An ordinary state-based peridynamic model for toughness enhancement of brittle materials through drilling stop-holes
Graphical abstract
Introduction
Over the last few centuries, a large collection of research studies has been dedicated to understanding the fracture mechanics of engineering materials, some can even be traced back to the 1770′s [1]. However, in today's engineering society, almost every engineering field still suffers from the lack of predicting the potential occurrence of crack propagations. Depending on the environmental/operational conditions of an engineering structure, load-bearing components of the structure may be subjected to extreme loading conditions, thus leading to emergence of the so-called micro-cracks. These micro-cracks can either grow independently or coalesce to form various macro-cracks, which eventually cause a complete failure of the structure. The formation of a complete rupture in a very short amount of time (e.g., microseconds) is a particular study case of fracture mechanics, which is referred to as “brittle fracture” and commonly present in brittle materials. This kind of failure bears a high potential risk to human safety, increase environmental pollution, and cause crucial financial losses [2], [3], [4]. Therefore, especially after the industrial revolution, fracture mechanics of brittle materials has gained a great deal of interest.
Numerous solutions have been suggested to reduce the crack growth rate of brittle fracture. One of which is to lessen the stress concentration at the crack tip through making a perforation, i.e., stop-hole. Among early studies available in the literature, Broek [5] experimentally investigated the effect of stop-holes on crack dynamics in a continuum by allocating a stop-hole on propagation path of a crack. According to his results, the arrestment effect of the hole on the crack is balanced by the crack growth acceleration caused by the hole. On the other hand, Miyagawa and Nisitani [6] later demonstrated the superior effect of holes on crack growth life by extending Broek's investigation to two and four hole combinations located ahead of a pre-existing crack. In fact, any growing crack tends to propagate with an increasing velocity towards a hole located in the vicinity of its tip or along its original propagation path. Moreover, the existence of holes in a body affects the stress intensity factor of growing cracks [7]. A crack running towards a hole has a considerably larger stress intensity factor, further accelerating the crack propagation. However, when a crack joins the hole, the hole arrests the growing crack for a significant amount of time causing the crack to release its strain energy accumulation. This eventually leads to a longer crack growth life. Hence, crack dynamics (i.e., acceleration and velocity of crack propagation and crack growth life) are substantially influenced if any discontinuity, such as a hole, exists in the continuum [8].
When a weak zone (i.e., so-called stop-hole) is introduced into a homogenous material, the mechanical response of the material will involve relatively non-homogeneous effects, especially near the weak zone. Therefore, crack dynamics in non-homogeneous regions of materials are different from the ones in homogenous regions. Recently, Carlsson and Isaksson [9] investigated the effect of this heterogeneity on crack dynamics using the dynamic phase-field method (DPF) introduced in [10]. To model the existence and propagation of a crack, the phase-field method requires an external differential equation to be solved along with the governing equations of classical continuum mechanics. This usage of the external equation is one of the main deficits of phase-field method [11]. A robust non-local continuum theory, originally introduced by Silling [12], named as peridynamics (PD), however, eliminates the shortcomings of classical continuum theory, especially the ones pertaining to modelling of solid continuums involving any discontinuity such as cracks. Although the peridynamic theory was first introduced in the 2000′s and applied to solid mechanics of isotropic materials [12], since then it has been further developed for other engineering applications including modelling of composites [13], multi-physics problems [14], heat transfer [15], etc. For example, De Meo et al. [16] introduced a computational PD model for fracture behaviour of cracked HSLA steel subjected to a corrosive environment. Recently, Kefal et al. [17] used peridynamic theory for structural topology optimizations of engineering structures with and without cracks. More recently, the PD-TO approach [17] has been extended for continuous density-based topology optimization [18]. Another work on the PD application is the failure model developed by Ghajari et al. [19] for orthotropic materials. And yet in another track, Wang et al. [20] developed a peridynamic formulation for thermo-visco-plastic deformation and impact fracture. Thus, with the help of these and other significant contributions not listed, PD approach has gained a capability to solve diverse applications of engineering problems for different materials [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39]. Reader may also refer to the study of Javili et al. [40] for a detailed review on PD theory.
In general, there are three different types of peridynamic theory, namely bond-based (BB) PD, ordinary state-based (OSB) PD, and non-ordinary state-based (NOSB) PD. In the bond-based PD formulation, it is assumed that the interaction forces between two material points are equal in magnitude and opposite in direction. Therefore, the bond-based PD can be applicable only to analyses of materials having Poisson's ratio of 1/4 in three dimensions and 1/3 in two dimensions [12]. However, this limitation was later circumvented by the introduction of the OSB and NOSB forms of peridynamic theory [26]. In OSB type of PD, it is assumed that the direction of the pair forces is along the interaction direction, but their magnitudes may vary. The NOSB, more generally, considers the pair forces between two material points to be different in both magnitude and direction [26]. In all of the PD types, the interactions between material points are independent of each other, enabling PD to be applied to materials with discontinuities, such as cracks, with no usage of complex mathematical formulations [12]. This ease of implementation makes the PD formulation advantageous for analysis of static and/or dynamic crack propagation problems.
Recently, Vazic et al. [41] used the bond-based (BB) PD formulation to investigate the effect of the presence of parallel micro-cracks on dynamics of a main crack and their effects on the toughness enhancement of the poly methyl methacrylate (PMMA) material. According to their results, both positive and negative influence rate of micro-cracks are observed on the velocity of main-crack propagation and material toughness. Most recently, Basoglu et al. [42] extended Vazic et al.’s work by examining different combinations of linear and curvilinear micro-cracks in order to determine the most effective intentionally applied micro-crack combination on material toughness by means of micro-macro crack dynamics. The crescent-like combinations of micro cracks in the vicinity of the main crack tip have proven to be effective in reducing the main crack propagation velocity [42]. However, the suggested combinations of micro-cracks can be difficult to implement to a real-world material. Furthermore, the usage of BB in the analysis of materials having a Poisson's ratio other than 1/3 (in two dimensions) or 1/4 (in three dimensions) can sometimes cause significant error.
In the present study, the OSB formulation of PD is used to address the above-mentioned issues and to simulate more realistic path of crack propagation in materials with different Poisson's ratios. In addition, instead of micro-cracks, the effects of stop-holes are investigated on crack dynamics and toughness of materials given that stop-holes can be realistically created. Since the presence of stop-holes can reduce the weight of the structure and is relatively easy to implement, it can be more economical to use as a toughening mechanism of the engineering parts.
Several researchers have investigated the arresting effect of stop-holes drilled at the vicinity of the pre-existing crack tip and have suggested drilling stop-holes as a repairing technique for the cracked parts of structures. For instance, Fu et al. [43] investigated the effect of one and two drilled stop-holes on fatigue life of steel bridge deck. They concluded that a higher fatigue life can be achieved if a stop-hole is drilled near the crack tip. They also indicated that a larger diameter of the drilled stop-hole causes a longer fatigue life by further reducing the stress concentration. Ghfiri et al. [44] used the hole-expansion method to introduce residual stresses that can reduce the effective stress around the crack tip. They drilled a stop-hole in aluminium sheet specimens and investigated its crack growth arresting behaviour under axial fatigue test. For some of the specimens, they cold-expanded the stop-hole diameter and compared its effect on fatigue life. According to their results, an expanded stop-hole had a higher crack growth arresting effect than that of non-expanded stop-holes. Similarly, Song et al. [45] used stop-hole drilling procedure to improve the fatigue life of aluminium alloys and stainless steel. Recently, Ferdous et al. [46] have investigated the effect of four uniformly distributed stop-holes on fatigue life of specimens. Aside the above listed experimental works, Ayatollahi et al. [47] numerically investigated the effect of drilling one stop-hole based on classical fracture mechanics model. However, the main variable of their numerical investigation was the size of the stop-hole. To optimize the shape of stop-hole, in a unique track, Fanni et al. [48] used structural optimization technique by utilizing finite element method.
Although several of the above researchers have numerically/experimentally studied the effects of one circular stop-hole on crack propagation path and fatigue life, to the best of authors’ knowledge, no literature has been dedicated to the investigation of the effects of the “linear” and/or “non-linear” combinations of more than one stop-hole on material toughness under “tensile” and “shear” loadings. Hence, the main novelty of the current study is to present a detailed knowledge about the various toughening effects of stop-hole combinations through the OSB-PD analyses, while introducing a new approach of implementing linear and/or nonlinear combinations of stop-holes as a material toughening technique.
The rest of the paper is structured as follows; a brief mathematical description of OSB-PD theory is given in Section 2. In Section 3, a computational application of the OSB-PD theory is performed and toughness enhancement in brittle materials via stop-holes under shear and tensile loading is investigated. It is shown that the relative positions of stop-holes have a notable influence on toughness enhancement. Moreover, the toughness of the material can be tailored through adjusting the locations of the stop holes. It is later indicated that every stop-hole has an influence range (µ-range) beyond which the crack dynamics are not affected. Furthermore, the experimental/numerical benchmark studies reported in [9] and [49] are revisited whereby the accuracy of the OSB approach is demonstrated and compared with the DPF models. Finally, the advantages, superior predictive capabilities, and concluding remarks of the current approach are elaborated in Section 4.
Section snippets
Formulation of ordinary-state based peridynamics
Peridynamics is a mesh-free approach originally introduced by Silling in 2000 [12]. The PD approach is referred to as the non-local form of classical continuum mechanics (CCM) or the shrunken form of molecular dynamics (MD). Peridynamic theory uses integro-differential equations rather than the classical spatial derivatives of stress components; thus, it is an ideal approach for fracture mechanics problems involving discontinuities (e.g., cracks). In PD, a continuum is introduced by a set of
Numerical examples
In this section, five different examples are investigated utilizing the aforementioned OSB peridynamic formulation. The OSB-PD formulation is implemented as an in-house code using the C++ programming language. All the OSB analyses are carried out for the PMMA material, which is considered to have linear-elastic and brittle properties. First, the original numerical and experimental work of Carlsson and Isaksson [9] is revisited. Through the analysis of this problem, we validate our OSB-PD
Conclusions
In the scope of this study, the effects of various linear/non-linear combinations of stop-holes on toughness enhancement in brittle materials are investigated under tensile and shear loadings by performing ordinary state-based peridynamic (OSB-PD) analysis. The numerical algorithm is assessed by solving benchmark cases with various facture modes and the results are compared with the experimental-numerical finding of literature [9,49]. It is demonstrated the OSB-PD can predict very similar final
CRediT authorship contribution statement
Mohammad Naqib Rahimi: Methodology, Software, Formal analysis, Validation, Visualization, Writing - original draft. Adnan Kefal: Conceptualization, Project administration, Funding acquisition, Writing - review & editing, Supervision. Mehmet Yildiz: Conceptualization, Writing - review & editing, Supervision. Erkan Oterkus: Conceptualization, Writing - review & editing, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The financial support provided by the Scientific and Technological Research Council of Turkey (TUBITAK) under the grant No: 217M207 is greatly acknowledged.
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