Fracture behaviour of notched round bars made of PMMA subjected to torsion at room temperature

Engineering Fracture Mechanics 90 (2012) 143–160

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Fracture behaviour of notched round bars made of PMMA subjected to torsion at room temperature F. Berto a, M. Elices b, P. Lazzarin a,⇑, M. Zappalorto a a b

Department of Management and Engineering, University of Padova, Stradella S. Nicola 3, 36100 Vicenza, Italy Department of Materials Science, Universidad Politécnica de Madrid, E:T:S: Ingegneros de Caminos, 28040 Madrid, Spain

a r t i c l e

i n f o

Article history: Received 28 November 2011 Received in revised form 27 March 2012 Accepted 3 May 2012

Keywords: Torsion Static failure Notched components Large scale yielding Strain energy density

a b s t r a c t The first part of the paper gives an account of more than 70 new fracture tests on notched specimens made of polymethyl-methacrylate. All static tests are carried out at room temperature under torsion loading conditions. Semi-circular notches as well as U- and V-notches (with an opening angle equal to 120°) are considered, with a root radius ranging from 0.1 to 7.0 mm. Plots of torque loads versus twist angles are recorded varying the notch root radius and the notch depth. In all cases static failure occurs under large scale yielding conditions. Such results can help in evaluating numerical and theoretical models of the fracture of notched components under mode III loading. The second part of the paper deals with a discussion on the experimental results and different approaches are applied to the new data. The notched specimens during the torsion tests present a substantial plastic behaviour and the influence of the effective resistant net area is found to be the predominant parameter with respect to the notch shape (i.e. notch opening angle and tip radius). A non-conventional application of the strain energy density is carried out showing a good agreement between experimental results and theoretical fracture assessments and it is used to justify the link between nominal and local fracture approaches. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Design based on damage tolerance criteria often deals with notched components giving rise to localised stress concentrations which, in brittle materials, may generate a crack leading to catastrophic failure or to a shortening of the assessed structural life. Modelling damage around notches has proven to be very difficult and strongly dependent on the microstructural aspects of each material. Therefore, proposed fracture criteria are based on critical values of some macroscopic stresses [1], critical virtual cracks [2–5], non-local averaged parameters [6], stress intensity factors [7–12], notch rounding approach [13], strain energy density [14–19], J-integral [20,21] and Cohesive Zone Model [22–27]. Under linear elastic conditions (or under small scale yielding conditions) when the stress concentrators are cracks, the stress intensity factors, SIFs, provided by the linear elastic fracture mechanics can be applied. Notch stress intensity factors, NSIFs, substitute SIFs in the case of sharp, zero radius, V-notches. As soon as the notch is blunted, i.e. the notch root radius R is not zero, the stress singularity disappears. The linear elastic fracture mechanics continues to be valid, but up to a some critical value of R which varies from material to material [28]. The problem becomes more involved if the loading symmetry is

⇑ Corresponding author. E-mail address: [email protected] (P. Lazzarin). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.05.001

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Nomenclature d E E0 G Kt,net K n R Rc Rref SED Torque Wref Greek 2a

c rnom,n smax,pl snom,n sn m

notch depth Young’s modulus generalised Young’s modulus shear modulus stress concentration factor referred to the net area tensile or torsion strength coefficient (MPa) torsion hardening exponent notch root radius SED critical length critical length for the non-conventional application of SED strain energy density criterion load at failure under torsion loading reference critical strain energy

notch opening angle shear strain nominal tensile stress referred to the net area maximum shear stress at the notch tip according to different elasto-plastic approaches nominal stress referred to the net area shear stress in the Ramberg–Osgood curve applied at a remote distance from the notch tip Poisson’s ratio

lost, i.e. when the notched structural component is subjected to mixed mode loading as widely discussed in previous works by other researchers [29–32] and also by the present Authors [33–37]. While systematic experimental data on fracture of blunted notched specimens (with notches of different root radii) loaded under mode I and mixed mode loading (I + II) have been recently provided [28–37], the results from notched specimens under torsion loading are relatively scarce since the research activity has mainly considered cracked bars under torsion [38–43]. Dealing with circumferentially cracked round bar specimens made of PMMA, it was evident in [41] that the fracture behaviour of superimposed modes I and III was strongly influenced by a non-linear deformation which caused the mixed mode toughness to be dependent on the sequence and type of loading. Conversely the mode I/II results were consistent with the mixed mode fracture response of a wide range of brittle materials [41]. The initiation of crack growth under a combination of opening and anti-plane shearing mode loading was investigated in [42]: the cracks did not grow through a continuous evolution of the crack surface but an abrupt fragmentation or segmentation of the crack front occurred. Within the classical framework of fracture mechanics, a recent contribution [43] has been focused on the theoretical prediction of the widely observed crack front instability in mode I + III, that causes both the crack surface and crack front to deviate from planar and straight shapes, respectively. Dealing with the specific case of blunt notches the only experimental results for PMMA samples under mixed mode (I + III) loading are provided in [44]. V-notched bars with a constant value of the opening angle (2a = 60°) and a root radius ranging from 0.2 to 1.2 mm have been considered. Some results from semicircular notches with a larger notch root radius (4.0 mm) have been also provided in that contribution. In Ref. [44], the behaviour showed by the tested PMMA under uniaxial loading was the typical one of a brittle material: the load vs. displacement curve was linear up to the complete failure. On the contrary, the material cracking behaviour showed by the notched specimens loaded in torsion was seen to be much more complex. In particular, the torque vs. twist angle curves were characterised by an initial linear-elastic stretch followed by an almost horizontal plateau preceding the final fracture. Finally, under combined tension–torsion the material cracking behaviour was seen to be in between the two extreme conditions discussed above. In particular, the observed mode I cracks preceding the final failure were always perpendicular to the maximum principal stress but the number of small cracks due to the coalescence phenomenon resulting in the final failure was seen to decrease as the tensile stress contribution to final failure increased, resulting in a unique mode I crack under uniaxial loading. In more detail the critical distance value under mode I and under mode III loading and the inherent material strengths were determined by means of some calibration results generated both under tension (by testing the V-notched flat specimens) and under torsion. Under plane stress mode I loading, the critical distance was equal to 0.054 mm and the material strength was equal to 113.9 MPa. On the other hand, under mode III loading the critical distance was much larger, equal to 0.2 mm, whilst the inherent material strength under

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mode III torsion was almost the same as under mode I loading, being equal to 111.7 MPa. For cylindrical un-notched specimens the value of the nominal stress to failure was equal to 30 MPa under pure torsion loading and 67 MPa under pure tension, both values referred to the net section. Keeping different values of critical distances under mode I and III loading the approach allowed to perform predictions falling in an error interval of about ±20% [44]. Other results are from ceramic notched components under combined tension and torsion and brittle glass under pure torsion [45,46]. In those papers the authors underlined that no other data were available in the literature. In parallel, many researchers have devoted strong efforts to investigate theoretically the stress distributions of sharp and blunt notches under torsion loading both under linear elastic conditions [47–51] and small scale yielding conditions [52–54]. The purpose of this research is twofold: a. To provide extensive experimental results of fracture from blunt notched specimens loaded under torsion loading, with different notch root radii, opening angles and notch depths which can serve as helpful for future researchers working in this area. b. To provide some indications and tools to predict rupture loads of structural components with notches of different severity loaded under torsion loading. Previous works by the authors have revealed that the strain energy density approach is successful for the assessment of brittle failure of notched samples loaded in mode I (sharp V-notched samples, blunt V-notched samples and U-notched specimens). Encouraged by these results, this method has been generalised to mixed mode problems (I + II) using a novel finite element developed for this purpose. Dealing with torsion loading and new available results, the strain energy density approach is applied in this paper in a non-conventional way. The paper is structured in the following way: in the first section, the experimental work is described (specimen geometries, testing procedures and experimental results). The second section deals with the discussion of the results under torsion loading, and shows some different assessments to explain the new data. The best assessment is based on a mean value of the linear elastic strain energy density averaged on a given control volume. The radius of the control volume will not be evaluated a priori on the basis of some fundamental properties of the material, but only a posteriori, on the basis of a best fitting of the experimental data. A single value of the control radius will be found able to summarise only experimental data from notched specimens with a notch tip radius smaller than 5 mm. This is due to the large amount of plasticity, which varies from case to case, and the different stress redistribution occurring for decreasing values of the net transverse section area and larger values of the radius in semicircular notches.

2. Experimental programme 2.1. Test samples Tests were performed with polymethyl-methacrylate (PMMA), an amorphous glassy polymer tested at room temperature. Mechanical properties of PMMA have been derived as explained below and are summarised in Table 1. The Table gives the average values of the longitudinal and transverse elastic modulus, the Poisson’s ratio, the tensile and torsion strength at room temperature, as measured by means of cylindrical samples with 12.5 mm diameter (see Fig. 1a). Different notched geometries were tested, as indicated in Fig. 1; plain specimens (1a), cylindrical specimens with U- and V-notches (1b) as well as with semicircular notches (1c), to explore a large range of geometrical configurations. Specimens were made from 2 m long round bars of 20 mm of diameter purchased to a PMMA manufacturer. All these bars were made from the same batch. Bars were cut into 200 mm long specimens and notches were machined in a turning lathe. Lathe speed was set to 800 r.p.m. and the specimens were cooled by liquid injection on the machining region to avoid material damage due to an excessive temperature increase. A specific chisel was made for each geometry (Fig. 2). These chisels were made of cobalt enriched steel and they all had the shape of its respective notch geometry.

Table 1 Material properties for the tested PMMA. Property

Value

Elastic modulus, E (MPa) Shear modulus, G (MPa) Poisson’s ratio, m Ultimate tensile strength (MPa) Ultimate torsion strength (MPa) Tensile hardening exponent, n Tensile strength coefficient, K (MPa) Torsion hardening exponent, n Torsion strength coefficient, K (MPa)

3600 1280 0.4 74 67 3.8 204 3.6 150

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(a)

Plain specimens φ 12.5

φ 20

R 40 60

80

60

2α=0°, 120°

(b)

U- and V-notches

φ 20 d

R Semicircular notch

R

(c)

φ 20 200 Fig. 1. Geometry of plain and notched specimens (a–c).

Fig. 2. Chisels used to fabricate the specimens.

For U-notched specimens (Fig. 1b), notches with four different notch root radii, R, were tested; R = 0.3, 0.5, 1.0, 2.0 mm. The effect of the net area was achieved by changing the notch depth d. Two values of the notch depth were used, d = 2 mm and d = 5 mm, with a constant gross diameter equal to 20 mm. For V-notched specimens (Fig. 1b), notches with five different notch root radii, R, were tested; R = 0.1, 0.3, 0.5, 1.0 and 2.0 mm. The notch opening angle was kept constant and equal to 120°. The effect of the net area was achieved by changing the notch depth d. Two values were used; d = 2 and 5 mm with a constant gross diameter equal to 20 mm. For semicircular notches (Fig. 1c), notches with seven different notch root radii, R, were tested R = 0.5, 1.0, 2.0, 4.0, 5.0, 6.0 and 7.0 mm. All in all, 23 different geometrical configurations were tested. Because each one was repeated at least three times, a total number of about 70 new tests were performed. Fig. 3 shows the specimens used in the tests while Table 2 gives the theoretical stress concentration factor of all geometries, as determined by FE analyses (ANSYS code, version 11). Previous tests on the PMMA, carried out under tension loads at room temperature, exhibited a behaviour very close to that ideally linear elastic. Varying the loading conditions, from traction to torsion, failures were expected to occur under small scale yielding, particularly in the presence of notches with small tip radii. 2.2. Testing procedure The tests were performed on a servocontrolled MTS biaxial testing machine (±100 kN/±110 Nm, ±75 mm/±55°). The load was measured with the MTS load cell with ±0.5 % error at full scale. A MTS strain gauge axial extensometer (MTS 632.85F-14), with a gauge length equal to 25 mm was used for measuring the tensile properties of the material from plain specimens. The torsion properties of the material were evaluated by a multiaxis extensometer (MTS 632.80F-04) with a gauge length equal to 25 mm (see Fig. 4).

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Fig. 3. Photos of the PMMA specimens.

Table 2 Values of the theoretical stress concentration factor (referred to the net area of the specimens).

U-notches

d (mm)

R (mm)

Kt,net

5

0.3 0.5 1.0 2.0 0.3 0.5 1.0

2.30 1.88 1.49 1.21 2.58 2.14 1.75

0.1 0.3 0.5 1.0 2.0 0.1 0.3 0.5 1.0

2.54 1.95 1.74 1.50 1.34 2.78 2.13 1.89 1.62

0.5 1 2 4 5 6 7

1.74 1.59 1.40 1.21 1.14 1.10 1.06

2

V-notches

5

2

Semicircular notches

0.5 1 2 4 5 6 7

Testing was done in two steps; during the first phase, the samples were set on the grippers of the hydraulic system under load control (axial or torsion loads). Then, all samples were tested up to failure under displacement control at rate of 2°/min. All tests have been carried out at room temperature.

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Fig. 4. Extensometers with gauge length of 25 mm applied to plain specimens tested under tension (a) and torsion (b).

Due to the rather high value of the notch depth, it has practically been impossible to analyse the early crack initiation and propagation during the tests by using an optical microscope. Despite the presence of large scale yielding conditions in all notched samples, sudden failures with fragmentation of small parts of the specimen embracing the highly stressed region systematically occurred for lower value of the notch depth, d = 2 mm, both for U- and V-notches, independent of R. Again, in the case of semicircular notches, fragmentation occurred when the notch depth (the notch radius) was lower than or equal

Fig. 5. SEM fracture surfaces, specimen with a semicircular notch (R = 4 mm).

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149

Fig. 6. Failure modes in some specimens with d = 5 mm: U-notch, R = 0.5 mm (a), V-notch, R = 2 mm, (b). Failure modes in some specimens with d = 2 mm: U-notch, R = 0.5 (c), V-notch, R = 1 mm (d).

to 2 mm. The same phenomenon occurred also in the V-notches with d = 5 mm, but only when the notch root radius R was greater than or equal to 1 mm. All samples broke suddenly without evident cracking before the final failure. For the specimens characterised by final fragmentation the fracture surface was irregular with a relevant detachment of material that compromised the original geometrical configuration not only on the notch bisector line but also along the notch flanks. On the other hand, the topography of the fracture surfaces of the specimens which did not explode at the failure was characterised by irregular inclined fracture planes with partial detachment of the material in the close neighbourhood of the notch tip. For these reasons it was not possible to understand precisely the actual crack path at fracture initiation. After the tests, the analysis of the fracture surfaces of the specimens, obtained by means of scanning electron microscopy, was carried out. The surfaces are documented in Fig. 5 for a semicircular notch and Fig. 6a–d for various notch configurations. The effects of the fragmentation at failure are particularly evident in Fig. 6b and d, whereas they are limited in Fig. 6c. The effect is absent in Fig. 6a.

3. Experimental results Table 3 lists all experimental results from U-notched specimens. For every notch geometry, the maximum torque and the nominal shear stress (referred to the net sectional area of the specimens) are reported. As can be realised, loads increase as the notch depth decreases, as expected, while they are almost independent of notch root radii. Table 4 shows all the experimental results from V-notched specimens. For every combination of geometrical parameters, maximum loads were summarised as a function of the chosen notch radii and depths. It is worth noting that also for Vnotches the load increases while decreasing of notch depths, whereas it is almost independent of the notch root radius. By comparing the results from U-and V-notches it is clear that also the influence of the notch opening angle on the maximum torque load is very limited if not absent at all. This behaviour can be explained by the large amount of plasticity occurring during the tests. Then the influence of the transverse sectional area is found to be largely predominant with respect to the notch shape (i.e. notch tip radius and notch depth).

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Table 3 Maximum torque measured from U-notched specimens and corresponding nominal shear stress evaluated on the net transverse section area. d (mm)

R (mm)

Torque (Nm)

snom,n (MPa)

d (mm)

R (mm)

Torque (Nm)

snom,n (MPa)

5 5 5 5 5 5 5 5 5 5 5 5

0.3 0.3 0.3 0.5 0.5 0.5 1 1 1 2 2 2

17.260 15.676 15.125 16.778 16.123 17.077 17.776 16.709 17.487 17.845 17.087 18.220

87.9 79.9 77.1 85.5 82.2 87.0 90.6 85.1 89.1 90.9 87.1 92.1

2 2 2 2 2 2 2 2 2

0.3 0.3 0.3 0.5 0.5 0.5 1 1 1

65.884 62.956 67.320 64.782 63.197 67.812 66.331 63.335 69.486

81.9 78.3 83.7 80.6 78.6 84.4 82.5 78.8 86.4

Table 4 Maximum torque measured from V-notched specimens (2a = 120°) and corresponding nominal shear stress evaluated on the net transverse section area. d (mm)

R (mm)

Torque (Nm)

snom,n (MPa)

d (mm)

R (mm)

Torque (Nm)

snom,n (MPa)

5

0.1 0.1 0.1

16.192 15.848 17.184

82.5 80.7 87.6

2

0.1 0.1 0.1

64.540 64.058 68.256

80.3 79.7 84.9

5

0.3 0.3 0.3

17.570 16.571 18.141

89.5 84.4 92.4

2

0.3 0.3 0.3

65.539 64.437 68.728

81.5 80.2 85.5

5

0.5 0.5 0.5

16.881 16.743 18.579

86.0 85.3 94.7

2

0.5 0.5 0.5

65.057 65.918 70.767

80.9 82.0 88.0

5

1 1 1

17.053 16.605 18.152

86.9 84.6 92.5

2

1 1 1

65.608 65.401 67.981

81.6 81.4 84.6

5

2 2 2

16.778 16.605 17.590

85.5 84.6 89.6

Table 5 summarises the results from semicircular notches in terms of maximum torque and nominal shear stresses. The nominal shear stress has been evaluated according to the Coulomb equation [55] applied to the net area of the specimen. Fig. 7 shows the typical torque–twist angle curves for U-notches with a depth d = 2 mm (Fig. 7a) and d = 5 mm (Fig. 7b). By observing the curves related to d = 2 mm, it is well visible that the maximum torque is almost independent of the notch radius, whereas the twist angle to failure depends on it. However, the twist angles are very high in all cases, ranging from about 40° to 55°. When d = 5 mm, the torque vs. angle curves were characterised by an initial small linear-elastic stretch followed by an almost horizontal zone preceding the final fracture. The maximum torque increases as R increases but the variation is evident only for the R = 0.3 mm case. Also the angles at failure increase as a function of the notch radius, but they are lower than those detected from the previous cases with d = 2 mm. Table 5 Maximum torque measured from specimens with semi-circular notches and corresponding nominal shear stress evaluated on the net transverse section area. R (mm)

Torque (Nm)

snom,n (MPa)

R (mm)

Torque (Nm)

snom,n (MPa)

0.5

99.700 95.809 99.545

74.1 71.2 74.0

5

14.729 15.359 15.786

75.0 78.2 80.4

1

92.572 88.267 94.800

80.9 77.1 82.8

15.831 7.095 7.045

80.6 70.6 70.1

2

68.019 64.368 68.570

84.6 80.1 85.3

7.198 7.411 2.632

71.6 73.7 62.0

4

29.416 29.347 27.625 30.263

86.7 86.5 81.5 89.2

2.625 2.635 2.673

61.9 62.1 63.0

6

7

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90

Torque (Nm)

U-notches d=2 mm

R=1 mm

(a)

R= 0.5 mm

60

R = 0.3 mm

30

0

20

60

40

Angle (degrees) 20

(b)

U-notches d=5 mm

R= 2 mm

Torque (Nm)

R= 0.5 mm

R= 1 mm

R= 0.3 mm

10

0

40

20

60

Angle (degrees) Fig. 7. Torque–twist angle curves from U-notched specimens with notch depth d = 2 mm (a) or d = 5 mm (b).

Torque (Nm)

80

60

Δ=6%

U-notches R = 0.3 mm d = 2 mm

40

20

0

20

40

60

Angle (degrees) Fig. 8. Repeatability of the static curves (same geometry of the specimen).

The repeatability of the results is shown is Fig. 8 where the results of three repeated tests on specimens with the same geometry (same notch radius and notch depth) are shown. The experimental scatter was found to be quite limited, the relative deviations being generally lower than ±10%. Fig. 9 shows the typical torque–angle curves for V-notches (2a = 120°) for a notch depth d = 2 mm (a) and d = 5 mm (b). The same remarks drawn for the U-notches can be straightforward transferred to V-notches. The only difference can be observed for the lowest values of the notch radius, R = 0.1 and 0.5 mm, where the plateau value is no longer present: the relevant curves reach the maximum value of torque and then progressively decrease until the final failure. Fig. 10a shows the typical torque–angle curves for semicircular notches. Being the radius equal to the notch depth in these cases, a variation of the notch radius provokes a change in the maximum value of the torque and when the radius decreases higher values of the torque moments are reached. The failure angle is very high in this case until a maximum of 100° for a radius equal to 0.5 mm. The axial load–displacement curve for the same notches is reported—in Fig. 10b. Non-linear

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90

(a)

Torque (Nm)

V-notches d=2 mm

R= 1 mm

R =0.3 mm

60 R= 0.1 mm

R= 0.5 mm

30

0

0

20

40

60

Angle (degrees) 20

Torque (Nm)

V-notches d=5 mm

R= 2 mm R= 0.5 mm

10

(b)

R= 1 mm

R= 0.3 mm R= 0.1 mm

0

20

40

60

Angle (degrees) Fig. 9. Typical torque–twist angle curves for V-notches (2a = 120°); notch depth d = 2 mm (a) and notch depth d = 5 mm (b).

120 Semicircular notches R = 0.5 mm

Torque (Nm)

100 80

R = 2 mm R= 1 mm

60 40

R = 4 mm R = 5 mm

20

R = 6 mm

(a)

R = 7 mm

0 0

20

40

60

80

100

120

Angle (degrees) 100 unnotched specimen

σ nom,n (MPa)

80 R=4 mm

60

R=2 mm

40 R=0.5 mm

20 0

(b) 0

1

2

3

4

5

Displacement (mm) Fig. 10. Torque–twist angle curves for semicircular notches (a), axial load-displacement curve for the same notches (b).

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R =1 mm, all specimens

τnom, n (MPa)

90

U-notch, d=5 mm

U-notch, d=2 mm

Semicircular

60

V-notch, d=5 mm

V-notch, d=2 mm

30

0

20

60

40

100

80

Angle (degrees) Fig. 11. Typical snom,n-twist angle curves for notches with different depth d.

effects are reduced with respect to the torsion loading and the influence of the notch radius remains strong with a reduction of the nominal stress at failure of about 50% from the case R = 4 mm to the case R = 0.5 mm. To confirm the idea that the net area is the predominant geometrical parameter, all the data from the different geometries tested in this work (U-notches, V-notches and semicircular notches) are summarised in terms of the nominal stress referred to the net area, varying the notch depth and keeping constant the notch radius (1 mm). It is well visible from Fig. 11 that the stress to failure referred to the net area is almost the same although the corresponding torsion angle strongly changes from specimen to specimen as a function of the geometry. This happens also for the other values of the notch radius. The fracture behaviour of round bars made of PMMA under torsion loading is then completely different from the brittle or quasi-brittle behaviour which was found to characterise notched plates under tensile loading [19,24,25]. Dealing with mode I static data, worth mentioning is also Ref. [56] where a sound discussion on low and high ductility effects in the presence of notches was carried out, distinguishing between PMMA glasses with oriented and unoriented molecular chains with respect to the specimen axis. 4. Different approaches for fracture assessment Due to the absence of linear elastic or small scale yielding conditions, static strength assessments of the notched components are far from easy. The results from different approaches, carried out over the theoretical limits, are documented herein. The first one is based on theoretical stress concentration factors, the second on the Neuber rule, the third on the equivalent strain energy density (ESED) criterion, the fourth on a modified version of the Neuber rule, the fifth on the nominal shear stress and, finally, the last one on the strain energy density averaged over a given control volume. 4.1. The maximum elastic stress at the notch tip The first approach is simply based on the maximum elastic shear stress present at the notch tip. This stress component has been evaluated by multiplying the nominal shear stress referred to the outer radius of the net area and the theoretical stress concentration factors Kt,net given in Table 2. Fig. 12 shows for all the data the maximum elastic stress at the notch tip as a function of the notch tip radius. The peak stress reaches values approximately equal to 240 MPa for the smaller values of the notch radius (0.1 and 0.3 mm) while it is 250 U-notches, d = 5 mm V-notches, d = 5 mm

K t, net τ nom,n

200

U-notches, d = 2 mm V-notches, d = 2 mm semicircular

150

100

50 0

1

2

3

4

5

6

7

R [mm] Fig. 12. Approach based on the maximum elastic shear stress at the notch tip.

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equal to 100 MPa for the case R = 4 mm. Moreover, at the same value of the tip radius but for different geometry the scatter of the diagram is very high and this is true in particular for lower values of the notch radius. The synthesis reported in Fig. 12 clearly shows that it is not the peak stress to govern the fracture behaviour of the considered specimens. 4.2. The maximum elastic–plastic stress at the notch tip determined by the Neuber’s rule A second approach is based on the plastic stress at the notch tip smax,pl. Neuber [57] analysed a strip with two symmetric notches under longitudinal shear and found that the geometrical mean value of the effective stress and strain concentration factors is equal to the elastic theoretical stress concentration factor, Ks  Kc = K 2t . Here Ks is the ratio between the maximum stress at the notch tip and the nominal stress while Kc is the ratio between the maximum strain and the nominal strain. The rule was considered independent of the stress–strain law by Neuber. The Neuber rule is very popular, so popular to be sometimes used out of its specific ambit, which should be confined to small scale yielding conditions [58,59], and to sharp notches, namely to notches with a small notch root radius. Without entering in the specific ambit of its analytical validity, the rule was later extended to loading conditions different from the anti-plane shear stress (as suggested, but never formalised, by Neuber himself) as well as to uniaxial and multiaxial fatigue problems [60,61]. The maximum plastic stress at the notch tip is obtained by using Neuber’s rule for the torsion loading as reported below [57]:

K 2t ¼ K s K c

ð1Þ

The elastic–plastic behaviour of the material under torsion loading is modelled by assuming a Ramberg–Osgood curve for PMMA. The hardening exponent and the torsion strength coefficient as obtained by plain cylindrical specimens tested under torsion by means of the bi-axial extensometer are summarised in Table 1. The following equation has been used to model the elastic–plastic behaviour of the material:

s  s n c¼ þ G

ð2Þ

K

By simply using Eqs. (1) and (2) in combination it is straightforward to obtain the maximum plastic stress at the notch tip by solving the following equation [57]:

ðK t sn Þ2 ¼ G



smax;pl G

þ

s

max;pl

n 

smax;pl

K

ð3Þ

The maximum plastic stress as a function of the notch tip radius is shown in Fig. 13 where a decreasing trend of the plastic stress as a function of the notch radius is shown. The trend of the curve is similar to that shown in Fig. 12 by using a linear elastic analysis based on the elastic peak stress but now the relative deviation is much more limited, the maximum value ranging from 110 MPa for smallest values of the notch radius to about 55 MPa for the largest root radius, R = 7 mm. Due to the large variability of yielding conditions, fracture assessments of the specimens based on the maximum plastic stress at the notch tip continues to give a strong scatter, with a ratio between the maximum to minimum value equal to 2. 4.3. The maximum elastic–plastic stress at the notch tip determined by the ESED criterion Alternative to the Neuber’s rule, Glinka’s Equivalent Strain Energy Density criterion, ESED, [62,63] is largely used to predict the elastic–plastic stress at the tip of a notch. It is acknowledged in the literature that the notch root strains are normally U-notches, d = 5 mm

110

U-notches, d = 2 mm

τ max, pl. (MPa)

V-notches, d = 5 mm V-notches, d = 2 mm

90

semicircular

70

50

0

1

2

3

4

5

6

7

R(mm) Fig. 13. Approach based on the maximum plastic stress obtained by using Neuber’s rule (Eq. (3)).

155

F. Berto et al. / Engineering Fracture Mechanics 90 (2012) 143–160

τ max, pl. (MPa)

150 U-notches, d = 5mm U-notches, d = 2 mm V-notches, d = 5mm V-notches, d = 2 mm Semicircular

100

50 0

1

2

3

4

5

6

7

R [mm] Fig. 14. Approach based on the maximum plastic stress obtained by using the ESED criterion.

overestimated by the Neuber rule and underestimated by the ESED criterion, which is superior under plane strain conditions [64,65]. The ESED concept has been later extended to torsion by Agnihotri [66], who achieved good notch tip strain predictions both for sharp and blunt notches. In all these cases however, the amount of plasticity effects was much lower than in the present analyses. The conditions were ascribable to small scale yielding, not to large scale yielding. Note also that the diffuse plasticity occurring on the entire section of the specimens makes it unusable also some theoretical solutions obtained by the present authors to describe the nonlinear behaviour of notches under torsion [53,54]. As suggested by Glinka [62] dealing with notches under traction and bending load conditions, the Ramberg–Osgood curve is used also to model the nominal behaviour of the material, far away from the notch root. Doing so, the maximum plastic stress at the notch tip can be estimated by the following equation:

K 2t



s2n 2G

þ

 s2max; pl n snþ1 n n ¼ þ n nþ1 K nþ1 2G

snþ1 max; pl

ð4Þ

Kn

The approach based on Eq. (4) is shown in Fig. 14. It is evident that a point-wise criterion (applied at the notch tip) is not able to condense all experimental data in a limited range of the maximum plastic stress. 4.4. The maximum elastic–plastic stress at the notch tip determined by a modified Neuber’s rule A fourth approach is based on a non-conventional application of Neuber’s rule where also the nominal behaviour of the material is assumed to follow the Ramberg–Osgood law and no longer a linear elastic law. Once again the parameter chosen for this approach is the maximum plastic stress, which is determined according to the following expression that can be seen as a modified version of that proposed in [62]:



s2n G

þ

snþ1 n Kn



 ¼

smax;pl G

þ

s

max;pl

K

n 

smax;pl

ð5Þ

150 U-notches, d=5 mm U-notches, d=2 mm V-notches, d=5 mm

τ max, pl.(MPa)

K 2t

V-notches, d=2 mm semicircular

100

50 0

1

2

3

4

5

6

7

R(mm) Fig. 15. Approach based on the maximum plastic stress obtained by using a modified Neuber’s rule (Eq. (5)).

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F. Berto et al. / Engineering Fracture Mechanics 90 (2012) 143–160

U-notches, d = 5 mm

110

U-notches, d = 2 mm V-notches, d = 5 mm

τ nom, n (MPa)

V-notches, d = 2 mm Semicircular

90

70 τ nom, n = 67 MPa

50 0

1

2

3

4

5

6

7

R (mm) Fig. 16. Approach based on the nominal shear stress referred to the net area.

Fig. 15 shows the results plotting the maximum plastic stress as a function of the notch radius. The trend is analogous to that already shown in Fig. 13 and based on the conventional Neuber’s rule: the relative scatter at the same value of the notch radius is found to be slightly reduced although the obtained plastic stress is higher than in the previous case. Note also that, due the particularly low values of the theoretical stress concentration factors, the results shown in Fig. 15 are almost coincident to those obtained with the ESED criterion, the maximum difference being within 2%. 4.5. Approach based on the nominal shear stress A simple approach is applied in this section by assuming as critical parameter the nominal stress on the net area according to the Coulomb equation [55] for round bars under torsion loading. As discussed when commenting Fig. 11 for different values of the notch depth and notch opening angle the nominal stress assumes approximately the same value for each considered geometrical configuration even if the trend of the torque–angle curve strongly varies from case to case. Fig. 16 shows the nominal stress as a function of the notch radius. This figure makes it evident that for U- and V-notches the variability is quite limited, from about 75 to 95 MPa, where as the behaviour of the semicircular notches is quite anomalous. The maximum value of about 85 MPa is found to be in correspondence of a notch root radius of 4.0 mm, whereas values a little higher or a little lower of the plain specimens, 75 and 65 MPa, are from specimens with notch radius of 0.5 and 7.0 mm, respectively. The stress redistribution due to the plastic behaviour of the material involves a large part of the transverse sectional area of the specimen as just stated above. The behaviour under torsion has nothing in common with that exhibited under tensile loading by the same material tested at room temperature (see Fig. 10b). Plasticity occurs very early under torsion loading confirming experimentally what discussed in a previous contribution [67] with reference to low and medium cycle fatigue of V-notched components. 5. Averaged strain energy density The averaged Strain Energy Density, SED, criterion, first proposed in [17], states that brittle failure occurs when the mean value of the strain energy density over a control volume (which becomes an area in two dimensional cases) is equal to a critical energy Wc. The SED approach is based both on a precise definition of the control volume and the fact that the critical energy does not depend on the notch sharpness. Such a method was formalised and applied first to sharp, zero radius, Vnotches [17] and later extended to blunt U- and V-notches under mode I loading [19]. The radius RC of the control volume over which the energy has to be averaged, depends on the ultimate tensile strength, the fracture toughness and Poisson’s ratio in the case of static loads. In the case of sharp V-notches the control volume is simply a circular sector having its centre on the point of stress singularity (Fig. 17). The volume assumes a crescent shape in the case of blunt notches. A precise definition of the control radius under linear elastic hypothesis is given in Ref. [19] for mode I and mixed, I + II, loading conditions. Under torsion loading, the control radius RC can be estimated by using Eq. (6) valid for sharp V-notches [68]:

RC ¼

rffiffiffiffiffiffiffiffiffiffiffiffi  1 e3 K 3c 1k3  1 þ m st

ð6Þ

where K3c is the mode III critical notch stress intensity factor to failure and st is the ultimate torsion strength of the unnotched material. Moreover, e3 is the parameter that quantifies the influence of all stresses and strains over the control volume and (1  k3) is the degree of singularity of the linear elastic stress field, which depends on the V-notch opening angle [68]. As soon as the control radius RC is known, the mean value of the strain energy density over the control volume can be easily

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F. Berto et al. / Engineering Fracture Mechanics 90 (2012) 143–160

r0 R c 2α

2α RC

(a)

RC

(b)

R

R C +r 0

(c)

Fig. 17. Shape of the control volume (area, in 2D) for V-sharp notches (a), cracks (b) and blunt V-notches (c). The outer radius of the crescent shape volume, RC + r0, intersects the semicircular notch root edge or the rectilinear flanks of the U- and V-notch. Distance r0 dependent on 2a (r0 = R/2 when 2a = 0°; r0 = R/ 4 when 2a = 120°).

evaluated of the basis the mode III notch stress intensity factor (sharp notches), the theoretical stress concentration factor (blunt notches) or directly from the finite element models. SED-based assessments for V-notched components under mode III or mixed, I + III, mode are carried out in Refs. [68–71] both under static and fatigue loading. For a blunt V-notch under mode III loading, the volume assumes the crescent shape shown in Fig. 17, where RC is the depth measured along the notch bisector line. The outer radius of the crescent shape is equal to RC + r0, being r0 the distance between the notch tip and the origin of the local coordinate system (Fig. 17), r0 ¼ Rðp  2aÞ=ð2p  2aÞ, which depends on the V-notch opening angle 2a and the notch root radius R. Due to the presence of large scale yielding, Eq. (6) cannot be directly applied. The crescent shape volume shown in Fig. 17 is not changed, but the reference radius is determined only a posteriori, by means of a best fit of the experimental data. The averaged value of the strain energy density (SED) over a well-defined control volume surrounding the notch tip has been used to summarise the static strength data of notched specimens under static loads [17,33–37] as well as notched components and welded joints under fatigue loading. Dealing with static loads, the SED approach has successfully been applied to assess the critical loads to failures of brittle or quasi-brittle materials weakened by U- and V-notches. About 1200 experimental data under mode 1 and mixed mode conditions (1 + 2) were summarised in Ref. [35]. Recently the SED approach was also applied to summarise a number of fatigue data from V-notched specimens subjected to uniaxial and multiaxial loading [69,70]. Under torsion loading a number of nonlinear elastic effects were detected resulting in a control volume dependent on the loading mode. The volume radius under torsion resulted to be much greater than the radius under tensile loading. A non-conventional approach, based on the ‘apparent’ linear elastic SED evaluated considering a different critical radius, allowed to overcome the problems tied to different extrinsic and intrinsic fracture mechanisms occurring under modes I and III loading and to summarise the main body of the data in a quite narrow scatterband. The term ‘apparent’ seemed to be appropriate to describe the SED value measured without any clear distinction between non-linear intrinsic and extrinsic mechanisms and based on a linear-elastic analysis of the stress distribution on the highly stressed zone ahead of the notch tip. Being the authors conscious that a synthesis based on the apparent value of the linear elastic SED is only an engineering tool for strength assessments, the SED is applied here to the new data from torsion loads, despite the presence of large scale yielding. The SED can be seen as a possible bridging between local and nominal approaches. A final synthesis has been carried out considering four different values of the control volume radius, Rref = 0.5, 0.75, 1.0 and 1.50 mm. Due to the presence of large scale yielding, it was not possible to establish a priori the value of the radius, as made in the past for quasi-brittle materials under static conditions and for welded joint in the high-cycle fatigue regime. Note that the trial values chosen for critical radius are very high in comparison with that found for the same material under tension loads, RC = 0.1 mm at room temperature [24,25], reduced to RC = 0.035 mm at 60 °C [33–37]. Fig. 18 summarises the results with reference to the values Rref = 0.5 mm (Fig. 18a) and Rref = 1 mm (Fig. 18b). The square root of the normalised value of the SED is plotted versus the notch tip radius normalised by the control radius Rref. The scatter approximately ranges from 0.8 to 1.2 in both proposed diagrams, but excluding from the scatterband the cases related to semicircular notches with R greater than 5 mm. The dispersion of the data with respect to the reference value Wref seems to be well-balanced for Rref = 0.5 mm, where the relative deviation reaches its minimum. The large critical radius necessary to summarise the data in a single scatterband supports the idea of a strong stress redistribution due to plasticity, which involves a large region of the net sectional area. The averaged value of the SED has not been estimated a priori on the basis of the unnotched specimens that are characterised by a lower value of SED. The present SEDbased proposal should not be extended to specimens with a root radius greater than 5 mm, where the stress concentration effect are much lower resulting in a shear stress redistributions totally different. It is evident from Fig. 18 that the data referred to notch root radii greater than 5 mm fall below the scatter band. This should not be surprising because such root radii are larger than the net radius of the specimen. Then the resistant net section is very limited resulting in very high geometrical non linearity which is coupled to the extensive plasticity due to the material. This anomalous trend could be overcome by increasing the gross diameter of the specimens keeping constant the notch depth.

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2

(a)

73 data Rref = 0.5 mm

1.6

(W/Wref)0.5

Wref = 3.51 Nmm/mm3 1.2 1 0.8

0.4

0.1

1

10

20

R/R ref 2

(b)

73 data Rref = 1.0 mm

1.6

(W/Wref)0.5

Wref = 2.21 Nmm/mm3 1.2 1 0.8

0.4

0.1

1

10

R/R ref Fig. 18. Approach based on the apparent SED; Rref = 0.5 mm (a) and Rref = 1.0 mm (b).

However, Fig. 19 shows a comparison between the assessed values of torques based on the apparent SED (considering Rref = 0.5 mm) and the experimental values from semicircular notches with a root radius ranging from 0.5 to 7 mm. The trend for torque seems to be satisfactory also for R > 5 mm, but for those cases the percentage deviation is not negligible, as implicitly shown in Fig. 18a and b. On the basis of the results obtained in the present work, it seems that the ‘apparent’ SED can be successfully applied to fracture assessments of notched components, from small to large notch root radii, but only when the net radius is greater than or equal to the notch depth. The specimens characterised by a net radius lower than the notch depth do not belong to the same scatterband, being systematically characterised by a lower local SED value. With the aim to strongly reduce (or remove at all) the non-linear effects observed at room temperature, some new tests at low temperature have been planned on the same material. A forthcoming contribution will present additional data from 120 Semicircular notches 0.5 < R < 7mm PMMA Rref = 0.5 mm Wref = 3.51 Nmm/mm3

Torque (Nm)

100 80 60 40 20 0

0

1

2

3

4

5

6

7

Notch radius, R (mm) Fig. 19. Comparison between experimental and theoretical results based on the mean SED for semicircular specimens.

F. Berto et al. / Engineering Fracture Mechanics 90 (2012) 143–160

159

PMMA specimens tested under torsion at low temperature. The same kind of notches will be considered to induce a wide range of stress concentrations. The specimens will be tested at 60 °C to have a behaviour closer to the linear elastic one, as previously made under modes I and II loadings [33–35]. The results obtained here shows how at room temperature the SED provides a good prediction of the maximum torques but only up to a limit value of the notch root radius (about 5 mm). At low temperature, this restriction should be removed, with a failure basically governed by the stress concentration effects. The averaged SED criterion is expected to provide good maximum torque assessments for all specimens, independent of the notch radius. The expected control volume radius and the critical SED values should depend on the temperature. 6. Conclusions This paper gives an account of about 70 fracture tests from notched specimens (with notches of different depth and radii), loaded under torsion loading. In all tests, maximum loads and failure angles were measured as a function of notch root radius and specimens geometrical configuration. Repeatability of measurements was rather good considering the unavoidable small differences in machining notch root radii. The most favourable results were achieved with U-notches characterised by a notch depth equal to 2 mm and small values of the notch radius. There, standard deviation was always below 2%. For the other specimens, s.d. was always below 11%. The least favourable results came from small values of the notch radius and larger values of the notch depth (U-notches, d = 5 mm and R = 0.3 mm) where machining and measurement of length presented more difficulties. The second part of the paper deals with the suitability of different approaches for predicting fracture loads under torsion loading. Here, as previously explained, the analysis is strongly influenced by the fact that the torsion loading behaves completely differently from PMMA tested under tensile loading; the notched specimens during the torsion tests presents a large plastic behaviour and the influence of the effective resistant net area is found to be the predominant parameter instead of the notch shape details (i.e. notch opening angle and tip radius). A non-conventional approach of the present data in terms of strain energy density is carried out showing a good agreement between experimental results and theoretical fracture assessment but only when the notch root radius is lower than or equal to 5.0 mm. Acknowledgements This work was supported by the Italian Research Program CPDA100715 entitled ‘‘Static and fatigue behaviour of structural notched components subjected to tension and torsion under small or large scale yielding’’. Prof. M. Elices would also like to express his gratitude to Fundación Agustín de Betancourt and to CICYT for financial support through Project CONSOLIDER ‘‘SEDUREC’’ CSD 2006 00060. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

Papadopoulos GA, Paniridis PI. Crack initiation from blunt notches under biaxial loading. Engng Fract Mech 1988;31:65–78. Leguillon D. Strength or toughness? A criterion for crack onset at a notch. Eur J Mech Phys A/Solids 2002;21:61–72. Leguillon D, Yosibash Z. Crack onset at a V-notch. Influence of the notch tip radius. Int J Fract 2003;122:1–21. Yosibash Z, Priel E, Leguillon D. A failure criterion for brittle elastic materials under mixed-mode loading. Int J Fract 2006;141:291–312. Leguillon D, Quesada D, Putot C, Martin E. Prediction of crack initiation at blunt notches and cavities—size effects. Engng Fract Mech 2007;74:2420–36. Seweryn A, Poskrobko S, Mróz Z. Brittle fracture in plane elements with sharp notches under mixed-mode loading. J Engng Mech 1997;123:535–43. Seweryn A. Brittle fracture criterion for structures with sharp notches. Engng Fract Mech 1994;47:673–81. Nui LS, Chehimi C, Pluvinage G. Stress field near a large blunted tip V-Notch and application of the concept of the critical notch stress intensity factor (NSIF) to the fracture toughness of very brittle materials. Engng Fract Mech 1994;49:325–35. Dunn ML, Suwito W. Cunningham S fracture initiation at sharp notches: correlation using critical stress intensities. Int J Solids Struct 1997;34:3873–83. Strandberg M. Fracture at V-notches with contained plasticity. Engng Fract Mech 2002;69:403–15. Lazzarin P, Filippi S. A generalised stress intensity factor to be applied to rounded V-shaped notches. Int J Solids Struct 2006;43:2461–78. Gomez FJ, Elices M, Berto F, Lazzarin P. A generalised notch stress intensity factor for U-notched components loaded under mixed mode. Engng Fract Mech 2008;75:4819–33. Berto F, Lazzarin P, Radaj D. Fictitious notch rounding concept applied to sharp V-notches: evaluation of the microstructural support factor for different failure hypotheses Part II: microstructural support analysis. Engng Fract Mech 2009;76:1151–75. Sih GC, Ho JW. Sharp notch fracture strength characterized by critical energy density. Theor Appl Fract Mech 1991;16:179–214. Glinka G. Energy density approach to calculation of inelastic strain–stress near notches and cracks. Engng Fract Mech 1985;22:485–508. Ellyin F, Kujawski D. Generalization of notch analysis and its extension to cyclic loading. Engng Fract Mech 1989;32:819–26. Lazzarin P, Zambardi R. A finite-volume-energy based approach to predict the static and fatigue behaviour of components with sharp V-shaped notches. Int J Fract 2001;112:275–98. Yosibash Z, Bussiba A, Gilad I. Failure criteria for brittle elastic materials. Int J Fract 2004;125:307–33. Lazzarin P, Berto F. Some expressions for the strain energy in a finite volume surrounding the root of blunt V-notches. Int J Fract 2005;135:161–85. Livieri P. Use of J-integral to predict static failures in sharp V-notches and rounded U-notches. Engng Fract Mech 2008;75:1779–93. Berto F, Lazzarin P, Matvienko YG. J-integral evaluation for U- and V-blunt notches under Mode I loading and materials obeying a power hardening law. Int J Fract 2007;146:33–51. Bazant ZP, Planas J. Fracture and size effect in concrete and other quasi-brittle materials. CRC Press; 1998 [Chapter 7]. Elices M, Guinea GV, Gómez FJ, Planas J. The cohesive zone model: advantages, limitations and challenges. Engng Fract Mech 2002;69:137–63. Gómez FJ, Elices M. Fracture of components with V-shaped notches. Engng Fract Mech 2003;70:1913-1. Gómez FJ, Elices M. A fracture criterion for blunted V-notched samples. Int J Fract 2004;127:239–64. Gómez FJ, Elices M, Planas J. The cohesive crack concept: application to PMMA at 60 °C. Engng Fract Mech 2005;72:1268–85. Gómez FJ, Guinea GV, Elices M. Failure criteria for linear elastic materials with U-notches. Int J Fract 2006;141:99–113.

160 [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71]

F. Berto et al. / Engineering Fracture Mechanics 90 (2012) 143–160 Atzori B, Lazzarin P, Meneghetti G. Fracture mechanics and notch sensitivity. Fatigue Fract Engng Mater Struct 2003;26:257–67. Theocaris PS. A higher order approximation for the T-criterion of fracture in biaxial fields. Engng Fract Mech 1984;19:975–91. Ayatollahi MR, Aliha MRM. Analysis of a new specimen for mixed mode fracture tests on brittle materials. Engng Fract Mech 2009;76:1563–73. Ayatollahi MR, Torabi AR. Determination of mode II fracture toughness for U-shaped notches using Brazilian disc specimen. Int J Solids Struct 2010;47:454–65. Ayatollahi MR, Torabi AR. Investigation of mixed mode brittle fracture in rounded-tip V-notched components. Engng Fract Mech 2010;77:3087–104. Gómez FJ, Elices M, Berto F, Lazzarin P. Local strain energy to asses the static failure of U-notches in plates under mixed mode loading. Int J Fract 2007;145:29–45. Berto F, Lazzarin P, Gomez FJ, Elices M. Fracture assessment of U-notches under mixed mode loading: two procedures based on the ‘equivalent local mode I’ concept. Int J Fract 2007;148:415–33. Gómez FJ, Elices M, Berto F, Lazzarin P. Fracture of V-notched specimens under mixed mode (I + II) loading in brittle materials. Int J Fract 2009;159:121–35. Lazzarin P, Berto F, Elices M, Gómez J. Brittle failures from U- and V-notches in mode I and mixed, I + II, mode. A synthesis based on the strain energy density averaged on finite size volumes. Fatigue Fract Engng Mater Struct 2009;32:671–84. Gomez FJ, Elices M, Berto F, Lazzarin P. Fracture of U-notched specimens under mixed mode: experimental results and numerical predictions. Engng Fract Mech 2009;76:236–49. Wang NM. Twisting of an elastic plate containing a crack. Int J Fract 1970;6(4):367–78. Raju KR. Effect of depth of side grooves in double torsion specimens on plane strain fracture toughness. Int J Fract 1981;17:R189–90. Cheung YK, Wang YH. The torsion of a bar with arbitrary section containing two edge cracks. Int J Fract 1991;47:307–17. Davenport JCW, Smith DJ. A study of superimposed fracture modes I, II and III on PMMA. Fatigue Fract Engng Mater Struct 1993;16:1125–33. Lin Bisen, Mear ME, Ravi-Chandar K. Criterion for initiation of cracks under mixed-mode I + III loading. Int J Fract 2010;165:175–88. Leblond JB, Karma A, Lazarus V. Theoretical analysis of crack front instability in mode I + III. J Mech Phys Solids 2011;59:1872–87. Susmel L, Taylor D. The theory of critical distances to predict static strength of notched brittle components subjected to mixed-mode loading. Engng Fract Mech 2008;75:534–50. Zheng X, Zhao K, Wang H. Failure criterion with given survivability for ceramic notched elements under combined tension/torsion. Mater Sci Engng A 2003;357:196–202. Zheng XL, Zhao K, Yan JH. Fracture and strength of notched elements of brittle material under torsion. Mater Sci Technol 2005;21:539–45. Tanaka K. Engineering formulae for fatigue strength reduction due to crack-like notches. Int J Fract 1983;22:R39–46. Qian J, Hasebe N. Property of Eigenvalues and eigenfunctions for an interface V-notch in antiplane elasticity. Engng Fract Mech 1997;56:729–34. Lazzarin P, Zappalorto M, Yates JR. Analytical study of stress distributions due to semi-elliptic notches in shafts under torsion loading. Int J Engng Sci 2007;45:308–28. Zappalorto M, Lazzarin P, Yates JR. Elastic stress distributions resulting from hyperbolic and parabolic notches in round shafts under torsion and uniform antiplane shear loadings. Int J Solids Struct 2008;45:4879–901. Zappalorto M, Lazzarin P, Filippi S. Stress field equations for U and blunt V-shaped notches in axisymmetric shafts under torsion. Int J Fract 2010;164:253–69. Zappalorto M, Lazzarin P. Analytical study of the elastic-plastic stress fields ahead of parabolic notches under antiplane shear loading. Int J Fract 2007;148:139–54. Zappalorto M, Lazzarin P. A new version of the Neuber rule accounting for the influence of the notch opening angle for out-of-plane shear loads. Int J Solids Struct 2009;46:1901–10. Zappalorto M, Lazzarin P. A unified approach to the analysis of nonlinear stress and strain fields ahead of mode III-loaded notches and cracks. Int J Solids Struct 2010;47:851–64. Coulomb M. Recherches théoriques et expérimentales sur la force de torsion et sur l’élasticité des fils de metal. Histoire de l’Académie Royale des Sciences; 1784. p. 229–69. Zheng XL, Wang H, Yan JH. Notch strength and notch sensitivity of polymethylmethacrylate glasses. Mat Sci Engng A 2003;349:80–8. Neuber H. Theory of stress concentration for shear–strained prismatical bodies with arbitrary nonlinear stress–strain law. J Appl Mech 1961;28:544–50. Rice JR. Mechanics of crack tip deformation and extension by fatigue. In: Fatigue crack propagation, ASTM STP 415. Philadelphia, PA: American Society for Testing and Materials; 1967. p. 247–311. Ellyin F. Fatigue damage, crack growth and life prediction. London: Chapman & Hall; 1997. Hoffmann M, Seeger T. A generalized method for estimating multiaxial elastic-plastic notch stresses and strains. Part I: theory/Part II: application and general discussion. J Engng Mater Tech (Trans ASME) 1985;107(10):250–60. Topper TH, Wetzel RM, Morrow J. Neuber’s rule applied to fatigue of notched specimens. J Mater 1969;4:200–9. Glinka G. Energy density approach to calculation of inelastic strain–stress near notches and cracks. Engng Fract Mech 1985;22:485–508. Glinka G. Calculation of inelastic notch-tip strain–stress histories under cyclic loading. Engng Fract Mech 1985;22:839–54. Shin CS. Fatigue crack growth from stress concentrations and fatigue life predictions in notched components. In: Carpinteri A, editor. Handbook of fatigue crack propagation in metallic structures. Amsterdam: Elsevier; 1994. p. 613–52. Guo W, Wang CH, Rose LRF. Elastoplastic analysis of notch-tip fields in strain hardening materials. Report DSTO-RR-0137 1998. Melbourne, Vic., Australia: DSTO Aeronautical and Maritime Research Laboratory. Agnihotri G. Calculation of elastic plastic strains and stresses in notches under torsion load. Engng Fract Mech 1995;51:823–35. Lazzarin P, Berto F. Control volumes and strain energy density under small and large scale yielding due to tensile and torsion loading. Fatigue Fract Engng Mater Struct 2008;31:95–107. Lazzarin P, Livieri P, Berto F, Zappalorto M. Local strain energy density and fatigue strength of welded joints under uniaxial and multiaxial loading. Engng Fract Mech 2008;75:1875–89. Berto F, Lazzarin P, Yates JR. Multiaxial fatigue of V-notched steel specimens: a non-conventional application of the local energy method. Fatigue Fract Engng Mater Struct 2011;34(11):921–43. Berto F, Lazzarin P. Fatigue strength of structural components under multi-axial loading in terms of local energy density averaged on a control volume. Int J Fatigue 2011;33:1055–65. Berto F, Lazzarin P, Ayatollahi MR. Brittle fracture of sharp and blunt V-notches in isostatic graphite under torsion loading. Carbon 2012;50:1942–52.