1. Introduction

Material selection is an important task in product design, selecting the most suitable material from the over 200,000 engineering materials available today (Reference AshbyAshby, 2016) each characterized by multiple attributes (density, strength, thermal conductivity, heat capacity, recyclability, cost, etc.). The material selection process does not begin at the detailed design phase of a product. According to Pahl & Beitz (Reference Bender and GerickeBender and Gericke 2021), a specific idea of the materials and their impact on the design should already be present in the concept phase. The aim is to specify the functional structures in more detail and to make the solution principles assessable. In order to avoid limiting the design space too early for an optimal solution, all materials that fulfill the requirements should be considered and not rejected by an early and too strict material selection. In many design cases, multiple criteria must be considered simultaneously, making the material selection process a complex task requiring a systematic approach. Multiple methods have been developed for that process (Reference Jahan, Ismail, Sapuan and MustaphaJahan et al., 2010; Reference Emovon and OghenenyerovwhoEmovon & Oghenenyerovwho, 2020). The selection processes are in general separated into 4 steps (Reference Thelandersson and LarsenRahim et. al., 2020): 1) Translating requirements to design criteria, 2) Screening: filtering for suitable materials, 3) Ranking of suitable materials and 4) Searching for additional knowledge for the top ranked materials. This allows for a systematic reduction of the materials, leaving back only the most suitable materials for the specified task.

For an efficient evaluation of the significant number of materials, the Ashby charts are commonly used. Here, materials are represented on logarithmic charts using a material property for each axis (e.g. Young's modulus over density). Materials of the same material classes appear as clusters for which material classes can be represented as areas for a better overview. In the screening process, attribute limits can be placed on such charts, dividing the chart into two areas, included and excluded materials for further selection. While in the ranking phase, a sloped line representing the derived objective function is placed on the chart and moved in parallel until only a few materials are left. (Reference AshbyAshby, 2016)

From a mechanical engineering point of view, one of the key concerns for product design is to ensure the structure can withstand critical loading including safety factors. Potential strength failure can be determined by evaluating the global stresses of the structure and comparing it with the material strength (e.g. yield strength). Failure by fracture is however more challenging. Fracture analysis requires the evaluation of how defects or cracks within the structure influence the structural integrity and load-bearing capability. Defects or cracks lead to strong localized stress concentrations and a stress and strength analysis is not applicable. At such stress concentrations the stress intensity factor K_I indicates severity of the stress concentration. If the stress intensity factor exceeds the material's fracture toughness (K _IC ), fracture will occur. Failure by fracture typically leads to fast-growing cracks without initial warnings that the failure is about to occur which can lead to sudden catastrophic collapse of the structure. Understanding failure by fracture and the behavior of brittle materials is important to utilize the full potential of all available materials in product design.

To avoid expensive experiments or fracture mechanics analysis at the end of the material selection brittle materials (materials prone to fail by fracture) should be identified early in the material selection process, ideally in the screening phase. In literature brittle materials are often removed in the screening phase by using a simple attribute limit on the magnitude of the fracture toughness of the material. To minimize the weight of metro vehicles Carruthers et al. (Reference Carruthers, Calomfirescu, Ghys and Prockat2009) considered a systematic material selection leading to a reduction in the energy consumption and the annual operating cost. They performed a case study on interior grab rails where they introduced the attribute limit of $$10\ MPa\sqrt m $$ for screening out brittle materials. In a paper by Rashedi et al. (2012), the material selection strategies of wind turbines were explored, focusing on cost, mass, carbon footprint and energy. The study was performed considering multiple constraints and conflicting objectives including an attribute limit of $$5MPa\sqrt m $$ for small wind turbine blades and for large blades $$15MPa\sqrt m $$ and the towers. From a critical raw material perspective Ferro and Bonollo (2019) proposed a material selection method for metallic materials. Using the proposed method, they performed a case study of a bicycle fork and used in that study an attribute limit of $$20MPa\sqrt m $$ . In his classical book on material selection Ashby (2016) proposes an attribute limit of $$15MPa\sqrt m $$ as a general rule of thumb for a filter against brittle materials.

Since the fracture toughness describes the material's resistance to fracture, it appears reasonable to use such a fracture toughness attribute limit as a filter for brittleness. However, considering the following example the limitation of the presented attribute limit becomes evident. Consider the ductile and non-brittle thermoplastic material polypropylene (PP) loaded near its yield limit by a stress of σ = 40Mpa. Introducing a small crack of 2a = 0.5 mm results in a stress intensity under mode I loading of

(1) $$K_I = \sigma \sqrt {\pi a} = 1.6MPa\sqrt m $$

With the fracture toughness of polypropylene $${K_{IC}} = 4.1MPa\sqrt m $$ well above the stress intensity factor (Equation 1) this analysis shows that the material will clearly fail due to yielding and not by fracture. However, an attribute limit on the fracture toughness of $$15MPa\sqrt m $$ would falsely filter out this material as too brittle.

In this paper, we propose a novel filtering method for brittle materials based on linear elastic fracture mechanics (LEFM). The filter is validated considering the failure mode of 35 materials. The effect of using the precented filtering method in the material selection screening phase is shown by the use of Ashby charts. Furthermore, we introduce a method for working with variable defect sizes on the material class level.

2. Screening using the fracture toughness attribute limit

To motivate a change in the filtering method for brittle materials, we perform a study of the currently used attribute limit by comparing it to the failure mode of 35 materials. For each material, the failure mode is evaluated considering an initial crack of a = 0.5, which is a typical assumption of defect size that is detectable with standard non-destructive inspection methods (Reference Lampman, Mulherin and ShipleyLampman, 2022) Strength failure is introduced when the global stresses exceed the material's strength σ > σf whereas brittle failure occurs when the stress intensity factor, K _I , exceeds the fracture toughness, K _IC , which happens when the global stress $$\sigma \ge {{{K_{Ic}}} \over {\sqrt {\pi a} }}$$ (Reference AndersonAnderson, 2017). For each of the 35 materials, it is evaluated which of the two limits is satisfied by the lowest load value, thereby determining the failure mode to be either strength failure or fracture failure. The material strength is in this paper defined according to the classical Ashby conventions: yield strength for metals and polymers, modulus of rupture for ceramics and glasses, and tensile failure for composites (Reference AshbyAshby, 2016).

In Figure 1, the 35 materials are represented on a fracture toughness-strength chart. As expected, the materials from the same class cluster into groups. The failure modes ductile and brittle failure are represented by a round and triangular symbol respectively. A fracture toughness attribute limit of $$15MPa\sqrt m $$ is represented, dividing the chart into two areas; materials there are included (white area) and excluded (grey area) for the further material selection process. As shown, the attribute limit excludes all materials except for 9 metals and 4 composite materials. However, comparing the attribute limit to the failure mode of ductile (round) and brittle (triangle) failure, the attribute limit does not predict the failure mode correctly. Multiple ductile polymers and natural materials are falsely assumed to be brittle by the attribute limit and thereby excluded from the further material selection process. Additionally, the three brittle metals (3,7,8) in the upper right corner of the chart are by the attribute limit falsely included and erroneously assumed to fail by ductile failure.

Figure 1. The material failure mode for 35 materials shown together with the often used K _IC filter

This study shows that the currently used attribute limit both excludes ductile materials and includes brittle materials. Falsely excluding whole material classes under the assumption of brittle failure limits the material selection process unnecessarily, excluding potentially good material candidates, thereby decreasing the possibilities for the overall final design solution. Including brittle materials can lead to unexpected brittle failure later in the design process, adding additional work by the need for redesign or reevaluation of the material.

The attribute limit is due to its simplicity the commonly used method today for filtering of brittle materials in material selection processes. An alternative way to detect brittle failure is a full fracture mechanics analysis. Multiple methods are available; however, those methods are complex, require additional knowledge, and are time-consuming and thereby not suitable for the early material selection process. A new simple filtering method correctly distinguishing between brittle and ductile failure is needed.

3. Analytical derivation of LEFM filter

To improve the filtering method for brittleness we aim to establish a filtering method capable of differentiating between ductile and brittle materials by evaluating the strength failure and fracture failure respectively. Strength failure is evaluated by comparing the global stresses to the material strength. For fracture, potential cracks and defects must be considered. The crack results in a strong local stress concentration around the crack tip which when exceeding the material fracture toughness will result in fracture. As the stresses in the crack tip exceed the yield stress of the material, a plastic zone in front of the crack tip will occur, limiting the maximum stresses and distributing the stress increase to a larger area. Brittle materials are characterized by a lack of ductility limiting the plastic zone to a small area. Such materials are described well by the linear elastic fracture mechanics (LEFM) theory assuming a global linear elastic behaviour of the materials, allowing for only small-scale yielding at the crack tip. When the materials become more ductile the assumptions of the LEFM break down and more complex theories are required (Anderson, Reference Anderson2017). However, as our filter is tailored to identify brittle materials the framework of LEFM is sufficient. The stress intensity factor describing the intensity of the singular stress field at the crack tip is given by:

(2) $$K_I = Y\sigma \sqrt {\pi a} $$

where Y is a geometric factor and a is the crack length. Facture occurs when the stress intensity factor exceeds the fracture toughness K _IC of the material. Hence, the fracture condition of LEFM can be given with stress intensity factors as:

(3) $$K_I \le K_{Ic} $$

The maximum stress level is bound by the material strength limiting the stresses to the elastic area of the material:

(4) $$\sigma = \sigma _f $$

By combining Equations (2) to (4) we get the following limit state:

(5) $${{K_{Ic} } \over {\sigma _f }} \ge Y\sqrt {\pi a} $$

which describes the transition between strength and fracture failure, allowing us to determine the material prone to fracture.

The limit defined in Equation (5) will in this paper be named as the LEFM filter. By specifying the values of Y and a, the filter limit value can be determined. For the following study of the filter, a conservative choice of these values is used. We consider an edge crack with Y = 1.1215 and a commonly used defect size of a = 0.5 mm (Reference AndersonAnderson, 2017). This results in the filter value of:

(6) $${{K_{Ic} } \over {\sigma _f }} \ge 0.044\sqrt m $$

4. Results

4.1. Validation of LEFM filter

The proposed LEFM filter from Equation (6) is presented together with the failure modes of the 35 materials discussed in Section 2. In Figure 2 the 35 materials are represented on the fracture toughness-strength chart where the symbols represent the material failure modes: strength failure (round) or fracture failure (triangle). The LEFM filter is represented on the plot by the sloped line where K _IC /σf > 0.044\sqrt{m}. As shown the LEFM filter correctly differentiates between the two failure modes such that materials exhibiting failure by fracture are filtered out, while materials limited by strength failure are included in the further selection process. The attribute limit K _IC > 15 MPa\sqrt{m} discussed in section 2 are represented in the plot by a horizontal line. The two filtering methods separated the chart into four distinct areas. In areas of high strength and low fracture toughness (area II) or low strength and high fracture toughness (area IV), the filtering methods considering the LEFM filter and the attribute limit agree on the inclusion/exclusion of materials in the screening process. For high strength and fracture toughness (area I) the K _IC filter falsely includes brittle materials while the LEFM filter correctly excludes those. And in the area of low strength and fracture toughness (area III) the K _IC filter incorrectly excludes a wide range of nonbrittle materials while the LEFM filter correctly includes those. The difference in the filtering methods can significantly alter the screening process. The attribute limit evaluates the material fracture toughness alone resulting in a horizontal line on the figure. In contrast, the LEFM filter takes the potential of ductile failure into account when evaluating the material failure mode resulting in a sloped line for the filter following the transition between the two failure modes.

Figure 2. The material failure mode for 35 materials is shown together with the LEFM filter and the attribute limit

4.2. Material classes

Material databases are a useful tool for evaluating the significant number of materials that are often required in the early stages of the material screening phase (Reference Ashby and CebonAshby, 1993; Reference Rahim, Musa, Ramesh and LimAshby, 2017; Rahim, 2020). By representing the LEFM filter on a fracture toughness-strength Ashby chart (Figure 3), along with the attribute limit K _IC ≥ 15 MPa\sqrt{m}, it becomes evident how the screening process is fundamentally altered by the proposed filtering method. While the attribute limit previously allowed metals and only a limited number of composite materials to proceed to the next stages of the material selection process, the new LEFM filter opens up the possibility for materials from all material classes to be considered, with the exception of ceramics. This broader inclusion of material types helps enhance the variety of potential materials available for selection. As shown in the validation of the filter in section 4.1, the LEFM filter is able to accurately distinguish between materials prone to brittle failure and those that are more ductile, ensuring that all ductile materials are included in the further material selection. The introduction of a wider range of material classes into the selection process does not compromise the safety of the design in any way. In fact, safety is improved through the exclusion of materials that possess both high strength and high fracture toughness found in area I of the chart. These materials, despite having high fracture toughness, have the potential to fracture under certain conditions, and their removal from consideration helps maintain a more reliable and safe design.

Figure 3. Comparison of the LEFM and the K _IC filter represented on an Ashby fracture toughness - strength chart

4.3. Variable filter value

The LEFM filtering value is proportional to the square root of the considered material defect size (see Equation 5). The defect size can alter significantly for different materials, given the variety of production and quality assurance methods. The LEFM filter can be extended to take material dependent defect size into account resulting in a material dependent filtering value. Take technical ceramics as an example: These materials are usually subjected to a proof test, where the materials are subjected to a load satisfying the crack growth condition (Equation 3) for cracks larger than 20 µm. For more sensitive applications such as biomedical implants, the proof test is performed at higher stress levels as defect size larger than 5 µm is not tolerated (Reference Bermejo, Danzer and SarinBermejo, 2014). Hence, for these materials the assumption of a = 0.5 mm is significantly underestimating the load bearing capability and will lead to an overly strict filter value.

In the following example, we consider a material class-dependent filtering value based on the defect sizes represented in Table 1. The LEFM filters for different material classes are illustrated in Figure 4. The defect size for ceramics is set to a value of 20 µm, which leads to a decrease in the filter value to 0.009\sqrt{m}. This value is significantly lower than the previously used filter value of 0.044\sqrt{m} for material defects of 0.5 mm (Figure 3). The reduction in the filter value to 0.009\sqrt{m} allows for a larger area of ceramic materials to become available for the subsequent material selection process.

When comparing the defect size of natural materials of 5 mm (Table 1), to the defect size of 0.5 mm, we observe that the defect size for natural materials is increased by a factor of 10, which leads to an increase in the filter value by a factor of approximately 3. This increase significantly limits the number of potential material candidates that come from natural materials, narrowing down the available options for the next stage of material selection. However, despite this reduction, the natural materials still maintain a considerable presence due to their high fracture toughness-to-strength ratio. As shown in Figure 4, a significant portion of the natural materials still meets the requirements and is included in the next phase of the material selection process. Furthermore, as Figure 4 demonstrates, by utilizing the filter values provided in Table 1, all material classes remain viable for further selection without compromising the safety of the final design.

Table 1. Material class dependent filter values

Figure 4. Material class dependent filter values represented on an Ashby chart