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A review on coupled modes in V-notched plates of finite thickness: a generalized approach to the problem
F. Berto
University of Padova, Vicenza, 36100, Italy
Mode 1, 2 and 3 cannot exist in isolation. One mode provokes the existence of a coupled mode which, in some conditions, can be more dangerous than the generating mode itself. This means that three-dimensional problems are automatically at least dual scale. While for a crack this effect was known to exist for a long period of time, it was largely ignored in theoretical studies of V-shaped notches subjected to in-plane and out-of-plane loading as well as in practical fracture problems associated with such geometries. Only recently, some numerical investigations confirmed that highly localized coupled modes do exist in the close vicinity of the notch tip. The present paper is aimed to briefly review important features of these recently identified singular coupled modes. The most significant results from a comprehensive three-dimensional numerical study are presented here to describe the contribution of these modes into the overall stress state in the close vicinity of the notch tip and discuss the implementation of these new results to the failure and integrity assessment of plate structures with sharp notches.
Keywords: couple modes, V-shaped notch, stress state, stress intensity factor
1. Introduction
The singular nature of the linear elastic stress states at re-entrant corners (sharp V-notches) was first demonstrated by Williams [1], who in 1952 pioneered an eigenvalue expansion method in which the local stress and displacement fields were expanded in power series and a substitution of this expansion resulted into an eigenvalue equation for the determination of the power of the stress singularities. The eigenvalue equation in the case of free from stresses radial edges can be represented as a product of two terms corresponding to symmetric and antisymmetric loading of the notch. The degree of the stress singularity for the symmetric (mode 1) and antisymmetric (mode 2) loading depends on the notch opening angle 2a only and decreases progressively with the increase of the notch opening angle from 1/2 for (crack), 2a = 0, to 0 for the notch opening angle 2a = 180° (mode 1) and 2a = 102.6° (mode 2). In addition to the strength (or power) of the singularity the coefficients at the singular terms in the asymptotic expansion play a very important role in the integrity and failure initiation. These coefficients are often referred to as the generalized (or notch) stress intensity factors. Williams explicitly quoted Coker and Filon’s treatise on photoelasticity [2].
Since Williams’ pioneering work, several studies have been carried out to determine the singular stress characteris-
tics at points where there is an abrupt change in geometry, material properties or boundary conditions leading to the unbounded stresses. The early investigations dealt mainly with the evaluation of the singular powers focusing on the effect of the local geometry and material properties (in the case of multimaterial structures) [3-5]. The practical implementation of these theoretical investigations in the design of various joints and structures against static or fatigue failure was developed only almost two decades later.
As mentioned above in the analysis of the strength of V-notched components one needs to know both the singular powers and the coefficients at singular terms in the asymptotic expansion or the notch stress intensity factors, which apart from the crack problems can be determined only using numerical techniques. The first contribution focusing on the definition and numerical investigation ofthe notch stress intensity factors, K1 and K2, for V-notched plane geometries is due to Gross and Mendelson [6]. The notch stress intensity factors introduced in this paper mirror the conventional definition of the stress intensity factors of the linear elastic fracture mechanics when the notch angle is equal to zero and the singular stress state is described by the inverse root singularity. In this case the units for notch stress intensity factor are MPa • m^2, in all the other cases, the units of the notch stress intensity factors depend on the notch
© Berto F., 2013
opening angle. However, the intensity of stress fields for a particular V-notch opening angle, 90°, is reported also in [7], where the stress distributions due to mode 1 and mode 2 are given.
A strength criterion based on the concept of the stress singularity, similar to the one used in the classical linear elastic fracture mechanics, states that, in two V-notched bodies of different geometries and loading, but with the same notch opening angle, the failure at the tip of the notch will occur at the same values of the notch stress intensity factor. This strength criterion was used in many studies of the notched plate components made of brittle materials [810], the static strength of bonded joints [12-15], the fretting fatigue behaviour of a sharp corner [16-20], and the high cycle fatigue behaviour of welded joints [21-25].
A recent approach, the strain energy density approach, to fracture assessment of notched components is based on the evaluation of the averaged strain energy density over a control volume. It allows a direct comparison of the strength or failure load of notches with different notch opening angles overcoming the problem of combining the intensity of single and different fracture modes into a single parameter [2640]. Due to its nature, the strain energy density is able to include all stress effect parallel to the plate thickness, the variability of the constraint effect and to include higher order terms [41-43]. While Sih’s criterion is a point-wise criterion [44, 45], the strain energy density, as first proposed in [26], is a volume based criterion.
The first systematic study on the three-dimensional stress states of a through-the-thickness crack subjected to mode 1 loading was conducted in [46-49]. Utilizing a variational principle, a system of simplified governing equations has been derived for the extension and bending deformations of an elastic plate with a through-the-thickness crack and investigated the three-dimensional stress states surrounding the crack tip.
In parallel, a possible influence on the strength of V-notched components of various three-dimensional singular effects was pointed by many researchers. Benthem, for example, noted the disappearance of the in-plane singularity when a corner front (crack front) intersects a free surface [50]. In that point a new three-dimensional corner singularity develops instead. The problem of such vertex singularities, also called corner point singularities, is well documented in a number of papers covering the last thirty years [51-54]. The case of a cracked plate subjected to shear loads was discussed also by Nakamura and Parks [55] who identified a new singular behaviour for the transverse shear stress components in plates subjected to in-plane loading.
This effect has been investigated for through-the-thick-ness cracks in finite thickness plates using analytical and numerical methods [56-60]. In particular, Kotousov formalised this three-dimensional singular effect for sharp notches with arbitrary notch opening angles based on the first order
plate theory [58]. This singular mode, which was called ‘the out-of-plane mode, or mode O’, was found to be coupled to the antisymmetric loading.
While the work by Nakumara and Parks was focused on cracked plates, the numerical study of out-of-plane mode in plates weakened by V-notches is more recent [61-67]. In these recent contributions it was demonstrated that the out-of-plane mode is provoked by the three-dimensional effects linked to the Poisson’s ratio of the material and related to the transverse shear stress components. The intensity of the out-of-plane mode has been discussed in detail considering three-dimensional plates subjected to nominal pure shear loading conditions. The local intensities of the singular modes which change through the thickness of the plate component have been investigated numerically using the finite element method. These contributions have proved that mode 1, 2 and 3 cannot exist in isolation. One mode provokes the existence of a coupled mode which, in some conditions, can be more dangerous than the generating mode itself. This means that three-dimensional problems are automatically at least dual scale [68].
Some recent numerical investigations of the coupled mode generated by anti-plane loading of V-notched plate with an opening angle ranging from 45° to 120° have been conducted in [69, 70]. The effect of the plate thickness as well as Poisson’s ratio on the intensity of the coupled mode, which is variable through the plate thickness, has been also studied. A simple dimensionless analysis has predicted a strong thickness effect, which was also verified by direct numerical calculations. The thickness effect on the intensity of the coupled modes is one of the important threedimensional effects, which needs to be investigated experimentally. It predicts a significant decrease in strength for very thick notched components loaded in mode 2 [61-67]. On the other hand, a significant decrease in strength for very thin notched plates under mode 3 loading is documented in [69, 70]. A useful analytical frame to describe the complex three dimensional stress field has been developed in [71] and successfully applied to different cases [72].
The main aim of the present paper is to review the main results recently obtained investigating coupled modes in V-notched plates subjected to in plane and out-of-plane shear loading.
2. Some analytical preliminaries
The stress fields in the close neighbourhood of a sharp V-notch (with the notch radius p = 0) was determined by Williams with the reference to plane theory of elasticity, which is a two-dimensional theory. As well known, stress components were found to be proportional to 1/rl~X, where r is the radial distance from the singular point (see Fig. 1) and 1 - X is the degree (or power) of singularity, which in the case of free radial edges depends on the notch angle and the type of loading, symmetric (mode 1) or antisymmetric (mode 2).
Fig. 1. V-notch in a plane and in a three-dimensional plate
As well known, the equations for the eigenvalues for the in-plane singular modes are [1]:
[sin(2 Xy) + X sin(2 y)] = 0 (mode 1), (1)
[sin(2 Xy) - X sin(2 y)] = 0 (mode 2), (2)
where y is the flank angle measured with respect to the notch bisector line. The eigenvalues X corresponding to Eqs. (1) and (2) are listed in Table 1 for various 2a and y angles. One should note that the eigenvalues X 2 result in unbounded stresses when the opening angle ranges from 0° to 102.4° whereas the mode 1 stays singular until the notch opening angle reaches 180°.
The stress field is singular also for a sharp V-notch subjected to the anti-plane mode, or mode 3. Similar to the in-plane singular modes, the power of singularity for mode 3 does depend on the V-notch opening angle according to the expression given in [73, 74] which can be written as cos (Xy) = 0 (mode 3). (3)
The corresponding eigenvalues X3 are listed in the last column of Table 1. Similar to mode 1, the transverse stress components due to mode 3 are always singular for 2a ranging from 0° to 180°.
Adopting the Kane and Mindlin’s theory [75] and introducing two different potential functions for stress and displacement components, a new characteristic equation was derived by Kotousov [58] for a three dimensional sharp notch subjected to in-plane loading utilising the classical eigen-
Table 1
Eigenvalues for different opening angles 2a
2a, rad Y, rad X1 X 2 X 3
0 n 0.5000 0.5000 0.5000
n 3 5 nl 6 0.5122 0.7309 0.6000
n 2 3 n 4 0.5445 0.9085 0.6667
2n/ 3 2 n/ 3 0.6157 1.1489 0.7500
3n/ 4 5 nl 8 0.6736 1.3021 0.8000
function expansion method pioneered by Williams for plane problems. For antisymmetric loading the equation can be written as:
[sin(2 Xy)-X sin(2 y)]cos(Xy) = 0. (4)
The expression within the square brackets on the left end side of Eq. (4) matches Eq. (2) obtained by Williams within plane theory of elasticity for mode 2 loading of a notch [1]. The term cos(Xy) = 0 corresponds to the out-of-plane singular mode, which gives the same eigenvalues as mode 3. Therefore, this mode remains singular for notch angles ranging from 0° to 180°. However, there are some differences between the out-of-plane mode and mode 3. The out-of-plane singular mode is not an independent mode but it can be coupled with antisymmetric in-plane loading or mode 2. The displacement field associated with this mode is symmetric with respect to the mid-plane z = 0. The intensity of this mode is strongly influenced by Poisson’s ratio v and this mode vanishing when v = 0. In contrast, mode 3 is an independent failure mode, independent of Poisson’s ratio, with an anti-symmetric distribution about the midplane.
3. V-notched plates under pure shear load
The problem considered here is a finite size plate containing a sharp V-notch, subjected to a remote shear stress [67]. The geometry of the problem is shown in Fig. 1. The V-notch is characterised by a notch opening angle 2 a and a depth a. The base and the height of the plate are 2 W and W, respectively, while the plate thickness is 21. To observe the dependence of the intensity and singular power of singular modes as a function of the notch angle, a range of finite element models are developed for notch angles of 2a = = 45°, 60°, 90°, 102.6 °, 120° and 135°.
A three dimensional model was drawn taking advantage of the double symmetry of the problem (Fig. 2). The origin of the Cartesian coordinate system (x, y, z) is located the middle plane of the plate as shown in Fig. 2. In the transverse
Fig. 2. Boundary conditions applied to the geometry of Fig. 1, taking advantage of double symmetry conditions
800
400
2a = 60o I
- t0 = 100 MPa \
I r = 0.05 mm - Txy (mode 2)
r = 0.1 mm
-o
p
r = 0.3 mm
o o o o o o oo
o o o o o o oo
o o o o o o oo
r = 0.05 mm
r = 0.1 mm X^o
fill*
7
r = 0.3 mm
0 5 10
Coordinate z, mm
Fig. 3. Mode 2 and mode O stress components plotted throughout the plate thickness (along the notch bisector line) at three different distances r from the point of singularity (r = x)
sectional area OABC, the displacements ux and uz are set equal to zero. Moreover, displacements uz = 0 are imposed to all nodes on the midplane z = 0, whereas displacements uy = 0 are applied to the lateral surface GFHI. In order to assure a nominal shear load of t0 = 100 MPa, the displacement boundary conditions ux0 and uy0 were applied to all nodes belonging to the lateral surfaces ABDE and DEGF of the finite element model, in accordance with the following equation:
T0 W
ux 0 = uy 0 = G ~2'
(5)
In all finite element analyses the same shear elastic modulus was used, G = E/(2(1 + v)), with a Young’s modulus E = 206 GPa and a Poisson’s ratio v = 0.3.
The stress component Tyx is plotted in Fig. 3 along the notch bisector line as a function of the transverse coordinate z for three radial distances from the V-notch tip (r = x = = 0.05, 0.1 and 0.3 mm). The notch opening angle is 2a = = 60°. This figure shows that there is a large central zone within the plate, with -18 < z < 18 mm, where T xy does depend only on the radial distance from the notch tip r and not on the vertical position, z-axis. A strong dependence on z-coordinate occurs in the vicinity of the free surface, when z > 18 mm and z < -18 mm, where the increase of the intensity of Txy stress component is particular strong. In that limited zone, which has a size in the z direction of about 5 % of the total plate thickness (the plate thickness is 40 mm), the out-of-plane stress component tyx reaches its minimum or maximum values. This stress component is zero at the midplane (z = 0) and on the free surfaces (z = = ±20 mm), as expected. This follows from the boundary and symmetry conditions of the problem under conside-
ration. Obviously, the previous two-dimensional studies couldn’t recover the presence of the out-of-plane mode (mode O), nor the variability of the Tyx in-plane stress component. Thus, the stress state in a notched plate component is essentially three-dimensional, due to the presence of tyx along the entire crack front, and the fast variation of tyx in the vicinity of the lateral surfaces of the plate. For the geometry considered in Fig. 4 the stress components Tyx are always much greater than the tyx. Due to the asymptotic nature of these stress components, their ratio strongly depends on the absolute size of the component, as it will be demonstrated later in this paper.
The singular behaviour for two modes, the in-plane and out-of-plane modes, are shown in Fig. 4 for an opening angle of 45°. The degree of singularity of the Txy stress component is 1 - X2 = 0.340 and exactly matches the value
Fig. 4. Shear stress distributions due to mode O and mode 2 on three different parallel planes with the z = 0 (midplane), z = 20 mm (lateral surface) and z = 18 mm
Fig. 5. Shear stress distributions due to mode O and mode 2 on three different parallel planes with: the z = 0 (midplane), z = = 20 mm (lateral surface) and z = 18 mm
Fig. 6. Mode O and mode 2 stress fields for three models scaled in geometrial proportion
predicted by Williams’s solution for mode 2 [1]. In parallel the out-of-plane stress components Tyx show a degree of singularity 1 - X O = 0.428, in agreement with Eqs. (3) and (4).
As expected from the theory, an increase of the opening angle 2a leads to the decrease of the power of the singular behaviour. When 2a = 102.4°, the in-plane shear stresses are no longer singular whereas the out-of-plane stresses remain singular.
When 2a = 120° only the mode O is singular, whereas the stresses Tyx tend to zero approaching the V-notch tip. The relevant plots are shown in Fig. 5. The intersection with the line representing the nominal shear stress T = = 100 MPa, does not depend only on the notch opening angle and the geometrical ratios involving notch depth, plate thickness and width, but also the absolute dimensions of the finite element model. This important feature will be discussed in detail in the next paragraph.
To compare the severity of the singular stress fields some assumptions are necessary to introduce, which should involve a material’s characteristic length or the microstructural support length [76-79]. Recent papers suggest to use as a characteristic of failure initiation the strain energy density averaged over a characteristic control volume [26]. This characteristic length is different for different materials and can range from 10-6 to 10-3 m. If the characteristic length would be constant and equal to 0.03 mm with the reference to the shear stress modes (mode 2 and mode O), the failure of the component made of a brittle material is expected to be controlled by the out-of-plane singular mode, or Tyx components since at that distance from the notch tip Tyx is much higher then Tyx. Conversely, with a characteristic length of order 0.1 mm, the most important component becomes Tyx which is expected to control the failure initiation.
4. Size effect for V-notched plates
The stress fields of three plates scaled in a geometrical
proportion are shown in Fig. 6. In these plates the notch
opening angle is kept constant and equal to 90°. The geometrical parameters of the base geometry (a = t = 20 mm and W = 200 mm) are simultaneously multiplied or divided by a factor 100. The plots are drawn considering a plane at a distance of 2 mm from the surface for the base geometry (z = 18 mm). This distance is increased or reduced according to the scale factor (= 100) in the other two cases. The increase or decrease of the in-plane shear stress components is according to the scale factor of (100)00915 = 1.52. The variability of the out-of-plane shear stress components is much more pronounced, and according to the scale factor (100)033 = 4.57. The intersection point between Tyz and T yx stress fields varies from case to case: this point is located at 1 mm from the V-notch tip when a = 2000 mm, at 10-2 mm when a = 20 mm and at 10-4 mm when a = 0.20 mm.
The plots of the corresponding notch stress intensity factors, KO and K2, are shown in Fig. 7. These factors are determined on the notch bisector line according to the following expressions
K2 = lim X yX (x )1_X 2,
x^0
(6)
Fig. 7. Notch stress intensity factors K and KO for three plates scaled in geometrical proportion
Fig. 8. Mode O and mode 2 stress fields for three models scaled in geometrial proportion
K O = V2^limx yz (x)
1-X O
x^0
(7)
where K2 is according to the definition suggested by Gross and Mendelson [6], and KO represents a natural extension of stress intensity factors for cracks.
The scale effect changes for 2a = 135° are due to the non-singular behaviour of the in-plane shear stress, see Fig. 8. Once again, the base geometry is scaled in geometrical proportion, by multiplying all geometrical parameters by a factor 4 or by a factor 8. The mode O stress field increases with an increase of t whereas, on the contrary, the mode 2 stress field decreases with t.
The values of KO obtained with reference to different values of the geometrical parameters a, t and W (and K2 applied) are reported in Table 2, all referred to a V-notch
angle of 135°. The results related to the first 5 rows are obtained by keeping constant the model dimensions a and W and varying the thickness t from 1 to 80 mm. There is a strong variability of the KO parameter while K2 is almost constant. Afterwards, the thickness t is kept constant whereas a is increased from 10 mm to 200 mm leaving unchanged the plate width to notch depth ratio, W /a = 10. Doing so, KO is found to decrease from 66.16 to 21.97 MPa • mm0 20. Finally, values of KO as determined from geometries scaled in geometrical proportion are summarised in the last rows of Table 2. It might be convenient to express KO in terms of the nominal shear t0 , a non-dimensional shape factor kO and a characteristic length L powered to the degree of singularity.
An appropriate expression might be
Ko = kO L~x° v (8)
A convenient choice might be L = t when t varies whereas the parameters a and W are kept constant. Conversely, one should assume L = a when a and W vary in geometrical proportion (a/W = const) and also t does vary. These choices results in the values of the non-dimensional factor kO reported in Table 2.
Values of the notch stress intensity factors KO, kO and K2 are listed in Table 3 for three opening angles, 2a = = 45°, 60° and 120° considering here only geometries scaled in geometrical proportion, all characterised by a notch depth a equal to t. Despite a strong variability of the opening angle 2a, the variations of kO, as directly determined from the numerical models, are found to be very limited, with kO ranging from about 0.560 to 0.588.
Table 2
Notch stress intensity factors related to the notched plate shown in Fig. 2 (nominal shear stress t0 = 100 MPa)
a, mm t, mm W, mm KO, MPa • mm02 kO K 2, MPa • mm0302
2 p II 3 o 20 1 200 9.79 0.0979 129.37
20 5 200 21.96 0.1592 129.62
20 20 200 46.80 0.2570 131.44
20 40 200 68.31 0.3267 137.47
20 80 200 96.14 0.4002 134.17
10 20 100 66.16 0.4174 162.23
20 20 200 46.80 0.2570 131.44
40 20 400 36.14 0.1728 106.77
80 20 800 28.98 0.1206 85.24
200 20 2000 21.97 0.0761 64.58
10 10 100 40.74 0.2571 162.04
20 20 200 46.80 0.2570 131.44
40 40 400 53.74 0.2570 106.55
80 80 800 61.75 0.2571 86.46
160 160 1600 70.91 0.2570 70.10
Table 3
Notch stress intensity factors related to the notched plate shown in Fig. 2 (nominal shear stress t0 = 100 MPa)
a, mm t, mm W, mm KO,MPa • mm1 11 ko K2, MPa • mm1 12
2 p II 4 o 20 20 200 204.46 0.566 723.96
40 40 400 276.74 0.569 919.42
80 80 800 371.61 0.568 1,161.48
160 160 1600 498.93 0.567 1,474.20
320 320 3200 672.14 0.567 1,860.96
2 P II 6 O o 20 20 200 194.18 0.5859 657.41
80 80 800 338.86 0.5872 956.85
160 160 1600 447.92 0.5882 1155.12
2 P II 9 O o 0.2 0.2 2 32.79 0.561 307.38
20 20 200 152.25 0.5609 469.93
2000 2000 20000 704.91 0.5595 716.18
5. V-notched plates under antiplane shear loading
Some recent numerical investigations of the coupled mode generated by anti-plane loading of V-notched plate with an opening angle ranging from 45° to 120° have been conducted in [69, 70]. A coupled mode is induced also for anti-plane loading of a notch. The stress-free boundary conditions on lateral surfaces lead to redistribution of stresses and generation of a coupled local mode with some characteristics of mode 2. The main results are discussed in this session.
Because the coupled singular modes are local modes and normally spread to the distance of approximately half of the plate thickness, the problem geometry is truncated to a disk with dimensions which avoid the effect of the finite boundaries on the stress state of the coupled mode. The anti-symmetric boundary conditions are utilized to further simplify the geometry. The anti-symmetric plane (z/t = = 0.5) is shown in Fig. 9, a.
The final geometry is shown in Fig. 9. Appropriate displacement boundary conditions corresponding to anti-plane loading were applied on the cylindrical surface as illustrated in the next paragraph. The origin of the Cartesian coordinate system is (x, y, z) located at the V-notch tip, at the midsurface where x direction was chosen to be the direction of the notch bisector line.
The displacement boundary conditions are applied to the outer cylindrical edge of the plate. The out-of-plane displacement field corresponding to the first singular term in the asymptotic expansion of the stress field, which is valid far from the notch tip (model boundaries), was applied using the equations that follow:
G = -
2 K
w = uz =-
3 r 3sin(A30),
(9)
gV2tc
ux = 0, uy = 0, where G is the shear modulus. It is linked to Young’s modulus by the following well-known relationship:
—. (10)
2(1 + v) ' '
In further numerical examples Young’s modulus was set at 200 GPa. However, the numerical results to be presented can be easily rescaled for other values of mechanical properties, loadings or plate thicknesses.
First the results for the out-of-plane shear stress components along the bisector line for different distances from the notch tip have been considered. The applied remote stress intensity factor, K3 = 1 MPa • mm1-13, Poisson’s ratio v = 0.3 and the thickness of the plate t = 20 mm, which are considered to be typical in many engineering applications.
Fig. 9. Geometry and coordinate system having the origin on the notch tip in the lower surface of the plate (a); finite element model (b)
0.0 0.1 0.2 0.3 0.4 0.5
z/t
Fig. 10. Distribution of the stress intensity factors (mode 3 and the coupled mode) along the plate thickness, at a distance x/t = = 0.02 from the notch tip
The stress intensity factor of the coupled mode can be defined similar to mode 2 as
K 2 = ^ lim TxyX1_12. (11)
x^0 '
In general there are many similarities between the coupled mode and mode 2 corresponding to shear loading of a crack and of a V-notch. However there are some essential differences between these two singular modes. The coupled mode is a local mode, which is concentrated in the vicinity of the plate free surfaces. It is generated due to the boundary conditions, which negate the out-of-plane shear stress components corresponding to the applied mode. The intensity of the coupled mode rapidly decays with the distance to the free edge.
The distributions of K3 and of the coupled mode K2 across the plate thickness are shown in Fig. 10 for 2a = = 90°. As it can be seen from this figure the coupled mode is localized in the close vicinity of the free surfaces and propagates into the thickness direction up to a distance of approximately 0.2t. There is a significant drop in the intensity of the applied mode in the vicinity of the free surface. In the very close vicinity of the free surface the intensity of the applied mode 3 is approaching to zero. It is interesting to note that the intensity of the coupled modes in all cases is higher than the intensity of the applied mode 3.
It is also found that the intensity of coupled mode 2, as due to the applied mode 3, is not significantly influenced
2a = 60o
^2.0 MPa-mm1-12 t = 2 mm
R = 1 mm
1^1.8 MPa- mm1-12 v = 0.1, 0.4
0.0...........0.1.............0.2 ^ 0.3 _ t 0.4 " 0.5
z/t
Fig. 11. Distribution of the stress intensity factors (mode 3 and the coupled mode) along the plate thickness, at a distance x/H = = 0.02 as a function of Poisson’s ratio
by Poisson’s ratio. The results of the calculations for different Poisson’s ratios are shown in Fig. 11. This is very interesting and worth of discussion.
The coupled mode generated by applying an external mode 2 loading is prevalently developed due to Poisson’s effect and is significantly affected by it as discussed in some recent references [58-67]. Moreover the intensity of the coupled mode 3 vanishes when v = 0. In contrast to previous findings, in the present investigation has been found that the intensities of the primary and coupled modes in the case of anti-plane loading (mode 3 loading) are not significantly affected by Poisson’s ratio as this coupled mode is generated by a mechanism associated with a redistribution of the transverse shear stresses close to free plate surfaces, see Fig. 11. These transverse stresses have to be negated due to the stress-free boundary conditions on the plate surfaces. The coupled mode 2 is generated also when v = 0.
To investigate the same problem an alternative model has been carried out. A plate weakened by a V-notch (2a = = 45°) has been modeled and two forces have been applied to generate mode 3 loading on the plate (Fig. 12). The stress components, tied to the generating and the coupled mode, have been plotted through the plate thickness for different values of the Poisson’s ratio (see Fig. 13, a). As it can be observed from the figure also the limit case v = 0 has been considered. The figure refers to a distance x = 1 mm from the notch tip. The analysis confirms the previous models obtained by applying mode 3 displacements far from the notch tip. In particular it has been found that the intensity of the coupled mode 2 is only slightly influenced by Poisson’s ratio and, more important, that the coupled mode arises also when v = 0 due to the stress redistribution near the stress-free surfaces, as discussed above. The same has been verified at different distances. Figure 13, b refers to a distance x = 0.3 mm. At the conclusion of the present work it seems clear that three dimensional effects are important
Ip
Fig. 12. Three-dimensional plate weakened by a V-notch under torsion loading. Two forces have been applied in z-direction while all displacements have been constrained on the dashed surface opposite to the notch
100
0
-100
-200
300
200
a | T^y ■ ~ O O 0 O 0 . e 8 3 B 9 8 8 S J Jxy □ 1 S • S 8 8 * 0 0 a 0 i jj 8 8 8 § S 6 • . {
Free surface s ® ^ . s 8 0\ - DO \ • ° ^xy :f ° , , , , Midplan e ^ o v = 0.0 □ v = 0.3 • v = 0.5 , , , ,
4 6
z, mm
10
Ph
100
&• 0 H^lOO -200 -300
b xzy s ■ Q o O O o o oN § "••••••« a J j T ° ' Lxy □ \ s • 1 S 2 8 8 8 8 8 • , L
§ ^ ^ Free surface @ 9 .800e ■ Ss\ <> O Lxy 1 ... . r ^ Midplane oV = 0.0 □ v = 0.3 • v = 0.5
0
10
Fig. 13. Distribution of the stress components (mode 3 and coupled mode) along the plate thickness, at a distance x = 1 mm (a) and x = 0.3 mm (b) from the notch tip. F = 2100 N, H = 10 mm, a = = 20 mm, W = 100 mm
and in principle not negligible if the aim is to assess the fracture crack propagation into the material. Similar conclusions have been drawn by other researchers also in the very recent literature [80, 81]. The importance of three dimensional finite element analysis and in particular the effects of thickness and notch radius on the fracture toughness of polycarbonate plates have been recently described in [80] explaining the brittle to ductile transition that usually occurs in polycarbonate decreasing the plate thickness and increasing the notch radius.
It is worth mentioning here, to give a complete state of the art, a forthcoming publication by Pook who reviews three dimensional effects at cracks and sharp notches [81] and will surely become a fundamental contribution for researchers facing with these problems. The overall aim will be to review developments over the past fifty years leading up to the current state of the art. The review shows that despite increased understanding three dimensional effects are sometimes ignored in situations where they may be important. This fact strongly penalizes the correct understanding of physical phenomena as well as the optimal design of structural components.
6. Conclusions
The main features of 3D coupled modes have been re-
viewed here dealing with plates of finite thickness weakened
by sharp V-notches. First, the induced out-of-plane singular mode, mode O, has been numerically investigated considering three dimensional models of V-notched plates under a nominal shear load conditions and varying opening angles. In the second part of the paper the induced in-plane shear generated by an applied anti-plane loading has been studied and the results deeply discussed. In more detail, the intensity of the mode O induced by the nominal in-plane shear has been discussed as a function of the opening angle and plate size, paying particular attention to the scale effect due to this singular mode, which is characterised by a degree of singularity matching that due to mode 3. However, some differences exist between the out-of-plane mode and the mode 3. The main difference is that the displacement field associated with mode O is symmetric with respect to the midplane. In contrast, mode 3 is an independent failure mode with an anti-symmetric distribution about the midplane. The numerical analysis has shown that the mode O is not independent but coupled with mode 2 stress fields. It is singular also when the plate subjected to in-plane shear loading presents a non-singular mode 2 stress distribution, as it happens when the opening angle is greater than 102.4°. For the V-notched plates the importance of the out-of-plane mode increases with the plate size, due to the different degrees of singularities. In the second part of the paper the coupled mode tied to an external applied anti-plane loading has been also investigated. In fact, anti-plane loading of a through the thickness V-notch is accompanied by a coupled mode, which has the same singular characteristics as mode 2. Also in this case, there are some essential differences between these modes. The coupled mode is a local mode, which is concentrated in the vicinity of the plate free surfaces and is generated due to Poisson’s effect and the boundary conditions, which negate the out-of-plane shear stress components corresponding to the applied mode. The coupled mode rapidly falls with distance from the crack tip. The intensity of the coupled mode 2 is only slightly affected by Poisson’s ratio contrary to what happens when the generating mode 2 induces a coupled mode 3. For considered examples, the intensities of the coupled modes for a range of notch opening angles were higher than the intensity of the applied mode 3. This could indicate that the coupled mode might play a very important role in fracture initiation at anti-plane loading. The intensity of the coupled mode is also affected by the plate thickness; with the decrease of the plate thickness there is a significant increase of the intensity of the coupled mode, which reaches maximum at the free surface. This phenomenon demonstrates that there is a strong scale effect of non-stochastic nature for notched plates. This can be very important for small scale structures, for example in microelectronics, where some components have sharp notch configurations.
All these theoretical findings, specifically the effect of the plate thickness on the stress intensity of the coupled
mode, have a direct implication to integrity and failure conditions of sharp notches stressed in mode 3. These findings demonstrate essential differences between classical twodimensional considerations and 3D fracture mechanics. For example, the generation of the coupled singular mode at anti-plane loading with K3 = 1 MPa • mm1-^3 indicates that contrary to the classical considerations, fracture under such loading conditions can be initiated due to the induced singular coupled modes. Such fracture is likely to take place close to the free surfaces. It is also recognized that much work needs to be done to understand the contribution of the coupled modes to fracture initiation and fatigue. The focus of future studies can be an experimental confirmation of the theoretically predicted phenomena.
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Berto Filippo, Prof., University of Padova, Italy, [email protected]
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