CC BY
79
The effect of the boundary conditions on in-plane and out-of-plane stress field in three dimensional plates weakened by free-clamped V-notches
F. Berto, P. Lazzarin, Ch. Marangon
University of Padova, Vicenza, 36100, Italy
Dealing with the material microstructure an analytical multiscale model has recently been developed by Sih. Physically, the different orders of the stress singularities are related to the different constraints associated with the defect thought as a microscopic V-notch at the tip of the main crack. Irregularities of the material microstructure tend to curl the crack tip being the clamped-free boundary conditions the more realistic and general representation of what occurs on the microscopic V-notch. As a result, mixed mode conditions are always present along the V-notch bisector line.
It is known for a long time that at the antisymmetric (mode II) stress distribution ahead of the crack tip generates a coupled out-ofplane singular mode. Recent theoretical and numerical analyses have demonstrated that this out-of-plane mode due to three-dimensional effects occurs also in the case of large V-notches where the mode II stress field is no longer singular. In addition, when the notch opening angle is non-zero, the three-dimensional singular stress state is strongly influenced by the plate thickness.
The aim of this paper is to investigate the effect of free-fixed boundary conditions along the notch edges in three dimensional plates weakened by pointed V-notches and quantify the intensity of the out-of-plane singularity occurring under this constraint configuration. For the sake of simplicity a macronotch is considered rather than a micronotch. A synthesis of the magnitude of the stress state through the plate thickness is carried out by using the mean value of the strain energy density over a given control volume embracing the notch tip. The capability of the strain energy density to capture all the combined effects due to the out-of-plane mode make it a powerful parameter at every scale levels.
Keywords: V-notches, strain energy density, critical radius
1. Introduction
Dealing with fracture assessment of cracked and notched components a clear distinction should be done between large and small bodies [1, 2]. The design rules applied to large bodies are based on the idea that local inhomogeneities, where material damage starts, can be averaged being large the volume to surface ratio. The concept of “elementary” volume and “structural support length” was introduced many years ago by Neuber [3]. It states that not the theoretical maximum notch stress is the static or fatigue strength-effective parameter in the case of pointed or sharp notches, but rather the notch stress averaged over a short distance normal to the notch edge.
In small bodies the low ratio between volume and surface makes strong any local discontinuities present in the material and the adoption of a multiscaling and segmentation scheme is the only way to capture what happens at pico-, nano- and microscopic levels [4-7]. In this scheme the crack tip has no dimension or mass to speak; it is the sink and source that absorbs and dissipates energy while
the stress singularity representation at every level is the most powerful tool to quantify the energy packed by an equivalent crack reflecting both material effect and boundary conditions. This new representation implies also a new definition of mass [7]. The distinction between large and small bodies should ever be considered by avoiding to transfer directly the design rules valid for large components to small ones under the hypothesis that all material inhomogeneities can be averaged [1, 2].
Other approaches considered the multiscale problems by the investigation of the nonlinear effects occurring at different scale levels [8-10].
Since Beltrami to nowadays, the strain energy density has been found being a powerful tool to assess the static and fatigue behavior of unnotched and notched components in structural engineering. The strain energy density factor S was defined by Sih as the product of the strain energy density by a critical distance from the point of singularity [11]. Failure was thought of as controlled by a critical value Sc, whereas the direction of crack propagation was determined
© Berto F., Lazzarin P., Marangon Ch., 2012
by imposing a minimum condition on S. The theory was extended to employ the total strain energy density near the notch tip [12], and the point of reference was chosen to be the location on the surface of the notch where the maximum tangential stress occurs. The material element was always kept at a finite distance from the crack or the notch tip, outside the “core region” where the inhomogeneity of the material due to microcracks, dislocations and grain boundaries precludes an accurate analytical solution. The criterion based on the factor S gave a sound theoretical basis to Gillemot’s experimental findings [13, 14] based on the absorbed specific energy.
A volume-based strain energy density approach has been applied to static and fatigue strength assessments of notched and welded structures [15-22]. The control volume radius, which depends on material and loading conditions, was found to vary between about 10-2 mm (brittle material under static loading conditions) to about 0.3 mm (steel welded joints under high cycle fatigue) or more.
Recently as stated above, the volume energy function has been scaled from macro to micro to take into account the microcracks with a stronger stress singularity [1, 2]. Positive definiteness of energy density can be a guide for precluding physical contradictions that are hidden at different scales of the material [2]. As widely described in [4] a mixed condition would be more realistic to represent a microcrack behaviour due to the fact that the microcrack surfaces are partly closed and partly open. The macrocrack has a stress singularity of the order r-12 while near the microcrack a stress singularity of the order r_3/4 is not unrealistic because the crack surfaces may be irregular and jag-shaped. The apex of the microcrack can be represented by a V-notch under clamped-free boundary conditions [1, 4].
Dealing with elastic stress singularities and corresponding generalized stress intensity factors for corners under various boundary conditions, the authors of [23] summarize the analytical description of the stress and displacement fields around a sharp angular corner subjected to various loading and displacement boundary conditions in plane problems of elasticity. The clamped-free conditions have been shown to correspond to a rigid punch with a strong friction applied to one face of an elastic corner or to a rigid inclusion connected with one side to elastic medium subjected to any load, while the second face of the medium remains free from traction.
An interesting three-dimensional effect was described in the case of a cracked plate subjected to shear loading by Nakamura and Parks [24] who first found a new singular behaviour for the transverse shear stress components in thin plates subjected to in-plane loading. This effect has also been studied for through-the-thickness cracks in finite thickness plates utilising analytical and numerical methods [2528]. In particular, Kotousov [28] formalized this three-dimensional singular effect for sharp notches with arbitrary
notch opening angles based on the first order plate theory. This singular mode, which was called “the out-of-plane mode”, or “mode O”, was found to be coupled with the classical antisymmetric stress distribution (mode II). Kotousov also demonstrated that the out-of-plane mode is provoked by the three-dimensional effects which increase as the Poisson’s ratio of the material increases. Important features of this recently identified singular mode, the out-ofplane singular mode, conduct a comprehensive three-dimensional numerical study of V-notched plates and welded lap joint-like geometries [29-31]. In [29] the intensity of the mode-O stress distributions 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 characterized by a degree of singularity matching that due to mode III. It is clear from the analysis that the intensity of mode O increases as the model size increases. This means that there exists the geometrical proportional factor at which the contribution into the failure initiation of the out-of-plane shear stresses will be always greater than the contribution from the generating in-plane shear stresses.
The influence on mode O of non-singular terms in cracked plates is investigated in [32, 33].
Dealing with a three-dimensional plate weakened by a sharp notch under clamped-free boundary conditions the mixed mode is automatically generated by the edge constraints as explained in [1, 23]. The presence of mode II should generate a singular mode O which has been neglected in all previous papers.
The aim of the paper is to investigate the contribution and the intensity of this mode into the overall three dimensional stress states in the close vicinity of the notch tip. Since the out-of-plane mode is not a point-wise effect but rather a volume-based effect, it is shown how the strain density over a control volume is sensitive to all the threedimensional effects occurring through the plate thickness and a parameter suitable for identifying the most critical zone. The results summarized in the paper are general and can be directly shifted to the mesomechanical approach proposed in [1, 2] which models the microcrack tip as a fixed-free sharp V-notch as well as to the cases discussed in [23] dealing with in-plane solution and extended to the threedimensional models.
2. Analytical background
The Fadle eigen-function expansion method was first used in the1930s to find the eigenvalues for a pointed notch with different boundary conditions along the notch edges [34]. However the modern literature usually refers mostly to [35, 36] in two dimensions and to [37, 38] in three dimensions. The results in [34] are for the plane stress case whereas the plane strain case has been solved in [1].
The biharmonic stress function x(r, 9) and the harmonic function (r, 9) can be expressed in the following form:
X [r, 0] = rX+1F[0],
Vi[r, 0] = rX-1G[0], where F(0) and G(0) are as follows:
(1)
G[0 ] =
X-1 4
x-1
B sin[(X-1)0 ] -D cos[(X-1)0 ],
(2)
F [0 ] = Acos[(X +1)0 ] + B cos[(X -1)0 ] +
+ C sin[(X +1)0 ] + D sin[(X -1)0 ].
The displacement in radial and circumferential direction, ur and u0, respectively, may be written in terms of X(r, 0) and ^1(r, 0) under plane stress [34, 35] or plane strain conditions [1]:
2\rnr = -d r X + (1 -CT)rd0^1,
2^u0 =- - 30X+ (1 -a)r 23 r ^1,
r
(3)
where |J.= E/(2(1 + v)) is the shear modulus and a = = v/ (1 + v) under plane stress conditions and v (Poisson’s ratio) under plane strain. The displacement field can be expressed also by the following equations:
Ur =— (-(X + 1)F [0 ] + (1 -v )30G[0 ]), 2^
r X
U0 = — (-90 F [0 ] + (1 - v )(X - 1)G[0 ]). 2^
(4)
a0 = r 1X(1 + X )(B cos [ (-1+ X )0] +
+ Acos[(1 + X)0] + Dsin [(-1+ X)0] + + C sin [ (1+ X )0]),
ar0 = r X-1X(C (1 + X)cos[(1 + X)0] -
(7)
- A(1 + X) sin [(1 + X)0] +
+ (X - 1)(D cos[(-1 + X)0] + B sin [(1 - X)]0)).
The same can be done for the radial and circumferential displacements under plane strain conditions:
1 X
ur =— r (-A(1 + X)cos[0(1 + X)]-2^
- (-3 + X + 4v )(B cos[0 (-1 + X )] +
+ D sin[0 (-1+ X )]) - C (1+ X ) sin[0 (1 -X )]),
1 X (8)
u0 =------rX (D(3 + X- 4v ) cos [0 (-1+ X )] +
2^
+ C (1+ X ) cos[0 (1+ X )] +
+ 4Bv sin[0(-1+ X )]) -
- A (1 + X ) sin [0 (1 + X )] +
+ B (3 + X ) sin [0 (1 -X )].
The free-fixed condition where the top edge is traction-free while the bottom edge is fixed from displacing can be expressed in the form:
a0 = ar0 = 0 for 0 = P*,
ur = u0= 0 for 0 = -P*.
(9)
Dealing with the free-fixed case and by calling k the eigenvalue of this particular case, as made in [1], the eigenvalue equation becomes:
x
cos[(K + 1)P*]
( K + 1)sin[( K + 1)P*] -(k + 1)cos[(K + 1)P* (k + 1)sin[( K + 1)P*]
cos[(K-1)P ]
(K-1)sin[(K - 1)P*]
-(-3 + k + 4v) cos[(K - 1)P*] (3 + k - 4v) sin[( k - 1)P*
sin[(K + 1)P ]
-( K + 1)cos[( K + 1)P* (k + 1)sin[(K + 1)P*] ( K + 1)cos[( K + 1)P*]
sin[(K-1)P ]
-(K-1)cos[(K - 1)P*] -(-3 + k + 4v) sin [(-K + 1)P* ] (3 + k - 4v) cos [( k - 1)PS
y 2 a 3
B
C
D
9- V /
(10)
The stresses are given as: ar = -2 +1 d r X, = d 2X,
r2 r
1 1 (5)
are =—ded r X + — deX, r r2
or, alternatively, as:
ar [r, e] = rx-10e (de F[e]) + (X + 1)F[e]),
ae [r, e] = rX-1 (X + 1)XF[e], (6)
are [r, e] = rX-1(-Xde F [e]).
The stress components can be explicitly expressed in terms of the unknown parameters:
ar = r X-1X(-B(-3 + X) cos [(X - 1)e] -
-(1 + X)(A cos [(1 + X)e] + C sin [(1 + X)e]) +
+D(-3 + X) sin [(1 -X)e]),
The non-trivial solution of Eq. (10) requires that the determinant of coefficients must vanish. This leads to the equation:
2(1 + k )[-5 + k 2 + 4(3 - 2v )v --K2 cos (4P*) + (4v- 3)cos(4P*K)] = 0. (11)
By neglecting the solution k = -1 which would generate infinite displacements at the notch tip, the eigen equation can be written as a function of the Poisson’s ratio:
-5 + k2 + 4(3 - 2v )v - k2 cos (4P* )+
+ (4v- 3)cos(4P*K) = 0. (12)
One should note that Eq. (12) gives the same eigenvalues of the equation
4(1 - v)2 - k2 sin2(2P*) - (3 - 4v)sin2(2KP*) = 0 (13) already reported in [1].
The link between the unknown parameters can be expressed by taking advantage of three auxiliary parameters e1; e2 and e3, where:
ex(P*, V) = A =
A
= {(1 + k )((-3 + k+ 4v)sin[P*(-1 +k )] --k sin[P*(3 + k)] + sin[P*(1 + 3k)])}x x {(-1 + K2)sin[P*(1 + k )] +
+ (-3 + k + 4v )(-k sin[P* (-3 + k )] +
+ sin [P* (1 - 3k )])}-1,
e2(P*, V) = C = {(-1+ K 2)cos[P*(1 +K )] +
A
+ (-3 + K+ 4v)(-kcos[P*(-3 + k )] +
+ cos[P*(1 - 3k )])} {(-1+ K 2)sin[P* (1+ K )] +
+ (-3 + K+ 4v)(-ksin[P*(-3 + k )] +
+ sin[p*(1 -3k )])}-1, (14)
ea(P*, V) = D =
A
= {(1 + k )(-(-3 + K+ 4v)cos[P* (-1 + k )] +
+ kcos[P* (3 + k )] + cos[P*(1 + 3k )])}x x{(-1 + K2)sin[P*(1 + K )] +
+ (-3 + k + 4v )(-k sin [P* (-3 + k )] +
+ sin[P*(1 - 3K )])}-!.
The factor KI/II can be defined in terms of A as:
K
k I/II=42ka.
The corresponding displacements are:
K
2^ur = -llM. r K [-(k + 1)cos(K+ 1)e -V2n
- ej(p*, v)((k +1) - 4(1 - v))cos (k - 1)e -
- (k + 1)e2 (P*, v) sin (k + 1)e -
- e3(P*, v)[(k +1) - 4(1 - v)] sin (k -1)0],
K
2^ue = r K [(k+ 1)sin(K + 1)e +
V2n
+ ej (P*, v)[(k -1) + 4(1 - v) sin (k - 1)e -
- (k + 1)e2 (P*, v) cos(K +1)0 -
- e3(P*, v)[(k -1) + 4(1 - v)] cos(K- 1)e]. In parallel, the stresses are:
(15)
K
i/ii
1-K
k[-(k + 1)cos (k + 1)e +
+ (3 - K)e1 (P , v)cos(K- 1)e -
- (k * + 1)e2(P*, v)sin (k* + 1)e + + (3 - k* )e3 (P*, v) sin (k* - 1)e],
(16)
=-
KI
i/ii
k(k+1) x
x[cos(K+ 1)e+ ej(P ,v)cos(K- 1)e +
+ e2(P*, v) sin (k + 1)e + e3(P*, v) sin (k -1) e)], (17)
a re ="
i/ii
1-K
k[(k + 1)sin (k + 1)e +
+ (k - 1)e1 (P , V)sin(K-1)e-
- (k + 1)e2 (P*, v)cos (k + 1)e -
- (k- 1)e3(P*, v)cos(K- 1)e]. (18)
Under plane strain conditions, the strain energy density evaluated over a control volume of radius R0 is as follows:
W =
KK 2 R 2(K-1) KKI/IIR0
{(1 -V-2v 2)x
2tcEP*(k-1) x (ej2 - e|) sin (2P* (k -1)) +
+ (k- 1)(1 + v)[(k2 -1)ej sin(2P*) +
+ P*(3 - (k - 2)k - 4v)e2 +
+ P*(1 +k)2(1 + e2) +*
+ (k2 - 1)e2e3 sin (2P*) +
+ P*(3 + (k-2)k- 4v)e32]}. (19)
The analytical frame described above allows us to consider the stress field at different scale levels as linear elastic. The singularity depends on the considered scale level, from macro to micro.
Alternative to the linear elastic multiscale models, other models consider the nonlinear effects occurring at different scale levels abandoning the field of the linear elasticity. A multilevel approach based on the physical mesomechanics of the material has been systematically used in [8] dealing with thin films. Nonlinear effects in surface layers under severe plastic deformation have been investigated experimentally in [9]. This issue is crucial for a large number of applications (nanomaterials and nanotechnology, multilayer materials for electronics, deposition of nanostructured protective and hardening coatings, catalysis and functional role of interfaces in biological objects). Experimental techniques mainly based on acoustic emission have been used to monitor the multiscale of localized plastic strain evolution stages in notched aluminum AA 2024 alloy tension specimens [10]. Digital Image Correlation (DIC) method for strain estimation and acoustic emission have been also used for investigating strain and fracture patterns at various scale levels.
3. Geometry and boundary conditions
The study considers an arbitrary semiinfinite plate of finite thickness 2h containing a V-notch subjected to a remote loading. In the finite element models, a denser nodal arrangement is created in the proximity of the crack notch tip where the mesh is very fine. The analysis is carried out using the ANSYS 11 code.
The mesh consists of an initial arrangement of 15-node trapezoidal elements at the notch tip, surrounded by a radial array of 20-node brick elements, where each element spans an angular sweep of 11.25°. The considered geometry is shown in Fig. 1.
Fig. 1. Geometry and coordinate system with the origin at the notch tip in the midplane of the plate
Table 1
Eigenvalues K and functions e1, e2 and e3 in terms of the notch opening angle
2a k 1 - k e1 e2 e3 ko
45° 0.253 0.747 1.5566 -1 -1.5566 0.2857
6 ° 0.257 0.743 1.4854 -1 -1.4854 0.3000
9 ° 0.2744 0.7256 1.3252 -1 -1.3252 0.3333
135° 0.3384 0.6616 1.1059 -1 -1.1059 0.4000
In two-dimensional models the eight nodes isoparametric element plane 82 is used under the plane strain hypothesis.
To simulate the remotely applied constraint conditions corresponding to the free-clamped boundary conditions, the displacements are applied to the nodes belonging to the outer cylindrical surfaces of the finite element models. The displacements exactly correspond to those of a far-field twodimensional plane strain distribution given by the solution [1] (Eqs. (15)).
In the beginning, in order to validate the developed finite element models and compare the results with previous findings, four opening angles are considered in two-dimensional models, 2a = 45°, 60°, 90o and 135°, all referred to a constant value of the radius (R = 100 mm) and to plane strain hypothesis.
In all these cases the notch stress intensity factor K I/n has been set equal to 1000 MPa- mm1-K*, whereas the eigenvalue k* and the functions e1, e2 and e3 have been updated as a function of the notch opening angle according to Eqs. (11)-(13) and Eq. (14), respectively (see Table 1). The Poisson’s ratio has been kept constant and equal to 0.49.
In the three-dimensional models the same notch opening angles and the same radius R have been considered. The thickness is 2h = 50 mm.
4. Results and discussion
4.1. Results from 2D numerical analysis
Two-dimensional models were carried out by imposing the displacements which exactly correspond to those of a far-field two-dimensional plane strain distribution given in [1]. For sake of brevity only the results from models with 2a = 45° and 135° are summarised below without loosing any significant information.
This preliminary analysis has been carried out to set up the model and to verify how the boundary conditions affect the two-dimensional stress field.
Figure 2 shows the stress distributions along the bisector line in a two-dimensional plate weakened by a V-notch with an opening angle 2a = 45°. Equations (15) have been used for calculating the displacements to be applied to the contour nodes of the model.
As shown in the figure, the applied boundary conditions corresponding to the free-clamped configuration generate not only mode I but also mode II in the plate. The stress field is typical of a mixed mode configuration. The degree of singularity of the stress components is 1 - k = 0.747 and exactly matches the value predicted by the solution given in [1] for the free-fixed configuration. In the case of 2a = 45° the intensity of the generated mode II remains much lower than that due to mode I. Along the notch bisector line it has been also possible to determine the factor KI/II by inverting Eq. (17). Doing so we have:
■j-kÆ°. (20)
K I/II =
k( k +1)[1 + e,(p*, V )]’
where for the specific case k = 0.253 and e1 = 1.5566 (see Table 1).
Figure 3 confirms that applying Eq. (20) along the notch bisector line KI/II results to be equal to 1000 MPa - mm0 747, which matches the value applied when imposing the boundary conditions. The correctness of the applied displacements is then verified.
Figure 4 shows the stress field corresponding to the opening angle 2a = 135°. Also in this case the degree of singularity matches the theoretical value 1 - K = 0.66 [1]. The intensity of the shear stress Txy corresponding to an applied stress intensity factor KI/II = 1000 MPa - mm0662 is higher than that found in the previous case, 2a = 45°. Comparing Figs 2 and 4 the main conclusion is that the mode II stress intensity increases with respect to the mode
I intensity as the notch opening angle increases.
100000
10000
CO
1000
100
10
0.1
! 2a = 45° x ax = 299.2x-07378
o ay = 324.67x-07408
a = 9.32 07X-0-7397
: R = 100 mm
> = 0.49
: K|/M = 1000 MPa ■ mm0-747
0.001
0.01
0.1
10
100
Distance from the notch tip x, mm
Fig. 2. Stress distributions in the two-dimensional plate with an opening angle 2a = 45°. Plane strain conditions are considered in the model
Fig. 3. Plot of KI/II along the crack bisector line, from the notch tip to
the outer radius R. Displacements ux of the lateral surface (R = 100 mm)
and uy applied only to the nodes
100000 10000 ro 1000 6" 100 10
2a = 135° x ctx = 2 32.2x-0 6588
o ay = 397.88x-0-6515
a Txy = 89.42x-06512
: R = 100 mm
[ v = 0.49
; K|/M= 1000 MPa-mm0-6616
i i i i 11 111 ii i i
0.001
0.01
0.1
10
100
Fig. 4. Stress distributions in the two-dimensional plate with an opening angle 2a = 135°. Plane strain conditions are considered in the model
Finally, Figure 5 confirms that also in this case the applied boundary conditions would generate along the notch bisector line the value of KI/II introduced in Eqs. (15).
4.2. Results from 3D numerical analysis and proof of mode O existence
The problem considered here is the three-dimensional finite size plate containing a sharp V-notch, subjected to remote displacements. The V-notch is characterised by a notch opening angle, 2a, and a depth a = R. The plate thickness is again equal to 2h. To observe the variability of the intensity of the singular modes as a function of the notch angle, different finite element models are created, characterised by a varying notch opening angle, 2a = 45 °, 60°, 90° and 135°.
The origin of the Cartesian coordinate system (x, y, z) is located in the middle plane of the plate as shown in Fig. 1. For a three-dimensional sharp notch subjected to in-plane loads it was shown in [24-28] that a singular mode, which was called the out-of-plane mode, or mode O, is coupled with the antisymmetric loading (mode II).
Similar to the in-plane singular modes, the power of singularity for mode O does depend on the V-notch opening angle according to the Eq. (21), valid for the free-fixed configuration [39]:
cos(2K OP*) = 0. (21)
The lowest real eigenvalue of Eq. (21) is:
KO =— : - <P* <1 (22)
O 4P*42
(see the last column of Table 1). The expression cos (2 k OP*) = = 0 corresponds to the out-of-plane singular mode, which results in the same eigenvalues of the mode III. Therefore, this mode remains singular for notch opening angles ranging from 0° to 180°. However, there are some differences between the out-of-plane mode and the mode III. The out-of-plane singular mode is not an independent mode but it is coupled with antisymmetric in-plane loading (mode II). The displacement field associated with this mode is symmetric with respect to the midplane z = 0. Moreover its intensity
strongly depends on the Poisson’s ratio v, vanishing when v = 0. Conversely, the mode III is independent of the Pois-son’s ratio, with an anti-symmetric distribution about the midplane.
As it was shown by Pook in [26] at a corner point the mode II and the out-of-plane mode cannot exist in isolation. If one of these modes is applied then the other is always induced. In order to describe the shape of crack displacements and to explain the link between mode II and the out-of-plane mode, Volterra’s distorsion in a ring element were used [26]. Dealing with the induced mode, Pook underlined that some controversies are still open: it is not clear over how best to characterize stress and displacement fields in the corner point region. He also wrote that ‘the crack tip surface displacement behaviour is not yet well understood. As a corner point is approached, there are conflicting theories which have to be resolved on the value of the out-ofplane stress intensity factor. Either it tends to infinity or drops to zero’ [26].
It is clear that the boundary conditions corresponding to the free-clamped constraints along the notch edge provoke a mixed mode state in the plate (I + II + O). The stress component T is plotted in Fig. 6 along the notch bisector line as a function of the coordinate x from the V-notch tip. The notch opening angle is 2a = 45°. This figure shows
Fig. 5. Plot of KlflI along the crack bisector line, from the notch tip to the outer radius R. Displacements ux and uy applied only to the nodes of the lateral surface (R = 100 mm)
Fig. 6. Three-dimensional stress distributions in a plate with an opening angle 2a = 45°
that also in a three-dimensional model the degree of singularity of the stress components ax, ay, a^ exactly matches the values predicted on the basis of Eqs. (12), (13) and already verified in the two-dimensional model with the same opening angle. Obviously, the previous two-dimensional studies could not recover the presence of the out-of-plane mode (mode O). Only the three-dimensional model, as shown in Fig. 6, permits to capture the presence of Tyz along the notch bisector line. This stress component is plotted in the plane at z = 24.38 mm where the mode O reaches its maximum intensity through the plate thickness. The out-of-plane stress component tyz shows a degree of singularity 1 -KO = 0.71, in fully agreement with Eq. (21). It is also worth noting that the intensity of the generating mode
II is lower than that of the coupled mode O. Once again, the notch stress intensity factor KI/n has been plotted in the middle plane of the plate to verify that the boundary conditions would correctly been applied to the model (Fig. 7).
The plot of the notch stress intensity factor KO is shown in Fig. 8. This factor is determined on the notch bisector line according to the expression
KO =42k lim t yZ (x)
1-K0
x—— 0
(23)
Fig. 8. Mode O stress intensity factor at a distance z = 24.38 mm as a function of the distance x from the notch tip
KO represents the natural extension to V-notches of the stress intensity factor for the crack case.
The stress component az, which has been omitted from the figures, respects the plane strain conditions, applied by means of contour nodal displacements, at least near the midplane of the model. On the other hand, it is negligible near the free surfaces where the mode O reaches its maximum intensity.
Figure 8 shows the variability of KO as a function of the distance from notch tip in the plane corresponding to the maximum intensity of the mode O. KO presents a significant value until a distance approximately equal to 1 mm.
Figure 9 shows the trend of KO through the plate thickness at a constant distance, x = 0.3 mm, from the notch tip. The intensity of KO varies through the thickness of the plate. It has zero value at the midplane of the plate where it changes sign (due to symmetry conditions) and has a maximum in an interior point prior to reaching the outer free surface of the plate where, theoretically, it should drop to zero. Surface stresses, obtained from the finite element analysis, are simple extrapolations since the 3D mesh used is not refined enough to provide shear stresses equal to zero on the free surfaces.
The stress intensity factor KI/II is plotted in Fig. 10 along the notch bisector line as a function of the transverse coor-
Fig. 7. Plot of KI/n along the crack bisector line in the middle plane of the plate, from the notch tip to the outer radius R. Displacements ux and uy applied only to the nodes of the lateral surface (R = 100 mm)
300
200
. 100
2a = 45° 8
■ R = 100 mm go
2h = 50 mm o c
' v = 0.49 Applied K|/n= ! O °°° o 1000 MPa-mm0747 ¿P C 000<fP o°° ooO°°°° 0 ° °
0d>-
0.0
0.1
0.2
z/(2h)
0.3
0.4
0.5
Fig. 9. Mode O stress intensity factor through the plate thickness at x = 0.3 mm
Fig. 10. Ki/ii through the plate thickness at different distances from the notch tip
Fig. 12. Mode O stress intensity factor through the plate thickness at x = 0.3 mm
dinate z for two radial distances from the V-notch tip (r = = x = 0.3 and 1.0 mm). This figure shows that there is a large central zone within the plate where K I/n 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 where the decrease of the txy stress component is particularly strong. In that limited zone, which has a size in the z-direction of about 5 % of the total plate thickness, the out-of-plane stress component tyz reaches its minimum or maximum values.
By increasing the notch opening angle from 45° to 60° the theoretical eigenvalue k changes from 0.253 to 0.257 whereas the mode O eigenvalue k o for the free-fixed configuration changes from 0.2857 to 0.3000. As expected from the theory, an increase of the opening angle 2a, leads to the decrease of the power of the singular behaviour. Both the in-plane stresses and the out-of-plane stresses remain singular and are characterized by a singularity stronger than that induced by a free-free notch configuration. It is worth noting that also the in-plane shear stress remains singular when opening the notch angle over the value 2 a = 102°, contrary to the free-free case.
Figure 11 summarizes the stress field obtained along the notch bisector line for the case of 2a = 60°. The figure
is almost coincident to that obtained for 2a = 45°. The plane where the intensity of mode O reaches its maximum value is at the coordinate z = 24.38 mm, very close to the free-surfaces of the plate. Figure 12 shows the trend of KO evaluated according Eq. (21) through the plate thickness at a distance x = 0.3 mm from the notch tip. Also in this case the intensity of KO is found to vary through the thickness of the plate. The stress intensity factor KI/II remains constant in a large central zone of the plate whereas a strong dependence on z-coordinate occurs in the vicinity of the free surface where the decrease of the intensity of Txy stress component is very strong. The figure is omitted for sake of brevity being the behaviour analogous to that just described for the case 2a = 45°. Also for the case 2a = 60° the notch stress intensity factor KI/II has been plotted in the middle plane of the plate to verify that the applied boundary conditions would correctly apply to the model.
While the case 2a = 90° does not present any novelty with respect to the previous two angles, the case corresponding to a notch opening angle equal to 135° presents some peculiarities. Figure 13 plots the in plane stress field along the bisector line and in the middle plane of the plate together with the shear stress component Tyz in the plane of maximum intensity (z = 24.38). The intensity of tyz is in
Fig. 11. Three-dimensional stress distributions in a plate with an opening angle 2 a = 60°
100000
10000
6 100
10
xax = 96.718x-°-6976 (z = 0)
o Gy = 313.16x-0-6565 (z = 0)
a Txy = 40.24x-°-6986 (z = 0)
o Tyz = 117.07x-° 6 (z = 24.38 mm)
’ 2a = 135°
! R = 100 mm
: 2h = 50 mm
I v = 0.49
7s _ II ' o : o - o "O Q) mm0-6616
0.001
0.01
0.1 x, mm
10
Fig. 13. Three-dimensional stress distributions in a plate with an opening angle 2a = 135°
Fig. 14. Mode O stress intensity factor through the plate thickness at x = 0.3 mm
Fig. 16. Shear stress tyz through the thickness for different opening angles from 2a = 45° to 135°
this case comparable to that of the in-plane stress component a x and then is stronger with respect to the previous cases. In parallel the behaviour of the mode O through the plate thickness confirms the previous trends, shown in Fig. 14. The main novelty is the variability of the stress intensity factor K I/n which increases its intensity approaching the free surfaces of the plate, contrary to the previously investigated cases (see Fig. 15).
Finally Figure 16 plots the shear stress component T yz through the thickness for the different notch angles considered in the present investigation. The stress component T yz is plotted at a distance x = 0.3 mm from the notch tip. It is evident that the notch opening angle does not influence the intensity of the out-of-plane stress contrary to the free-free notch configuration.
The main conclusion of this section is that the existence of the out-of-plane mode has been proofed for a V-notch under a free-fixed configuration.
By investigating different notch opening angle has been shown that the singularity of the new investigated mode matches that obtained by using the theoretical solution of a free-fixed notch under antiplane loading [39]. While the stress intensity factor KI/II is equal to the applied value on the midplane it varies along the plate thickness decreasing near the free surfaces for small opening angle (2a = 45 o, 60o and 90o) and increasing for larger values (2a = 1350).
o x = 0.3 mm
x x =1 mm <
> s ® 8 ® g ® 8 S 6 8s'
_ 2a = 135°
" R = 100 mm
. 2h = 50 mm
v = 0.49
" Applied K|/M = 1000 MPa ■ mm0 6616
0.0 0.1 0.2 0.3 0.4 0.5
z/(2h)
Fig. 15. Ki/n through the plate thickness at different distances from the notch tip
It has also shown that, contrary to the authors’ expectations, the intensity of the out-of-plane shear stress through the plate thickness is not influenced by the notch opening angle for the same applied value of KI/II.
5. Strain energy density in a control volume surrounding the notch tip
To compare the severity of the singular stress fields some assumptions are necessary to introduce, which should involve a material characteristic length or the microstructural support length. Recent papers suggest to use as a characteristic of failure initiation the strain energy density averaged over a characteristic control volume [15-22, 40-45]. This characteristic length varies from material to material and can range from 10-6 to 10-3 m. However some recent applications [46, 47] have demonstrated that the elastic energy evaluated in a control volume of radius R0 (as defined in [15]) remains a powerful tool also at lower scale levels. In [46], taking R0 as the diameter of Au atoms and a as the crack length, the elastic energy has been used to explain the different scenarios observed in propagation of two subcracks under similar loading conditions. A bi-crystal model showed that the elastic energy (measured over a volume of radius 0.3 nm) decreased with increasing the notch angle, which implies that cracks with small angles could propagate along the grain boundaries more easily [47].
A possible synthesis of the magnitude of the stress state through the plate thickness is carried out here by using the mean value of the strain energy density over a control volume embracing the notch tip. The control volume can be though of as a cylinder having a radius and a height equal to R0 (see Fig. 1). Recently the approach based on the strain energy density has been used to summarize a large bulk of experimental data from fatigue failures of seam and spot welded joints and from static tests under mode I and mode II loading [18, 40-41]. As discussed in [17, 42, 43] the strain energy density can be evaluated directly with a coarse mesh also in complex cases where an analytical formulation is far from easy. One of the most important advantages of the
2h = 50 mm □ 2a = 45°
- K|/M = 1000 MPa ■ mm1 -K • 2a = 60°
v = 0.49 o 2a = 90°
■ 2a = 135° /• o
■
• • • - _ D ° 0
• o • o • o » o o o o > ■ •o •o • o • o
0.0 0.1 0.2 0.3 0.4 0.5
z/(2h)
Fig. 17. Local strain energy density (SED) averaged over a cylindrical volume having radius R0 and height H, with H about equal to R0
strain energy density approach is that to provide a mean value which is substantially mesh independent. In fact, contrary to some stress parameters integrated in the local criteria (e.g. maximum principal stress, hydrostatic stress, deviatoric stress), which are mesh-dependent, the strain energy density averaged over a control volume is insensitive to the mesh refinement. As widely documented in [17, 42, 43] dealing with sharp V-notches, refined meshes are not necessary, because the mean value of the strain energy density on the control volume can be directly determined via the nodal displacements, without involving their derivatives. As soon as the average strain energy density is known, the notch stress intensity factors, quantifying the asymptotic stress distributions, can be calculated a posteriori on the basis of very simple expressions linking the local strain energy density and the relevant notch stress intensity factors. This holds true also for the stress concentration factors, at least when the local stress distributions ahead of the blunt notch are available for the plane problem. The extension of the strain energy density method to three-dimensional cases is also possible as well as its extension to notched geometries exhibiting small scale yielding [17].
Figure 17 plots the local strain energy density and makes it evident that the maximum value is close to the lateral surface where the maximum intensity of the mode O takes place. This behavior can be explained by analyzing the complex stress state through the plate thickness. The strain energy density is able to naturally capture and unite the features of the combined effects. The results summarised in the paper are general and can be directly translated to the mesomechanic approach proposed by Sih which models the microcrack tip as a free-fixed sharp V-notch as well as to the cases discussed in [1] dealing with in-plane solution and straight extended to the three-dimensional models.
Moreover it is strongly confirmed the fact that the notch opening angle has a negligible influence in the vicinity of the free surface and that the variability is slight also in the midplane. For the V-notched plates the importance of the out-of-plane mode has been investigated in [29] by increasing the plate size. Also for this geometry and boundary con-
ditions the importance of mode O has been strongly expected to be dependent on the main plate thickness.
6. Conclusions
Irregularities of the material microstructure tend to curl the crack tip being the clamped-free boundary conditions the more realistic and general representation of what occurs on the microscopic V-notch. As a result, mixed mode conditions are always present along the V-notch bisector line. Physically, the different orders of the stress singularities are related to the different constraints associated with the defect thought as a microscopic V-notch at the tip of the main crack.
Dealing with a three-dimensional plate weakened by a sharp notch under clamped-free boundary conditions the mixed mode is automatically generated by the edge constraints. The presence of mode II has been proofed to generate a singular mode O which has been neglected in all previous papers. Different values of the notch opening angle have been considered to show the variability of the intensity and singularity of the new detected mode. It has been found that the singularity of mode O matches that obtained by using the theoretical solution of a free-fixed notch under antiplane loading.
Dealing with three-dimensional model it has been highlighted that the influence of the notch opening angle on the intensity of the mode O is very low even for large values of 2a.
Finally the local energy has been used to capture the features of the combined effects through the plate thickness. Despite the complexity of the stress state, a simple scalar parameter, i.e. the strain energy density over a given control volume, makes easy the identification of the critical zones of the notched plate. It has been found that the most critical zone of the plate is close to the free surface and that the notch opening angle does not affect significantly the value of the strain energy density when the notch is modelled by using a clamped-free configuration.
The strain energy density can be applied at different scale levels keeping into account the boundary conditions that should be updated at every level. In fact the constraint conditions are sensitive to the environments and cannot be captured or simulated by using a model with the same conditions at nano-, micro- or macrolevels. The change of boundary conditions at different scale level permits to overcome a complex elastoplastic formulation and bypass the problem by maintaining a multiscale model in the field of elasticity. This is enormously advantageous and powerful.
References
1. TangX.S., Sih G.C. Weak and strong singularities reflecting multiscale damage: Micro-boundary conditions for free-free, fixed-fixed and free-fixed constraints // Theor. Appl. Fract. Mech. - 2005. - V. 43. -No. 1. - P. 5-62.
2. Sih G.C., Tang X.S. Scaling of volume energy density function reflecting damage by singularities at macro-, meso- and microscopic level // Theor. Appl. Fract. Mech. - 2005. - V. 43. - No. 2. - P. 211-231.
3. Neuber H. Kerbspannungslehre. - Berlin: Springer-Verlag, 1958. -225 p.
4. Sih G.C. Multiscaling in Molecular and Continuum Mechanics: Interaction of Time and Size from Macro to Nano. - Dordrecht: Springer,
2007. - 460 p.
5. Sih G.C. Crack tip mechanics based on progressive damage of arrow: Hierarchy of singularities and multiscale segments // Theor. Appl. Fract. Mech. - 2009. - V. 51. - No. 1. - P. 11-32.
6. Sih G.C. Ideomechanics of transitory and dissipative systems associated with length, velocity, mass and energy // Theor. Appl. Fract. Mech.- 2009. - V. 51. - No. 3. - P. 149-160.
7. Ne A&.N. Ia9iiaoaieea a9aeiiaaenoaey yiadaee e iannu a aenne-iadeaiuo nenoaiao // Oe£. ia9iiao. - 2010. - O. 13. - 1 5. - N. 27-
40.
8. Idiei A.I., Idiei A.A., Ndddddd A.I., 0ododid A.D. Yooaeou neaeeeiaa a nodoeoodii-oat^iaie naiiidaaie9a6ee ia eioadoaena «oiieay ieaiea - iiaefeea» // Oe9. ia9iiao. - 2007. - O. 10. -13. - C. 9-21.
9. Idiei A.I., lenoeidd O.O., Ididooeei A.I., Adoeeid I.P., ii+edd-eid P.E. laeeiaeiua aieiiaua yooaeou nieeoiiia edeae9m a iiaadoimomo neiyo iieeedenoaeeia auniei^enoiai ae^ieiey ide eioaineaiie ieanoe+aneie aaoidiaoee. I. Yeniadeiaio // Oe9. ia9iiao. - 2007. - O. 10. - 1 6. - C. 21-32.
10. Idiei N.A., Ayeid A.A., Addied A.A., 0deedid E.A., Pnneo N.A.E.
liiaiianooaaiia enneaaiaaiea noaaeeiinoe eieaee9iaaiiie ieanoe+aneie aaoidiaoee ide danoy^aiee iada9oia nieaaa
A16AO n iaada9aie aeonoeei-yienneiiiui e ii6eei-6aeaae9eii-
iui iaoiaaie // Oe9. ia9iiao. - 2009. - O. 12. - 1 6. - N. 63-72.
11. Sih G.C. Strain-energy-density factor applied to mixed mode crack problems // Int. J. Fract. - 1974. - V. 10. - No. 3. - P. 305-321.
12. Sih G. C., Ho J.W Sharp notch fracture strength characterized by critical energy density // Theor. Appl. Fract. Mech. - 1991. - V. 16. -No. 3. - P. 179-214.
13. GillemotL.F. Brittle Fracture of Welded Materials // Proc. 2nd Commonwealth Welding Conference. - London, 1965. - C. 7. - P. 353.
14. Gillemot L.F., Czoboly E., Havas I. Fracture mechanics applications of absorbed specific fracture energy: Notch and unnotched specimens // Theor. Appl. Fract. Mech. - 1985. - V. 4. - No. 1. - P. 39-45.
15. 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. - V. 112. - No. 3. - P. 275298.
16. 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. - V. 135. - No. 1-4. - P. 161-185.
17. Lazzarin P., Berto F Control volumes and strain energy density under small and large scale yielding due to tensile and torsion loading // Fatigue Fract. Eng. Mater. Struct. - 2008. - V. 31. - No. 1. - P. 95107.
18. Berto F., Lazzarin P. A review of the volume-based strain energy density approach applied to V-notches and welded structures // Theor. Appl. Fract. Mech. - 2009. - V. 52. - No. 3. - P. 183-194.
19. Lazzarin P., Berto F., Radaj D. Fatigue-relevant stress field parameters of welded lap joints: pointed slit tip compared with keyhole notch // Fatigue Fract. Eng. Mater. Struct. - 2009. - V. 32. - No. 9. - P. 713735.
20. Radaj D., Berto F., Lazzarin P. Local fatigue strength parameters for welded joints based on strain energy density with inclusion of small-size notches // Eng. Fract. Mech. - 2009. - V. 76. - No. 8. - P. 11091130.
21. Gomez F.J., Elices M., Berto F., Lazzarin P. Fracture of V-notched specimens under mixed mode (I plus II) loading in brittle materials // Int. J. Fract. - 2009. - V. 159. - No. 2. - P. 121-135.
22. Lazzarin P., Berto F., Elices M., Gomez 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. Eng. Mater. Struct. - 2009. - V 32. - No. 8. - P. 671-684.
23. Seweryn A., Molski K. Elastic stress singularities and corresponding generalized stress intensity factors for angular corners under various boundary conditions // Eng. Fract. Mech. - 1996. - V. 55. - No. 4. -P. 529-556.
24. Nakamura T, Parks D.M. Antisymmetrical 3D stress field near the crack front of a thin elastic plate // Int. J. Solids Struct. - 1989. -V. 25. - No. 12. - P. 1411-1426.
25. Jin Z.H., Batra R.C. A crack at the interface between a Kane-Mindlin plate and a rigid substrate // Eng. Fract. Mech. - 1997. - V. 57. -No. 4. - P. 343-354.
26. Pook L.P. Finite element analysis of corner point displacements and stress intensity factors for narrow notches in square sheets and plates // Fatigue Fract. Eng. Mater. Struct. - 2000. - V. 23. - No. 12. -P. 979-992.
27. Huang C.S. Corner stress singularities in a high-order plate theory // Comput. Struct. - 2004. - V. 82. - No. 20-21. - P. 1657-1666.
28. Kotousov A. Fracture in plates of finite thickness // Int. J. Solids Struct. - 2007. - V. 44. - No. 25-26. - P. 8259-8273.
29. Berto F., Lazzarin P., Kotousov A., Harding S. Out-of-plane singular stress fields in V-notched plates and welded lap joints induced by inplane shear load conditions // Fatigue Fract. Eng. Mater. Struct. -2011. - V. 34. - No. 4. - P. 291-304.
30. Kotousov A., Lazzarin P., Berto F, Harding S. Effect of the thickness on elastic deformation and quasi-brittle fracture of plate components // Eng. Fract. Mech. - 2010. - V. 77. - No. 11. - P. 1665-1681.
31. Harding S., Kotousov A., Lazzarin P., Berto F. Transverse singular effects in V-shaped notches stressed in mode II // Int. J. Fract. - 2010.-V. 164. - No. 1. - P. 1-14.
32. Berto F, Lazzarin P., Kotousov A. On higher order terms and out-ofplane singular mode // Mech. Mater. - 2011. - V. 43. - No. 6. - P. 332341.
33. Berto F., Lazzarin P., Kotousov A. On the presence of the out-ofplane singular mode induced by plane loading with K(II) = K(I) = 0
// Int. J. Fract. - 2011. - V. 167. - No. 1. - P. 119-126.
34. Fadle J. Die Selbstspannungs-Eigenwertfunktionen der quadratischen Scheibe // Ing.-Arch. - 1941. - V. 11. - No. 2. - P. 125-149.
35. Williams M.L. Stress singularities resulting from various boundary conditions in angular corners on plates in tension // J. Appl. Mech. -1952. - V. 19. - No. 4. - P. 526-528.
36. Coker E.G., Filon L.G.N. A Treatise on Photoelasticity. - Cambridge: Cambridge University Press, 1931. - 720 p.
37. HartranftR.J., Sih G.C. The use of eigenfunction expansions in gen-
eral solution of three-dimensional crack problems // J. Math. Mech. -1969. - V. 19. - No. 2. - P. 123-138.
38. Hartranft R.J., Sih G.C. Alternating Method Applied to Edge and
Surface Crack Problems // Method of Analysis and Solutions to Crack Problems, 1 / Ed. by G.C. Sih. - Netherlands: Noordhoff Int. Publishing, 1973.
39. Sih G.C., Tang X.S. Equilibrium mechanics model of multiscaling by segmentation: Asymptotic solution for macro-meso-micro damage in anti-plane shear deformation // Theor. Appl. Fract. Mech. - 2005. -V. 44. - No. 1. - P. 1-15.
40. 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. - V. 148. -No. 4. - P. 415-433.
41. Gomez F.J., Elices M., Berto F, Lazzarin P. Local strain energy density to assess the static failure of U-shaped notches in plates under mixed mode loading // Int. J. Fract. - 2007. - V. 145. - No. 1. - P. 2945.
42. Lazzarin P., Berto F., Gomez F.J., Zappalorto M. Some advantages derived from the use of the strain energy density over a control volume in fatigue strength assessments of welded joints // Int. J. Fatigue. -
2008. - V. 30. - No. 8. - P. 1345-1357.
43. Lazzarin P., Berto F., Zappalorto M. Rapid calculations of notch stress intensity factors based on averaged strain energy density from
coarse meshes: Theoretical bases and applications // Int. J. Fatigue. -2010. - V. 32. - No. 10. - P. 1559-1567.
44. 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. - V. 33. - No. 8. - P. 10551065.
45. Berto F., Lazzarin P., Yates J. Multiaxial fatigue behaviour of quasisharp V-notched specimens made of 39NiCrMo3 Steel: A non-con-ventional application of the local energy // Fatigue Fract. Eng. Mater. Struct. - 2011. - V. 34. - No. 11. - P. 921-943.
46. Wang H., Nie A., Liu J., Wang P., Yang W., Chen B., Liu H., Fu M. In situ TEM study on crack propagation in nanoscale Au thin films // Scr. Mater. - 2011. - V. 65. - No. 5. - P. 377.
47. Liu J., Nie A., Dong C., Wang P., Wang H., Fu M., Yang W Grain boundary structure dependent fracture in nanocrystalline Au films // Mater. Lett. - 2011. - V. 65. - No. 17-18. - P. 2769-2771.
nocTynnna b pe^aKunro 18.01.2012 r.
CeedeHUH 06 aemopax
Berto Filippo, Prof., Assistant Professor, University of Padova, Italy, berto@gest.unipd.it Lazzarin Paolo, Prof., Full Professor, University of Padova, Italy, plazzarin@gest.unipd.it Marangon Christian, Dr., PhD student, University of Padova, Italy, marangon@gest.unipd.it
