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Stress singularity transition of generic wedges due to nanoelement layers
Engineering Fracture Mechanics 78 (2011) 2789–2799
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Stress singularity transition of generic wedges due to nanoelement layers Taisuke Sueda a, Insu Jeon b,⇑, Takashi Sumigawa a, Takayuki Kitamura a a b
Department of Mechanical Engineering and Science, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan School of Mechanical Systems Engineering, Chonnam National University, 300 Yongbong-dong, Buk-gu, Gwangju 500-757, Republic of Korea
a r t i c l e
i n f o
Article history: Received 6 October 2010 Received in revised form 29 July 2011 Accepted 3 August 2011
Keywords: Nanoelement layer Stress singularity transition Generic wedges Bi-material Discreteness
a b s t r a c t The effect of a nanoelement interfacial layer on the stress singularity transitions of generic wedges is analyzed using the finite element method. The singularity transitions corresponding to changes in the vertical stiffness and the lateral stiffness of the nanoelement are examined. In general, high vertical stiffness and high lateral stiffness yield singularities close to those of wedges without nanoelements while low vertical stiffness and low lateral stiffness cause the elimination of the stress concentration. Under high vertical stiffness and low lateral stiffness the singularity is dependent on wedge shapes and the dependence is analyzed in terms of the constraint condition. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction A nanoelement layer, which comprises of a great number of nano-scale helical springs, bars, or zigzags, can be fabricated by the Dynamic Oblique Deposition (DOD) method that exploits the atomic self-shadowing effect during physical vapor deposition with a highly oblique angle [1–3]. The size and geometric shape of the nanoelements can be controlled for engineering applications by the number of turns of the substrate and the incident angle of the vapor flux to the substrate during deposition [2,3]. In particular, Hirakata et al. [4] show that a helical nanoelement layer with various heights and radii can be freely fabricated by the DOD method and the lateral and vertical stiffness of the nanoelement can be controlled by its geometric shape. This nanoelement layer has high potential for use as a low dielectric layer in microelectronic devices, microelectronic packaging, and nano or micro electromechanical (NEMS/MEMS) systems [5–9]. Recently, a unique role of the nanoelement interfacial layer was found by Sumigawa et al. [10,11], which is the elimination of the stress concentration around an interfacial crack tip or a right-angled interfacial notch root. In fact, such stress concentration may be a cause of failure of electronic devices and systems because these are fabricated by the deposition of many thin film layers of different materials and include unavoidably numerous generic wedges such as interfacial cracks and notches. Therefore, a clear understanding of the nanoelement interfacial layer as a discrete material for eliminating the conditions for stress concentration is very helpful for fabricating highly reliable electronic devices and systems using the layer. Moreover, such inquiry helps to obtain the background of fracture mechanics for bridging continuous and discrete materials and extends our knowledge about fracture mechanics for continuous materials to that for discrete materials. In this study, using the finite element method, we analyze the effect of the nanoelement interfacial layer on the transition of the stress singularity of generic wedges, which governs the order of the stress concentration. For the numerical analysis,
⇑ Corresponding author. Tel.: +82 62 530 1688; fax: +82 62 530 1689. E-mail address: [email protected] (I. Jeon). 0013-7944/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2011.08.004
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Nomenclature kv kl v k k l E
r0 rij x, y K r, h k fij(h) r W0
ryy
r yy Py dn tn k k0
the vertical stiffness of the beam element the lateral stiffness of the beam element the non-dimensional vertical stiffness the non-dimensional lateral stiffness elastic modulus the applied uniform normal stress the stress tensor, i, j = x, y the cartesian coordinates centered at the wedge tip the stress intensity the polar coordinates centered at the wedge tip the stress singularity the function of solution of the stress field the non-dimensional distance from a wedge tip the width of wedge specimens the y-direction stress along the interface the normalized y-direction stress along the interface the force of each nanoelement the distance between nanoelements the unit thickness of the nanoelement layer the normalized singularity the stress singularity of Direct-connected model for each generic wedge
an actual helical nanoelement is modeled using a 2D beam element with vertical and lateral stiffness and all the other parts of the generic wedge specimens are modeled with 2D plane strain solid elements. The suitability of the finite element modeling is confirmed through an experimental compression test and numerical analysis for the test. Then, numerical analyses are performed using modeled specimens with two representative generic wedges, viz., an interfacial crack and a right-angled interfacial notch including the nanoelement interfacial layer. The stress singularity transition corresponding to changes in the vertical and lateral stiffness of the nanoelement, as well as the difference in material between the upper and lower parts of the specimens, is examined mainly by using the numerical results.
2. Modeling for the nanoelement layer The helical nanoelement layer with the turns of 5.5 shown in Fig. 1a and b are fabricated using Ta2O5 on a Si substrate by the DOD method under an incident vapor flux angle of 84°. The height of the nanoelement obtained from this fabrication process is 900 nm and the helical coil radius and wire diameter are 50 nm and 40 nm, respectively. The areal number density of the nanoelements is 110 per lm2. On the nanoelement layer, a solid Ta2O5 film of 575 nm is deposited as the upper layer. To confirm the suitability of numerical analysis method for an arbitrary model that includes the nanoelement layer, we have performed an experimental compression test using a specimen introduced by Sumigawa et al. [11] (see Fig. 2a) and a numerical analysis using the finite element method (see Fig. 2b). The test specimen is fabricated by attaching a stainless steel cantilever on the top surface of the solid Ta2O5 in Fig. 1a using an epoxy adhesive (see Fig. 2c). A vertical compression load with a displacement rate of 50 lm/s is applied at a point on the cantilever surface using a micro material test machine, MMT100N, with a low capacity load cell of 100N (see Fig. 2d). For the numerical analysis, each helical nanoelement is modeled using 2D beam elements of 900 nm height that has a vertical stiffness of kv and lateral stiffness of kl, and the beam diameter and the distance between beams are set as 40 nm and 55 nm, respectively (see Fig. 1b). All the other parts of the specimen are modeled as 2D plane strain solid elements. A 2D beam element with two nodes is used for the nanoelement and plane strain elements with eight nodes are used for all the other parts. The vertical and lateral stiffness of each nanoelement are 2.05 N/m and 6.89 N/m, respectively [4]. Note that these stiffness are converted from the experimentally measured values by taking into account the difference in the heights of 690 nm while applying the same helical coil radius and wire diameter of this research (‘Material A’ in Hirakata et al. [4]). The elastic modulus and Poisson’s ratio of all the other parts are listed in Table 1. The software package, ABAQUS of Dassault Systems, is used for the numerical analysis. Fig. 3 shows the load–displacement curves obtained from both the experiment and numerical analysis. Although the numerical analysis uses 2D element models, both results show excellent agreement in terms of the overall load–displacement behavior. From this figure, the suitability of the numerical analysis method in this research using 2D element modeling for the nanoelement layer and for each part of the actual specimen is confirmed.
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575 nm
900 nm
525 μm
300 nm
(a)
40 nm
575 nm
900 nm
55 nm
(b) Fig. 1. (a) The helical nanoelement layer and (b) its 2D modeling.
Table 1 Material properties for numerical calculation. Materials
Elastic modulus (GPa)
Poisson’s ratio
Stainless steel Epoxy Ta2O5 Si
200 1.2 117 130
0.3 0.3 0.23 0.29
3. Numerical analysis 3.1. Models for specimens with generic wedges We prepare the modeled test specimens with two different generic wedges, viz., an interfacial crack and a right-angled interfacial notch, which include the nanoelement interfacial layer between the upper and lower parts (see Fig. 4a and c) for examining the effect of the nanoelement interfacial layer on the stress singularity transition of generic wedges. In order to investigate the combination effect of dissimilar materials, two different materials are selected for the lower part, such as a rigid solid, and the same material, Ta2O5, for the upper part, which yields either the maximum or minimum difference in the material properties. Also, we prepare the modeled specimen without the nanoelement interfacial layer for comparing the analysis results. For convenience in distinguishing all the cases, each specimen is named following the type of wedge, i.e., ‘Crack’ and ‘Notch’, and material combination, i.e., ‘T/R’ and ‘T/T’. For example, ‘Crack-T/R’ is a specimen with an interfacial crack in a bi-material of Ta2O5 and a rigid solid with the nanoelement interfacial layer. ‘Notch-T/T’ is a specimen with a right-angled interfacial notch in two components of Ta2O5 that are connected by the nanoelement interfacial layer. Table 2 shows the names of all specimen types. The specimen that does not include the nanoelement interfacial layer is named as ‘Direct-connected model’.
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T. Sueda et al. / Engineering Fracture Mechanics 78 (2011) 2789–2799 Table 2 Specimens simulated. Specimen name
Wedge type
Upper material
Lower material
Crack-T/R Notch-T/R Crack-T/T Notch-T/T
Crack Notch Crack Notch
Ta2O5 Ta2O5 Ta2O5 Ta2O5
Rigid solid Rigid solid Ta2O5 Ta2O5
b Load P t L
l
Stainless steel Epoxy Ta2O5 Ta2O5 nanoelement Si
(a)
(c)
Load P Stainless steel Si wafer y z
x
Stainless steel Epoxy Ta2O5 solid film
Ta2O5 nanoelement Si wafer
(d)
(b) Fig. 2. (a) The structure of a compression test specimen, (b) its finite element model, (c) an actually fabricated specimen, and (d) the experimental compression test.
Fig. 4b and d shows the finite element models for the specimens. Although the results are presented in a dimensionless form in the following section, the actual calculation is carried out under Wo = 100 lm. The 2D plane strain elements with eight nodes are used for the upper and lower parts of the specimens. Each nanoelement is modeled using a 2D beam element with two nodes that is a suitable model for the actual helical nanoelement, as confirmed in Section 2. The height of the beam is 500 nm, and the beam diameter and distance between beams are selected as 50 nm and 50 nm, respectively. This is a model of a helical nanoelement named as ‘Material B’ that is fabricated by Hirakata et al. [4]. 3.2. Material properties and boundary condition In this simulation, Ta2O5 is used for the upper part though the results are extended to other materials though normalization. For the lower part, a rigid solid or Ta2O5 is selected. Here, all the materials are assumed to be linear elastic and isotropic, and the elastic constants of Ta2O5 are shown in Table 1.
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2
Load P, N
1.5
1
0.5
0
Experiment FEM
0.005
0.01
0.015
0.02
Displacement uy, mm Fig. 3. Load–displacement curves obtained from both the experimental test and numerical analysis.
3W0 3
σ0
W0 y
Crack
x
(a)
(b)
σ0
W0
(c)
3W0
y x
(d)
Fig. 4. Two representative generic wedge models and their finite element models: (a) and (b) are for an interfacial crack, while (c) and (d) are for a rightangled interfacial notch.
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Stress singularity
Stress singularity 101
0.486
Crack-T/R
0.453
Non-dimensional normal stress along the interface, σyy
Non-dimensional normal stress along the interface, σyy
101 1 1 0.276 1
100
0.012 k v = 8.8 × 10 -7 , k l = 1.4 × 10 -7 k v = 8.8 × 10 -5 , k l = 1.4 × 10 -5 k v = 8.8 × 10 -3 , k l = 1.4 × 10-3
1 0.261
1
100
0.011 kv
10-3
10-2
10-1
8.8 × 10 -7 , k l = 1.4 10 -7
k v = 8.8 × 10 -5 , k l = 1.4 10 -5 k v = 8.8 × 10 -3 , k l = 1.4 × 10 -3
Without nanoelemnt (λ= 0.5)
10-1 -4 10
Crack-T/T 1
Without nanoelement (λ= 0.5)
100
10-1 -4 10
10-3
10-2
100
Non-dimensional distance from crack tip, r
Non-dimensional distance from crack tip, r
(a)
(c)
Stress singularity
Stress singularity
101
1
10 Notch-T/R
Notch-T/T
Non-dimensional normal stress along the interface, σyy
Non-dimensional normal stress along the interface, σyy
10-1
0.262 1 1 0.166
0
10
1 0.013 k v = 8.8 × 10-7 , kkll = 1.4 × 10-7 k v = 8.8 × 10-5 , k l = 1.4 × 10-5 k v = 8.8 × 10-3 , k l = 1.4 × 10 -3
Without nanoelement (λ= 0.274)
10-1 -4 10
-3
10
-2
10
-1
10
0.402 1 1 0.230 0
10
1 0.008 kv
8.8 × 10 -7 , k l = 1.4 × 10 -7
k v = 8.8 × 10 -5 , k l = 1.4 × 10 -5 k v = 8.8 × 10 -3 , k l = 1.4 × 10 -3
Without nanoelement (λ= 0.461) 0
10
10-1 -4 10
-3
10
10-2
10-1
100
Non-dimensional distance from notch tip, r
Non-dimensional distance from notch tip, r
(b)
(d)
Fig. 5. Stress distribution with stress singularities near the tip of generic wedges: (a) Crack-T/R, (b) Notch-T/R, (c) Crack-T/T, and (d) Notch-T/T.
At first, the vertical stiffness, kv, and lateral stiffness, kl, of the beam element are selected as 10.3 N/m and 1.65 N/m, respecv ¼ kv ¼ 8:8 107 and k ¼ kl ¼ 1:4 107 ; tively, which are measured by Hirakata et al. [4]. This means the magnitude of k l pffiffiffiffiffiffiEW o pffiffiffiffiffiffi EW o O . Then, we extend the magnitudes to 10 times, 10 times, 10 10 times, 100 times, here, E is the elastic modulus of Ta 2 5 pffiffiffiffiffiffi pffiffiffiffiffiffi v and k for examining the effect on 100 10 times, 1000 times, 1000 10 times and 10,000 times, respectively, of each k l v and k pairs, only 45 cases, which yield similar behavthe stress singular field near the tip of the crack or notch. Of all the k l iors to vertical springs under a uniaxial tension state, are used for the numerical calculations. For the boundary condition, a uniform normal stress of r0 = 10 MPa is applied to the upper edge of the specimen while the y-direction displacement at the bottom edge of the specimen is constrained as shown in Fig. 4a–d. For all the numerical analyses, the software package, ABAQUS is used. 4. Results and discussion 4.1. Stress singularity From linear elastic fracture mechanics, it is well known that the stress field near the tip of generic wedges often has the following form:
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10-7 1.0
1.0
Crack-T/R
C 0.5
0.5
0
C’
0.5
0.5
0
B’
-6 -7
10
-7
10
al sio n 10 men ess, k v i d Non l st iffn ica vert 10-5
-6
10-7
10-4
10-2
1.0
Crack-T/T
l 10 na -5 10 sio k l -4 en ess, 10 -3 10 dim fn n- st if No eral lat
B
-6
l 10 na -5 10 sio k l -4 en ess, 10 -3 10 dim fn n- st if No eral lat
0 10-3
-4 10-5 10 10
10-6
10-7
Nond latera imensio na l l st iff ness, 10-2 10-3 kl 10-4 -5 10 10-6 10-7 A 1.0
Notch-T/R
0.5
0.5
0
10-2
10-5
-4 10-5 10
Nond latera imensio na l l st iffn -2 ess, k 10-3 10 10-3 10-4 l 10-5 -6 10 10-7 A’ 1.0
Notch-T/T
C’
0.5
0.5
0
B’ -7
(b)
10-6
10
-7
10
al sio n 10 men ess, k v i d Non l st iffn ica vert -6
10-7
10-4
10-3
10-7 1.0
-6 l 10 na -5 sio k l 10 -4 en ess, 10 -3 dim fn 10 n- st if No eral lat
0
-6
l 10 -5 na 10 sio k l -4 10 en ess, -3 10 dim fn n- st if No eral lat
C
B
10-5
o nal Non-dimensi s, k v es fn if vert ical st
Normalized r, λ singular orde
-4 10-5 10 10
10-6
0 10-2
(c)
Normalized singular order, λ
Normalized r, λ singular orde
10-7 1.0
10
10-4
10-3
al sio n men ess, k v i d Non l st iffn ica vert
-6
(a) o nal Non-dimensi s, k v es fn vert ical st if -3
Normalized singular order, λ
10
10
0 10-2
al sio n men ess, k v i d Non l st iffn ica vert
-6
10-7
10-4
10-3
Normalized singular order, λ
10-5 110
10-6
Nond latera imensio na l l st iff -2 ness 10 10-3 k 10-4 -5 l 10 10-6 10-7 A’
o nal Non-dimensi s, k v es fn if st al vert ic -3
Normalized singular order, λ
Normalized r, λ ng si ular orde
10-7 1.0
Nond latera imensio na l l st iffn ess, k 10-2 10-3 10-4 -5 l 10 10-6 10-7 A
Normalized r, λ singular orde
o nal Non-dimensi s, k v es fn vert ical st if -4 -3
10-5
(d)
Fig. 6. The stress singularity transitions of (a) Crack-T/R, (b) Notch-T/R, (c) Crack-T/T, and (d) Notch-T/T.
rij ¼ K rk fij ðhÞ ði; j ¼ x; yÞ;
ð1Þ
rij ¼ K rk ði; j ¼ x; yÞ on h ¼ 0
ð2Þ
or
where r and h are the polar coordinates centered at the wedge tip and k > 0 is the order of the stress singularity. The parameter, K, represents the stress intensity. The detailed solutions for fij(h) depending on the geometry of the generic wedges are given in Rice et al. [12], Im and Kim [13] and Jeon and Im [14]. Moreover, generic wedges on the bi-material interface sometimes bring about several complexities regarding the stress field with double singularities or an oscillatory singularity. The double singularities, which consist of a strong and a weak singularity, give combined singular effects on the stress field. The complex k brings about not only the steep gradient but also oscillation in the stress field because of its imaginary part, viz., an oscillatory singularity. In this paper, our interest is confined in the effect of nanoelement layer on the steep stress concentration near the generic wedge as a first inquiry. Thus, for simplification, we set the origin of the x, y coordinate at the wedge tip in nanoelement layer and examine the stress field near the tip of 5 103 < r < 5 102 (see Fig. 4b and d). The detailed analysis of the exact stress filed is out of scope of this paper and it remains for future investigation.
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Fig. 7. (a) and (d) correspond to Crack-T/R and Notch-T/R at the point C; (b) and (e) represent simplified boundary conditions of (a) and (b); (c) and (f) are their transformed models due to symmetry.
r
yy ¼ ryy . The solid lines show the ones in Fig. 5a–d shows the log–log plots of the normalized stress along the interface, r 0 the direct mount material (Substrate)/(Ta2O5 film) without the nanoelement layer and indicate clearly the existence of
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Fig. 8. (a) An acute-angled notch specimen, (b) its transformed model, (c) an obtuse-angled notch specimen, and (d) its transformed model.
singular field presented by Eq. (1). The others are obtained from numerical analysis with the nanoelement layer of represenv and k , and the normalized distance from the wedge tip, r ¼ r . Here, the stress, r , is the interfacial stress tative pairs of k l yy W0 Py on the boundary that is defined as dn t , where Py is the force of each nanoelement, dn ¼ 50 nm is the distance between n nanoelements and tn ¼ 1 nm is the unit thickness of the nanoelement layer. It should be noted that the stress distributions along upper and lower boundaries are the same because Py is the force along the element axis. v ¼ 8:8 103 , k ¼ 1:4 103 ) show almost straight relation, the triangles While the symbol of squares (k l 5 5 (kv ¼ 8:8 10 , kl ¼ 1:4 10 ) suggest the transition from stress singularity to non-singularity or the oscillated singularity where the field depart from the straight line in the region very near the wedge. As a measure representing the steepness of v stress field near the wedge, the slope of each curve, k, in the region of 5 103 < r < 5 102 is calculated for each case of k and kl and the fundamental behavior is discussed in the following sections. v ¼ 8:8 107 , k ¼ 1:4 107 (circles), the nanoelement layer almost eliminate the stress concentration In the case of k l near the wedge. For the real material (Si substrate)/(Ta2O5 nano-spring)/(Ta2O5 film) shown in Fig. 2b, the FEM analysis predicts no stress singularity near the corner along the interface as well. The experiments using several specimens with different sizes were carried out in the previous study [11], and the crack initiation behavior was carefully observed. It clarified that the analyzed stress without the singularity agreed well for all cases. This verifies that the nano-spring layer actually eliminates the stress concentration near the corner and proves the validity of the FEM analysis for a real material. 4.2. Transition of the stress singularity v and k , where k0 is the stress singularity of Direct-con ¼ k , is calculated in 45 cases of k The normalized singularity, k l k0 nected model for each generic wedge, i.e., k0 ¼ 0:5 for the crack and k0 ¼ 0:274 for the right-angled notch, respectively. v and k . Fig. 6a and b shows the normalized singularity transitions of Crack-T/R and Notch-T/R depending on k l k is nearly v and high k ). The stiff nanoelements make strong constraint the displacements at the upper and unity at the point A (high k l lower ends. Though the interfacial layer is discrete, densely disposed nanoelements give adhesion condition close to that of Direct-connected model. Thus, they keep strong singularity. v and low k ), the stress singularity is close to zero. Because the nanoelement allows large displaceAt the point B (low k l ment between the upper and lower part, the layer acts like a gel phase. Thus, the stress on the interface becomes uniform and stress concentration is eliminated. Since the stress singularities of the point A and point B do not depend on the wedge types, these can be general characteristics in various generic wedges. v and low k ), there is a clear difference in the singularity transition between Crack-T/R and Notch-T/ At the point C (high k l R. This means that the stress distribution is highly dependent on the wedge shape. In order to understand the difference, we examine the boundary condition of the upper part of the specimen along the interface. Fig. 7a shows a simplified model v and low k , the bottom boundary of the upper material around the nanoelement interfacial layer of Crack-T/R. Under high k l is approximated as the condition shown in Fig. 7b where the displacement of y-direction is perfectly restricted. Because the boundary condition implies a symmetric condition, this (Fig. 7b) is the same as the homogeneous double edge-cracked
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Fig. 9. The stress singularity transitions caused by differences in material between the upper and lower parts of: (a) crack specimens and (b) notch specimens.
specimen illustrated in Fig. 7c. In fact, the stress field around the crack tip shows r0.5 distribution. Thus, the singularity of Crack-T/R at the point C is close to that of the homogeneous crack problem, viz., k ¼ kk0 1. The case of Notch-T/R at point C can be examined using a similar approach to that of Crack-T/R. The model (Fig. 7d) approximately has the boundary condition shown in Fig. 7e, which is the same as the one illustrated in Fig. 7f due to symmetry. Because the homogenous plate (Fig. 7f) does not have singular stress fields, the stress singularity of Notch-T/R disappears at the point C. Based on the above considerations, it is possible to estimate the singularities of various wedges at point C. For example, a bi-material with an acute-angled notch, as shown in Fig. 8a, can be regarded as the model in Fig. 8b, which clearly possesses the stress singularity. On the other hand, an obtuse-angled notch specimen shown in Fig. 8c, which can be regarded as the model illustrated in Fig. 8d, does not have the singularity. 4.3. Effect of material combinations v and k . The change in Fig. 6c shows the normalized singularity transitions of Crack-T/T corresponding to changes in k l singularity is similar to that of Crack-T/R. Particularly, at point C0 , the normalized singularity, k, is almost ‘1’ although the v is the dominant parameter lower part is of a different material from that of Crack-T/R. From Fig. 6a and c, it is found that k for controlling the stress concentration near the crack tip.
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of 1.4 107. To see more clearly the dependence of the sinFig. 9a shows the changes in singularity under a constant k l gularity transition on the combination of materials, the result for another computed set for a bi-material (upper: Ta2O5; lower: Young’s modulus of 250 GPa; this denoted as the –T/T0 specimen) is shown in the figure as well. Although there are large differences in the material properties, all the singularity transitions are in good agreement. This shows that the singular orv , regardless of the material combination. der, k, near the crack tip is solely governed by k Although the trends in the singularity transitions are similar, there is a difference at the point C0 in Fig. 6d (Notch-T/T) compared with that at the point C in Fig. 6b (Notch-T/R). In the case of large differences in material properties between strongly affects the stress concentration of the notch specimens. Fig. 9b shows the singularity the upper and lower parts, k l v of 8.8 103. Another computed set is also included in the figure. The transitions of the notch specimens under a constant k case of the largest difference in material properties between the upper and lower parts (Notch-T/R) shows a lower bound of the singularity transition and the case where the material properties are the same (Notch-T/T) shows an upper bound thereof. Therefore, the singularity transition of Notch-T/T0 lies between those of Notch-T/R and Notch-T/T. So far, we have discussed about the stress singularity transition of a wedge problem using the nanoelement interfacial layer. Actually, as a criterion for the fracture of a wedge specimen, the stress intensity K needs to be defined for each corresponding singularity. The stress intensity is not discussed here because it highly depends on the types of applied loading and various specimen geometries. As the stress intensity should be an important factor for engineering applications, it needs to be studied as a future study in design of real micro-components. 5. Conclusions The effect of the nanoelement interfacial layer on the transition of the stress singularity near a generic wedge tip is analyzed by using the finite element method. The suitability of the modeling is confirmed by an experimental compression test and numerical analysis for the test. Then, modeled specimens with two representative generic wedges, viz., an interfacial crack and a right-angled interfacial notch including the nanoelement interfacial layer, are selected and analyzed in relation to changes in the vertical and lateral stiffness of the nanoelement. v and high k , the singularities are close to those of Direct-connected model in general, while the sinIn the case of high k l v and low k . Under a high value of k v and a low value of k , the crack problems yield gularities are eliminated under low k l l singularities that are close to those of Direct-connected model and the right-angled notch problems result in lower singularities than those of the model. v causes decrease in the stress concentration regardless of k as well as the Particularly, in the cracked body, a lower k l material properties. For a notched body, the kl value can be the dominant parameter for decreasing the stress concentration. Acknowledgments This paper was supported in part by a Grant-in-Aid for Scientific Research (S) (No. 21226005), of the Japan Society of the Promotion of Science, and by a Grant-in-Aid for Young Scientists (A) (No. 21686013) of the Ministry of Education, Culture, Sports, Science and Technology, Japan. Moreover, this work was supported in part by Nuclear R&D Program through the National Research Foundation of Korea funded by the Ministry of Education, Science, and Technology (2009-0093620). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Robbie K, Brett MJ, Lakhtakia A. First thin film realization of a helicoidal bianisotropic medium. J Vac Sci Technol 1995;A13:2991–3. Robbie K, Brett MJ. Sculptured thin films and glancing angle deposition: growth mechanics and applications. J Vac Sci Technol 1997;A15:1460–5. Suzuki M, Taga Y. Integrated sculptured thin films. Jpn J Appl Phys 2001;40:L358–9. Hirakata H, Matsumoto S, Takemura M, Suzuki M, Kitamura T. Anisotropic deformation of thin films comprised of helical nanosprings. Int J Solids Struct 2007;44:4030–8. Nitta SV, Pisupatti V, Jain A, Wayner PC, Gill WN, Plawsky JL. Surface modified spin-on xerogel films as interlayer dielectrics. J Vac Sci Technol 1999;B17:205–12. Doshi DA, Huesing NK, Lu M, Fan H, Lu Y, Simmons-Potter K, et al. Optically defined multifunctional patterning of photosensitive thin-film silica mesophases. Science 2000;290:107–11. Fan H, Bentley HR, Kathan KR, Clem P, Lu Y, Brinker CJ. Self-assembled aerogel-like low dielectric constant films. J Non-Cryst Solids 2001;285:79–83. Plawsky JL, Achanta R, Cho W, Rodriguez O, Saxena R, Gill WN. Mechanical and transport properties of low-k dielectrics. In: Mikhail RB, Martin LG, Karen M, editors. Dielectric films for advanced microelectronics. London: Wiley and Sons; 2007. p. 137–90. Plawsky JL, Jain A, Rogojevic S, Gill WN. Nanoporous dielectric films: fundamental property relations and microelectronics applications. In: Murarka SP, Eizenberg M, Sinha AK, editors. Interlayer dielectrics for semiconductor technologies. London: Elsevier Academic Press; 2003. p. 7–36. Sumigawa T, Hirakata H, Takemura M, Matsumoto S, Suzuki M, Kitamura T. Disappearance of stress singularity at interface edge due to nanostructured thin film. Engng Fract Mech 2008;75:3073–83. Sumigawa T, Sueda T, Futamura Y, Suzuki M, Kitamura T. Effect of interface layer consisting of nanosprings on stress field near interface edge. Engng Fract Mech 2009;76:1336–44. Rice JR, Suo Z, Wang JS. Mechanics and thermodynamics of brittle interfacial failure in bimaterial systems. In: Rühle M, Evans AG, Ashby MF, Hirth JP, editors. Metal ceramic interfaces. New York: Pergamon Press; 1990. p. 269–94. Im S, Kim KS. An application of two-state M-integral for computing the intensity of the singular near-tip field for a generic wedge. J Mech Phys Solids 2000;48:129–51. Jeon I, Im S. The role of higher order eigenfields in elastic–plastic cracks. J Mech Phys Solids 2001;49:2789–818.
Contents lists available at SciVerse ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Stress singularity transition of generic wedges due to nanoelement layers Taisuke Sueda a, Insu Jeon b,⇑, Takashi Sumigawa a, Takayuki Kitamura a a b
Department of Mechanical Engineering and Science, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan School of Mechanical Systems Engineering, Chonnam National University, 300 Yongbong-dong, Buk-gu, Gwangju 500-757, Republic of Korea
a r t i c l e
i n f o
Article history: Received 6 October 2010 Received in revised form 29 July 2011 Accepted 3 August 2011
Keywords: Nanoelement layer Stress singularity transition Generic wedges Bi-material Discreteness
a b s t r a c t The effect of a nanoelement interfacial layer on the stress singularity transitions of generic wedges is analyzed using the finite element method. The singularity transitions corresponding to changes in the vertical stiffness and the lateral stiffness of the nanoelement are examined. In general, high vertical stiffness and high lateral stiffness yield singularities close to those of wedges without nanoelements while low vertical stiffness and low lateral stiffness cause the elimination of the stress concentration. Under high vertical stiffness and low lateral stiffness the singularity is dependent on wedge shapes and the dependence is analyzed in terms of the constraint condition. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction A nanoelement layer, which comprises of a great number of nano-scale helical springs, bars, or zigzags, can be fabricated by the Dynamic Oblique Deposition (DOD) method that exploits the atomic self-shadowing effect during physical vapor deposition with a highly oblique angle [1–3]. The size and geometric shape of the nanoelements can be controlled for engineering applications by the number of turns of the substrate and the incident angle of the vapor flux to the substrate during deposition [2,3]. In particular, Hirakata et al. [4] show that a helical nanoelement layer with various heights and radii can be freely fabricated by the DOD method and the lateral and vertical stiffness of the nanoelement can be controlled by its geometric shape. This nanoelement layer has high potential for use as a low dielectric layer in microelectronic devices, microelectronic packaging, and nano or micro electromechanical (NEMS/MEMS) systems [5–9]. Recently, a unique role of the nanoelement interfacial layer was found by Sumigawa et al. [10,11], which is the elimination of the stress concentration around an interfacial crack tip or a right-angled interfacial notch root. In fact, such stress concentration may be a cause of failure of electronic devices and systems because these are fabricated by the deposition of many thin film layers of different materials and include unavoidably numerous generic wedges such as interfacial cracks and notches. Therefore, a clear understanding of the nanoelement interfacial layer as a discrete material for eliminating the conditions for stress concentration is very helpful for fabricating highly reliable electronic devices and systems using the layer. Moreover, such inquiry helps to obtain the background of fracture mechanics for bridging continuous and discrete materials and extends our knowledge about fracture mechanics for continuous materials to that for discrete materials. In this study, using the finite element method, we analyze the effect of the nanoelement interfacial layer on the transition of the stress singularity of generic wedges, which governs the order of the stress concentration. For the numerical analysis,
⇑ Corresponding author. Tel.: +82 62 530 1688; fax: +82 62 530 1689. E-mail address: [email protected] (I. Jeon). 0013-7944/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2011.08.004
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Nomenclature kv kl v k k l E
r0 rij x, y K r, h k fij(h) r W0
ryy
r yy Py dn tn k k0
the vertical stiffness of the beam element the lateral stiffness of the beam element the non-dimensional vertical stiffness the non-dimensional lateral stiffness elastic modulus the applied uniform normal stress the stress tensor, i, j = x, y the cartesian coordinates centered at the wedge tip the stress intensity the polar coordinates centered at the wedge tip the stress singularity the function of solution of the stress field the non-dimensional distance from a wedge tip the width of wedge specimens the y-direction stress along the interface the normalized y-direction stress along the interface the force of each nanoelement the distance between nanoelements the unit thickness of the nanoelement layer the normalized singularity the stress singularity of Direct-connected model for each generic wedge
an actual helical nanoelement is modeled using a 2D beam element with vertical and lateral stiffness and all the other parts of the generic wedge specimens are modeled with 2D plane strain solid elements. The suitability of the finite element modeling is confirmed through an experimental compression test and numerical analysis for the test. Then, numerical analyses are performed using modeled specimens with two representative generic wedges, viz., an interfacial crack and a right-angled interfacial notch including the nanoelement interfacial layer. The stress singularity transition corresponding to changes in the vertical and lateral stiffness of the nanoelement, as well as the difference in material between the upper and lower parts of the specimens, is examined mainly by using the numerical results.
2. Modeling for the nanoelement layer The helical nanoelement layer with the turns of 5.5 shown in Fig. 1a and b are fabricated using Ta2O5 on a Si substrate by the DOD method under an incident vapor flux angle of 84°. The height of the nanoelement obtained from this fabrication process is 900 nm and the helical coil radius and wire diameter are 50 nm and 40 nm, respectively. The areal number density of the nanoelements is 110 per lm2. On the nanoelement layer, a solid Ta2O5 film of 575 nm is deposited as the upper layer. To confirm the suitability of numerical analysis method for an arbitrary model that includes the nanoelement layer, we have performed an experimental compression test using a specimen introduced by Sumigawa et al. [11] (see Fig. 2a) and a numerical analysis using the finite element method (see Fig. 2b). The test specimen is fabricated by attaching a stainless steel cantilever on the top surface of the solid Ta2O5 in Fig. 1a using an epoxy adhesive (see Fig. 2c). A vertical compression load with a displacement rate of 50 lm/s is applied at a point on the cantilever surface using a micro material test machine, MMT100N, with a low capacity load cell of 100N (see Fig. 2d). For the numerical analysis, each helical nanoelement is modeled using 2D beam elements of 900 nm height that has a vertical stiffness of kv and lateral stiffness of kl, and the beam diameter and the distance between beams are set as 40 nm and 55 nm, respectively (see Fig. 1b). All the other parts of the specimen are modeled as 2D plane strain solid elements. A 2D beam element with two nodes is used for the nanoelement and plane strain elements with eight nodes are used for all the other parts. The vertical and lateral stiffness of each nanoelement are 2.05 N/m and 6.89 N/m, respectively [4]. Note that these stiffness are converted from the experimentally measured values by taking into account the difference in the heights of 690 nm while applying the same helical coil radius and wire diameter of this research (‘Material A’ in Hirakata et al. [4]). The elastic modulus and Poisson’s ratio of all the other parts are listed in Table 1. The software package, ABAQUS of Dassault Systems, is used for the numerical analysis. Fig. 3 shows the load–displacement curves obtained from both the experiment and numerical analysis. Although the numerical analysis uses 2D element models, both results show excellent agreement in terms of the overall load–displacement behavior. From this figure, the suitability of the numerical analysis method in this research using 2D element modeling for the nanoelement layer and for each part of the actual specimen is confirmed.
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575 nm
900 nm
525 μm
300 nm
(a)
40 nm
575 nm
900 nm
55 nm
(b) Fig. 1. (a) The helical nanoelement layer and (b) its 2D modeling.
Table 1 Material properties for numerical calculation. Materials
Elastic modulus (GPa)
Poisson’s ratio
Stainless steel Epoxy Ta2O5 Si
200 1.2 117 130
0.3 0.3 0.23 0.29
3. Numerical analysis 3.1. Models for specimens with generic wedges We prepare the modeled test specimens with two different generic wedges, viz., an interfacial crack and a right-angled interfacial notch, which include the nanoelement interfacial layer between the upper and lower parts (see Fig. 4a and c) for examining the effect of the nanoelement interfacial layer on the stress singularity transition of generic wedges. In order to investigate the combination effect of dissimilar materials, two different materials are selected for the lower part, such as a rigid solid, and the same material, Ta2O5, for the upper part, which yields either the maximum or minimum difference in the material properties. Also, we prepare the modeled specimen without the nanoelement interfacial layer for comparing the analysis results. For convenience in distinguishing all the cases, each specimen is named following the type of wedge, i.e., ‘Crack’ and ‘Notch’, and material combination, i.e., ‘T/R’ and ‘T/T’. For example, ‘Crack-T/R’ is a specimen with an interfacial crack in a bi-material of Ta2O5 and a rigid solid with the nanoelement interfacial layer. ‘Notch-T/T’ is a specimen with a right-angled interfacial notch in two components of Ta2O5 that are connected by the nanoelement interfacial layer. Table 2 shows the names of all specimen types. The specimen that does not include the nanoelement interfacial layer is named as ‘Direct-connected model’.
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T. Sueda et al. / Engineering Fracture Mechanics 78 (2011) 2789–2799 Table 2 Specimens simulated. Specimen name
Wedge type
Upper material
Lower material
Crack-T/R Notch-T/R Crack-T/T Notch-T/T
Crack Notch Crack Notch
Ta2O5 Ta2O5 Ta2O5 Ta2O5
Rigid solid Rigid solid Ta2O5 Ta2O5
b Load P t L
l
Stainless steel Epoxy Ta2O5 Ta2O5 nanoelement Si
(a)
(c)
Load P Stainless steel Si wafer y z
x
Stainless steel Epoxy Ta2O5 solid film
Ta2O5 nanoelement Si wafer
(d)
(b) Fig. 2. (a) The structure of a compression test specimen, (b) its finite element model, (c) an actually fabricated specimen, and (d) the experimental compression test.
Fig. 4b and d shows the finite element models for the specimens. Although the results are presented in a dimensionless form in the following section, the actual calculation is carried out under Wo = 100 lm. The 2D plane strain elements with eight nodes are used for the upper and lower parts of the specimens. Each nanoelement is modeled using a 2D beam element with two nodes that is a suitable model for the actual helical nanoelement, as confirmed in Section 2. The height of the beam is 500 nm, and the beam diameter and distance between beams are selected as 50 nm and 50 nm, respectively. This is a model of a helical nanoelement named as ‘Material B’ that is fabricated by Hirakata et al. [4]. 3.2. Material properties and boundary condition In this simulation, Ta2O5 is used for the upper part though the results are extended to other materials though normalization. For the lower part, a rigid solid or Ta2O5 is selected. Here, all the materials are assumed to be linear elastic and isotropic, and the elastic constants of Ta2O5 are shown in Table 1.
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2
Load P, N
1.5
1
0.5
0
Experiment FEM
0.005
0.01
0.015
0.02
Displacement uy, mm Fig. 3. Load–displacement curves obtained from both the experimental test and numerical analysis.
3W0 3
σ0
W0 y
Crack
x
(a)
(b)
σ0
W0
(c)
3W0
y x
(d)
Fig. 4. Two representative generic wedge models and their finite element models: (a) and (b) are for an interfacial crack, while (c) and (d) are for a rightangled interfacial notch.
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Stress singularity
Stress singularity 101
0.486
Crack-T/R
0.453
Non-dimensional normal stress along the interface, σyy
Non-dimensional normal stress along the interface, σyy
101 1 1 0.276 1
100
0.012 k v = 8.8 × 10 -7 , k l = 1.4 × 10 -7 k v = 8.8 × 10 -5 , k l = 1.4 × 10 -5 k v = 8.8 × 10 -3 , k l = 1.4 × 10-3
1 0.261
1
100
0.011 kv
10-3
10-2
10-1
8.8 × 10 -7 , k l = 1.4 10 -7
k v = 8.8 × 10 -5 , k l = 1.4 10 -5 k v = 8.8 × 10 -3 , k l = 1.4 × 10 -3
Without nanoelemnt (λ= 0.5)
10-1 -4 10
Crack-T/T 1
Without nanoelement (λ= 0.5)
100
10-1 -4 10
10-3
10-2
100
Non-dimensional distance from crack tip, r
Non-dimensional distance from crack tip, r
(a)
(c)
Stress singularity
Stress singularity
101
1
10 Notch-T/R
Notch-T/T
Non-dimensional normal stress along the interface, σyy
Non-dimensional normal stress along the interface, σyy
10-1
0.262 1 1 0.166
0
10
1 0.013 k v = 8.8 × 10-7 , kkll = 1.4 × 10-7 k v = 8.8 × 10-5 , k l = 1.4 × 10-5 k v = 8.8 × 10-3 , k l = 1.4 × 10 -3
Without nanoelement (λ= 0.274)
10-1 -4 10
-3
10
-2
10
-1
10
0.402 1 1 0.230 0
10
1 0.008 kv
8.8 × 10 -7 , k l = 1.4 × 10 -7
k v = 8.8 × 10 -5 , k l = 1.4 × 10 -5 k v = 8.8 × 10 -3 , k l = 1.4 × 10 -3
Without nanoelement (λ= 0.461) 0
10
10-1 -4 10
-3
10
10-2
10-1
100
Non-dimensional distance from notch tip, r
Non-dimensional distance from notch tip, r
(b)
(d)
Fig. 5. Stress distribution with stress singularities near the tip of generic wedges: (a) Crack-T/R, (b) Notch-T/R, (c) Crack-T/T, and (d) Notch-T/T.
At first, the vertical stiffness, kv, and lateral stiffness, kl, of the beam element are selected as 10.3 N/m and 1.65 N/m, respecv ¼ kv ¼ 8:8 107 and k ¼ kl ¼ 1:4 107 ; tively, which are measured by Hirakata et al. [4]. This means the magnitude of k l pffiffiffiffiffiffiEW o pffiffiffiffiffiffi EW o O . Then, we extend the magnitudes to 10 times, 10 times, 10 10 times, 100 times, here, E is the elastic modulus of Ta 2 5 pffiffiffiffiffiffi pffiffiffiffiffiffi v and k for examining the effect on 100 10 times, 1000 times, 1000 10 times and 10,000 times, respectively, of each k l v and k pairs, only 45 cases, which yield similar behavthe stress singular field near the tip of the crack or notch. Of all the k l iors to vertical springs under a uniaxial tension state, are used for the numerical calculations. For the boundary condition, a uniform normal stress of r0 = 10 MPa is applied to the upper edge of the specimen while the y-direction displacement at the bottom edge of the specimen is constrained as shown in Fig. 4a–d. For all the numerical analyses, the software package, ABAQUS is used. 4. Results and discussion 4.1. Stress singularity From linear elastic fracture mechanics, it is well known that the stress field near the tip of generic wedges often has the following form:
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10-7 1.0
1.0
Crack-T/R
C 0.5
0.5
0
C’
0.5
0.5
0
B’
-6 -7
10
-7
10
al sio n 10 men ess, k v i d Non l st iffn ica vert 10-5
-6
10-7
10-4
10-2
1.0
Crack-T/T
l 10 na -5 10 sio k l -4 en ess, 10 -3 10 dim fn n- st if No eral lat
B
-6
l 10 na -5 10 sio k l -4 en ess, 10 -3 10 dim fn n- st if No eral lat
0 10-3
-4 10-5 10 10
10-6
10-7
Nond latera imensio na l l st iff ness, 10-2 10-3 kl 10-4 -5 10 10-6 10-7 A 1.0
Notch-T/R
0.5
0.5
0
10-2
10-5
-4 10-5 10
Nond latera imensio na l l st iffn -2 ess, k 10-3 10 10-3 10-4 l 10-5 -6 10 10-7 A’ 1.0
Notch-T/T
C’
0.5
0.5
0
B’ -7
(b)
10-6
10
-7
10
al sio n 10 men ess, k v i d Non l st iffn ica vert -6
10-7
10-4
10-3
10-7 1.0
-6 l 10 na -5 sio k l 10 -4 en ess, 10 -3 dim fn 10 n- st if No eral lat
0
-6
l 10 -5 na 10 sio k l -4 10 en ess, -3 10 dim fn n- st if No eral lat
C
B
10-5
o nal Non-dimensi s, k v es fn if vert ical st
Normalized r, λ singular orde
-4 10-5 10 10
10-6
0 10-2
(c)
Normalized singular order, λ
Normalized r, λ singular orde
10-7 1.0
10
10-4
10-3
al sio n men ess, k v i d Non l st iffn ica vert
-6
(a) o nal Non-dimensi s, k v es fn vert ical st if -3
Normalized singular order, λ
10
10
0 10-2
al sio n men ess, k v i d Non l st iffn ica vert
-6
10-7
10-4
10-3
Normalized singular order, λ
10-5 110
10-6
Nond latera imensio na l l st iff -2 ness 10 10-3 k 10-4 -5 l 10 10-6 10-7 A’
o nal Non-dimensi s, k v es fn if st al vert ic -3
Normalized singular order, λ
Normalized r, λ ng si ular orde
10-7 1.0
Nond latera imensio na l l st iffn ess, k 10-2 10-3 10-4 -5 l 10 10-6 10-7 A
Normalized r, λ singular orde
o nal Non-dimensi s, k v es fn vert ical st if -4 -3
10-5
(d)
Fig. 6. The stress singularity transitions of (a) Crack-T/R, (b) Notch-T/R, (c) Crack-T/T, and (d) Notch-T/T.
rij ¼ K rk fij ðhÞ ði; j ¼ x; yÞ;
ð1Þ
rij ¼ K rk ði; j ¼ x; yÞ on h ¼ 0
ð2Þ
or
where r and h are the polar coordinates centered at the wedge tip and k > 0 is the order of the stress singularity. The parameter, K, represents the stress intensity. The detailed solutions for fij(h) depending on the geometry of the generic wedges are given in Rice et al. [12], Im and Kim [13] and Jeon and Im [14]. Moreover, generic wedges on the bi-material interface sometimes bring about several complexities regarding the stress field with double singularities or an oscillatory singularity. The double singularities, which consist of a strong and a weak singularity, give combined singular effects on the stress field. The complex k brings about not only the steep gradient but also oscillation in the stress field because of its imaginary part, viz., an oscillatory singularity. In this paper, our interest is confined in the effect of nanoelement layer on the steep stress concentration near the generic wedge as a first inquiry. Thus, for simplification, we set the origin of the x, y coordinate at the wedge tip in nanoelement layer and examine the stress field near the tip of 5 103 < r < 5 102 (see Fig. 4b and d). The detailed analysis of the exact stress filed is out of scope of this paper and it remains for future investigation.
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Fig. 7. (a) and (d) correspond to Crack-T/R and Notch-T/R at the point C; (b) and (e) represent simplified boundary conditions of (a) and (b); (c) and (f) are their transformed models due to symmetry.
r
yy ¼ ryy . The solid lines show the ones in Fig. 5a–d shows the log–log plots of the normalized stress along the interface, r 0 the direct mount material (Substrate)/(Ta2O5 film) without the nanoelement layer and indicate clearly the existence of
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Fig. 8. (a) An acute-angled notch specimen, (b) its transformed model, (c) an obtuse-angled notch specimen, and (d) its transformed model.
singular field presented by Eq. (1). The others are obtained from numerical analysis with the nanoelement layer of represenv and k , and the normalized distance from the wedge tip, r ¼ r . Here, the stress, r , is the interfacial stress tative pairs of k l yy W0 Py on the boundary that is defined as dn t , where Py is the force of each nanoelement, dn ¼ 50 nm is the distance between n nanoelements and tn ¼ 1 nm is the unit thickness of the nanoelement layer. It should be noted that the stress distributions along upper and lower boundaries are the same because Py is the force along the element axis. v ¼ 8:8 103 , k ¼ 1:4 103 ) show almost straight relation, the triangles While the symbol of squares (k l 5 5 (kv ¼ 8:8 10 , kl ¼ 1:4 10 ) suggest the transition from stress singularity to non-singularity or the oscillated singularity where the field depart from the straight line in the region very near the wedge. As a measure representing the steepness of v stress field near the wedge, the slope of each curve, k, in the region of 5 103 < r < 5 102 is calculated for each case of k and kl and the fundamental behavior is discussed in the following sections. v ¼ 8:8 107 , k ¼ 1:4 107 (circles), the nanoelement layer almost eliminate the stress concentration In the case of k l near the wedge. For the real material (Si substrate)/(Ta2O5 nano-spring)/(Ta2O5 film) shown in Fig. 2b, the FEM analysis predicts no stress singularity near the corner along the interface as well. The experiments using several specimens with different sizes were carried out in the previous study [11], and the crack initiation behavior was carefully observed. It clarified that the analyzed stress without the singularity agreed well for all cases. This verifies that the nano-spring layer actually eliminates the stress concentration near the corner and proves the validity of the FEM analysis for a real material. 4.2. Transition of the stress singularity v and k , where k0 is the stress singularity of Direct-con ¼ k , is calculated in 45 cases of k The normalized singularity, k l k0 nected model for each generic wedge, i.e., k0 ¼ 0:5 for the crack and k0 ¼ 0:274 for the right-angled notch, respectively. v and k . Fig. 6a and b shows the normalized singularity transitions of Crack-T/R and Notch-T/R depending on k l k is nearly v and high k ). The stiff nanoelements make strong constraint the displacements at the upper and unity at the point A (high k l lower ends. Though the interfacial layer is discrete, densely disposed nanoelements give adhesion condition close to that of Direct-connected model. Thus, they keep strong singularity. v and low k ), the stress singularity is close to zero. Because the nanoelement allows large displaceAt the point B (low k l ment between the upper and lower part, the layer acts like a gel phase. Thus, the stress on the interface becomes uniform and stress concentration is eliminated. Since the stress singularities of the point A and point B do not depend on the wedge types, these can be general characteristics in various generic wedges. v and low k ), there is a clear difference in the singularity transition between Crack-T/R and Notch-T/ At the point C (high k l R. This means that the stress distribution is highly dependent on the wedge shape. In order to understand the difference, we examine the boundary condition of the upper part of the specimen along the interface. Fig. 7a shows a simplified model v and low k , the bottom boundary of the upper material around the nanoelement interfacial layer of Crack-T/R. Under high k l is approximated as the condition shown in Fig. 7b where the displacement of y-direction is perfectly restricted. Because the boundary condition implies a symmetric condition, this (Fig. 7b) is the same as the homogeneous double edge-cracked
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Fig. 9. The stress singularity transitions caused by differences in material between the upper and lower parts of: (a) crack specimens and (b) notch specimens.
specimen illustrated in Fig. 7c. In fact, the stress field around the crack tip shows r0.5 distribution. Thus, the singularity of Crack-T/R at the point C is close to that of the homogeneous crack problem, viz., k ¼ kk0 1. The case of Notch-T/R at point C can be examined using a similar approach to that of Crack-T/R. The model (Fig. 7d) approximately has the boundary condition shown in Fig. 7e, which is the same as the one illustrated in Fig. 7f due to symmetry. Because the homogenous plate (Fig. 7f) does not have singular stress fields, the stress singularity of Notch-T/R disappears at the point C. Based on the above considerations, it is possible to estimate the singularities of various wedges at point C. For example, a bi-material with an acute-angled notch, as shown in Fig. 8a, can be regarded as the model in Fig. 8b, which clearly possesses the stress singularity. On the other hand, an obtuse-angled notch specimen shown in Fig. 8c, which can be regarded as the model illustrated in Fig. 8d, does not have the singularity. 4.3. Effect of material combinations v and k . The change in Fig. 6c shows the normalized singularity transitions of Crack-T/T corresponding to changes in k l singularity is similar to that of Crack-T/R. Particularly, at point C0 , the normalized singularity, k, is almost ‘1’ although the v is the dominant parameter lower part is of a different material from that of Crack-T/R. From Fig. 6a and c, it is found that k for controlling the stress concentration near the crack tip.
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of 1.4 107. To see more clearly the dependence of the sinFig. 9a shows the changes in singularity under a constant k l gularity transition on the combination of materials, the result for another computed set for a bi-material (upper: Ta2O5; lower: Young’s modulus of 250 GPa; this denoted as the –T/T0 specimen) is shown in the figure as well. Although there are large differences in the material properties, all the singularity transitions are in good agreement. This shows that the singular orv , regardless of the material combination. der, k, near the crack tip is solely governed by k Although the trends in the singularity transitions are similar, there is a difference at the point C0 in Fig. 6d (Notch-T/T) compared with that at the point C in Fig. 6b (Notch-T/R). In the case of large differences in material properties between strongly affects the stress concentration of the notch specimens. Fig. 9b shows the singularity the upper and lower parts, k l v of 8.8 103. Another computed set is also included in the figure. The transitions of the notch specimens under a constant k case of the largest difference in material properties between the upper and lower parts (Notch-T/R) shows a lower bound of the singularity transition and the case where the material properties are the same (Notch-T/T) shows an upper bound thereof. Therefore, the singularity transition of Notch-T/T0 lies between those of Notch-T/R and Notch-T/T. So far, we have discussed about the stress singularity transition of a wedge problem using the nanoelement interfacial layer. Actually, as a criterion for the fracture of a wedge specimen, the stress intensity K needs to be defined for each corresponding singularity. The stress intensity is not discussed here because it highly depends on the types of applied loading and various specimen geometries. As the stress intensity should be an important factor for engineering applications, it needs to be studied as a future study in design of real micro-components. 5. Conclusions The effect of the nanoelement interfacial layer on the transition of the stress singularity near a generic wedge tip is analyzed by using the finite element method. The suitability of the modeling is confirmed by an experimental compression test and numerical analysis for the test. Then, modeled specimens with two representative generic wedges, viz., an interfacial crack and a right-angled interfacial notch including the nanoelement interfacial layer, are selected and analyzed in relation to changes in the vertical and lateral stiffness of the nanoelement. v and high k , the singularities are close to those of Direct-connected model in general, while the sinIn the case of high k l v and low k . Under a high value of k v and a low value of k , the crack problems yield gularities are eliminated under low k l l singularities that are close to those of Direct-connected model and the right-angled notch problems result in lower singularities than those of the model. v causes decrease in the stress concentration regardless of k as well as the Particularly, in the cracked body, a lower k l material properties. For a notched body, the kl value can be the dominant parameter for decreasing the stress concentration. Acknowledgments This paper was supported in part by a Grant-in-Aid for Scientific Research (S) (No. 21226005), of the Japan Society of the Promotion of Science, and by a Grant-in-Aid for Young Scientists (A) (No. 21686013) of the Ministry of Education, Culture, Sports, Science and Technology, Japan. Moreover, this work was supported in part by Nuclear R&D Program through the National Research Foundation of Korea funded by the Ministry of Education, Science, and Technology (2009-0093620). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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