## Abstract

By using a generalized maximum tensile stress (MTS) criterion to predict onset of brittle fracture, it is shown that the presence of T-stress can have a significant effect on mode I and mode II toughness. The most prominent influence of T-stress on toughness occurs for mode II conditions. However, earlier tests concentrated on near mode I and results were masked by scatter. New experiments, using combinations of mode II loading and T-stress, support conclusively the generalized MTS criterion. This criterion is shown to be very robust and applicable to predicting probability of brittle failure. The criterion is also relevant to other experimental results where combinations of mode II loading with high values of T-stress can lead to values of mode II toughness that are greater than mode I toughness.

## 1. Introduction and overview

### (a) Scope of the paper

Brittle fracture is a major mode of failure in materials, components and structures that contain cracks. The consequences of brittle failure are enormous and costly (Anderson 1995). The focus of much research for over 100 years has been to provide a scientific description of the conditions that lead to failure. This research has developed many fracture models to describe brittle fracture in linear elastic materials. These models usually require information about the applied load, crack geometry and the fracture toughness of the material.

The principal aim of this paper is to explore the consequences of the T-stress on brittle fracture in elastic solids subjected to mode I and II loading. Of special interest to this paper is the robustness of the application of a simple model to conditions where T-stress influences the initiation of brittle fracture. First, failure criteria used to describe the onset of brittle fracture, particularly, for mixed mode fracture and in the presence of T-stress are summarized. There follows a short summary of earlier experiments used to verify the theories. In §2, an overview of a theoretical framework for predicting brittle fracture in linear elastic solids is provided. This is called the generalized maximum tensile stress (MTS) criterion and includes a summary of recent results to examine particular loading cases for pure mode I and mode II. New experimental results are then presented with a special emphasis on using a single specimen to examine the effects of both negative and positive T-stress. A discussion of the results in the context of the theoretical framework follows with the aim of highlighting the robustness of a simple failure criterion using statistical analysis.

### (b) Failure criterion

A crack can deform according to three different modes known as mode I, mode II and mode III. Mode I refers to tensile loading in which the crack flanks tend to open. In mode II, the crack is subjected to an in-plane shear load and its flanks slide without opening. Mode III corresponds to out-of-plane shear loading. Because of symmetry, crack extension in mode I is expected to initiate along the line of the initial crack. A criterion for pure mode II or mixed mode I/II loading is more complicated than for pure mode I, because both the direction of fracture initiation and the onset of fracture must be determined.

One simple model for describing the micromechanism of brittle fracture is due to Ritchie, Knott & Rice (1973), often known as the RKR model. The model proposes that crack growth in brittle fracture takes place when the tensile stress at a critical distance *r*_{c} ahead of the crack tip attains a critical value *σ*_{c}. The critical tensile stress is determined via the stress intensity factor, *K*.

The conventional relation for onset of mode I fracture is(1.1)The mode I stress intensity factor *K*_{I} describes the state of stress near the crack tip and fracture toughness *K*_{Ic} is a material parameter representing the critical conditions required for the initiation of fracture.

When the crack is subjected to external loads such that there are other modes of crack deformation, e.g. modes II and III, the fracture criterion is more complicated. For example, Erdogan & Sih (1963) developed the MTS criterion, and the maximum energy release rate criterion was introduced by Hussain *et al*. (1974). Another criterion, called the minimum strain energy density criterion, was developed by Sih (1973). Most of these mixed mode fracture criteria agree with each other for the tensile (mode I) condition and deviate from each other for cracks under combinations of tensile and shear loading (mixed modes I and II).

In the MTS criterion, brittle fracture initiates radially from the crack tip along the angle of maximum tangential stress, *θ*_{m}. In addition, fracture occurs when the tangential stress, *σ*_{θθ}, at some distance *r*_{c} in the direction defined by *θ*_{m} exceeds the critical stress *σ*_{c}. For combinations of mode I and mode II loading (often called mixed mode loading), the stresses near the crack tip depend on both mode I and mode II stress intensity factors, *K*_{I} and *K*_{II}. An earlier model by Erdogan & Sih (1963) (called the MTS criterion) showed that the onset of fracture is predicted by(1.2a)where *θ*_{m} is the direction for fracture initiation found by solving(1.2b)By combining equations (1.1)–(1.2*b*), the onset of fracture, irrespective of the mode of loading, requires the definition of a critical stress, *σ*_{c}, and a critical distance *r*_{c}. The MTS criterion predicts that mode II fracture occurs at an angle of −70.5° relative to the crack plane and the mode II fracture toughness is equal to 0.87 times the mode I fracture toughness.

The sole use of a stress intensity factor to describe the near crack tip stress assumes that singular stresses dominate the near crack tip stresses. Williams (1957) showed that the elastic stresses in a cracked body can be expressed as infinite series expansions with the first terms in the expansions providing singular stresses. At an important micro-mechanical distance from the crack tip, equations (1.1)–(1.2*b*) both assume that singular stresses alone characterize the onset of failure. Although singular stresses dominate very close to the crack tip, at the critical distance, *r*_{c}, the singular stress may not be sufficient to describe the onset of fracture and the higher-order terms in the Williams' expansions are also important. In particular, the second term, often known as the T-stress, can influence significantly the processes of crack extension. The T-stress is a constant stress parallel to the crack.

Under the predominantly mode I condition, the consequence of T-stress for the initiation of brittle fracture has been assumed to be small. This is mainly because the T-stress has assumed to provide no contribution to the opening stress ahead of mode I cracks. For mode I cracks, T-stress effects in linear elastic materials have been investigated mainly for determining the trajectory of fracture rather than the initiation angle of fracture. For instance, theoretical and experimental results presented by Cotterell (1966), Leevers *et al*. (1976) and Cotterell & Rice (1980) show that for positive values of T-stress, the fracture trajectory deviates gradually from the line of the initial crack. In contrast, specimens with negative T-stress exhibit a stable fracture path. Finnie & Saith (1973) and Streit & Finnie (1980) have also studied the influence of T-stress on the stability fracture initiation angle in mode I. More recent work on the effect of T-stress for linear elastic materials has been investigated by Chao & Zhang (1997), Selvarathinam & Goree (1998) and Chao *et al*. (2001). Recent theoretical studies by Ayatollahi *et al*. (2002*a*) also suggest that brittle fracture in mode I is influenced significantly when the T-stress exceeds a critical value. This is explored further in this paper.

It is now widely accepted that, for brittle fracture under conditions of constrained yielding, a single fracture parameter, such as *K*_{Ic}, is insufficient. Here, fracture is assumed to take place inside a process zone of constrained yielding, and T-stress has been used as a measure of the constraint. Larsson & Carlsson (1973) illustrated that for mode I, positive or negative values of T-stress influence the shape and size of the plastic zone around the crack tip, so that specimens with a negative T-stress have lower constraint compared to those with a positive T-stress. For elastic–plastic material behaviour, it is known that the crack tip constraint can be quantified by the T-stress but only for small to moderate scale yielding. This has been studied extensively for mode I (e.g. Betegon & Hancock 1991; O'Dowd & Shih 1991, 1992) as well as for mode II (Ayatollahi *et al*. 2002*b*). This paper focuses on conditions where plasticity does not play a role in governing the conditions for onset of brittle fracture.

### (c) Earlier experiments

Brittle fracture experiments fall into two categories: experiments to validate a failure criterion and experiments to determine the fracture toughness of the material. For the former, Erdogan & Sih (1963) first provided experimental validation of the MTS criterion by considering only the singular term in the tangential stress around the crack tip. Later, Williams & Ewing (1972), Finnie & Saith (1973), Ewing & Williams (1974) and Ueda *et al*. (1983) showed that an improved prediction of test results can be achieved for an angled central crack specimen if, in addition to the singular term, the effect of T-stress is also accounted for in the MTS criterion. However, these studies were confined only to angled central cracks and are unable to examine the influence of mode II loading alone. Shetty *et al*. (1987) also proposed possible effects from the non-singular stress term in brittle fracture of the cracked Brazilian disc (CBD) specimens, but did not provide an acceptable consistency between the experimental and theoretical results.

Smith *et al*. (2001) recently reviewed a range of experimental results. The results are shown in figure 1 and exhibit a large amount of scatter. In figure 1, the effective stress intensity factor is given by(1.3)Also shown in figure 1 is a prediction using a non-dimensional critical distance *α* discussed later in this paper.

The second category of experiment is that used to determine the fracture toughness of a material. For mode I loading, there are many well-defined test configurations, (Anderson 1995), such as the compact tension, C(T) and single edge notch bend, SEN(B) specimens, which are known to provide fracture conditions in elastic brittle solids, where *T* is zero or close to zero. Similarly, there are test specimens that provide fracture conditions in combinations of tension and shear (mixed mode loading).

Many different test specimens have also been designed for determining the mode II fracture toughness for various engineering materials. Some of these specimens are summarized here. Erdogan & Sih (1963) used a centre-cracked specimen subjected to two skew-symmetric point loads applied at holes located near the crack edges. Ueda *et al*. (1983) employed the angled centre-cracked specimen under biaxial loading. Royer (1986, 1988) designed a Y-shape specimen containing two edge cracks. Mode II could be achieved in Royer's specimen by applying the load at two top branches of the specimen at appropriate angles. Richard (1981), Banks-Sills & Arcan (1986) and Mahajan & Ravi-Chandar (1989) made use of rather complicated configurations, in which a compact-tension-shear specimen was located inside a fixture. The specimen can be subjected to mode II by applying the load in relevant holes in the fixture.

Davenport & Smith (1993) used a similar test apparatus. They replaced the compact-tension-shear specimen with a single edge crack specimen. A few researchers have conducted mode II fracture tests on a single edge crack specimen subjected to antisymmetric four-point bending (e.g. Maccagno & Knott 1989; Suresh *et al*. 1990). Ayatollahi *et al*. (2002*a*,*b*) used a S-shape crack specimen to perform mode II fracture experiments. A disc specimen containing a centre crack and subjected to a diametral compressive load has also been used frequently for fracture tests in brittle materials (e.g. Awaji & Sato 1978; Atkinson *et al*. 1982; Shetty *et al*. 1987; Liu *et al*. 1998; Krishnan *et al*. 1998; Khan & Al-Shayea 2000; Scherre *et al*. 2000; Chang *et al*. 2002). Mode II is provided in this specimen, often called the CBD specimen, when the crack line is oriented in an appropriate angle relative to the loading direction. Any of the specimens mentioned above can be used for conducting a mode II fracture test on brittle materials.

Many experimental fracture results have been reported for mode II loading that do not agree with equations (1.2*a*) and (1.2*b*), in that the measured mode II toughness is not equal to the mode I toughness multiplied by 0.87. For example, Ayatollahi & Aliha (2005) review experimental results generated using the Brazilian disc specimen and show that the fracture toughness ratio, , is always significantly higher than 0.87 predicted by the conventional MTS criterion. The reported values for typically commence from 1.09 for graphite SM-124 (Awaji & Sato 1978) and increase to a figure as high as 2.2 for Saudi Arabian limestone (Khan & Al-Shayea 2000). Although it can be assumed that friction between the crack faces provides a contribution, we will show later that the higher than predicted mode II fracture toughness values can be explained because of the presence of the T-stress.

Earlier experimental work demonstrates that the presence of T-stress plays a role in changing the conditions for fracture, particularly for mixed mode loading. The evidence, however, is sparse and somewhat prone to high levels of scatter as shown in figure 1. In addition, there is considerable evidence indicating that T-stress may also influence fracture behaviour in circumstances where no account of T-stress was considered. In some instances, claims are made that a mode II value of toughness can be higher than mode I.

Section 2 provides the theoretical basis for establishing the consequence of T-stress on brittle fracture before considering experimental validation for the theory.

## 2. Theoretical framework

### (a) Generalized criterion

The in-plane linear elastic stresses around the tip of a crack can be described by symmetric and antisymmetric fields, called mode I and mode II, respectively. The stresses, in polar coordinates, can be written as an eigen series expansion (Williams 1957), for any homogeneous and isotropic body, as(2.1)(2.2)(2.3)where *σ*_{rr}, *σ*_{θθ} and *σ*_{rθ} are the polar stresses. The higher-order terms *O*(*r*^{1/2}) can be considered negligible near the crack tip. It is seen that *T* has a contribution to all three stress components in polar coordinates, unlike in Cartesian coordinates. The crack tip parameters *K*_{I}, *K*_{II}, and *T* depend on the geometry and loading configurations and can vary considerably for different specimens.

The direction of the maximum tangential stress, *θ*_{m}, is determined from(2.4)where *r*_{c} is a critical distance for onset of failure.

Once the direction of initiation of fracture, *θ*_{m}, is found from equation (2.4), the value of *θ*_{m} is replaced into equation (2.2) to determine the onset of brittle fracture, so that(2.5)where (*σ*_{θθ,c})=*σ*_{c} is the critical value of the tangential stress at the critical radius *r*_{c}. For pure mode I, where *K*_{II} and *θ*_{m} are equal to zero and *K*_{I} can be replaced by the mode I fracture toughness *K*_{Ic} and then equation (2.5) reduces to equation (1.1).

Replacing the left-hand side of equation (2.5) with equation (1.1) and rearranging gives(2.6)Equations (2.4) and (2.6) describe a generalized MTS criterion. When *K*_{I}, *K*_{II}, and *T* are known, the condition for brittle fracture is predicted for any combination of in-plane loading. Rearranging equation (2.6) by collecting the singular stress terms to the left-hand side gives(2.7)It is appropriate to consider two special cases from this analysis, modes I and II alone. The former has been studied in detail by Ayatollahi *et al*. (2002*a*) and summarized here. The cases for modes I and II act as bounds for any combination of mode I and mode II and also as a basis for experimental validation of the MTS criterion, together with the RKR model for brittle fracture.

### (b) Mode I loading

The conventional solution from equation (2.4) for mode I loading is that the maximum angle, *θ*_{m}, is zero. However, this is only correct (Kosai *et al* 1996, 1999; Chao *et al*. 2001 and Ayatollahi *et al*. 2002*a*) when the second derivative of *σ*_{θθ} at *θ*_{m}=0 is negative, which occurs when(2.8)where , and *a* is crack length. Note that *α* corresponds to a non-dimensional distance and *B*_{I} is a non-dimensional T-stress.

*B*_{I} is the biaxiality ratio proposed by Leevers & Radon (1982). With conditions satisfied by equation (2.8), equation (1.1) describes the fracture condition.

For values of *Bα* greater than 0.375, Kosai *et al*. (1996, 1999), Chao *et al*. (2001), Ayatollahi *et al*. (2002*a*) and Liu & Chao (2003) show that the maximum tangential stress is given by(2.9)With *Bα* greater than 0.375, equation (2.7) describes the fracture condition with *K*_{II} equal to zero, and the fracture angle *θ*_{m} given by equation (2.9).

The product *B*_{I}*α* is also related to *T*/*σ*_{θθ,c} by(2.10)and equation (2.10) can be re-expressed in terms of *Bα* to give(2.11)

The variation of the fracture angle and mode I stress intensity at fracture relative to *K*_{Ic} are shown in figures 2 and 3, respectively. When a stress-based mechanism of fracture is used, for values of *Bα* (or *T*/*σ*_{θθ,c}) less than 0.375, T-stress has no influence on the fracture angle and the mode I toughness, whereas conversely for *Bα* greater than 0.375 the T-stress dramatically reduces the mode I toughness with the fracture angle deviating away from the plane of the initial crack. In figure 3, the vertical and horizontal axes represent, respectively, the contributions of the singular term and the T-stress term to the initiation of brittle fracture under mode I loading alone.

### (c) Mode II loading

For mode II loading alone, the fracture angle *θ*_{m} is provided from equation (2.4), so that(2.12)where B_{II} is the mode II biaxiality ratio, so that . The product *B*_{II}*α* is related to *T*/*σ*_{θθ,c} by(2.13)The variation of the fracture angle with respect to *T*/*σ*_{θθ,c} is shown in figure 2, noting that for pure mode II (with *T*/*σ*_{θθ,c}=0), the fracture angle is −70.5° as predicted by the conventional MTS criterion. For negative values of *T*/*σ*_{θθ,c}, the fracture angle changes to values greater than −70.5°, and for positive values of *T*/*σ*_{θθ,c} the fracture angle to values less than −70.5°.

Equation (2.7) describes the fracture condition for mode II with *K*_{I} equal to zero. Alternatively, by using equation (2.10), the fracture condition for mode II alone is given by(2.14)The variation of the mode II fracture toughness relative to *K*_{Ic} is shown in figure 3. When *T*=0, the conventional MTS criterion predicts that *K*_{II}/*K*_{Ic}=0.87. Unlike the influence of T for mode I, for mode II loading, negative values of T can lead to values of mode II toughness that are greater than the mode I toughness. Conversely, for positive values of *T*, the mode II can be lower than values for *T*=0.

The generalized criterion, equation (2.7), demonstrates that the consequence of the T-stress on values of brittle fracture toughness is marked, more so for mode II conditions under moderate levels of T-stress. Equation (2.7) also shows that the effect of *T* on *K*_{II} is important. For example, Banks-Sills & Arcan (1986) use a definition for mode II that assumes *T* should be zero. This results from William's (1957) special approach for solving the crack tip stresses. However, Ayatollahi *et al*. (1996, 1998) show that in mode II, *T* vanishes only for purely antisymmetric loading. Furthermore, ideal antisymmetric loading occurs rarely for real engineering components and *T* can be considerable in some practical cases in the shear loading of cracked bodies. This suggests most of the experimental results presented in the literature for mode II brittle fracture can be used only for a limited set of real applications, where the crack tip is subjected to conditions very close to antisymmetric loading. It should be noted that Banks-Sills & Arcan (1986) and Banks-Sills & Bortman (1986) also attempted to consider the effect of the higher-order stress terms in their mixed mode specimen. However, their solution method only accounted for the and terms and *T* was omitted in their analyses.

Section 3 presents experimental work to confirm the theoretical results.

## 3. Experimental validation

Earlier we summarized existing data that provide limited validation for the consequences of T-stress on brittle fracture. Figure 1 illustrates results reviewed by Smith *et al*. (2001) collated from Williams & Ewing (1972) and Ueda *et al*. (1983). There is only limited evidence that substantiate the consequences of T-stress and the robustness of the simple fracture criterion. This is because there is substantial scatter in these earlier results. The analysis in §2, however, reveals that the greatest influence on fracture by the T-stress occurs for mode II conditions, where there is an almost linear inverse relationship between T-stress and fracture toughness. In the following, we describe a set of experiments where the combination of pure mode II loading and T-stress is explored for brittle fracture in perspex. These experiments used a sufficient number of specimens to perform a statistical analysis of results and to allow a more comprehensive appraisal of the consequences of T-stress.

### (a) Experiments and material

Previous pure mode II test specimens that give antisymmetric loading (e.g. Richard 1981; Maccagno & Knott 1989) do not provide a T-stress. However, most common mode II specimens do not give *T*=0, and earlier work by the authors (Ayatollahi *et al*. 1996, 1998) suggests that there are particular geometric and loading configurations where considerable values of the T-stress exist with different combinations of mode I and mode II. To test the theory described in §2, our attention focused on developing a simple test specimen where *K*_{I} is zero but a significant value of *T* is present in conjunction with *K*_{II}. There are other mode II specimens which provide either negative T-stress (e.g. Brazilian disc specimen) or positive T-stress (like semi-circular bend (SCB) specimen). The major advantage of our specimen is that it can provide both +T and −T just by changing the loading direction. Thus, we could study mode II fracture in one specific material but under two different conditions (+T and −T). This is not the case for other specimens.

Figure 4 shows the design of the mode II specimen used for the experiments. The specimen is, in general, a modified version of the crack specimen used by Williams & Birch (1976) for mode II fracture tests in wood. When either a tensile or a compressive load is applied at the loading holes, the central part of the specimen near the crack is subjected to pure shear. A high ratio of cross-section area to length *L* prevents possible buckling when subjected to a compressive load.

The conventional elastic crack tip stresses for mode II relate to a positive shear loading. This is shown in figure 4*b*, where positive *θ* is counter-clockwise. However, the same equations for the crack tip stresses can be used for a negative shear loading, provided a clockwise direction is considered as positive *θ* (see figure 4*b*). Tensile and compressive loads on the mode II specimen shown in figure 4*b* correspond to positive and negative shear loads, respectively. The appropriate direction for *θ* as described above is used for all the results presented in this research for the mode II shear specimens.

Finite-element analyses were conducted to determine stress intensity factors and T-stress for tensile and compressive loading. This is described later. With reference to the sign of the T-stress, in the present analysis the mode II specimen is called a +T shear specimen for tensile loading and a −T shear specimen for compressive loading.

The test material was polymethyl methacrylate (PMMA). This material is known to behave in a brittle fashion in both mode I and mode II (Williams 1983; Davenport & Smith 1993). Mode II specimens were machined from a cast perspex sheet of 20 mm thickness. According to the manufacturer, the PMMA had a molecular weight of *M*_{w}=10^{6} g mol^{−1}. An earlier study by Davenport & Smith (1993) determined the basic material properties, where Young's modulus *E*=2800 MPa, Poisson's ratio *ν*=0.38. All the specimens were cut from one sheet of material and with the same orientation for the loading axis to reduce possible orientation effects on material properties associated with the manufacturing process.

To produce the crack in the specimens, a slit of almost 9 mm length was introduced using a fret saw of 0.35 mm thickness. A sharp tip was then produced by pushing a razor blade for another 1 mm inside the slit to make the total crack length around 10 mm. The crack length was measured optically.

A set of mode I tests were also conducted using the conventional single edge notched tension, SEN(T), specimen. By using the maximum tangential stress criterion, the mode II fracture toughness was predicted from the mode I test results. The length, width and thickness of the SEN(T) specimen were 76, 20 and 20 mm, respectively.

Tests were conducted at room temperature (20 °C), under displacement control at 0.5 mm min^{−1} using a testing machine of 10 kN capacity. Ten mode I tests, 10 mode II +T shear tests and 10 mode II −T shear tests were conducted. A data processing unit attached to the machine recorded the peak load at fracture and the corresponding displacement.

### (b) Finite-element analysis

The finite-element code ABAQUS (1997) was used to determine the stress intensity factors and the T-stress for the mode II specimen. The material model was linear elastic, with the same properties for the experimental tests. The crack tip zone consisted of 30 rings of straight-sided elements, where each ring had 36 eight-noded elements circumferentially. A quarter point scaling was used between the circumferential rows of nodes surrounding the crack tip to produce a square root singularity in the strain field at the crack tip.

Similar to the specimen used by Williams & Birch (1976), the centre points of loading holes were first considered to be along the crack line. However, the finite-element results revealed that there were contributions from mode I in deformations around the crack tip. Williams & Birch (1976) reported the same difficulty. This mode I contribution is attributed to a slight bending in the two arms of the specimen. The locations of loading holes were then changed to remove the mode I effects. It was finally found through a set of finite-element analyses that mode II without any mode I contribution could be achieved if the loading holes were 4 mm from the crack line (see figure 4*a*). With such a shift, the ratio of *K*_{I}/*K*_{II} is 0.002 and assumed to be negligible. This ratio was calculated from displacements at two respective nodes along the upper and lower crack faces as(3.1)where *u* and *v* are displacements parallel and normal to the crack line. The signs (+) and (−) refer to upper and lower crack faces.

The specimen was subjected to compressive and tensile reference loads. The *J*-integral was obtained directly from ABAQUS, which uses the modified virtual crack extension method proposed by Li *et al*. (1985). The *J*-integral for both cases of tensile and compressive loading was equal. Apart from the first contour, the *J*-integral was path independent for the remaining 30 contours surrounding the crack tip. *T* was determined by using the displacements along the crack faces (Ayatollahi *et al*. 1998). The biaxiality ratio *B*_{II} is given by . For plane strain conditions . *B*_{II} was calculated by using the results for *J* and *T* from the FE analysis into the equation for *B*_{II}. For the specimen in tension *T*>0 and *B*_{II}=+2 and in compression *T*<0 and *B*_{II}=−2. Unlike *J* and *T*, the magnitude of the biaxiality ratio *B*_{II} is independent of the magnitude of load. The finite-element results showed that the very slight differences in the lengths of manufactured cracks in different specimens had negligible influence on the crack tip parameters *J*, *B*_{II} and *K*_{I}/*K*_{II}. Therefore, the crack tip parameters obtained for the crack length *a*=10 mm were considered in the theoretical studies for all specimens.

The mode II stress intensity factor *K*_{II} is written as(3.2)where *P* is the load applied whether compressive or tensile, *t* and *W* are the specimen thickness and width, respectively, and *f*_{II}(*a*/*W*) is a geometry factor determined from FE analysis to be 0.141.

### (c) Experimental results

All the specimens exhibited a linear load–displacement diagram prior to fracture, confirming the predominantly linear elastic behaviour of the material. There was no stable crack growth and the specimens all failed by fast fracture.

None of the compressively loaded −T shear specimens failed due to buckling. Visual inspection showed that the crack faces did not meet each other and remained parallel in both tensile and compressive tests. After testing, the fracture initiation angles for the shear specimens were measured from the magnified picture of each broken specimen. However, for some of the specimens, the fracture initiation angles varied slightly along the crack front. Therefore, the fracture angle was measured along the centre line of the surface at mid-thickness. For mode I specimens, fracture took place along the direction *θ*_{o}=0 and the variation in the direction of fracture was negligible.

The measured fracture angles, together with corresponding mean values, are illustrated in figure 5 for the 20 mode II tests. The mean values of fracture initiation angle in mode II tests are −78.6° for the +T specimens and −55.2° for the −T specimens. The scatter in the results for the −T specimens is slightly more than that of the +T specimens. The initial crack planes and the trajectories of crack growth in the +T and −T shear specimens are illustrated in figures 6 and 7. It can be seen that the plane of fracture in the +T specimen is only slightly curved in front of the crack across the ligament, whereas the curvature of the fracture plane across the ligament in the −T specimen is considerable. The plane of fracture was flat for all mode I specimens.

Figure 6 shows the fracture surface for a +T shear specimen. This was indicative of the fracture faces of all the +T shear specimens. The fracture surface consisted of three different zones: a mirror area (i.e. very flat and smooth), an area with ridge markings and an area with hyperbolic features. The mirror area initiates from the middle section of the original crack front (portion ‘aa’ indicated in figure 6*c*) and extends to the central zone of the fracture surface. The area of ridge markings occurs next to the edge of the original crack. The ridges are inclined such that at one end they tend towards the central part of crack tip (points ‘a’ shown in figure 6*c*) and at the other end they tend towards the two lateral surfaces (surfaces B). The ridges closer to the surfaces B are longer than those near the points ‘a’. However, the strip of the ridge markings is limited to a small band next to the crack front. The area of hyperbolic features is observed in the zone surrounding the central mirror area as shown in figure 6*c*. Each hyperbolic marking is related to a local secondary crack growth initiated from the pole of the hyperbola (Doll 1989; Bhattacharjee & Knott 1995).

The fracture surface for the −T shear specimen is shown schematically in figure 7. The fracture surface exhibits two main areas: a band of mirror zone and two strips of ridge markings. The mirror band is seen in the middle part of the specimen, which extends all along the fracture surface. The ridges extend in two strips on the sides of the central mirror area. The heights of ridges near the edge of the initial crack are higher than those on the +T shear specimens. However, the heights of the ridges decrease considerably when the curved fracture surface becomes almost flat, further away from the edge of the initial crack. No hyperbolic markings are seen on the fracture faces of the −T shear specimens.

To summarize, the fracture surface features of +T specimens are similar to those found in previous PMMA fracture studies (Bhattacharjee & Knott 1995). For −T specimens, the mirrored fracture region is bounded by ridges rather than an area of hyperbolic features, suggesting that compressive loading of the −T specimens plays a role in preventing the initiation of secondary crack growth.

Fracture toughness values for all the mode II and mode I tests are shown in figure 8. The mean value toughness for the mode I test was 1.94 MPa(m)^{1/2}, and 2.43 and 1.27 MPa(m)^{1/2} for the mode II −T and +T shear specimens, respectively. Mode II fracture toughness values were calculated from the fracture load, *P*_{cr}, substituted into equation (3.2).

Mode I fracture toughness values were calculated using(3.3)The geometry factor *f*_{I}(*a*/*W*) is given by Tada *et al*. (1985) for single edge notched tensile specimens.

The average value for the mode I fracture toughness is 1.95 MPa(m)^{1/2}, which is in good agreement with 1.87 MPa(m)^{1/2} given by Maccagno & Knott (1989).

Chao & Zhang (1997) have shown that at room temperature, the size of the plastic zone in PMMA in front of the crack tip at fracture load is less than the critical distance *r*_{c}. This implies that the plasticity effects around the crack tip are negligible and the use of linear elastic stresses for predicting the onset of crack growth is justified. In mode I, *T* has no effect on the elastic tangential stress in front of the crack tip where *θ*=0. Therefore, the mode I fracture toughness obtained here for PMMA can be considered to be independent of *T* if a fracture criterion based on the critical tensile stress is used. It is suggested that the mode II fracture toughness calculated from mode I test results (by using the MTS criterion) can be attributed to a mode II fracture test with *T*=0.

Figures 5 and 8 demonstrate that the tests on the +T and −T specimens provide different fracture loads as well as fracture initiation angles. These results, as we will discuss in §4, provide confirmation of the effects of T-stress on mode II fracture toughness.

## 4. Discussion

### (a) Consequences of a simple model

The theoretical framework, in §2, demonstrates that the onset of brittle fracture in a cracked body subjected to external loading is a function of the mode I and mode II stress intensity factors and the T-stress. In particular, we explored the consequences of T-stress on loading conditions for modes I and II alone. Notably, figures 2 and 3 show that the T-stress has a significant effect only on mode I toughness when the T-stress is greater than about 38% of the critical fracture stress. In contrast, for mode II loading, the consequences of the T-stress are more significant; where mode II toughness increases for negative T-stress and decreases for positive T-stress.

A key feature throughout the framework is the simple assumption that fracture occurs when the critical tangential stress is achieved at a critical distance, *r*_{c} (1.1). Consequently, local parameters determine the onset of fracture, and the applied loading determines the nature of the stress field. While for mixed mode in-plane loading there are three parameters *K*_{I}*, K*_{II} and T-stress describing the stress field, the onset of fracture remains a simple model. The emphasis here is that both singular and non-singular stresses contribute to the onset of brittle fracture. This was well demonstrated by the early work of Williams & Ewing and others.

To what extent does the simple assumption apply? Earlier experimental work, predominantly using angled cracks in plates and as illustrated in figure 1, show considerable scatter although demonstrating decreasing toughness with increasing *T*. The predicted variation of the effective stress intensity factor with the T-stress using equation (2.7) follows the same trend as the experimental results. Note that the prediction depends on the non-dimensional length . However, as pointed out by Smith *et al*. (2001), the agreement with experimental results at higher values of T-stress is rather poor. Furthermore, the results correspond to combinations of *K*_{I} and *K*_{II} with *T*, and because the angled internal crack problem cannot provide pure shear, results in figure 1 do not reveal the possible effect of *T* on mode II fracture of brittle materials.

For experiments using PMMA, Williams & Ewing (1972) suggest a non-dimensional distance *α*=0.2. For the test specimen shown in figure 4, with *a*=10 mm and with *α*=0.2, gives *r*_{c} =0.2 mm. To obtain values of the fracture angle, *θ*_{m}, equation (2.12) is used, but values of *K*_{II} and *B*_{II} are required. These were evaluated by finite-element analysis in §3. The corresponding predicted values of the fracture angles, *θ*_{m}, are −77° for the +T shear specimen and −59.1° for the −T shear specimen. These results are also shown in figure 5 and are in excellent agreement with the measured values. The predictions depend on the choice of the non-dimensional distance *α*. Other values of *α* do not give the agreement shown in figure 5 between the experiments and the predictions shown in the figure. Consequently, *α* is a fitted parameter that gives the best results shown in figure 5.

Finally, experimental mean values of *K*_{II}/*K*_{Ic} for the +T and −T shear specimens are shown in figure 8. Using the predicted fracture angle, *θ*_{m}, and *B*_{II} in equation (2.14), the corresponding predicted values of *K*_{II}/*K*_{Ic} are 0.653 and 1.214 for −T and +T shear specimens, respectively. Figure 8 shows the predictions. There is excellent agreement between experiments and predictions with the predictions for +T shear specimens agreeing exactly with the experimental results. However, the results depend on the choice of *α*, with the same value used for both modes I and II.

As well as predicting an average failure load, the experimental results also provide validation of a method of determining the statistical distribution of toughness where T-stress influences fracture. This is examined in §4*b*.

### (b) Statistical analysis

The failure probabilities, *P*_{f}, determined from the experimental results illustrated in figure 8, are plotted in figure 9 and shown as a function of the measured fracture toughness. Failure probability, *P*_{f}, was determined from(4.1)where *N* is the total number of experiments.

Wallin (1984) proposed a probability function for brittle fracture using a general statistical model suggested by Weibull (1951). In general, the micromechanism of brittle fracture in PMMA for mode I is similar to that for mode II. Therefore, a similar model is used here to explore a statistical description of the fracture toughness data shown in figure 9 for both mode I and II conditions.

The Wallin probability function for mode I is(4.2)where *K*_{min,I} and *K*_{0,I} are variables, which are found by fitting the model to experimental data for mode I. In keeping with Wallin's model, the exponent in equation (4.2) is equal to 4. Fitted values of *K*_{min,I} and *K*_{0,I} using data shown in figure 9 for mode I fracture are 1.47 and 1.99 MPa(m)^{1/2}, respectively. Figure 9 also shows the fitted curve to mode I results.

The failure probability for any other loading condition, such as mode II with ±T, was determined using an extension of the Wallin model developed by Fowler *et al*. (1997) and Hadidimoud *et al*. (2002). The failure probability for mode II with ±T is given by(4.3)The fitted parameters *K*_{min,I} and *K*_{0,I} were modified to determine *K*_{min,II} and *K*_{0,II}, assuming that(4.4)where *K*_{II}/*K*_{Ic} is determined from equation (2.14) (or figure 3) and depends on the value of *T*. As before, *α*=0.2 was selected and leads to *T*/*σ*_{c} equal to 0.26 and −0.5 for the +T and −T shear specimens, respectively. The corresponding values of *K*_{II}/*K*_{Ic} are 0.653 and 1.214 for the +T and −T shear specimens, respectively. The values of *K*_{min,II} and *K*_{0,II} were calculated using equation (4.4). The revised failure probabilities were determined from equation (4.3), with the resulting curves for *P*_{f} as a function of fracture toughness illustrated in figure 10. Overall, there is a good agreement between the distributions and experimental data, although the predicted function for the −T shear tests is slightly conservative.

### (c) Other materials

The mean values from the main experiments described in this paper are shown in figure 10, together with the predicted variation in mode II toughness as a function of T-stress. However, these results are confined to a small range of T-stress (+0.26≤*T*/*σ*_{c}≤−0.5). Results from Brazilian disc tests summarized in §1 reveal mode II toughness values well in excess of the mode I toughness. Ayatollahi & Aliha (2005) provide an analysis that demonstrates that there is a strong influence of the T-stress. Their reanalysis of earlier test results for geomaterials, such as limestone (Khan & Al-Shayea 2000) and soft sandstone (Krishnan *et al*. 1998), shows that the consequences of T-stress were substantial. Values of the non-dimensional distance *α* were for limestone 1.11 and for soft sandstone 0.37. The fracture toughness values for these geomaterials, obtained from Brazilian disc tests, are shown in figure 10, where the increase in toughness because of a negative T-stress can be as high as about 2.2 times the mode I fracture toughness. The experimental values from these tests agree very well with the predicted curve from equation (2.12). This demonstrates that the consequences of T-stress are substantial for this test condition.

Recent results for mode II tests with high positive T-stress are also shown in figure 10. Ayatollahi & Aliha (in press) explored the use of the SCB specimen to obtain data for mode II fracture. As with the Brazilian disc, the position of the crack in the specimen and the specimen dimensions give rise to combinations of *K*_{II} and T-stress, but notably with positive values of T-stress. The average of recent results for PMMA, using the SCB specimen produced by Ayatollahi & Aliha (in press), is shown in figure 10 using *α*=0.093. Again, there is excellent agreement between theory and experiment.

## 5. Conclusions

A theoretical framework, called the generalized MTS criterion, demonstrates that brittle fracture under mode I and mode II loading conditions is influenced strongly by the presence of T-stress.

The most substantial consequence of T-stress occurs for mode II loading with an almost inverse linear relationship between T-stress and fracture toughness. For example, a negative T-stress increases mode II fracture toughness.

Experiments using PMMA specimens, subjected to mode II loading, together with negative and positive T-stress have provided excellent validation for the generalized MTS criterion. An extension of a statistical analysis for predicting the failure probability for mode II, including T-stress, is supported by the experimental findings.

Collated data for brittle fracture in other materials also substantiates the predicted influence of T-stress on mode II brittle fracture. The experiments and theory demonstrate that a combination of mode II loading with a large negative T-stress can lead to mode II fracture toughness greater than twice the mode I toughness.

## Footnotes

- Received April 21, 2005.
- Accepted December 8, 2005.

- © 2006 The Royal Society