1 Introduction

The strength of mechanical components is strongly affected by the presence of cracks, which give rise to local stress concentrations and can result in the brittle collapse of the structure. The interaction between cracks leads to enhancing/shielding phenomena, which have great influence on the failure mechanism and cannot be neglected (Mahadevan and Shi 2001; Tan and Chen 2015; Shu et al. 2017). Focusing on crack shielding, for instance, due to the proximity of the cracks, the “effective crack-driving force” actually experienced at their tips is locally reduced (Ritchie 1988). The driving force for crack growth is generally described by a field-characterizing parameter, the stress intensity factor (SIF), which characterizes the stress and strain fields in the singular region close to the crack tip. Accordingly, the SIF at the tips of the shielded cracks is lower, and hence the failure load is higher.

Aim of this study is to analyse the crack shielding and failure size effect regarding a tensile slab presenting one, two or three parallel edge cracks. For this purpose, two experimental campaigns are conducted by tensile testing polymethyl methacrylate (PMMA) cracked plates. The initial crack length and their distance are both set equal to 0.1 times the sample width w for reasons of practicality. Size effects are highlighted by varying w in the slab presenting two interacting cracks. Note that different catastrophic failures are related to the interaction between parallel edge cracks. To cite but a few, let us mention those related to aero-engine turbines (Martín-Meizoso et al. 1995), actuator piston rods (Rütti and Wentzel 1998) and pressure vessels (Parker 1999).

The theoretical framework is developed by means of the Finite Fracture Mechanics (FFM) approach. FFM is a coupled fracture criterion which allows to provide strength predictions based on the simultaneous fulfilment of a stress condition and the energy balance. Originally proposed in the static framework (Leguillon 2002), (Cornetti et al. 2006), it was recently applied in the fatigue limit regime (Sapora et al. 2020), (Liu et al. 2020). The approach rests on the assumption of finite crack advance in contrast to Linear Elastic Fracture Mechanics (LEFM), which assumes crack growth to develop continuously. Note that LEFM fails to deal with very short cracks, according to which the SIF tends to vanish and the approach provides an infinite failure load. On the other hand, FFM is able to overcome this drawback catching the transition from a strength- to a toughness-governed regime, as the crack length varies (Cornetti et al. 2006), (Cornetti and Sapora 2019). Similar arguments hold when dealing with ligament size effect in largely cracked tensile structures (Sapora et al. 2023). As regards the notch size effect, FFM has been applied to brittle failure in presence of circular holes (Doitrand and Sapora 2020), (Doitrand and Leguillon 2021), (Leite et al. 2021), square holes (Doitrand et al. 2021) and spherical or spheroidal voids (Chao Correas et al. 2021), (Ferrian et al. 2023). A computational coupled implementation was proposed in (Wegert et al. 2022) to estimate the strength of a plate with centre crack, circle, diamond, and hexagon perforations of different sizes. Finally, let us mention the study carried out in (Doitrand et al. 2017) to assess the influence of a crack on the initiation and propagation of other cracks in woven composites.

In order to implement FFM, the knowledge of the stress field and SIF expressions is required. This task is here accomplished numerically by parametric Finite Element Analysis (FEA) through ANSYS® code. Note that different works focused on the evaluation of the SIF for parallel edge cracks configurations, exploiting analytical approaches (Freese 1976), (Civelek and Erdogan 1982), numerical simulations (Jiang et al. 1990), (Jiang et al. 1992), (Gope et al. 2014), (Bisht et al. 2015), (Mariggiò et al. 2023) and experimental methods (Farahmand and Raftopoulos 1981), (Raftopoulos and Farahmand 1982).

The paper is organized as follows. The experimental campaign on PMMA cracked samples and related results are presented in Sect. 2, whereas the FFM criterion and its numerical implementation are introduced in Sect. 3. Crack shielding and failure size effect are described in detail. Section 4 presents the comparison with experimental data, whereas Sect. 5 is voted to conclusions.

2 Experimental investigation

In order to analyse the crack shielding and failure size effect, uniaxial tensile tests are carried out on cracked plates made of PMMA. Samples are machined by laser cutting PMMA sheets and sharpening the crack tips through a fresh razor blade. The sheet thickness t is equal to 5 mm to ensure plane strain conditions, according to the relationship t ≥ 2.5 lch (Taylor 2007), where lch = (KIc/σc)2 is the well-known Irwin’s length and it is generally comprised between 0.2 and 1 mm for PMMA (Seweryn 1994), (Taylor 2007). Here, KIc denotes the plain fracture toughness and σc refers to the tensile strength.

Two different experimental campaigns are performed at a constant displacement rate of 0.5 mm/min. The first one is related to the crack shielding investigation. The samples are weakened by the presence of one, two or three edge cracks perpendicular to the loading direction (Figs. 1a, b, c). The width w of the samples is set equal to 30 mm, whereas the parameters a0 and d representing the initial crack length and the distance between cracks, respectively, are both taken equal to 3 mm (hence, a0/w = d/w = 0.1), to point out the crack shielding phenomenon. Furthermore, the length L is considered equal to 150 mm, five times higher than w. Four samples are considered for each configuration, for a total of 12 tested geometries. From the images of the broken samples (Fig. 2a), it is possible to state that the crack propagation is asymmetric (involving always one external crack tip) in the configurations with two and three parallel cracks. Moreover, the failure is of brittle character and plastic deformation is negligible around the crack tip, as evident from the force-extension curves represented in Fig. 2b.

Fig. 1
figure 1

Tensile slab presenting one a, two b or three c edge cracks

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Fig. 2
figure 2

a Cracked configurations analysed in the study and b related force-extension curves

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The second one is devoted to analyse the failure size effect. Referring to the configuration presenting two cracks and keeping fixed a0/w = d/w = 0.1, two additional geometries are considered by varying the size as w = 10 mm (a0 = d = 1 mm, L = 50 mm) and w = 50 mm (a0 = d = 5 mm, L = 250 mm), see Fig. 3a. Again, during these tests, the failure is of brittle character (Fig. 3b), showing the increase of the failure load Pf = σf (w × t) at which the crack starts to propagate as the size w increases.

Fig. 3
figure 3

Failure size effect: a Tested geometries and b corresponding force-extension curves

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For all tested cracked geometries, the recorded failure loads Pf are reported in Table 1: the scatter of recorded values, always lower than 3.5%, proves the significant test repeatability.

Table 1 Tensile tests on PMMA cracked samples: characteristic dimensions and recorded failure loads
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As concerns the crack propagation, as previously mentioned, an asymmetric one was observed for all the analysed configurations (Figs. 2 and 3). It is important to note that this outcome may be influenced by sample machining and micro-imperfections in the material. In order to minimize the uncertainty related to this aspects, crack lengths were measured carefully before the test. Moreover, to ensure the accuracy and repeatability of the experimental results, each test was repeated four times. As presented in Table 1, the scatter of the experimental results is very small, indicating that sample preparation and testing were performed with very good accuracy.

The mechanical properties of PMMA are presented in Table 2. The ultimate tensile strength σc as well as Young’s modulus E and Poisson’s ratio v were estimated following ASTM D638-14 standard code. Instead, the fracture toughness KIc was evaluated in order to minimize the standard deviation between the experimental data and avg-FFM failure estimations for the former experimental campaign (see Sect. 4). Accordingly, KIc = 1.52 MPa√m and lch = 0.51 mm. These values fall within the typical ranges measured for PMMA (Taylor 2007), (Sapora et al. 2015), (Cicero et al. 2018).

Table 2 PMMA mechanical properties obtained experimentally except from the value of KIc that is fitted on avg-FFM failure estimations
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3 Finite Fracture Mechanics (FFM)

The theoretical framework is developed by means of the FFM approach. Accordingly, finite crack propagation occurs when a stress condition and the energy balance are simultaneously fulfilled. The stress requirement, considering the approach proposed by (Leguillon 2002), imposes that the normal stress σy over a finite distance l must be larger than the ultimate tensile strength σc of the material. On the other hand, the energy balance ensures that the integration of the strain energy G over a finite crack increment l must exceed Gcl, where Gc is the material fracture energy. This last condition can be recast exploiting Irwin’s relationship, thus introducing the SIF KI = √(GE′) and fracture toughness KIc = √(GcE′), E′ being Young’s modulus of the material under plane strain conditions. Particularizing the FFM formulation to the (positive) geometries analysed in this study and to the Cartesian frame of reference (x, y) (Fig. 1), yields:

$$ \left\{ {\begin{array}{*{20}l} {\sigma_{y} \left( {x = a_{0} + l} \right) = \sigma_{c} } \hfill \\ {\mathop \smallint \nolimits_{{a_{0} }}^{{a_{0} + l}} K_{I}^{2} (a^{\prime})da^{\prime} = K_{Ic}^{2} l} \hfill \\ \end{array} } \right. $$
(1)

The approach can be also developed considering an average stress condition (Cornetti et al. 2006), (Carpinteri et al. 2008), requiring that the average normal stress σy over a distance l must exceed σc. Accordingly, the avg-FFM approach writes:

$$ \left\{ {\begin{array}{*{20}l} {\int_{{a_{0} }}^{{a_{0} + l}} {\sigma_{y} \left( x \right)dx = \sigma_{c} l} } \hfill \\ {\mathop \smallint \nolimits_{{a_{0} }}^{{a_{0} + l}} K_{I}^{2} (a^{\prime})da^{\prime} = K_{Ic}^{2} l} \hfill \\ \end{array} } \right. $$
(2)

In Sect. 3.3 it will be shown that, in critical conditions (σ = σf), the two unknowns of FFM systems (1) or (2) are represented by the remote failure stress σf, implicitly embedded in the stress field and the SIF functions, and the critical crack increment lc. This latter quantity will reveal to be a structural parameter, since dependent on both material and geometric characteristics.

3.1 Finite element analysis (FEA)

The stress field and the SIF functions related to the geometries under investigation, necessary for the FFM implementation as shown above, can be expressed as:

$$ \sigma_{y} = \sigma \,S(x/w) $$
(3)
$$ K_{I} = \sigma \sqrt {\pi a} F_{I} \left( {a/w} \right) $$
(4)

where S(x/w) and FI(a/w) are the shape functions related to the stress field and SIF, respectively.

In this work, for each geometry, S(x/w) and FI(a/w) are determined through parametric FEAs by ANSYS ® code. Quadratic elements, with 6 nodes and characterized by a reduced integration solution scheme are implemented under plane strain conditions. To obtain accurate results, following a convergence analysis, the mesh is refined within a circle of radius 0.5a0 centered at the crack tip (Fig. 4): therein, the minimum dimension of the elements is set equal to 0.01a0.

Fig. 4
figure 4

Finite element model for the geometry presenting two interacting cracks. The model refers to symmetric crack propagation. It is assumed that a0 = d = 0.1 w, coherently with experimental data

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3.2 Symmetric vs asymmetric crack propagation

In critical conditions, considering an isotropic and homogeneous material, a crack is expected to initiate from the crack tip along the x-axis for the single edge cracked configuration (Fig. 1a). Instead, taking into account the other two geometries, two different cases need to be analysed: (i) Symmetric and (ii) Asymmetric crack initiation. More in detail, referring to the double cracked configuration (Fig. 1b), cracks could initiate simultaneously from the two crack tips (once d/w = 0.1 is fixed) or asymmetrically from only one crack tip (once d/w = a0/w = 0.1 are fixed), Fig. 5b. Comparing the maximum values of KI, i.e. KImax, the most critical scenario results to be the asymmetric one. Indeed, the corresponding SIF is nearly 8% higher considering a/w = 0.11. This result is in agreement with the experimental observations reported in Sect. 2 (Figs. 2a and 3a).

Fig. 5
figure 5

Symmetric a vs. asymmetric b crack propagation for the tensile plate presenting two interacting cracks; c most critical scenario regarding the triple cracked configuration

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Analogously, for the third configuration (Fig. 1c), comparing KImax related to all possible scenarios, the most critical one results to be the asymmetric crack initiation from one of the two external cracks (Fig. 5c). This result is once again in agreement with the experimental evidence (Fig. 2a). Indeed, considering a/w = 0.11, the corresponding SIF is 18% higher than KImax related to symmetric initiation from the three crack tips. Taking into account the propagation from the central crack or the simultaneous initiation from two cracks, the value of KImax related to the most critical scenario is 13% and 9% higher, respectively. Thus, henceforth only the asymmetric crack initiation from one of the two outer cracks will be analysed.

The trend of the SIF shape functions FI(a/w), related to the three analysed cracked configurations, is shown in Fig. 6 as the ratio a/w varies between 0.1 and 0.5. As regards asymmetric propagation, significant differences are present as long a/w < 0.2, i.e., when a < 2a0. For the geometry with two interacting cracks (Fig. 1b), both symmetric and asymmetric crack initiation cases are reported according to our simulations and to the analytical approximating expression proposed by Jiang et al. (1990), Jiang et al. (1992). The deviation from the values hereby computed in this work is less than 3 and 1% in the symmetric and asymmetric cases, respectively.

Fig. 6
figure 6

Shape functions FI related to single, double or triple edge cracked geometries

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Finally, it is worth observing that, for the configuration presenting two cracks and for the two outer cracks in the triple cracked geometry, KII is not zero, as it can be easily argued since these cracks do not lie on a symmetry axis. However, it can be easily checked that mode mixity angle is really small. Hence, throughout our investigation, we have disregarded the mode II contribution. Note that this hypothesis was assumed also in (Jiang et al. 1990) and is confirmed experimentally by our tests, characterized by an almost collinear crack growth (Figs. 2 and 3).

3.3 FFM implementation

FFM can now be implemented, in critical conditions (σ = σf), for each of the three configurations analysed. Introducing the stress field (3) and the SIF (4) into Eq. (1), yields:

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\sigma_{f} }}{{\sigma_{c} }} = \frac{1}{{S[(a_{0} + l_{c} )/w]}}} \hfill \\ {\frac{{\sigma_{f} }}{{\sigma_{c} }} = \sqrt {\frac{{l_{ch} }}{\pi }\frac{{l_{c} }}{{\int_{{a_{0} }}^{{a_{0} + l_{c} }} {a^{\prime}F_{I}^{2} (a^{\prime}/w)da^{\prime}} }}} } \hfill \\ \end{array} } \right. $$
(5)

FFM failure estimations are obtained by equalling the right-hand sides of system (5) to get lc through the solution of an implicit equation. This value is then introduced into one of the two equations in (5) to get the corresponding dimensionless failure stress.

On the other hand, as concerns avg-FFM, introducing Eqs. (3) and (4) into Eq. (2), yields:

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\sigma_{f} }}{{\sigma_{c} }} = \frac{{l_{c} }}{{\int_{{a_{0} }}^{{a_{0} + l_{c} }} {S(x/w)dx} }}} \hfill \\ {\frac{{\sigma_{f} }}{{\sigma_{c} }} = \sqrt {\frac{{l_{ch} }}{\pi }\frac{{l_{c} }}{{\int_{{a_{0} }}^{{a_{0} + l_{c} }} {a^{\prime}F_{I}^{2} (a^{\prime}/w)da^{\prime}} }}} } \hfill \\ \end{array} } \right. $$
(6)

which can be solved analogously to Eq. (5).

Strength estimations according to FFM are reported in Fig. 7a. As evident, the normalized failure stress σf /σc decreases as the dimensionless width w/lch increases catching the typical failure size effect. For sufficiently large sizes, coupled approaches revert to LEFM and the single edge cracked plate results to be the most critical configuration. The deviation in terms of failure stress from the other two geometries tends to 30% (two cracks) and 33% (three cracks) as w/lch → ∞. This behaviour can be justified by the crack shielding phenomenon that characterizes the configurations with more than one crack.

Fig. 7
figure 7

Strength estimations σf /σc a and critical crack advancements lc/lch b provided by FFM (thick curves) and avg-FFM (thin curves) for single, double and triple cracked configurations

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It is worthwhile to underline that, for vanishing w/lch, the normalized failure stress σf /σc converges to 0.9 according to avg-FFM, coherently with the initial fixed ratio a/w = 0.1 (Fig. 1). On the other hand, FFM provides a dimensionless failure load σf /σc ≈ 1.1. As observed in other works (Sapora et al. 2023), this is a questionable FFM trend as the crack advance approaches the ligament width.

In Fig. 7b the critical crack advancement lc/lch is plotted as a function of the normalized width w/lch, thus revealing a structural parameter and not just a material one, as it happens for Theory of Critical Distance (TCD) approaches (Taylor 2007). Avg-FFM provides higher lc/lch values with respect to FFM, as already established in past studies (e.g. (Ferrian et al. 2023)). Indeed, for all cracked configurations analysed, the failure mechanism is governed by the fracture toughness for large sizes and the stress condition only provides the critical crack advancement. It converges to 1/(2π) or 2/π taking into account FFM (Eq. (5)) or avg-FFM (Eq. (6)), respectively: in the former case, the value coincides with the distance according to TCD point method, in the latter case it matches the estimation by TCD line method (Taylor 2007), (Cicero et al. 2012), (Ayatollahi et al. 2016).

4 FFM estimations vs experimental results

FFM failure predictions are now compared with experimental data presented in Sect. 2. As concerns the first experimental campaign related to crack shielding, results are reported in Table 3.

Table 3 Failure stress related to tensile PMMA cracked samples (w = 30 mm): experimental data, FFM and LEFM predictions
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Indeed, taking into account avg-FFM predictions, the maximum deviation from the average value of the experimental failure load is around 3% for the configurations with one and three cracks, whereas is lower than 1.5% for the plate with two interacting cracks. It should be underlined, as observed preliminarily in Sect. 2, that the value of fracture toughness for PMMA was fitted on avg-FFM predictions. Also FFM results are accurate and deviations always keep below 6.5%. Similar arguments hold for LEFM, which provides the highest predictions.

On the other hand, as concerns the second experimental campaign, results are depicted in Fig. 8: the difference between FFM predictions according to symmetric and asymmetric crack propagation decreases as the size increases, both converging to LEFM estimations. It is worth emphasizing that the ability to discriminate between symmetrical and asymmetrical crack propagation is a strength point of the present FFM approach. In fact, lying on the assumption of an infinitesimal crack growth, LEFM provides the same crack onset load value if one or both cracks propagate in the configuration presenting two interacting cracks (and one or both outer cracks in the three-edge crack configuration). Note that TCD shows the same drawback as LEFM does, since TCD depends only on the stress field before crack growth. Finally, note that avg-FFM always reveals the most conservative approach, as typically observed in the field of brittle failure of cracked/notched structures (Sapora et al. 2023), (Ferrian et al. 2023).

Fig. 8
figure 8

Size effect on the plate containing two interacting cracks: comparison between experimental data and theoretical FFM results. The dotted line represents LEFM predictions

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Basically, the curves related to the asymmetric case are the most critical ones confirming experimental observations (Fig. 2a) and numerical results (Sect. 3.2). Taking into account avg-FFM predictions obtained for asymmetric crack initiation, the maximum deviation from the average value of the experimental failure stress is equal to 26.6% for the first configuration (w = 10 mm) and 17.5% for the third one (w = 50 mm). Instead, considering FFM, the maximum deviation decreases to 18% and 14.8% for the first and the third geometry, respectively.

5 Conclusions

In this study, three different tensile configurations were analysed presenting one, two or three parallel edge cracks. Both the length and the distance between cracks were set equal to 0.1 times the width of geometry. Experimental tests were carried out on PMMA cracked samples and the results were compared with theoretical failure estimations obtained by means of the coupled FFM approach, which requires the simultaneous fulfilment of a stress condition and the energy balance. The stress field and SIF functions -necessary for FFM implementation- were obtained through parametric FEAs. It was shown that, due to crack shielding, propagation always occurred asymmetrically and increasing the number of cracks increased the strength of the structure. Indeed, due to the proximity of the shielded cracks, the lower the SIFs at their tips, the higher the failure load. Furthermore, also the failure stress size effect was analysed varying the sample dimension for the configuration presenting two interacting cracks. Unfortunately, the range investigated experimentally was limited and it was not possible to fully investigate the range where LEFM predictions are meaningless (w < 1–2 mm, Fig. 8). This drawback could be overcome testing materials which are less brittle than PMMA. In this regard, let us mention that the coupled FFM approach was recently applied to catch the size effect related to single cracked geometries (Cornetti et al. 2006; Baldassari et al. 2023), validating the results by considering experimental data on concrete and rock samples present in the Literature.