Asymptotic Crack-tip Fields Research Articles

Crack propagation simulation and overload fatigue life prediction via enhanced physics-informed neural networks

The fatigue crack growth simulation and life prediction of structures are implemented in this paper based on the physics-informed neural networks (PINNs). Firstly, the enhanced PINNs are proposed by introducing the crack tip asymptotic displacement fields, so that the crack tip stress intensity factors can be calculated accurately even when the number of collocation points is small and the distribution grid is regular. The enhanced PINNs essentially transform the solution of elastic body containing crack into the optimization of minimizing the constructed loss functions, and can invert the fracture parameters. Then, an automatic crack propagation simulation method is developed based on the enhanced PINNs. The network architecture and the overall node distribution can be unchanged during the crack propagation process, and only new crack surfaces need to be processed and corresponding loss functions need to be modified. Because the nodal refinement around crack-tip is not required, this simulation method is convenient and can accurately predict the mixed-mode crack propagation path. Finally, the fatigue crack growth life algorithm considering overload is developed, where the effect of each overload can be captured by the cycle-by-cycle method. Based on this algorithm, the retardation behavior can be characterized and the fatigue life of structure under the load spectrum with periodic overloads can be accurately predicted. The sufficient examples are given to verify the feasibility and accuracy of the method proposed in this paper.

A predicting model for three-dimensional crack growth in power-law creeping solids

Creep fracture mechanics has been extensively studied in the past half a century, but the gap between the creep crack-tip asymptotic field and crack growth prediction has not been effectively bridged so far, hindering the development of high temperature damage tolerance design. Here a predicting model is developed for three-dimensional crack growth in power-law creeping solids with the C(t)PC-Tz asymptotic solution being applied to quantify the creep crack growth driving force. The local resistance for creep crack growth in three-dimensional cracked structures is then described and evaluated based on the micro-cavity growth and constraint theories. Validation against forty-four crack growth experimental data for eleven creeping materials with deferent properties and specimen geometries and subjected to different load level in the literature shows that the proposed model can effectively predict the creep crack growth rate and provide the upper and lower limits. In all cases, the predicted crack growth life is within a triple error band around the experimental data. All the needed material parameters can be obtained from tension tests of standard uniaxial round bar specimens. Therefore, the proposed model can bridge the C(t)PC-Tz asymptotic solution with creep crack growth in practical materials and specimens, paving a road to predict damage tolerance of three-dimensional cracked structures at high temperature.

Weak form quadrature element analysis of crack-tip asymptotic field coefficients

Coefficients of crack-tip asymptotic fields are computed using the weak form quadrature element method (QEM). The entire domain is divided into several integrable subdomains and expressions of the crack tip asymptotic displacement field are introduced into those near the crack tip. Algebraic equations with coefficients of the crack-tip asymptotic field and nodal displacements are established using the minimum potential energy principle. Coefficients of arbitrary order can be extracted directly by solving the equations. The first ten order coefficients for several typical examples are given in this paper. As compared with the finite element method (FEM) and the boundary collocation method (BCM), fewer degrees of freedom are needed in the present approach to achieving the same accuracy. In addition, the merits of succinctness and directness of the weak form quadrature element method for fracture analysis are demonstrated.

Crack tip fields in a fiber-reinforced hyperelastic sheet: Competing roles of fiber and matrix stiffening

Fiber-reinforced materials are widely observed in biological tissue such as human arteries and tendons. Characterizing the influence of fibers on crack-tip fields in such nonlinear materials undergoing large deformation is an important precursor to understanding damage evolution and fracture or tearing. In this paper, we present asymptotic crack tip fields in a fiber-reinforced hyperelastic sheet and explore the competing roles of fiber and matrix stiffening. A generalized neo-Hookean model and a power-law model are employed to characterize the behavior of the matrix and fibers, respectively. We show that the asymptotic crack-tip fields are mainly determined by the phase with largest degree of stiffening. For example, when the stiffening effect of fibers is much larger than that of the matrix, the crack tip fields are determined by the constitutive behavior of fibers, and the wire model and the fabric model reasonably characterize the deformation and stress near the crack tip. On the contrary, if the matrix stiffening is larger than that of the fibers, we show that the crack tip fields approach those of a pure matrix material. The asymptotic results are complemented by finite element simulations which show good agreement. These findings may shed light on material damage at a crack tip in fiber reinforced materials.

Analysis of bimaterial interface cracks using the localized method of fundamental solutions

This short communication makes the first attempt to apply the localized method of fundamental solutions (LMFS), a newly-developed meshless collocation method, for fracture mechanics analysis of bimaterial interface crack problems. The asymptotic crack-tip field for bimaterial interface cracks exhibits an oscillatory behavior which is quite different from that for in-plane cracks in homogeneous materials. This paper describes an enriched LMFS approach whereby a set of enrichment functions are embedded in the classical LMFS approximation to account for the presence of interface cracks. This method automatically incorporates the oscillatory behavior of the near-tip fields and thus the complex stress intensity factors (SIFs) can be solved accurately with no or very little remeshing. The results presented show excellent accuracy for a range of two-dimensional (2D) bimaterial with interface cracks, where the complex SIFs for interface cracks are computed with relatively errors less than 0.6 per cent.

Finite deformation near a crack tip terminated at an interface between two neo-Hookean sheets

A crack terminating at an arbitrary angle to the interface between two neo-Hookean sheets is investigated under plane stress conditions using finite deformation theory. The asymptotic crack-tip deformation and stress fields are analyzed as a function of the ratio of the moduli and the angle of the crack relative to the interface. Full-field numerical calculations and experimental studies validate the analytical results. A stretch-based crack growth criterion is developed using crack-tip field solutions. Such criterion can predict the delay of crack growth through the bi-material interface observed in experiments and can be extended to any heterogeneity and material.

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.

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.

Crack tip asymptotic field coefficients analyses based on extended finite element method using over-deterministic displacement field fitting method

The evaluation of the stress intensity factors (SIFs), T-stress, and higher-order coefficients in the crack tip asymptotic field in the extended finite element method (XFEM) is investigated based on the over-deterministic displacement field fitting method. Numerical examples of central cracked and edge cracked plate under uniaxial tension exhibit the accuracy of the evaluated coefficients. A convergence study is conducted to explore the influence of fitting parameters: the number of nodes used for fitting, the number of terms in the asymptotic field series used for fitting, and the distance from selected nodes to the crack tip. It indicates that the distance of selected nodes affects the accuracy of coefficients above 3rd order. Besides, the study of enrichment function and strategy demonstrates that adopting higher-order enrichment terms or more enriched layers of elements can improve the accuracy of coefficient evaluation. Particularly, the higher-order coefficients can be evaluated without the enrichment terms of the corresponding order. The analyses verify that the crack tip asymptotic field coefficients can be accurately evaluated in XFEM using this over-deterministic displacement field fitting method with the appropriate setting of enrichment and fitting parameters.

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.

Crack tip asymptotic fields in anisotropic planes: Importance of higher order terms

This paper introduces the series expansion of the crack tip asymptotic fields in an anisotropic plane. Using stress-based definitions of the fracture loading modes, we present the series solutions of stresses, strains and displacements associated with the modes I and II in a cracked anisotropic plane. Special care is given to the first non-singular terms of the stress field, deriving expressions of the T-stress and the crack rigid body rotation. We then present the elasticity solution of an infinite anisotropic plane containing an inclined central crack subjected to uniaxial or biaxial loading. Under such conditions, exact analytical solutions are obtained for all coefficients of the crack tip series expansion. Our results show that the T-term is the only stress parameter that depends on the elastic constants of the anisotropic material. Finally, we evaluate the dependency of the T-stress on the material properties, and show that how significantly the higher order terms, particularly the T-stress, can contribute to the stresses near the crack tip.

Novel special crack-tip elements for interface crack analysis by an efficient boundary element method

This paper presents a new set of novel special crack-tip elements for analysis of interface cracks in linear elastic composite bimaterials by using the boundary element method (BEM). Such interface crack problems present some simulation difficulties based on the conventional numerical methods because the asymptotic crack-tip fields in this case exhibit an oscillatory behavior which is quite different from that in a cracked homogeneous material. The novel special crack-tip elements developed in this study contain the correct local asymptotic behavior of the displacement and stress fields at the tip of an interface crack, and can thus appropriately describe the oscillatory near-tip displacement and stress fields for cracked composite bimaterials. The novel special crack-tip elements are implemented into a BEM, which can significantly improve the computational accuracy of the complex stress intensity factor (SIF) for an interface crack even with a relatively coarse mesh. The proposed computational strategy for interface crack analysis based on the novel special crack-tip elements also enables a direct and accurate evaluation of the complex SIF at the collocation point extremely close to the crack-tip. Several numerical examples are presented and discussed to demonstrate that the present method based on the novel special crack-tip elements is pretty simple, highly accurate, and relatively robust with respect to the size of the used crack-tip elements.

Crack tip asymptotic field and K-dominant region for anisotropic semi-circular bend specimen

This study reports the coefficients of the crack tip asymptotic field for the semi-circular bend (SCB) specimen made of anisotropic rocks when subjected to pure mode I loading. The finite element over-deterministic method is employed to determine the singular and higher order terms of the crack tip asymptotic field for a wide range of geometry and anisotropy parameters associated with the Mode I SCB test. These parameters are helpful when analysing the fracture path and failure mechanism of anisotropic rocks in cases where the process zone is large compared to the crack length. We also apply an energy-based criterion on the crack tip fields to evaluate the size of the K-dominant region as a function of geometry configuration and anisotropy. It is concluded that the ISRM-suggested size requirement for the SCB samples of rock can yield underestimated values of fracture toughness due to the constraints applied to the FPZ development.

Asymptotic path-independent integrals for the evaluation of crack-tip parameters in a neo-Hookean material

In this paper, we develop new asymptotic path-independent integrals for the evaluation of the crack tip parameters in a 2D neo-Hookean material. The new integrals are of both J-integral and interaction energy integral type and rely on the separation of the asymptotic boundary value problem into independent problems for each of the deformed coordinates. Both the plane stress and plane strain cases are considered. The integrals developed are used to compute the amplitude parameters of the asymptotic crack tip fields, which allows for direct extraction of these parameters from numerical results. A long strip with an edge crack under mixed loading modes is considered for both homogeneous and biomaterial cases. It is found that the asymptotic J-integrals produce good results for the first-order parameters while the interactions integrals produce good results for both the first and second-order parameters.

The finite element over-deterministic method to calculate the coefficients of crack tip asymptotic fields in anisotropic planes

This article applies the finite element over-deterministic (FEOD) method to determine the coefficients of the crack tip asymptotic fields in an anisotropic plane. This method relies on finding the best fit of the crack tip displacement equations to the displacements taken from a large number of nodes around the crack tip. We first revisit the asymptotic solution of crack tip fields in an anisotropic plane, followed by giving the exact solution of the crack tip parameters for an infinite anisotropic plane with a central crack. Using the derived displacement field, we then formulate the FEOD method to determine the crack tip parameters from a set of finite element nodal displacements. Numerical examples of cracks in finite and infinite plates are then investigated, where the FEOD results manifest a good agreement with exact or alternative numerical solutions. A convergence study is performed to examine the robustness of the methodology and the reliability of the results. By implementing a least squares error function, we also calculate the corresponding error in approximating the displacement field. The low values of errors demonstrate the high precision of the FEOD method.

Interfacial toughness: dependence on surface roughness and test temperature

Interfacial toughness quantifies resistance to crack growth along an interface and in this investigation the toughness of an aluminum/epoxy interface was measured as a function of surface roughness and test temperature. The large strain response of the relatively ductile epoxy adhesive used in this study was also characterized. This epoxy adhesive exhibits intrinsic strain-softening after initial compressive yield and then deforms plastically at a roughly constant flow stress until it rapidly hardens at large compressive strains. We find that interface toughness scales as the product of the temperature dependent epoxy yield strength and a length scale that characterizes surface roughness. The proposed scaling is based upon dimensional considerations of a model problem that assumes that the characteristic length scale of both the roughness and the crack-tip yield zone is small relative to the region dominated by the linear elastic asymptotic crack-tip stress field. Furthermore, the model assumes that interfacial failure occurs only after the epoxy begins to harden at large strains. The proposed relationship is validated by our interfacial toughness measurements.

Poroelastic effects on steady state crack growth in polymer gels under plane stress

Fracture experiments on polymer gels are often conducted with thin specimens, which are close to plane stress in two-dimensional models. However, many of the previous theoretical and numerical studies on fracture of polymer gels have assumed plane strain conditions. The subtle differences between the plane stress and plane strain conditions are elucidated in this paper based on a linear poroelastic formulation for polymer gels, including the asymptotic crack-tip fields and finite element simulations of steady-state crack growth in long strip specimens. Moreover, a poroelastic cohesive zone model is adopted to study the rate-dependent fracture process of polymer gels. It is found that, without the cohesive zone model, the normalized crack-tip energy release rate at the fast crack limit is greater than the slow crack limit, suggesting reduced poroelastic toughening for fast crack growth under plane stress conditions, while the two limits are identical under plane strain conditions. With a solvent-permeable cohesive zone for the case of immersed specimens, solvent diffusion within the cohesive zone enhances the poroelastic toughening significantly as the crack speed increases, leading to a rate-dependent traction-separation relation. On the other hand, with no solvent diffusion in the cohesive zone for the not-immersed case, the poroelastic toughening effect diminishes as the crack speed increases. Based on the present study, the intrinsic steady-state fracture toughness of a poroelastic gel can be determined using long-strip pure-shear specimens, which in general is smaller than the applied energy release rate.

Asymptotic crack-tip fields in perfect plasticity under mixed mode loading

In the paper the refined presentation of the asymptotic stress fields in the vicinity of the crack tip in perfectly plastic Mises materials under mixed mode loading for the full range of the mode mixities is given. This objective of the paper is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities both for plane strain and plane stress conditions to grasp stress tensor components behaviour in the vicinity of the crack tip as the mixity parameter is changing from 0 to 1. To gain a better understanding of the stress distributions all values of the mixity parameter to within 0.1 were considered and analyzed. The asymptotic solution to the statically determinate problem is obtained by the eigenfunction expansion method. Steady - state stress distributions for the full range of the mode mixities are found. The type of the mixed mode loading is controlled by the mixity parameter changing from zero for pure mode II loading to 1 for pure mode I loading. It is shown that the analytical solution is described by different relations in different sectors, the value of which is changing from 7 sectors to 5 sectors. At loadings close to pure mode II, seven sectors determine the solution whereas six and five sectors define the solution for the mixity parameter higher 0.33 and less than 0.89 and higher than 0.89 respectively for plane strain conditions and seven sector determine the asymptotic solution for the mixity parameter less than 0.39 and five sectors determine the solution for other values of the mixity parameter for plane stress conditions. The number of sectors depends on the mixity parameter. The angular stress distributions are not fully continuous and radial stresses are discontinuous for some values of the mixity parameter. It is interesting to note that the characteristic feature of the asymptotic solution obtained is the presence of a segment of values of the mixity parameter for which the solution does not depend on the mixity parameter (the solution does not depend on the mixity parameter for the mixity parameter from 0.89 to 1 and the solution coincides with the solution for mode I crack in perfect plastic materials for plane strain conditions). Thus, the salient point of the study is that the asymptotic solution is described by the same formulae for all values of the mixity parameter from 0.89 to 1 for plane strain. For plane stress conditions this segment can’t be observed. The solution in each sector corresponds to the certain value of the mixity parameter. The obtained solutions for plane strain and plane stress conditions can be considered as the limit solution for power law hardening materials and creeping power law materials.

On higher order parameters in cracked composite plates under far‐field pure shear

AbstractThis paper presents the analytical solution of the crack tip fields as well as the crack parameters in an infinitely large composite plate with a central crack subjected to pure shear loading. To this end, the complex variable method is employed to formulate an asymptotic solution for the crack tip fields in an anisotropic plane. Using a stress‐based definition of the crack tip modes of loading, only the mode II crack parameters are found to be non‐zero under pure shear load. Special focus is given to the determination of the higher order parameters of the crack tip asymptotic field, particularly the first non‐singular term, ie, the T‐stress. Unlike the isotropic materials, in which the T‐stress is zero under pure shear, it is found that the T‐stress is non‐zero for the case of anisotropic materials, being the only material‐dependent crack tip stress parameter. The veracity of our exact crack tip fields is assessed and verified through a comparison made with respect to the finite element (FE) solution. Finally, we demonstrate the significance of the T‐stress on stresses near the crack tip in composite plates under pure shear loads.

A unified method to solve higher order asymptotic crack-tip fields of mode I, mode II and mixed mode I/II crack in power-law creeping solids

A higher order asymptotic analysis with separable variable form is presented in this paper for crack tip mechanics fields under mixed mode creep conditions. By defining a second order mode mixity factor, higher order term solutions are investigated by solving the boundary value problem. The solution procedure is validated first and then the characteristics of various kinds of second order asymptotic solutions for mixed I/II creep crack tip field are presented and discussed. The angular distribution functions of the second order stress and the stress exponent for various cases are given. Detailed relations between the amplitudes of the second order term and the first order term are also discussed. The method and solutions given in this paper encompass the higher order term solutions of pure mode I, pure mode II and mixed mode I/II.