Asymptotic Stress Fields Research Articles

Asymptotic stress field for the blunt and sharp notches in bi-material media under mode III loading

Abstract This study introduces a novel approach to analyze the stress and displacement fields around blunt notches in bi-material media, focusing on mode III loading conditions. The eigenfunction expansion method is used to derive a simplified yet accurate solution, satisfying the boundary conditions for bi-material blunt V-notches. The robustness of the proposed asymptotic solution is validated through several finite element analyses, encompassing a range of notched geometries such as blunt V-notches, VO-notches, and circular holes. Notably, it is demonstrated that when the notch tip radius approaches zero, the solution coincides with the existing sharp V-notch model, illustrating the robustness of the methodology.

An analytical approach for evaluation of linear elastic stress fields around sharp V-shaped notches in plates

In the context of the theory of elasticity, the stress state at a sharp notch is singular, and the degree of stress singularity depends on the notch opening angle. Most existing analytical approaches for evaluating asymptotic linear elastic stress fields ahead of V-notches rely on notch stress intensity factors (N-SIFs). This paper presents an alternative analytical approach for evaluating notch stress fields by taking advantage of traction-based structural stress and Williams’ eigenfunction methods. To better represent notch stress fields, the first order and higher order terms of eigenfunction are included in the present study. The analytical solutions are then compared to the stress fields resulting from finite element analysis for validation purposes. The stress contours evaluated by the analytical approach are nearly identical to the numerical results corresponding to three distinguishing notch opening angles under pure Mode I loading, which signifies the effectiveness and robustness of the analytical solution methodology developed in the present study. Subsequently, the approach is extended to accommodate Mode II loading by introducing a skew-symmetric term in the stress function. The generalized method is then applied on a welded cruciform joint, which precisely captures the notch fields compared to FEA results.

A general interaction integral for dynamically propagating cracks in anisotropic materials

In this paper, some relevant tools for numerical analysis of dynamically propagating cracks in anisotropic materials are discussed within the framework of the eXtended Finite Element Method (XFEM). A general interaction integral that is domain-independent is proposed to determine the Stress Intensity Factors (SIFs) for both stationary and dynamically propagating cracks in anisotropic materials. Besides, a set of functions for enriching the nodes of the tip element, which are applicable for modeling both quasi-stationary and dynamically propagating cracks in anisotropic materials, are also proposed. In addition, a modified form of the Maximum Tangential Stress (MTS) criterion is manipulated for estimating the direction of the propagating crack which takes into account the asymptotic stress field near a propagating crack tip in anisotropic materials under mixed mode plane loading. Finally, the accuracy of the suggested method is then confirmed using some numerical examples that involve both stationary and dynamically propagating cracks. Additionally, a thorough investigation is done into the impacts of material anisotropy and material orientation on the path of dynamically propagating cracks in orthotropic structures.

Dynamically propagating cracks in anisotropic plates subjected to hyperbolic thermal shock

ABSTRACT In this paper, dynamically propagating cracks in anisotropic plates exposed to a generalised thermal shock are investigated. Here, the governing equations are considered according to the Lord-Shulman (LS) model as a fully coupled generalised thermoelasticity theory. The crack is modelled in the context of the eXtended Finite Element Method (XFEM). To evaluate Stress Intensity Factors (SIFs), a technique based on the J integral and crack tip opening and sliding displacements, previously applied for stationary cracks, is developed for a dynamically propagating crack in orthotropic solids. Besides, a variety of functions for enriching the displacement field are derived relying on the behaviour near the tip of a dynamically propagating crack in anisotropic materials. The maximum hoop stress criterion is also modified by using the asymptotic stress field for a dynamically propagating crack. In addition, the effect of the angular variation of fracture toughness in orthotropic materials is accounted in computing the crack propagation speed. In some examples, dynamic crack growth in complex cracked orthotropic structures subjected to LS thermal shock is studied in detail. The impacts of fibre orientations and LS relaxation time are also investigated in these examples. In conclusion, applying non-Fourier thermal shock for modelling dynamically propagating cracks in orthotropic structures is recommended to obtain more realistic results. In addition, by increasing the LS relaxation time, the crack grows less oscillatory and is more aligned along the material fibres.

On the T-stress extraction method used by current version of Abaqus

In the present note, it is shown that in the current version of Abaqus (6.10 and later), there is an error in the method used for the determination of T-stress. In particular, errors will occur in cases when there are tractions applied to the crack faces. Since T-stress is an important parameter characterizing the crack tip stress field, used widely in fracture mechanics analysis, it is important to point out this to the users of Abaqus. The reason for this error is due to the incorrect interpretation of the asymptotic crack tip stress fields. To obtain correct T-stress results, the users need to remove this error manually.

Asymptotic stress field around the blunt and sharp notches in bimaterial media under mixed mode I/II loading

This paper provides a simple approximate solution for the stress and displacement fields around the blunt notches in bimaterial media under in-plane loading. The Kolosov-Muskhelishvili's method is used to determine the stress and displacement components according to the complex potential functions. Then, by imposing the boundary conditions of bimaterial blunt V-notches, the asymptotic solutions of the stress and displacement fields are obtained. Due to the convenient selection of the boundary conditions, the solution is shown to be the same as the one proposed by Williams’ for sharp V-notches, in case the notch tip radius becomes zero. Finally, finite element method is employed to benchmark the precision of the suggested asymptotic solution. Some sample examples are selected, and the results obtained by the proposed analytical solution are compared with the numerical results of the finite element method. It is demonstrated that the proposed stress field is not only applicable to obtain the stress distribution around the blunt V-notches but also is suitable for stress field calculation around some other types of bimaterial notches.

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.

Mixed mode equilibrium analysis of asymptotic fields around stationary sharp V-notches in polymer gels: A linear poroelastic study

In this paper, the equilibrium asymptotic stress and solvent concentration fields around stationary sharp V-notches are extracted for in-plane mixed mode loadings employing the linear poroelasticity theory. It is shown that at thermodynamic equilibrium, mechanical and chemical equilibrium equations are uncoupled and thus can be solved independently. The mechanical equilibrium equations are then solved using the Airy stress function technique. This technique replaces the coupled mechanical equilibrium relations with the compatibility biharmonic equation which can be solved using the separation of variables method. Applying traction free boundary conditions on the notch edges leads to an eigenvalue problem which gives both mode I and mode II eigenvalues. These eigenvalues can be real or complex numbers depending on the notch opening angle. Since there are infinite eigenvalues for both modes of deformation, series solutions will be obtained for the equilibrium fields. From these asymptotic solutions, it is found that the obtained stress fields are similar to their corresponding linear elasticity solution. Furthermore, from the solutions, it can be observed that the opening and shear deformations near the notch tip lead to cosine and sine variations of the solvent concentration field with respect to the angular coordinates, respectively. The accuracy of these asymptotic results is finally verified using finite element analyses of a single-edge notched (SEN) specimen subjected to far field applied small displacements. The numerical analyses are performed for two notch opening angles of 30° and 60° for both plane stress and plane strain conditions. The comparative study between the finite element results and the asymptotic solution proves that the present solution can accurately capture the near notch tip stress and solvent concentration fields by considering only the first few terms of the series solutions.

Analysis of Linear Elastic Fracture Mechanics for Cracked Functionally Graded Composite Plate by Enhanced Local Enriched Consecutive-interpolation Elements

This paper reports the application of consecutive-interpolation procedure into four-node quadrilateral elements for analysis of two-dimensional cracked solids made of functionally graded composite plate. Compared to standard finite element method, the recent developed consecutive-interpolation has been shown to possess many desirable features, such as higher accuracy and smooth nodal gradients it still satisfies the Kronecker-delta property and keeps the total number of degrees of freedom unchanged. The discontinuity in displacement fields along the crack faces and stress singularity around the crack tips are mathematically modeled using enrichment functions. The Heaviside function is employed to describe displacement jump, while four branch functions being developed from asymptotic stress fields are taken as basis functions to capture singularities. The interesting characteristic of functionall graded composite plate is the spatial variation of material properties which are intentionally designed to be served for particular purposes. Such variation has to be taken into account during the computation of Stress Intensity Factors (SIFs). Performance of the proposed approach is demonstrated and verified through various numerical examples, in which SIFs are compared with reference solutions derived from other methods available in literatures.

Asymptotic stress fields near the crack tip in perfect plastic materials under mixed mode loading (plane strain conditions)

In the paper 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 are presented. This objective is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities for plane strain 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 analysed. 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 0.89 respectively for plane strain 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.

Mixed mode fracture in perfect plastic materials for plane stress conditions

In the study the asymptotic stress fields in the neighbourhood of the crack tip in perfectly plastic Mises materials under mixed mode loading for the full range of the mode mixities are presented. This objective is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities for 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 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.

Nanofracture of stretched hexagonal boron nitride strip with an edge crack

The tensile fracture behavior in hexagonal boron nitride (h-BN) strip with an introduced zigzag edge crack is investigated by molecular dynamics (MD) using a modified Tersoff potential. The far field remote stress σremote at crack initiation for various original crack lengths is examined, which shows that Griffith criterion overestimates σremote for short cracks < 2.5 nm while underestimates that for longer cracks. Thus we propose a modified exponent factor 0.28 for Griffith criterion rather than 0.5. The asymptotic stress field around nano-crack is calculated by MD atomic virial stress theory, comparing with continuum theory of stress intensity factor K and stress concentration factor, which indicates that elastic continuum overestimates local stresses around the crack front as the bond breaking stress remains ~ 54.4 GPa at the tip. The critical energy release rate Gc at initiation for small cracks < 1.5 nm is also much higher than theoretical toughness GIc. Our results provide a clear understanding of the nano-crack fracture in a h-BN strip for the continuum limitation in discrete hexagonal lattices.

Asymptotic stress fields in the vicinity of the crack in perfectly plastic solids under mixed mode loading

In the paper presents the asymptotic stress fields in the vicinity of the crack tip in perfectly plastic Mises materials under mixed mode loading for a full range of the mode mixities. This objective 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 within 0.1 were considered and analyzed. The asymptotic solution to the statically determinate problem is obtained using 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 0.89 respectively for plane strain conditions and seven sectors determine the asymptotic solution for the mixity parameter less than 0.39, while 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.

A new unified asymptotic stress field solution for blunt and sharp notches subjected to mixed mode loading

This paper presents a new relation for the stress field around blunt and sharp notches in a linear elastic body. The stress components are determined by using Kolosov-Muskhelishvili's approach and considering proper complex potential functions. Then, to reduce the number of free parameters in the stress components, the boundary conditions of blunt V-notches are satisfied without applying a conformal mapping method. After replacing the free parameters, the stress components are calculated in the form of two series expansions related to mode I and mode II loading. In the case of sharp V-notches, it is demonstrated that the eigenvalues of the problem are the same as those of the well-known Williams’ solution. Afterwards, the stress intensity factors of blunt V-notches are presented by utilizing two different approaches. In order to evaluate the accuracy of the proposed relation, finite element analysis is conducted on a number of different types of notches and the results are compared with those of the presented and classical solutions. It is shown that the proposed relation is more accurate than the best available solutions for blunt V-notches. Furthermore, the proposed stress field solution is capable of specifying the stress distribution around some other types of notches with proper accuracy. Finally, some advantages of the proposed stress relations in comparison with other stress field solutions are presented and discussed.

Dynamic crack modeling and analytical stress field analysis in single-crystal silicon using quantitative fractography

Single-crystal silicon is the fundamental building block enabling today’s plethora of integrated electronic components. However, complex mechanical stresses, originating from either direct mechanical loading or thermal cycling, can result in fracture of the constituent silicon, one of the leading causes of semiconductor device failure. Although phenomenological relationships to estimate the fracture strength in silicon have been proposed in the past, no quantitative fractographic method addressing the intrinsic anisotropy of crystals exists. In this work, a fractographic approach using optical confocal microscopy and atomic force microscopy is developed to identify and analyze the cleavage planes associated with dynamic instabilities in single-crystal silicon. We analytically determined the dynamic crack propagation behavior and asymptotic stress field at the crack-tip for unstable, anisotropic, circular cracks in silicon. The fractographic features predicted by the analytical model are consistent with experimental observations, and correctly predict how the {111} planes define the fractographic mirror region (including mirror constant), as well as the {112} planes associated with fractographic Wallner lines. These findings have important consequences in reducing mechanical and thermomechanical failure in semiconductor devices, where the mechanical strength is highly dependent on crystallographic anisotropy.

Large deformation near a crack tip in a fiber-reinforced neo-Hookean sheet

The asymptotic fields at the tip of a crack in a fiber-reinforced neo-Hookean sheet are derived. The investigation is carried out for the case of a strain energy function for a fiber-reinforced hyperelastic material motivated by composite mechanics (Guo et al., 2006, 2007ab), where the fibers are also neo-Hookean. The resulting asymptotic deformation and stress fields depend qualitatively and quantitatively on the degree of fiber reinforcement. For suitable choice of parameters, the strain energy potential for the material reduces to that of a pure neo-Hookean material and the corresponding asymptotic fields to those obtained by Knowles and Sternberg (1983). The result obtained may prove useful in providing a framework for future exploration in modeling and assessing the mechanical behavior near a slit or tear in soft biological tissue reinforced by collagen fibers and in other applications of fiber-reinforced soft materials.

On applicability and uniqueness of the correspondence principle to pie-shaped wedge (“wedge paradox”) with various boundary conditions

A new paradigm for re-examination of the concept of the correspondence principle (originally due to Dundurs) in association with the much-studied issue of “wedge paradox” is presented. This study connects for the first time two parallel streams of research: (i) Carothers-Sternberg-Koiter’s wedge paradox study and (ii) Williams' eigenfunction approach. First, a three-dimensional eigenfunction expansion approach for determination of the asymptotic (singular) stress fields in the vicinity of the fronts of infinite wedges of finite thickness, subjected to various wedge-flank boundary conditions, is presented. This analysis is then exploited to establish rigorous rules or conditions, which would depend on the prescribed wedge-flank boundary conditions, the related state of deformation and the elastic property (Poisson’s ratio), for validity (or lack thereof) of the assumptions implicit in the heuristic enunciation of the correspondence principle. More important, a hitherto unavailable analysis for uniqueness (or lack thereof) of the prevalence of such a heuristically postulated correspondence is also developed. Finally, the present study offers both mathematical as well as (heretofore missing) physical explanations behind the working of the correspondence principle.

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.

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.

Elastic stress analysis of blunt V-notches under mixed mode loading by considering higher order terms

This paper presents the in-plane asymptotic displacement and stress fields for blunt V-notched components based on Kolosov–Muskhelishvili's approach. In the first part, the displacement and stress components in the polar coordinate system are determined by choosing appropriate complex potential functions. In order to construct the notch geometry, the Neuber's mapping relation is utilized. Then, the notch boundary conditions are imposed to calculate the free parameters of the stress distribution. Eventually, the stress and displacement components are calculated in the Cartesian and polar coordinates in the forms of series expansion. In the second part, the coefficients of series expansions are computed by using the least square method (LSM). The blunt V-notched Brazilian disk (BV-BD) specimen under mixed mode loading is used as an example to verify the proposed procedure. The stress components in arbitrary distances and directions are determined for different blunt V-notches in order to evaluate the accuracy of the calculated stress series solutions and their associated coefficients. The numerical results indicate that a single-term solution can lead to considerable errors, and to achieve good accuracy in the stress field calculation, one should take account of at least three terms in the stress series solution.