Linear Elastic Stress Field Research Articles

Using fractal voids to understand Mode I compressive fracture in brittle materials – A two-dimensional analysis

Unlike cases of direct or indirect tensile loading, analysis of Mode I fracture in cases of compressive loading must necessarily consider two-dimensional flaw geometries. The consequent analytical complications have resulted in little theoretical evolution towards understanding Mode I fracture in compressive loading. Researchers have recently observed that the linear elastic (LE) stress fields surrounding two-dimensional flaws in compression appear to have indicative value regarding Mode I fracture, but no rational framework has yet been proposed for their interpretation. Herein it is demonstrated that by idealizing void surfaces as fractal, and using what will be called the “small flaw assumption,” LE stress fields can be interpreted to obtain significant insight regarding brittle Mode I compressive fracture. Using fractal voids, a rational and consistent explanation for the satisfaction of the stress and energy criteria at the surface of two-dimensional flaws is presented. The discussion resolves many typical theoretical issues in the literature in a manner consistent with fundamental Griffith theory, and accounts for size and shape effects. The framework further enables the computationally efficient prediction of peak KI values associated with the propagation of line cracks from two-dimensional flaws from a brittle medium subject to macroscopic uniaxial compression via LE stress fields.

Assessing notch fatigue performance of aluminium alloy AA2024-T4 after uni-axial and equi-biaxial tensile pre-straining: Experimental observations and predictive models

This study delves into the effect of different tensile pre-straining (uni-axial and equi-biaxial pre-strain) modes on the notch fatigue performance of AA2024-T4. By conducting a series of fully reversed stress-controlled high cycle fatigue (HCF) experiments, pre-strained samples demonstrated notable improvements in HCF resistance, surpassing those without pre-strain. A stress-based fatigue life prediction approach, namely the theory of critical distance (TCD), is applied. We propose an experimentally validated physics-based method for predicting notch fatigue life in the HCF regime, utilising the cyclic plastic zone (CPZ) concept and TCD’s point method. The CPZ commonly plays a significant role in both crack initiation and propagation at the notch roots, thereby altering the stress in the vicinity under cyclic loading. However, TCD considers a linear-elastic stress field around the notch root for fatigue life calculation. The model successfully predicts HCF life based on experimental results and gives better prediction than the conventional TCD approach in terms of prediction accuracy, which falls within a range of ± 2 the experimental results for all the pre-strain conditions.

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 local approach for fracture analysis of sharp notches under Mode I loading

On the basis of the linear elastic notch stress fields, the local approach for fracture analysis of sharp notches under Mode I loading is proposed in this study. In the local approach, the stress concentration factor eigenvalue is introduced. For any sharp notch (V-notch), the geometric material characteristic parameters (GMPs) corresponding to the pointed V-notch are calculated by using an approach of determining the eigenvalue and models such as notch angle effect on the eigenvalue, notch depth effect on the eigenvalue and different material effect on the eigenvalue. It is seen from this study that fracture analysis for the sharp notches is well performed by using the obtained GMPs and the local stress field failure model proposed in this paper with a high accuracy within an error interval of a few percent. Taking into account that the fracture toughness is unknown at some circumstances, an empirical equation of determining is proposed in this study. By using the empirical equation, the local approach proposed in this study has a wide of applications such as fracture analysis of the rounded-tip V-notched Brazilian disc (RV-BD) specimens and fracture analysis of center circular hole plates made of metallic materials.

A local approach for fracture analysis of V- notch specimens under Mode-I loading

By using the concepts of the critical stress and the critical distance by Taylor [Taylor D.,The theory of critical distances: a new perspective in fracture mechanics. Oxford, UK: Elsevier; 2007], in this study, a local approach for fracture analysis of V- notch specimens is proposed based on the linear-elastic stress field ahead of V- notch. In the local approach, the size of the critical distance is not taken to be a material constant, whereas it is assumed that the critical distances of different notches are different and are connected closely with the variation of notched geometry. By using fracture test data of V- notches made of both brittle and ductile materials from the literature, the local approach has been verified to be both simple in computation form and highly accurate. Concerning that the fracture analysis for V- notched specimens made of ductile materials can be well performed by the local stress field failure model proposed in this study, an explanation description given in this study is that, if fracture analysis for V- notched specimens made of ductile materials can be well performed by a stress-based intensity parameter, the stress-based intensity parameter can be calculated by the Linear-Elastic Analysis without the need for carrying out complex and time-consuming Elastic-Plastic analysis and in addition some experimental verifications given recently by the author are stressed.

On the Internal Enrichment Implementation for Non-Convex Paths, Discontinuities and Crack Problems

Studies on the cracked plates have shown high variations in thestress values around the crack tip. On the other hand, micro-cracks are observed in man-made pieces. Thus, analysis of stress fields as well as displacement at the crack tip will be inevitable and very important. Mesh-free methods are new techniques that do not require applying the communication-based concept on what is presented in the finite element method. The discretization of the problem domain is done by a set ofnode points. In the present study, the Element-Free Galerkin (EFG) method was used to analyze the problems of linear elastic stress field in cracked bodies. Two important and essential measures (steps) were done for increasing the accuracy of the results obtained from the analysis of the problems. In the first step, the standard moving least squares shape function was enriched for capturing discontinuity, using some extra basis functions obtained from analytical solutions. In the next step, some consideration was applied in the case of confronting non-convex paths and discontinuities. For this purpose, the diffraction method was used to generate suitable shape functions. Finally,the accuracy of the results and proper efficiency of the proposed extended EFG method were assessed by the standard problem analysis and the results of numerical analysis were compared with the theoretical results.

Combined notch and size effect modeling of metallic materials for LCF using plasticity reformulated critical distance theory

Fatigue assessment of notched parts plays imperative performance in structural integrity and reliability of major equipment. The theory of critical distance (TCD) involves determining the fatigue strength of notched parts from the linear elastic stress field surrounding the notch apex. In addition, the fatigue strength of engineering part is also size dependent. Accordingly, by coupling a plasticity reformulated TCD with the weakest link theory, this study elaborates a novel scheme for probabilistic low cycle fatigue (LCF) analysis under notch and size effects. Results of fatigue tests and complementary FE analysis with the proposed model is utilized to compute highly stressed volume (HSV). Furthermore, test data of titanium alloys TC4 (Ti-6Al-4V) and TC11 (Ti-6.5Al-3.5Mo-1.5Zr-0.3Si) as well as aluminum alloy 7075-T6 are utilized for model verification. Results specify that the model predicted P–S–N curves are in good agreement with experimental data.

Effect of plastic deformation on the elastic stress field near a crack tip under small-scale yielding conditions: An extended Irwin's model

The effective stress intensity factor derived from Irwin’s model is greater than the linear elastic counterpart. This is in contradiction to the shielding effect of crack-tip plastic zone. Besides, the classic Irwin’s model underestimates the size of the crack-tip plastic zone compared with Dugdale’s model and FE analysis. In this paper, an extended Irwin’s model for elastic perfectly-plastic material under small-scale yielding conditions is developed through imposing static equivalence between the plasticity-affected stress field and linear elastic stress field in the plasticity-affected zone. The extended Irwin’s model is capable of quantifying the effect of the crack-tip plastic zone on the crack-tip elastic stress field. It reduces to the classic Irwin’s model in the limiting case. This new version of Irwin’s model provides new insights in elucidating the effect of the crack-tip plastic zone on fracture behavior.

Linear-elastic stress field of notched concrete beam: An application of finite element in theory of critical distances

Fatigue failures occur in all structures including brittle and ductile structures. However, the study on fatigue in ductile material like metal and steel has tremendously developed compared to brittle material such as concrete. Fortunately, the Theory of Critical Distance (TCD) is witnessed successfully assess fatigue fracture in concrete. In obtaining one of the outputs using TCD which is critical distance, fatigue limit of concrete that is obtained through laboratory testing and stress field generated using computational analysis engineering software (CAE) are required. In this article, the concern will be on producing the valid and reliable stress field data since inaccurate input into the CAE will result unreliable output that is exposed to errors. In order to guarantee the result is accurate, validation works were conducted in preprocess and post-process phase while analysing finite model using ABAQUS. As the outputs comply accordingly based on the validation works, the critical distance is confidence to be consumed for the subsequent research related to TCD.

Evaluation of generalized stress intensity factors at blunt V-notch tip in polymer materials using transmitted caustics

The paper deals with the notch stress intensity factors (NSIFs) at a blunt V-notch tip in polymer materials using transmitted caustics. First, the governing equations of transmitted caustics which relate to the experimental measurements and the linear elastic stress fields at blunt V-shaped notches are established. Second, a series of caustic curves at the blunt V-notch tip under mode I loading are simulated. Finally, fracture experiments of polymer materials with a blunt V-shaped notch under three-point-bend loading are performed. The NSIFs obtained from caustic measurements show a good agreement with numerical results.

Assessing variable amplitude multiaxial fatigue lifetime of notched components based on the notch critical plane approach

In the present paper, the notch critical plane method proposed by the authors is extended from 2D- to 3D-cases. The plane passing through the assumed critical point and experiencing the maximum variance of the resolved shear stress is defined as the the notch critical plane. The linear-elastic stress fields around the notch root are fitted via polynomial functions. The relationship between critical distance, l, and fatigue damage, Db, under variable amplitude loading blocks is proposed. Susmel’s parameter and a specific multiaxial cycle counting method are incorporrated into this approach to estimate multiaxial variable amplitude fatigue life. Numerous fatigue data generated by testing five different notched metallic materials were collected from the literature to check the accuracy of the proposed approach. This validation exercise allowed us to demonstrate that this method is highly accurate, with the majority of the predicted experimental results falling within an error band of 2.

Estimating static/dynamic strength of notched unreinforced concrete under mixed-mode I/II loading

The Theory of Critical Distances (TCD) is a powerful design tool capable of estimating the strength of notched/cracked materials, with this being done by directly post-processing the linear-elastic stress fields ahead of the stress raisers being assessed. In the present study, an advanced formulation of the TCD is devised to specifically predict static and dynamic strength of notched unreinforced concrete subjected to Mixed-Mode I/II loading. The reliability and accuracy of the design approach being proposed was checked against a large number of experimental results generated by testing plain concrete containing notches of different sharpness, with these experiments being run not only under various degrees of Mode mixity, but also under different values of the nominal displacement rate (i.e., in the range 0.002–35 mm/s). The predictions made by this advanced version of the TCD were seen to fall mainly within an error interval of ± 30%, that is, within an error band as wide as the band characterizing the intrinsic scattering of the calibration data. This suggests that that the TCD philosophy can effectively be extended also to the assessment of notched plain concrete subjected to in-service static/dynamic Mixed-Mode loading, with the relevant stress fields being determined by modelling concrete as a linear-elastic, homogeneous and isotropic material.

A novel tearing residual strength model for architectural coated fabrics with central crack

This study concerns the tearing residual strength models of architectural woven coated fabrics. Firstly, central crack tearing tests were conducted, the evolution of tearing residual strength with the increase of initial crack length is investigated. Secondly, the commonly used tearing residual strength models, including linear elastic fracture mechanics (LEFM) models and stress field models, are summarized and analyzed. Due to their inherent defects in theoretical foundation, the prediction precision is merely determined by curve fitting, no physical meaning can be drawn from these models. Finally, an asymptotic model based on fracture mechanics theory is proposed. The relative size of fracture process zone (FPZ) is adopted to distinguish the transition from strength-controlled fracture to quasi-brittle fracture and finally to toughness-controlled fracture. The prediction results agree very well with the test data, and most of test data lies in nonlinear fracture range which has never been considered.

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.

Failure Prediction of AISI 420 Martensitic Stainless Steel Using the Theory of Critical Distances

This work aims to evaluate the capability of the theory of critical distances (TCD) to predict the static failure of U-notched AISI 420 martensitic stainless steel specimens with different geometric features under pure bending loading. Theoretical estimates of the stress intensity factor during fracture onset were calculated according to the line (LM) and point methods (PM), which consider the characteristic length L, inherent strength σ0, and notch tip radius ρ. Initially, L and σ0 were determined on the basis of the material’s properties (i.e., fracture toughness KIc and ultimate tensile strength σu,t), resulting in imprecise estimates. Conversely, L and σ0 determined using the appropriate analysis of linear–elastic stress fields ahead of notches with different sharpness provided highly accurate predictions. The microscopic study of fractured specimens ensured better comprehension of the results. Moreover, the accurate values of L and σ0 were used to predict the failure of V-notched specimens.

The Theory of Critical Distances to Predict Static and Dynamic Strength of Notched Plain Concrete under Mixed-Mode I/II Loading

The Theory of Critical Distances (TCD) is a powerful design tool capable of estimating the strength of notched/cracked materials by post-processing the linear-elastic stress fields ahead of the relevant stress raisers. The purpose of this paper is to reformulate the TCD to make it suitable for predicting the static/dynamic strength of notched unreinforced concrete subjected to Mixed-Mode I/II loading. The accuracy and reliability of the new extension of the TCD were checked using a large number of experimental results generated by testing plain concrete containing different geometrical features and tested under different loading rates and loading multiaxility. The predictions based on the proposed approach were seen to be within an error interval of ±30%. This level of accuracy is acceptable because it is within the scattering level of the experimental results used to calibrate the approach. These findings are promising and proving that this new reformulation of the TCD can be used to design notched plain concrete by modelling concrete as a linear-elastic, homogeneous, and isotropic material.

A gradient-based multiaxial criterion for fatigue crack initiation prediction in components with surface roughness

The current study presents methods to predict the governing crack initiation site and fatigue crack initiation life of components with surface roughness. The surface topography is measured with white light interferometry and explicitly accounted for in detailed finite element models. The micro-notch stress fields are used in multiaxial and uniaxial crack initiation criteria where the relative stress gradient is included. The numerical predictions are compared with test results for cylindrical aluminum specimens with axi-symmetric surface roughness. Damage parameters based on the average stress fields over a certain distance were found to be highest in the micro-notches where cracks grew to failure. Lifetime predictions using a multiaxial damage criterion with a gradient correction and elastic–plastic stress fields showed good correlation with the experiments. Uniaxial criteria, criteria without gradient correction, and criteria based on linear elastic stress fields were found to be overly conservative. In some specimens, the failure location could not be identified by the proposed damage criterion. This is likely due to the presence of microstructural weaknesses near the micro-notches, leading to shorter initiation lives that cannot be described by geometry alone. It is concluded that resolving the detailed surface topography and accounting for this geometry in a detailed finite element model provide a predictive approach when multiaxial stresses are accounted for, but the importance of microstructure needs further attention.

The peak stress method to calculate residual notch stress intensity factors in welded joints

AbstractAccording to the recent literature, the intensity of linear elastic residual stress fields near the toe region of a welded joint can be quantified by the residual notch stress intensity factors (R‐NSIFs). The computational effort required to compute the R‐NSIFs implies strong limitations of applicability in practice, owing to the very refined meshes needed and to the non‐linear transient nature of welding process simulations, especially in 3‐dimensional numerical models of large structures. The peak stress method (PSM) is a design approach that takes care of the industrial needs of rapidity and ease of use. According to the PSM, it is possible to evaluate the R‐NSIFs by using the peak stress calculated at the point of singularity with coarse finite element (FE) models. While the PSM was originally calibrated by using the Ansys FE code, in the present contribution, the PSM has been calibrated to rapidly estimate the R‐NSIFs in the Sysweld FE environment.

The Theory of Critical Distances to assess failure strength of notched plain concrete under static and dynamic loading

The Theory of Critical Distances (TCD) is a design method that is widely used in situation of practical interest to estimate the strength of notched/cracked components subjected to either static, dynamic, or fatigue loading. The TCD makes use of a characteristic length to post-process the linear-elastic stress fields damaging the material in the vicinity of the stress concentrators being designed. The employed length scale parameter depends on the specific microstructural features of the material under investigation. By making the most of the TCD's unique features, the present paper summarises an attempt of reformulating this powerful theory to make it suitable for assessing static and dynamic strength of notched plain concrete. The accuracy and reliability of the proposed reformulation of the TCD is checked against a number of experimental results that were generated by testing, under different displacement rates, square section beams of plain concrete containing notches of different sharpness. This validation exercise allowed us to demonstrate that the proposed reformulation of the TCD, which is based on the use of simple power laws, is capable of accurately assessing the static and dynamic strength of the notched un-reinforced concrete being tested, with the estimates falling within an error interval of ± 20%. The obtained level of accuracy is certainly satisfactory, especially owing to the fact that static and dynamic strength is predicted without explicitly modelling those non-linearities characterising the stress vs. strain dynamic behaviour of concrete.

Creep behavior of V-notched components

Geometrical discontinues such as notches play a significant rule in structural integrity of the components, especially when the component is subjected to very severe conditions, such as the high temperature fatigue or creep. In this paper, a generalized form of the existing notch tip creep stressstrain analysis method developed by Nuñez and Glinka, is developed and extended to a wide variety of blunt V-notches. Assuming the generalized Lazzarin-Tovo solution that allows a unified approach to the evaluation of linear elastic stress fields in the vicinity of both cracks and notches is the key in getting the extension to blunt V-notches. Numerous cases have been analysed and the stress fields obtained according to the proposed method were compared with proper finite element data, showing a very good agreement.