Expressions For Stress Intensity Factors Research Articles

Strength analysis due to thermal loading and tensile loading when metals are bonded by heat-curing adhesives

Heat-curing adhesives are widely used after being cured by heating to a temperature higher than room temperature. To evaluate the adhesive strength, therefore, it is necessary to consider both the thermal stress generated during heat curing and external loads such as tensile stress. Butt joint specimens are essential for evaluating tensile adhesive strength but also thermal strength. The interfacial strength can be discussed from the stress intensity factor (SIF) of a fictitious edge interfacial crack assumed at the interface end. This is because the SIF is controlled by the intensity of singular stress field (ISSF) at the crack-free interface end and a constant term associated with the thermal load. In this paper, a useful thermal SIF solution is proposed by superposing the SIF under tensile stress and the SIF under uniform interface stress associated with thermal loading. This general SIF expression provided under arbitrary material combination can be applied for predicting the tensile strength σc and critical temperature change ΔT without performing new FEM calculations. The usefulness of the expression is confirmed through the adhesive strength of Aluminum/Epoxy butt joint experimentally obtained. Once the critical SIF K1C can be obtained from the tensile strength σc and the temperature change ΔT, the adhesive strength can be expressed as K1C = constant of an assumed fictitious interface, and this can be used to predict critical σc for various temperature change ΔT and for various adhesive bondline thickness h.

A closed form expression of stress intensity factor for an arbitrarily shaped planar crack in 3-D under tensile loading

A closed-form equation was developed in this work to quantify the stress intensity factor (SIF) along an arbitrarily shaped planar crack under mode-I loading by substituting the ellipse-related parameters in Irwin’s solution with geometry-related parameters fi of the crack. This allowed incorporation of the local crack curvature effect on SIF at each discrete point along the crack front and its relative weighted contribution to SIF of the neighboring crack point i, where either stress concentration or shielding took place, into computation of the SIF distribution along the crack front. Good agreement was obtained between the KI values derived in the current work and previous numerical simulation-based studies. The current method could capture the KI distributions for surface and embedded cracks having the same shape, with their maximum values being consistent with Murakami’s area method. This solution of KI allows for highly efficient calculation of ΔKI for quantification of the 3D growth behaviors of a fatigue crack with a complex shape, paving the way for studying the statistical nature of short fatigue crack growth.

Random Dynamic Responses of Two Parallel Interfacial Cracks Between A Functionally Graded Material Strip And Two Dissimilar Elastic Strips

Abstract An analytical approach is presented in this article for the random dynamic study of two parallel interfacial cracks in a functionally graded material (FGM) strip that is bonded between two distinct elastic strips. One of the parallel cracks is placed at the interface of the elastic strip I and the FGM strip, and another is at the interface of the FGM strip and elastic strip II. A stationary stochastic process of time is used to model the dynamic loadings that are applied to the crack faces. To find the solution, the FGM strip is splited into a number of sub-strips, and using an average method, the material properties of each sub-strip are reduced to random variables. A fundamental problem is formulated to find the solution. The boundary conditions are reduced to a set of singular integral equations employing the Fourier sine, Fourier cosine, and Laplace transforms, which are solved by using the Collocation method. Further, the analytical expressions of dynamic stress intensity factors (DSIFs) about the crack tips in the time domain are obtained with the help of the Improved Dubner and Abate's method. Finally, the Monte Carlo method is used to obtain the mathematical expectation and standard deviation of DSIFs. The outcomes of the present study are also verified. The unique aspect of this study is the pictorial illustration of mathematical expectation and standard deviation as functions of the number of sub-strips, functionally graded parameter, thickness of the strips, and length of parallel interfacial cracks.

Effect of quadratically varying electric potential on the multiple cracks in piezoelectric materials

A basic solution to the problem of three collinear equal cracks in a transversely isotropic piezoelectric material plate is obtained under the influence of electromechanical forces. Stroh formalism and complex variable methods are used to derive the mathematical expressions for SIF (stress intensity factor), COD (crack opening displacement), and COP (crack opening potential). The length of the saturation zones is investigated at each crack tip under the influence of quadratically varying electric displacement. Furthermore, a numerical study is carried out to discuss the effect of quadratically varying electrical potential on the saturation zone, COD, and COP. The results obtained numerically are reported graphically.

Magnetoelastic effect on semi‐infinite moving crack in an orthotropic structure influenced by anti‐plane wave

AbstractA two dimensional boundary value problem of magnetoelasticity for a semi‐infinite moving crack situated between an orthotropic half‐space and an orthotropic layer has been considered. Also, it has been considered that the crack is agitated by an anti‐plane shear wave. The contemplated problem has been reduced to a typical Wiener–Hopf equation by employing Fourier integral transform to the convenient boundary condition. The accomplished Wiener–Hopf equation has been solved only for asymptotic case to get the analytical expression of stress intensity factor (SIF) and crack tearing displacement (CTD). The dependence of several parameters including the magnetoelastic coupling factor, material constants, crack propagation velocity and crack depth from the upper surface on the SIF and CTD have been illustrated using variety of graphs.

The Two-Parameter Fracture Criterion Taking into Account Two-Dimensional Deformation Constraints at the Front of a Mixed-Type Crack

In this article, a new fracture criterion for a mixed-type crack (type I + type II) is proposed. It is based on the assumption that the maximum tangential stresses in the prefracture region are equal to the local strength of the material. In this case, the size of the prefracture area and the local strength are determined taking into account nonsingular Тхx and Тzz stresses included in the expansion of the Williams stress function. The use of Тхx and Тzz stresses in the calculations allows one to describe the two-dimensional local constraint of the deformation along the crack front in three-dimensional bodies. In addition to KI and KII, the expression for the effective stress intensity factor (SIF) includes the ratios of the Тхx and Тzz stresses to the yield strength. This makes it possible to take into account the constraint of deformations in the transverse and longitudinal directions of the crack front, respectively. An example of the implementation of the developed criterion in relation to the determination of the fracture load of a tensile-stressed plate with a through inclined crack is given. Dependences of the Тxx and Тzz stresses on the plate thickness are presented for various crack slopes and plate thicknesses. It is shown that an increase in the plate thickness and a decrease in the crack slope lead to a decrease in the fracture load.

The Two-Parameter Fracture Criterion Taking into Account Two-Dimensional Deformation Constraints at the Front of a Mixed-Type Crack

In this article, a new fracture criterion for a mixed-type crack (type I + type II) is proposed. It is based on the assumption that the maximum tangential stresses in the prefracture region are equal to the local strength of the material. In this case, the size of the prefracture area and the local strength are determined taking into account nonsingular Тхx and Тzz stresses included in the expansion of the Williams stress function. The use of Тхx and Тzz stresses in the calculations allows one to describe the two-dimensional local constraint of the deformation along the crack front in three-dimensional bodies. In addition to KI and KII, the expression for the effective stress intensity factor (SIF) includes the ratios of the Тхx and Тzz stresses to the yield strength. This makes it possible to take into account the constraint of deformations in the transverse and longitudinal directions of the crack front, respectively. An example of the implementation of the developed criterion in relation to the determination of the fracture load of a tensile-stressed plate with a through inclined crack is given. Dependences of the Тxx and Тzz stresses on the plate thickness are presented for various crack slopes and plate thicknesses. It is shown that an increase in the plate thickness and a decrease in the crack slope lead to a decrease in the fracture load.

Interaction of anti‐plane shear waves with two collinear cracks in 1D hexagonal piezoelectric quasicrystals

AbstractThe present article presents the interaction of anti‐plane shear waves by two collinear cracks in one dimensional (1D) hexagonal piezoelectric quasicrystals (PQCs). With the aid of Fourier transform, the mixed boundary value problem (MBVP) is transformed into three pairs of dual integral equations, which are solved analytically by Hilbert transform. The explicit expressions for dynamic stress intensity factors (DSIFs) of phonon and phason fields, crack opening displacement (COD) and electric displacement (ED) are derived in closed form and some special cases are studied. Numerical values of DSIFs of phonon and phason fields, COD and ED are plotted to show the effect of crack length and electric boundary condition. Moreover, the DSIFs of phonon field and COD are represented graphically for single crack.

SH-wave scattering by the interface crack of piezoelectric ceramic polymer composites

The scattering of SH waves by interfacial crack in piezoelectric ceramic polymer composites is investigated. The mixed boundary condition are reduced to dual integral equations using Fourier transforms, and the series and Schmidt methods are used to solve an unknown quantity in crack displacement, and the expression of stress intensity factor at crack tip is obtained. The method was verified by degrading the piezoelectric ceramic polymer to the piezoelectric ceramic material. Finally, the crack propagation under different conditions such as frequency and angle are analyzed, as well as the influence of different piezoelectric ceramic polymer materials on crack propagation. It is of great significance for fracture propagation. This method can also be used as a guide for SH wave scattering of different interface cracks in real life, such as multi-cracks, rectangular crack and penny-shaped crack, so it can also be used as a starting point for future research.

A logarithmic model for predicting fracture trajectory of pre-cracked rock specimen under compression

Researches on the fracture trajectory of pre-cracked specimen are mostly focused on experiments and numerical simulations, and theoretical studies are usually limited to the fracture criterion, crack initiation angle and load, with few theoretical studies on the prediction of the complete fracture trajectory. This work first derives stress intensity factor (SIF) and T-stress expressions related to crack geometry, friction and deformation parameters, and physical properties of the rock. Then, the classical maximum tangential stress (MTS) fracture criterion is modified, followed by the development and validation of a logarithmic model for predicting the fracture trajectory of pre-cracked specimen made of diverse brittle or quasi-brittle rock materials under compression. The results shown that the crack geometry, friction and deformation parameters have significant effects on SIF and T-stress under compression. The present study demonstrates a strong concurrence between the anticipated fracture trajectory of pre-cracked specimens subjected to compressive loading and diverse experimental outcomes. Furthermore, the fracture trajectory is predominantly influenced by the crack geometry, friction and deformation parameters, and the mechanical characteristics of the specimen.

Mechanisms of coupled mode I-II fracture initiation in rocks subjected to heat flow-induced fracture gas-steam pressure and heat flow stress

The construction of tunnel projects in the Sichuan-Tibet region has led to the occurrence of rock fractures caused by high temperature and high humidity, which are inevitable challenges. When there is fracture water in rock fractures and it encounters a heat flow, the water vaporizes into fracture steam, which undergoes thermal expansion to generate fracture gas-steam pressure. The heat flow direction is at an angle to the crack, causing heat flow stress in the rock. To understand the mechanism of crack initiation and propagation in rocks under the joint action of uniform heat flow-induced fracture gas-steam pressure and heat flow stress, the control equation for fracture gas-steam pressure is derived and expressions for stress intensity factor, crack temperature field, stress field, and crack initiation strength criterion are established based on the theory of complex variable functions. The key parameters are analyzed and discussed, finding that under low heat flux density, small incidence angles, and large crack inclination angles, KI values are higher, and the crack initiation is mainly in tension. Under high heat flow density, large incidence angles, and crack inclination angles near 45°, KII values are higher, and the crack initiation is mainly in shear. The coupling effect of KI and KII is more sensitive to heat flow density and inclination angle. The critical initiation angle and initiation strength increase with the increase of heat flow density and incidence angle, with heat flow density being the determining factor.

A Fracture Criterion with Biaxial Constraints of Deformations along the Front of a Normal Rupture Crack

In this article, a new fracture criterion is formulated for a normal rupture crack, the most common in practice, based on the assumption that the tangential stresses in the prefracture zone are equal to the local strength of the material. In this case, the size of the prefracture area and the local strength are determined taking into account the nonsingular Тхx and Тzz stresses included in the asymptotic stress distribution according to Williams and characterizing the two-dimensional local constraint of deformation along the crack front in three-dimensional bodies. An expression for the effective stress intensity factor is obtained. In addition to the classical stress intensity factor, it includes the ratios of the Тxx and Тzz stresses to the yield strength. This makes it possible to take into account the restriction of deformations in the transverse (due to Тxx stresses) and longitudinal (due to Тzz stresses) directions in the vicinity of the crack front. Verification of the developed software tools and the proposed fracture criterion has been carried out. Examples of the implementation of the developed criterion for assessing crack resistance of a plate stretched in one or two directions with a coaxial transverse crack are given.

The Improved Irwin’s Model for Crack Problems in Power-Law Hardening Materials

In this paper, an improved Irwin’s model for power-law hardening materials is established through the static equivalence between the linear elastic stress solution and the stress redistribution in the plasticity-affected zone (PAZ) under small-scale yielding (SSY) conditions. HRR solution is adopted to describe the stress field within the crack-tip plastic zone (CTPZ). Analytical expressions of the effective stress intensity factor (SIF) and the size of PAZ are derived to illustrate the effect of CTPZ on the fracture behavior for power-law hardening materials. Influences of external load and hardening exponent on fracture behavior are explored. The proposed model can assess the toughening effect of CTPZ accurately using the SIF ratio, which is always smaller than 1. This model can be applied to various materials obeying power-law hardening constitutive equations. It will reduce to the extended Irwin’s model for elastic-perfectly plastic materials in the limiting case.

Coupled crack tip fields under the action of an electric current

Coupled fields of crack tip under the action of electric current is studied by means of theoretical analysis, experimental investigation and numerical simulation. The distribution of electric field, temperature field and thermal stress field near the crack tip is analyzed theoretically when electric current is applied to a metal plate with an edge crack, and the theoretical calculation expression of thermal stress intensity factor is derived. The current treatment process of Al 5052-H32 alloy plate with fatigue crack is investigated by experiment and finite element simulation. The results show that current density and crack length have significant effects on temperature, thermal compressive stress and thermal stress intensity factor near the crack tip. Through the current treatment experiment of Al 5052-H32 alloy plate with fatigue crack, it is found that micro welding is occurred due to temperature concentration and thermal compressive stress around the crack tip. The simulation results also show that there is a high heat flux at the crack tip due to the concentration effect of current flow around. The annular thermal compressive stress caused by temperature gradient is formed around the crack tip to prevent crack propagation and help crack healing. The simulation results are in good agreement with the theoretical and experimental results.

A Method on Estimates of Stress Intensity Factors of Outer Circle and Inner Ellipse Submarine Pipeline Under Bending Moment

The slender circle submarine pipeline possesses both shell and beam characteristics, which are widely used in practical engineering. Unfortunately, for some reason, there will be some geometric defects in the cross-section of the pipeline (such as machining errors and seawater corrosion, etc.), resulting in the stiffness of the circular submarine pipeline being different. The cracked variable stiffness pipeline is actually a typical three-dimensional complex crack problem. In this paper, taking the outer circle and inner ellipse submarine pipeline as an example, a method based on elementary mechanics and the concept of the conservation law is proposed to determine the stress intensity factors (SIFs) of variable stiffness pipeline, and a series of closed-form expressions of SIF are derived for the cracked pipeline. The results from the present method are shown to agree well with available solutions and FEM. In particular, the present method can also be easily applied to other similar variable stiffness submarine pipelines.

A modified formula for cyclic compression crack growth model considering compressive load effect

Fatigue crack growth under cyclic compression has been confirmed by experiments and widely exists in engineering applications, with the expression of stress intensity factor often serving as milestones. A typical difficulty in calculating the stress intensity factor under cyclic compression is the effect of compressive load on fatigue crack growth. Compressive load effect, resulting in the change of plastic zone and residual stress near crack tip, is a long-term challenge for predicting fatigue crack growth life. We modify the expression of effective stress intensity factor to predict fatigue crack growth life based on the distribution of residual stress, and analyze the effect of load characteristics. It considers the characteristic of material and load, and calculates the driving force of crack growth by integrating the residual stress in plastic zone. In the study of fatigue crack growth experiments under cyclic compression, the modified formula is proved to improve, by comparative analysis, the predictive ability on the whole range of experiment data. Consequently, the modified formula is conducive to calculate the fatigue crack growth.

Prediction of multiple fatigue crack growth based on modified Paris model with particle filtering framework

This paper presents a novel method based on modified Paris model with particle filtering (PF) framework to prognosis fatigue multi-cracks in metal structures. Since conventional Paris model is only applicable for the single crack, modified Paris model is proposed by introducing the mutual interaction between multiple cracks into stress intensity factor (SIF) expression and utilized to portray multiple fatigue crack growth process. To accurately monitor multi-cracks lengths, Lamb waves are periodically acquired during the fatigue growth of multiple cracks and fed into a deep autoencoder (DAE) network to automatically track response signals variations. Online monitoring model of multi-cracks lengths is then constructed by fitting the mapping relationship between the deep damage feature extracted by the bottleneck layer of DAE network and crack length. PF framework is further adopted to strategically fusing the prediction results of modified Paris model and real-time measurements of online monitoring model to reduce the uncertainties of material parameters and obtain more reliable prognosis results of multi-cracks lengths. The proposed method is demonstrated on center-hole metal specimens with two fatigue cracks. Experimental results show that the proposed method can accurately prognosis two fatigue cracks lengths, in which modified Paris model could more accurately describe multiple fatigue crack growth than conventional Paris model, and online monitoring model based on deep damage feature could more accurately track the fatigue crack length than artificial damage feature.

Basis for calculation the residual mechanical properties of corroding bars

AbstractCorrosion of steel reinforcement is known to affect the residual bearing capacity of the structure. The loss of metal supposes a decrease of the sectional area of the reinforcing bar, either homogeneously distributed or concentrated in some zones. The morphology of the corroded area is a critical issue because while the uniform loss is influencing the steel strength, when it is localized also the ductility is reduced due the local stress concentration. There are previous publications showing the impact of corrosion on some mechanical aspects of the steel as on the ductility, the strength or the elastic modulus, but still we lack of a comprehensive framework of all types of corrosion. In this work a classification is made of the types of damages for better assessing their consequences. Further, through application of basic classical mechanics are given expressions to calculate the “corrosion factor” related to the yield strength, the ultimate strength, the elastic modulus and the elongation (ductility). Finally, through fracture mechanics and an energetic approach are given the expressions for the critical stress intensity factor and the “apparent” residual section “loss” of the corroded steel bars.

Linear crack initiation analysis on rock surface under the combined action of sub-elevated temperature stress and fracture air-vapor pressure

Near the surface of the Sichuan-Tibet line tunnel surrounding rock under construction is in a sub-hot and humid environment, and long-term accumulation will produce damp-heat diseases. The significant temperature difference leads to the thermal stress of the surrounding rock, and the surrounding rock fracture water vaporizes to create fracture air-vapor pressure, which induces crack initiation. The state equation of gases, thermoelastic mechanics, and the theory of linear elastic fracture mechanics are used to study the crack initiation law of rock under the joint action of sub-elevated temperature stress and fracture air-vapor pressure. The stress intensity factor's expression under the two joint actions is derived from the superposition principle. Then the initiation criterion considering both internal influencing factors (rock properties, crack geometry parameters (inclination angle)) and external influencing factors (far-field stress, temperature, humidity factors) is derived. The theory is used to quantitatively analyze the distribution law of tangential stress at the crack tip and the crack initiation form. The effects of each parameter on critical crack initiation angle and crack initiation strength are discussed. The theoretical results are further verified by numerical simulation. The results show that in the sub-elevated temperature range, the thermal stress and the fracture air-vapor pressure jointly affect the crack initiation law, and thermal stress is the dominant factor. With the temperature increase, KI >KII, the critical crack initiation angle increases gradually, the crack initiation form changes from shear to compression-shear, and finally tends to tension. The fracture air-vapor pressure promotes the tension part. The maximum tangential stress increases linearly with temperature. The maximum tangential stress and critical crack initiation angle increase first and then decrease with the rise of crack inclination angle and reach the maximum at 45°. The research results supply a theoretical basis for preventing and treating the rock crack propagation surrounding the tunnel under sub-elevated temperatures and high humidity.

A novel coupling approach for determination of stress intensity factor for bi-material Reissner plates under bending or twisting

A novel coupling approach is proposed to determine stress intensity factors (SIFs) for interface cracks in bi-material Reissner plates under bending or twisting. The Reissner plate theory is adopted to derive asymptotic solutions around the crack tip and T-spline functions are introduced to establish the model. The approach has two major steps. Firstly, the overall cracked plate is divided into two regions (near and far fields) and is discretized by the T-spline based isogeometric analysis (IGA). Subsequently, the large number of fundamental unknowns of control points in the near field is reduced into a small set of undetermined coefficients of asymptotic solutions obtained by the eigenfunction expansion method. Therefore, explicit expressions of SIFs and singular stress components near the crack tip are obtained simultaneously. A number of numerical examples for various cracks laying at the interface are presented, and are found to be in good agreement with available solutions.