A concise review about fracture assessments of brittle solids with V-notches

Abstract

A concise review of recent studies about the fracture assessments of elastic brittle solid materials containing V-notches is presented. In this preliminary and brief survey, elastic stress distributions in V-notched solids are discussed first. The concept of notch stress intensity factor is introduced. Combine the digital image correlation method with numerical computation techniques to analyze the stress distribution near the notches. Fracture criteria such as strain energy density, J-integral, theory of critical distance are used.

However, various new materials are developed in different engineering fields, thus, the establishment of reliable and accurate material strength theory or failure criterion is imperative. Therefore, predicting fracture for various modern materials would require more experiments to infer material dependent parameters in the local fracture model.

Keywords

fracture / assessment / brittle solids / V-notches / review

Introduction

Fracture assessment is one of the key concerns in structural analysis as well as machine element design. It is well known that the accurate estimates of the crack tolerance within the structures is of the vital importance for not only the safety but also the economic requirements [¹–³].

It is difficult to discuss fracture mechanics research. Identifying major breakthroughs usually requires the passage of time, what seems important today may be obsolete later, while a major discovery may be overlooked when it is first published. It is possible, however, to identify a few trends in recent works [⁴].

Current fracture mechanics research about solids containing V-notches, will be briefly reviewed.

Up until now, the problems of elastic solids containing sharp cracks are basically completed. However, for elastic bodies containing V-notches, the investigations are in an infant stage [⁵]. Methods of elastic solutions about V-notches are described below. And fracture criteria of cracked bodies are discussed, a preliminary and short survey of research including strength criteria applications in rock mechanics and continuous fiber-reinforced composite materials is presented as well.

Stress distribution in elastic solids containing V-notches

Fracture research about notch vertex was started by Wieghardt [⁶] and followed by Brahtz [⁷,⁸]. A completed solution about the wedge vertex stress distribution under various conditions was obtained by Williams [⁹] via eigenfunction expansion technique. Later, the same problem was studied by other researchers, such as: Karal and Karp [¹⁰], Kalandiya [¹¹], Rösel [¹²], Vasilopoulos [¹³], Savruk et al. [¹⁴], Seweryn and Molski [¹⁵].

The Airy stress function was used by Mori [¹⁶] to study the keyhole problem, and the stress solutions were expressed in the form of an infinity series [¹⁷]. Neuber [¹⁸] studied the keyhole problem and presented the theoretical stress concentration factor. In 1992, Kullmer [¹⁹] revisited the Neuber analysis and obtained the sharp crack stress intensity factor (SIF). The Mode-I stresses of a keyhole notch was also studied by Radaj et al. [²⁰]. Smith [²¹] analyzed the keyhole notch under Mode-III conditions. Also, in 2006, Kullmer and Richard [²²] studied structural components made of brittle materials with keyholes based on the Creager and Paris Eqs. [²³].

Using the Kolosov-Muskhelishvili [²⁴] approach, Zappalorto and Lazzarin [¹⁷] presented a closed form solution for the Mode-I and Mode-III stress distributions of V-notches with end holes.

At the notch tip, notch stress intensity factor (NSIF) is used to express Mode-I stress distributions. The stress components are a function of the NISF in Mode-II case. In engineering applications, the approximate solutions are good enough. Using the Kolosov-Muskhelishvili [²⁴] approach, the Mode-III problem has been solved and the maximum shear stress as well as the Mode-III NSIF are involved the Mode-III V-notch stress distributions.

Fracture assessments of V-shaped notches

Benthem [²⁵] studied the parabolic and hyperbolic notches by using classic Sherman-Lauricella integral equation [²⁴] to understand the asymptotic relation between the stress intensity and the stress concentration factors.

The NSIF is used to describe the asymptotic stress intensity fields ahead of sharp V-notches with zero radius and linear elastic conditions. Based on the consideration that the NSIFs as the extension of the conventional SIFs, both the Mode-I loading [⁹] and the Mode-III loading [¹⁵] indicated that the NSIFs with the units depend on the opening angle of V-notch and their intensity depend on the overall geometry of the component as well as the applied loads [²⁶].

After Gross and Mendelson [²⁷], the investigation of the NSIFs are reported in a number of papers, such as: Zhao and Hahn [²⁸], Chen [²⁹], Dunn et al. [³⁰,³¹], Lazzarin and Tovo [³²], Strandberg [³³,³⁴], Noda and Takase [³⁵], Zappalorto et al. [³⁶], etc. And the applications of elastic NSIFs to evaluate fracture toughness about components made of brittle materials are presented, such as: Knésl [³⁷], Nui et al. [³⁸], Gogotsi [³⁹], Gómez and Elices [⁴⁰,⁴¹], Gómez et al. [⁴²,⁴³]. In addition, in the estimation of fatigue crack initiation at notches and weld toes [⁴⁴,⁴⁵], the use of elastic NSIFs is reported as well.

In 2006, Gómez and Elices [⁴⁶] studied deep round notched components under static conditions, by combined the concept of generalized SIF with the cohesive zone model, the master curve for estimating fracture loads is established. The curve indicated the non-dimensional generalized fracture toughness as function of the non-dimensional notch radius. It shows that the experimental data from alumina, zirconia, and silicon ceramics (18 materials) including PMMA, are within the same scatter band [²⁶]. Ayatollahi and Torabi used the Mode-I and Mode-II NSIFs, to study the blunt V-notches and obtained experimental data from Brazilian disc specimens made of PMMA and polycrystalline graphite [⁴⁷]. In addition, at the notch tip, the Mode-II NSIFs is obtained by Lazzarin and Filippi [⁴⁸].

In general, V-notches are either sharp or blunt. Even though there is no perfectly-sharp notches, but, if the blunt V-notches containing very small notch tip radii, then the blunt ones can be accepted as the sharp V-notches.

In notched elements, there are two major mechanisms of failure: brittle fracture and ductile rupture. Near the notch tip, moderate or large scale yielding will be developed for the rupture of ductile materials, and for brittle and quasi-brittle materials, the instantaneous fracture without any significant plastic deformation is the failure pictures. Because of the catastrophic consequences, a reliable failure criterion to evaluate fracture for the notched brittle components is an excited research topic for years [⁴⁹].

Under pure Mode-I loading, the notched brittle components are extensively investigated, such as: Carpinteri [⁵⁰], Knésl [³⁷], Nui et al. [³⁸], Dini and Hills [⁵¹] and Taylor [⁵²]. And, for example, Seweryn [⁵³] used the mean stress (MS) criterion, in addition, NSIFs are used by Dunn et al. [⁵⁴], Lazzarin and Zambardi [⁵⁵], Gómez and Elices [⁴⁰], Livieri [⁵⁶], Leguillon and Yosibash [⁵⁷], and Ayatollahi and Torabi [⁵⁸], etc.

For mixed mode problems, Dunn et al. [⁵⁹] failed in the extension of their earlier criterion, but a significant discrepancy with the testing data exhibited.

To estimate failure of structural elements under multi-axial loading, a non-local stress failure criterion was suggested by Seweryn and Mróz [⁶⁰,⁶¹]. For the fracture study of brittle sharp V-notches problems, Priel et al. [⁶²] used the strain energy density (SED) criterion. The mixed mode cracking in layered materials is reviewed and investigated by Hutchinson and Suo [⁶³] based on the Griffith criterion. For the case of sharp-notch mixed mode failure estimation, It is important to point out that very complicated mathematical derivations is required for almost all of these investigations.

For the mixed mode failure problem with sharp cracks, Erdogan and Sih [⁶⁴] proposed the maximum tangential stress (MTS) criterion. Ayatollahi et al. [⁴⁹] used test samples called sharp V-notches Brazilian disk (SV-BD) specimen made of PMMA to study mixed mode brittle failure and the SV-MTS criterion was proposed. Very good agreement was obtained between the theoretical estimation and the SV-BD testing data.

Based on some macroscopic stresses, fracture criteria are developed [⁶⁵], such as: critical virtual cracks [⁶⁶], SIFs [⁴⁸], notch rounding approach [⁶⁷], SED [⁶⁸], J-integral [⁶⁹], and cohesive zone model [⁷⁰,⁷¹], etc. It is understood that the SIFs, are available under small scale yielding conditions. The NSIFs are used for the case of sharp, zero radius, V-notches. However, if the notch radius R is not zero, (i.e. blunt notch), there is no stress intensity, therefore, the linear elastic fracture mechanics theory will be valid up to some critical value R which depends on different materials [⁷²]. If the notched component is under mixed mode loading condition, since there is no more loading symmetry, [⁷³,⁴³], the problem becomes more involved.

Berto et al. [⁷⁴] studied the round bars made of PMMA with notches under torsion loading including low temperature investigations [⁷⁵].

Under various loading conditions at different temperatures, the fracture and crack growth problems about brittle graphite materials have been studied, such as: Lomakin et al. [⁷⁶], Sato et al. [⁷⁷], Yamauchi et al. [⁷⁸], Nakhodchi et al. [⁷⁹], etc. and including the notch sensitivity investigation about different graphite materials, for example: Bazaj and Cox [⁸⁰] and Kawakami [⁸¹].

In recent years, failure analysis of notched brittle and quasi-brittle materials have been studied and the notch fracture mechanics (NFM) is developed. Ayatollahi and Torabi [⁸²] using NFM to investigated V-notched graphite components. They measured the notch fracture toughness data experimentally and compared the predicted values from the MS model successfully. In addition, to assess mixed mode fracture of graphite materials, the SED fracture criterion was utilized as well. To study the failure of disk-type graphite plates with U-notches under not only pure Mode-I, mixed Mode-I/II, and pure Mode-III loading conditions, Torabi and Berto [⁸³] also used the SED fracture criterion and obtained good results.

For the case of mixed-mode loading, not only the maximum tangential stress concept but also the MS concept are used to investigate the failure about brittle key-hole notches by Torabi and Pirhadi [⁸⁴] and found both criteria could estimate the experimental results successfully.

In engineering applications, to handle small size cracks, it is common to drill a circular hole with the radius equal to the crack length and the crack is removed with the changed notch geometry. For V-notches, this drilling approach changes the initial V-notch to a V-notch with end hole, i.e., VO-notch, thus, different fracture behavior is developed. Therefore, for brittle and quasi-brittle materials, the fracture estimation of VO-notched members is more complicated. The problem of brittle fracture in V-notches with end holes was investigated by Torabi and Amininejad [⁸⁵] using both the point stress and the MS failure concept. Total 36 failure tests on the Brazilian disk with central V-notch containing end hole (VO-BD specimen) made of polymethyl-metacrylate were performed. Good agreement between the experimental and theoretical predictions was reported.

To study the Brazilian disk made of PMMA specimen weakened by a central dumbbell-shaped slit with two key-shape ends under mixed mode I/II loading, Torabi et al. [⁸⁶] reported that the sudden fracture estimated by the local SED criterion, agreed well with the testing data.

Both the “elementary” volume and the “micro structural support length” concepts were proposed in 1958 by Neuber [⁸⁷]. He argued that if there exists a sharp notch, the material will be sensitive to a fictitious root radius and that characteristic value will be correlated not only to the “micro structure support length” but also to the multi-axial stress state.

For the local elastic-plastic fields in the neighborhood of a crack, Rice [⁸⁸] proposed the J-integral to handle the crack initiation and propagation. Atluri [⁸⁹] developed numerical methods to evaluate the J-integral. Under Mode-I loading, Berto and Lazzarin [⁹⁰] developed a connection between Rice’s J-integral and SED to study the problems involving sharp crack tips and blunt V-notches.

Under Mode-I bending loading, a plate made of linear elastic material containing U-notches was studied by Barati et al. [⁹¹] to perform the J-integral calculations.

In the past few decades, to measure the full field material deformation, the digital image correlation (DIC) has been developed [⁹²]. Therefore, fracture parameters can be directly evaluated via experimental-numerical computations provided by DIC, such as: the crack opening displacement [⁹³], energy release rate [⁸⁸], and SIFs [⁹⁴].

A combined experimental-numerical technique proposed by Becker et al. [⁹⁵] for the J-integral calculation of both elastic and elastic-plastic materials. They indicated that as long as the small-scale yielding condition is presented, there is no need to measure the crack length and is not elastic fracture mechanics restricted either.

Recently, using the numerical methods to analyze problems of cracks and notches is very popular by engineers. The use of refine meshes to obtain reliable results is a major drawback of numerical modeling [⁹⁶]. Therefore, people developed special singular finite elements and boundary elements to handle the singularities [⁹⁷,⁹⁸].

In addition, there are some other numerical techniques including degenerated asymptotic finite elements [⁹⁹,¹⁰⁰], hybrid finite elements [¹⁰¹,¹⁰²], and analytical finite elements [¹⁰³], etc.

Sheppard [¹⁰⁴] proposed that in a material volume, consider the averaged strain energy (SED parameter) as a single parameter to efficiently handle not only the static notch effects but also the fatigue structural strength problems.

To utilize the SED theory, the fundamental step is to determine the provisional (temporary) crack paths in engineering materials. Since the continuum mechanics failed at a short distance from the crack tip, therefore, Sih [¹⁰⁵] proposed the concept of “core region” surrounding the crack tip using a radius to describe the core region. Then the product of the energy density by a critical distance from the point of singularity is defined as the SED factor S [⁶⁸]. Thus, failure is controlled by a critical value of S and the SED fracture criterion was refined and extensively summarized [¹⁰⁶,¹⁰⁷].

Engineering applications of fracture assessments

Consider the fracture mechanics applications, the welding residual stresses based on experimental and theoretical data are overviewed including a comprehensive discussion by Hensel et al. [¹⁰⁸]. Song and Dong [¹⁰⁹] investigated the repaired welding geometry effects on residual stress distributions. Some major findings are: the dominance of repair-induced residual stresses over those generated by initial welds, suggests that initial weld residual stresses can be assumed negligible in repair weld modeling. A weld repair should be designed as long as possible, as narrow as possible, and as shallow as possible.

Martínez-Pañeda [¹¹⁰] investigated both mechanism-based and phenomenological strain gradient plasticity theories by using the numerical calculations.

In general, the notch fracture behavior can be analyzed via two criteria: local failure and the global failure [³⁸,¹¹¹]. Even though both criteria are with reliable results, in engineering applications, the local criteria are more popular. To study the failure of solid materials, Neuber [⁸⁷] and Peterson [¹¹²] proposed the theory of critical distance (TCD). Based on the TCD, the point method, the line method and the finite fracture mechanics (FFM) [¹¹³] are proposed, especially for the failure study of fiber glass reinforced composites.

Recently, the apparent fracture toughness of notched short glass fiber-reinforced polyamide 6 (SGFR-PA 6) was studied by Ibáñez-Gutiérrez [¹¹³] and Cicero using the TCD. For fiber reinforced materials, fracture assessment is closely related to the strength theories or failure criteria of the materials. There are two approaches: macromechanics and micromechanics in performing the failure analysis of composite materials.

Based on macromechanics, Jones [¹¹⁴] and Tsai-Wu [¹¹⁵] anisotropic strength criteria are very popular for the reliable structural design of composite materials. Start from the Mohr’s stress circle, the yielding criterion of a homogeneous, isotropic, linear elastic material is proposed by Kim and Yeh [¹¹⁶], and then use the macromechanics approach, the Yeh-Stratton strength criterion for continuous fiber-reinforced composite materials is developed [¹¹⁷] and compared with the Tsai-Wu failure criterion as well as the micromechanics strength theory established by Zhang et al. [¹¹⁸].

For rock-like materials, strength criteria such as: the Mohr-Coulomb Theory [¹¹⁹] and the Hoek-Brown failure criterion [¹²⁰,¹²¹] are used. A new rock failure characteristic parameter is determined by Zuo et al. [¹²²], and the related equation agreed with the original Hoek-Brown empirical strength criterion [¹²³,¹²⁴].

It is understood that under certain pressure, a critical condition exists for the brittleness to ductility transition of rocks. Based the linear elastic fracture mechanics, a 3-D crack model is employed by Zuo et al. [¹²⁵] to derive the empirical Hoek-Brown criterion theoretically, and a failure characteristic factor is proposed and is validated by experimental observations.

Finally, three-dimensional effects in V-notches is investigated by Campagnolo, Pook and other researchers [¹²⁶–¹²⁹] in recent years. Even though the existence of three-dimensional effects at cracks has been known for many years, the increasingly powerful computers made it possible to study three-dimensional effects numerically in detail.

Lazzarin et al. [¹³⁰] proposed a criterion based on the SED averaged over a material-dependent control volume surrounding the notch tip. When the control volume is small enough to make negligible the influence of higher order terms of William’s solution, the SED can be theoretically linked to the NSIF. And the expression of the critical Mode-I NSIF at failure is compared with those given by two different versions of the FFM criterion suggested by Leguillon [⁶⁶] and Carpinteri et al. [¹³¹], respectively.

In modern views, fatigue crack growth and crack initiation are considered as directly related phenomena with the formal being repeated instances of the latter. Taking advantage of these views, Glinka and other researchers [¹³²–¹³⁶] studied the fracture mechanics-based estimation of fatigue lives of welded joints.

Conclusions

1) In brittle fracture, there is still plastic defamation which can be viewed by the X-ray technology. Therefore, if the size of plastic zone is very small in comparing with the crack length, based on the energy dissipation consideration, we can assume the elastic stress solutions are acceptable, i.e., the SIFs derived from the linear elastic fracture mechanics can be used as physical parameters in fracture analysis [¹³⁷].

2) Since in brittle materials, the damage around notches is a very complicated physical phenomenon, therefore, it is definitely depend on both the macrostructure and microstructure of the materials. To attack this complicated and difficult problem, we have developed various fracture models for the failure investigations, such as: SIF, NSIF, J-integral, SED, TCD, MS, and the maximum tensile stress, etc.

In addition, microstructure research of various materials will help us not only to understand the material damage pictures in much better way, but also to establish more reliable fracture assessments criterion.

3) To measure the full field material deformation, the DIC has been developed. Therefore, fracture parameters can be directly evaluated via experimental-numerical computations provided by DIC and the finite element methods, such as the crack opening displacement, energy release rate, and SIFs. And this is one of the very important research development in the last few decades.

4) Because of the mathematical difficulties, nowadays, the numerical approaches combined with computation techniques such as finite element methods are very popular in performing the fracture assessments of various structural materials and machine components. Although computational methods are very useful in the research of fracture mechanics, they cannot replace experiments. It is worthwhile pointing out that stress distributions obtained with finite elements are of limited accuracy. A numerical analysis of cracked body can provide information on stresses and strains at the crack tip as well as global fracture parameters. Even though fracture can be modeled, however, a separate failure criterion is required. Predictions of fracture or fracture assessment would require more experiments to infer material-dependent parameters in the local fracture model. Since experiments are obliged to obey all laws of nature, thus, an experiment often conveys important information that a simulation overlooks.

5) In engineering applications, fracture assessments of various materials is one of the key concerns in safety designs. Thus, to establish a reliable and accurate material strength theory or failure criterion is imperative. A failure criterion must therefore provide a convenient means for predicting the loss of integrity that could lead to fracture or failure. However, with the rapid development of modern technology, various new materials, such as: composite materials, biomedical materials, etc. are provided and applied in different engineering fields. Obviously it is not realistic to adopt a single failure criterion to perform the failure analysis of various materials. Directly usable by the designers, a failure criterion must provide results at the macroscopic level. Thus, the continuing development of various material failure criterion is one of the core research subject in fracture assessment of safety design works.

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