magnetic conditions. Using an iterative scheme and the explicit solutions of COD, COP, and COI, the exact semipermeable crack-face conditions are evaluated at the center of the crack, and the same applies throughout the crack-surface for the study of semipermeable modified EMPS models. A comparative analysis of the fracture parameters has also been carried out between the results of exact semipermeable crack-face conditions and the results of semipermeable crack-face conditions of the center crack problem to analyze the impact of polynomial varying saturated conditions on the crack-face conditions. Numerical studies performed under different electromagnetic-mechanical loadings, volume fractions, and crack-face conditions demonstrate the increasing effects in the fracture parameters' values with the degree of the polynomial varying saturated conditions.

Higher-order asymptotic crack-tip fields in simplified strain gradient elasticity

Explicit representation for the higher-order in-plane crack-tip fields is derived using Papkovich–Neuber stress functions within the framework of simplified strain gradient elasticity (SGE). In the considered theory, strain energy density depends on both the infinitesimal strain tensor and its spatial gradient. The constitutive equations of simplified SGE contain an additional length scale parameter that can be correlated with the characteristic size of the material microstructure. Presented asymptotic solution has separable form and reduces to classical Williams’ series when the gradient effects are negligible. The leading terms in the derived solution coincide with the previously known asymptotic solutions for crack problems in SGE. The higher-order terms have coupled amplitude factors and modified definitions for the angular distribution as compared to the classical solution. Derived asymptotic fields are compared to the full-field numerical solution for the Mode I crack problem to quantify the amplitude factors and the zones of dominance for up to eight terms. The presented solution has practical applications in the development of advanced numerical methods, particularly concerning enriched finite elements for crack problems in SGE.

Wiener–Hopf method to solve the anti-plane problem of moving semi-infinite crack in orthotropic composite materials

This paper contains the solution to the problem of a semi-infinite moving crack situated in an orthotropic strip bonded between two identical strips. The crack moves with a constant velocity, and the surface is under shear wave disturbance. We first examine the equations of elasticity, which include equilibrium equations and stress and displacement constitutive relations with the model-specific continuity and boundary conditions. Using the Fourier integral transform, the standard form for the Wiener–Hopf (W-H) equation is obtained, which is solved using the W-H method. The analytical expressions for the considered crack problem have been obtained for stress intensity factor (SIF), normalized stress intensity factor (NSIF), and stress magnification factor (SMF). The behavior of NSIF has been graphically presented for particular cases of composite materials for different crack propagation velocities and various depth ratios of the strips. The novelty of this paper lies in the analytic solutions using the W-H method for the semi-infinite moving crack problem under the influence of anti-plane shear waves. The pictorial presentations of normalized SIF clearly show their dependency on strip depths, crack propagation velocities and elastic constants.

Stress Intensity Factors for Thermoelectric Bonded Materials Weakened by an Inclined Crack

Thermoelectric Bonded Materials (TEBM) weakened by an Inclined Crack Problems (ICP) subjected to remote stress was presented in this study. The problems are addressed by employing the Modified Complex Variable Function (MCVF) method, which incorporates the Continuity Conditions (CC) for the Resultant Electric Force (REF) and Displacement Electric Function (DEF) to formulate the Hypersingular Integral Equations (HSIEs) associated with these problems. By applying the curved length coordinate method, the unknown functions of Crack Opening Displacement (COD), Electric Current Density (ECD), and Energy Flux Load (EFL) are mapped onto the square root singularity function. The resulting equations are then numerically solved using appropriate quadrature formulas, with the traction along the crack utilized as the right-hand term. The obtained COD, ECD and EFL functions is then used to compute the dimensionless Stress Intensity Factors (SIFs) in order to determine the stability behavior of TEBM weakened by an ICP. The numerical results provided demonstrate the dimensionless SIFs at the crack tips. These results exhibit excellent agreement with previous studies conducted on the subject. Additionally, it is observed that the dimensionless SIFs at the crack tips are influenced by factors such as the ratio of Elastic Constants (ECR), the geometry of the cracks, and the coefficients associated with the Electric Current Density (ECD).

Thermal shock fracture in a cracked strip: Incorporating convective heat transfer between lateral surfaces and ambient environment

This paper presents a comprehensive analysis of transient temperature and thermal stress around a thermal insulated crack in a finite thermoelastic strip subjected to thermal shock based on the dual-phase-lag (DPL) heat conduction. A detailed investigation of the effect of convective heat exchange between the lateral surfaces of the strip and the ambient environment is conducted. By employing integral transform technique, the thermoelastic crack problem is reduced to a set of singular integral equations, and solutions for the transient temperature distribution and dynamic stress intensity factors (SIFs) at the crack tips are then obtained. Numerical results reveal a considerable temperature gradient near the heating surface when both the high convective heat transfer coefficient and thin strip are involved. Consequently, the SIFs in such scenarios where cracks are located close to the heating surface are significantly larger compared to those calculated by neglecting the convective heat loss. Furthermore, during the initial period following a thermal shock, the SIFs may exhibit higher magnitudes than the steady-state values. These findings emphasize the importance of accounting for the influence of convective heat exchange with the ambient environment in material design and optimization to enhance fracture resistance under transient thermal loading.

Singular integral method based closed form solution for moving EMPS model in semipermeable two‐equal‐collinear cracked MEE material

AbstractThis paper presents the analytical closed‐form solutions of moving two equal collinear semipermeable cracks in magneto‐electro‐elastic material considering the strip electric‐magnetic polarization saturation (EMPS) model. To simulate the dynamic/moving cracks problem, two equal collinear Yoffe type cracks moving with a constant subsonic velocity along the plane of the cracks is considered. The problem is solved by considering semipermeable crack face conditions and under the in‐plane electro‐magnetic‐mechanical loading. Applying distributed dislocation technique and symmetry of the collinear cracks, the problem is reduced into simultaneous singular integral equations which have been solved in closed form expressions using finite Hilbert approach. The explicit expressions for dynamic distributed dislocation densities, crack opening displacement (COD), crack opening potential (COP), crack opening induction (COI), and local stress intensity factors (LSIFs) are evaluated for both the cases of the EMPS model (magnetic zone is smaller or greater than electric zone). For evaluation of inner and outer electric and magnetic zone lengths (MZLs), an iterative approach is applied to solve the developed two non‐linear equations. Also, to implement the semipermeable crack‐face conditions, a bisection iterative method has been applied. Based on the developed closed form solutions, numerical studies are presented for standard fracture parameters with respect to crack propagation velocity, electrical loading and inter‐crack space distance. This study shows that electric displacement and magnetic induction defined over the crack‐surface for semipermeable conditions decrease whereas the numerical values of saturated zone lengths, COD, COP, COI, and LSIFs increase with increasing crack propagation velocity. Moreover, for both static and dynamic cracks problems, the significant crack‐interaction effects have been observed on all the studied fracture parameters except the crack‐face conditions.