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Analyses of steady-state interface fracture of elastic multilayered beams under four-point bending
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Scripta Materialia 60 (2009) 721–724 www.elsevier.com/locate/scriptamat
Analyses of steady-state interface fracture of elastic multilayered beams under four-point bending C.H. Hsueh,a,b,c,* W.H. Tuana and W.C.J. Weia a
Department of Materials Science and Engineering, National Taiwan University, Taipei 106, Taiwan b Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA c Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Received 9 September 2008; accepted 6 January 2009 Available online 13 January 2009
The closed-form solution for the steady-state interface energy release rate of elastic multilayered beams subjected to four-point bending is presented. The beam can have an arbitrary number of layers and fracture can occur at any interface. Specific results are calculated for alternate layered systems to elucidate the essential trends of the dependences of the normalized steady-state interface energy release rate on the thickness ratio and the modulus ratio for each interface. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Multilayers; Bending test; Layered structures; Analytical methods
Multilayered systems have been used extensively in microelectronic, optical and structural components, and protective coatings. Their functionality and reliability are strongly influenced by bonding at interfaces, and considerable efforts have been devoted to devise tests to characterize interface bonding [1,2]. A noticeable technique of four-point bending for measuring the fracture resistance of bimaterial interface was developed by Charalambides et al. [3], and is shown schematically in Figure 1(a). In this technique, a notch is cut through one layer to provide a point for the initiation of the interface crack under four-point bending, and the closed-form solution for the steady-state energy release rate for interface cracking in bilayered systems is obtained [3]. Klingbeil and Beuth [4] extended this technique to trilayered systems and derived a corresponding closed-form solution. While many engineering systems involve a number of layers, the purpose of the present study is to derive the general closed-form solution for the steady-state interface energy release rate of multilayered systems subjected to four-point bending. The solution is applicable to elastic multilayered systems with any number of layers and cracking at any interface.
* Corresponding author. Address: Department of Materials Science and Engineering, National Taiwan University, Taipei 106, Taiwan. Tel.: +886 2 33661307; fax: +886 2 23634562; e-mail: [email protected]
The cross-section of a multilayered beam subjected to four-point bending is shown schematically in Figure 1(b). The beam consists of n layers with individual thickness ti, where the subscript i denotes the layer number, with layer 1 being at the bottom. The coordinate system is defined such that the bottom of the system is located at z = 0, the interface between layers i and i + 1 is located at hi, and the top surface of layer n is located at z = hn. With these definitions, hn is the thickness of the beam, and the relation between hi and ti is described by hi ¼
i X
tj
ði ¼ 1; . . . nÞ
ð1Þ
j¼1
Under four-point bending, the beam is subjected to a moment, M, of Pl/2 between the two inner loading lines, where P is the total load and l is the horizontal distance between the inner and outer loading lines. For the loading configuration shown in Figure 1(b), the top and bottom surfaces of the beam are subjected to tension and compression, respectively. A center notch is cut through a certain depth underneath the top surface, such that symmetric interface cracking occurs between layers m and m + 1 (with m 6 n 1) under loading and the interface crack is located at interface m, z = hm. The energy-based criterion is adopted to analyze the critical load required for interface cracking. It should be noted that multilayered systems can be subjected to residual stresses because of the thermomechanical
1359-6462/$ - see front matter Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2009.01.001
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C. H. Hsueh et al. / Scripta Materialia 60 (2009) 721–724
II, the interface debonds at z = hm and only layers 1to m are subjected to bending, while layers m + 1 to n are stress-free. The bending of multilayered beams has been analyzed previously [8,9]. When an n-layered beam is subjected to bending, the in-plane stress distribution in each layer, ri, is linearly proportional to the distance from the neutral axis, which is located at z = zn, and inversely proportional to the radius of curvature, rn, of the system during bending, such that ri ¼ Ei ðz zn Þ=rn
ðfor i ¼ 1; . . . nÞ
ð2Þ
where Ei is Young’s modulus of layer i. It should be noted that plane stress is considered in formulating Eq. (2). For the plane strain condition, Ei should be replaced by Ei/(1 mi2) in Eq. (2), where mi is Poisson’s ratio of layer i. Also, the subscript, n, used in parameters zn and rn is to denote the number of layers subjected to bending. Using the force and the moment equilibrium conditions, the two unknowns, zn and rn, in Eq. (2) have been solved, such that [8,9] , n n X X Ei ti ð2hi1 þ ti Þ 2 Ei ti ð3aÞ zn ¼ i¼1
Figure 1. Schematics showing four-point bending to evaluate the interface energy release rate: (a) bilayer [3], (b) cross-section of multilayer with the coordinate system used in modeling, and (c) steadystate interface cracking for an n-layered system with cracking at interface m.
mismatch among layers. While residual stresses-induced delamination in multilayers has been analyzed elsewhere [5,6], the present study focuses on interface cracking by external loading and the effects of residual stresses are not considered to simplify the analysis. Also, the present analysis is limited to an elastic system and plasticity is not considered; studies of the effects of plasticity on interface fracture can be found elsewhere [7]. Hence, for the interface fracture problem considered in the present study, the following energy terms are involved: (i) U, the elastic strain energy resulting from the applied moment; (ii) G, the energy release rate for interfacial fracture; and (iii) W, the work done by the applied moment. To obtain the closed-form solution, the steady-state interface cracking is considered. Because of the symmetric geometry, only half of the system is shown in Figure 1(c). The crack extends along interface m. Under an applied moment, M, the crack advances a distance, Da. For steady-state cracking, the stress at the crack front remains unchanged during crack extension, and the stresses far behind and ahead of the crack front also remain unchanged. Hence, the change in the elastic strain energy due to crack extension is the difference in elastic strain energy between two strips, which are respectively far behind and ahead of the crack front, of length Da. These two strips are shown by the two shaded regions in Figure 1(c), which are designated as regions I and II, respectively, for the regions far ahead of and behind the crack front. In region I, the interfaces remain bonded and all layers are subjected to bending. In region
rn ¼b
i¼1
n X
Ei ti 6h2i1 þ 6hi1 ti þ 2t2i
i¼1
3zn ð2hi1 þ ti Þ=6M
ð3bÞ
where b is the width of the beam. Also, when i = 1, hi1 (i.e., h0) in Eqs. (3a) and (3b) is defined as zero. The elastic strain energy in region I, UI, is n Z hi X r2i dz ð4Þ U I ¼ bDa hi1 2E i i¼1 Substitutions of Eqs. (2) and (3b) into Eq. (4) yield , n X 2 b Ei ti 6h2i1 þ 6hi1 ti þ 2t2i U I ¼ 3DaM i¼1
3zn ð2hi1 þ ti Þ
ð5Þ
When an m-layered beam is subjected to bending, the strain energy in region II, UII, can be obtained from Eq. (5) by replacing n with m, such that , m X Ei ti 6h2i1 þ 6hi1 ti þ 2t2i U II ¼3DaM 2 b i¼1
3zm ð2hi1 þ ti Þ where zm is given by zm ¼
m X
ð6Þ ,
Ei ti ð2hi1 þ ti Þ
i¼1
2
m X
Ei ti
ð7aÞ
i¼1
The corresponding radius of curvature, rm, in region II is m X rm ¼ b Ei ti 6h2i1 þ 6hi1 ti þ 2t2i i¼1
3zm ð2hi1 þ ti Þ=6M The change in the elastic strain energy, dU, is
ð7bÞ
C. H. Hsueh et al. / Scripta Materialia 60 (2009) 721–724
dU ¼ U II U I
ð8Þ
The increase in the interface energy, dG, due to crack extension is dG ¼ bDaG
ð9Þ
Also, because of the crack extension, region I, with a radius of curvature rn, is replaced by region II, with radius of curvature rm. This change in curvature would induce additional deflection of the beam under a constant moment, M, and work is done due to this deflection. For an elastic system, the work done, dW, is twice of the change in elastic strain energy [10,11], i.e. dW ¼ 2dU
ð10Þ
The critical moment required for steady-state interface cracking can be obtained from the energy balance relation, such that dW ¼ dU þ dG
ð11Þ
Substitutions of Eqs. (5), (6), (8)–(10) into Eq. (11) yield
( 3M 2 1 2 G ¼ 2 Pm 2 b E t 6h þ 6h t i i i1 i þ 2ti 3zm ð2hi1 þ ti Þ i1 i¼1 ) 1 2 Pn 2 i¼1 E i t i 6hi1 þ 6hi1 t i þ 2ti 3zn ð2hi1 þ t i Þ
ð12Þ
where zm and zn are given by Eqs. (7a) and (3a), respectively. The solution given by Eq. (12) is valid for multilayered beams with any number of layers and fracture at any interface. Also, each layer has its individual elastic properties and thickness. However, to elucidate the essential trends of the dependences of steady-state interface energy release rate on the layer thickness and modulus, the number of layers, and the location of the fracture interface, specific results are calculated for alternate layered systems in the present study. A layer arrangement of ABABA— is considered, with layer A being at the bottom of the system. All odd-number layers, A, have the same thickness and modulus, tA and EA. Similarly, all even-number layers, B, have the same thickness and modulus, tB and EB.
Figure 2. The normalized steady-state interface energy release rate, GEA b2 h3n =P 2 l2 , as a function of the thickness ratio, tB/tA, at different modulus ratios, EB/EA, for bilayered systems.
723
For bilayered systems, the normalized steady-state interface energy release rate, GEA b2 h3n =P 2 l2 , as a function of the thickness ratio, tB/tA, is shown in Figure 2 at different ratios of EB/EA. This corresponds to n = 2 and m = 1 in the closed-form solution, Eq. (12). The results shown in Figure 2 recover Charalambides et al.’s solutions (Fig. 3 in Ref. [3]). For a constant moment and fixed tA and EA, U1 decreases with increasing tB and EB while U2 remains constant. As a result, the normalized G increases with increasing thickness ratio, tB/tA, and modulus ratio, EB/EA, as shown in Figure 2. For trilayered systems (n = 3), the results are shown in Figure 3(a) and (b), respectively, for cracking at interface 2 (m = 2) and interface 1 (m = 1). It should be noted that the ratio of the total thickness of layer B to the total thickness of layers A is defined as the thickness ratio, tB/2tA, in plotting Figure 3(a) and (b). When tB = 0, the system cracks at the mid-plane of a monolithic material and the normalized G is 10.5, which can be observed from the solutions for (i) tB/tA = 1 and EB/EA = 1 in Figure 2 and (ii) tB = 0 in Figure 3(a) and (b). For n = 3 and m = 2, both U1 and U2 decrease with increasing tB when the other parameters are fixed. Depending upon the balance between these two decreasing rates, the normalized G can go through a maximum and then decrease with increasing thickness ratio, e.g. see the two lines for EB/EA = 0.1 and 0.2 in Figure 3(a). Also, both U1 and U2 decrease with increasing EB when other parameters are fixed; however, U2 has
Figure 3. The normalized steady-state interface energy release rate, GEA b2 h3n =P 2 l2 , as a function of the thickness ratio, tB/2tA, at different modulus ratios, EB/EA, for trilayered systems with cracking at (a) interface 2 and (b) interface 1.
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U2 is constant and, while U1 is insensitive to EB for small values of tB, it becomes much smaller than U2 for large values of tB. As a result, the normalized G becomes insensitive to the modulus ratio, and the two lines for EB/EA = 0.1 and 5 are hard to distinguish in Figure 3(b). For four-layered systems (n = 4), the results are shown in Figure 4(a)–(c), respectively, for cracking at interface 3 (m = 3), interface 2 (m = 2) and interface 1 (m = 1). For m = 3, the discussion for Figure 2 can be applied to Figure 4(a). For m = 2, the discussion for Figure 3(a) can be applied to Figure 4(b). Although local maxima are not shown in Figure 4(b), the line for EB/ EA < 1 will go through a maximum and then decrease with increasing tB/tA when tB/tA is sufficiently large (>1). Also, when EB/EA = 1, the system becomes cracking at the mid-plane of a monolithic material and the normalized G is a constant, 10.5. For m = 1, the discussion for Figure 3(b) can be applied to Figure 4(c). In conclusion, we derive the general closed-form solution for the determination of steady-state interface energy release rate of multilayered systems from fourpoint bending tests. Fracture can occur at any interface. We apply the solution to alternate bilayered, trilayered and four-layered systems, with the results shown, respectively, in Figures 2–4. The physical meanings of the trends of the dependences of the normalized steady-state interface energy release rate on the thickness ratio and modulus ratio are discussed for each location of the fractured interface. This research was sponsored by the National Science Council, Taiwan under Contract No. NSC962811-E-002-022.
Figure 4. The normalized steady-state interface energy release rate, GEA b2 h3n =P 2 l2 , as a function of the thickness ratio, tB/tA, at different modulus ratios, EB/EA, for four-layered systems with cracking at (a) interface 3, (b) interface 2, and (c) interface 1.
the higher decreasing rate. As a result, the normalized G decreases with increase in the modulus ratio, EB/EA. For n = 3 and m = 1 (see Fig. 3(b)), U2 is constant and U1 decreases with increasing tB when the other parameters are fixed. Hence, the normalized G increases with increasing thickness ratio. Also, for fixed tA and EA,
[1] W.D. Nix, Metall. Trans. A 20 (1989) 2217. [2] R.H. Dauskardt, M. Lane, Q. Ma, N. Krishna, Eng. Fract. Mech. 61 (1998) 141. [3] P.G. Charalambides, J. Lund, A.G. Evans, R.M. McMeeking, J. Appl. Mech. 56 (1989) 77. [4] N.W. Klingbeil, J.L. Beuth, Eng. Fract. Mech. 56 (1997) 113. [5] C.H. Hsueh, C.R. Luttrell, S. Lee, T.C. Wu, H.Y. Lin, J. Am. Ceram. Soc. 89 (2006) 1632. [6] M.D. Drory, M.D. Thouless, A.G. Evans, Acta Metall. 36 (1988) 2019. [7] I.E. Reimanis, B.J. Dalgleish, A.G. Evans, Acta Metall. Mater. 39 (1991) 3133. [8] C.H. Hsueh, J. Appl. Phys. 91 (2002) 9652. [9] C.H. Hsueh, S. Lee, T.J. Chuang, J. Appl. Mech. 70 (2003) 151. [10] J. Aveston, A. Kelly, J. Mater. Sci. 8 (1973) 352. [11] Y.C. Gao, Y.W. Mai, B. Cottrell, J. Appl. Math. Phys. 39 (1988) 550.
Scripta Materialia 60 (2009) 721–724 www.elsevier.com/locate/scriptamat
Analyses of steady-state interface fracture of elastic multilayered beams under four-point bending C.H. Hsueh,a,b,c,* W.H. Tuana and W.C.J. Weia a
Department of Materials Science and Engineering, National Taiwan University, Taipei 106, Taiwan b Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA c Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Received 9 September 2008; accepted 6 January 2009 Available online 13 January 2009
The closed-form solution for the steady-state interface energy release rate of elastic multilayered beams subjected to four-point bending is presented. The beam can have an arbitrary number of layers and fracture can occur at any interface. Specific results are calculated for alternate layered systems to elucidate the essential trends of the dependences of the normalized steady-state interface energy release rate on the thickness ratio and the modulus ratio for each interface. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Multilayers; Bending test; Layered structures; Analytical methods
Multilayered systems have been used extensively in microelectronic, optical and structural components, and protective coatings. Their functionality and reliability are strongly influenced by bonding at interfaces, and considerable efforts have been devoted to devise tests to characterize interface bonding [1,2]. A noticeable technique of four-point bending for measuring the fracture resistance of bimaterial interface was developed by Charalambides et al. [3], and is shown schematically in Figure 1(a). In this technique, a notch is cut through one layer to provide a point for the initiation of the interface crack under four-point bending, and the closed-form solution for the steady-state energy release rate for interface cracking in bilayered systems is obtained [3]. Klingbeil and Beuth [4] extended this technique to trilayered systems and derived a corresponding closed-form solution. While many engineering systems involve a number of layers, the purpose of the present study is to derive the general closed-form solution for the steady-state interface energy release rate of multilayered systems subjected to four-point bending. The solution is applicable to elastic multilayered systems with any number of layers and cracking at any interface.
* Corresponding author. Address: Department of Materials Science and Engineering, National Taiwan University, Taipei 106, Taiwan. Tel.: +886 2 33661307; fax: +886 2 23634562; e-mail: [email protected]
The cross-section of a multilayered beam subjected to four-point bending is shown schematically in Figure 1(b). The beam consists of n layers with individual thickness ti, where the subscript i denotes the layer number, with layer 1 being at the bottom. The coordinate system is defined such that the bottom of the system is located at z = 0, the interface between layers i and i + 1 is located at hi, and the top surface of layer n is located at z = hn. With these definitions, hn is the thickness of the beam, and the relation between hi and ti is described by hi ¼
i X
tj
ði ¼ 1; . . . nÞ
ð1Þ
j¼1
Under four-point bending, the beam is subjected to a moment, M, of Pl/2 between the two inner loading lines, where P is the total load and l is the horizontal distance between the inner and outer loading lines. For the loading configuration shown in Figure 1(b), the top and bottom surfaces of the beam are subjected to tension and compression, respectively. A center notch is cut through a certain depth underneath the top surface, such that symmetric interface cracking occurs between layers m and m + 1 (with m 6 n 1) under loading and the interface crack is located at interface m, z = hm. The energy-based criterion is adopted to analyze the critical load required for interface cracking. It should be noted that multilayered systems can be subjected to residual stresses because of the thermomechanical
1359-6462/$ - see front matter Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2009.01.001
722
C. H. Hsueh et al. / Scripta Materialia 60 (2009) 721–724
II, the interface debonds at z = hm and only layers 1to m are subjected to bending, while layers m + 1 to n are stress-free. The bending of multilayered beams has been analyzed previously [8,9]. When an n-layered beam is subjected to bending, the in-plane stress distribution in each layer, ri, is linearly proportional to the distance from the neutral axis, which is located at z = zn, and inversely proportional to the radius of curvature, rn, of the system during bending, such that ri ¼ Ei ðz zn Þ=rn
ðfor i ¼ 1; . . . nÞ
ð2Þ
where Ei is Young’s modulus of layer i. It should be noted that plane stress is considered in formulating Eq. (2). For the plane strain condition, Ei should be replaced by Ei/(1 mi2) in Eq. (2), where mi is Poisson’s ratio of layer i. Also, the subscript, n, used in parameters zn and rn is to denote the number of layers subjected to bending. Using the force and the moment equilibrium conditions, the two unknowns, zn and rn, in Eq. (2) have been solved, such that [8,9] , n n X X Ei ti ð2hi1 þ ti Þ 2 Ei ti ð3aÞ zn ¼ i¼1
Figure 1. Schematics showing four-point bending to evaluate the interface energy release rate: (a) bilayer [3], (b) cross-section of multilayer with the coordinate system used in modeling, and (c) steadystate interface cracking for an n-layered system with cracking at interface m.
mismatch among layers. While residual stresses-induced delamination in multilayers has been analyzed elsewhere [5,6], the present study focuses on interface cracking by external loading and the effects of residual stresses are not considered to simplify the analysis. Also, the present analysis is limited to an elastic system and plasticity is not considered; studies of the effects of plasticity on interface fracture can be found elsewhere [7]. Hence, for the interface fracture problem considered in the present study, the following energy terms are involved: (i) U, the elastic strain energy resulting from the applied moment; (ii) G, the energy release rate for interfacial fracture; and (iii) W, the work done by the applied moment. To obtain the closed-form solution, the steady-state interface cracking is considered. Because of the symmetric geometry, only half of the system is shown in Figure 1(c). The crack extends along interface m. Under an applied moment, M, the crack advances a distance, Da. For steady-state cracking, the stress at the crack front remains unchanged during crack extension, and the stresses far behind and ahead of the crack front also remain unchanged. Hence, the change in the elastic strain energy due to crack extension is the difference in elastic strain energy between two strips, which are respectively far behind and ahead of the crack front, of length Da. These two strips are shown by the two shaded regions in Figure 1(c), which are designated as regions I and II, respectively, for the regions far ahead of and behind the crack front. In region I, the interfaces remain bonded and all layers are subjected to bending. In region
rn ¼b
i¼1
n X
Ei ti 6h2i1 þ 6hi1 ti þ 2t2i
i¼1
3zn ð2hi1 þ ti Þ=6M
ð3bÞ
where b is the width of the beam. Also, when i = 1, hi1 (i.e., h0) in Eqs. (3a) and (3b) is defined as zero. The elastic strain energy in region I, UI, is n Z hi X r2i dz ð4Þ U I ¼ bDa hi1 2E i i¼1 Substitutions of Eqs. (2) and (3b) into Eq. (4) yield , n X 2 b Ei ti 6h2i1 þ 6hi1 ti þ 2t2i U I ¼ 3DaM i¼1
3zn ð2hi1 þ ti Þ
ð5Þ
When an m-layered beam is subjected to bending, the strain energy in region II, UII, can be obtained from Eq. (5) by replacing n with m, such that , m X Ei ti 6h2i1 þ 6hi1 ti þ 2t2i U II ¼3DaM 2 b i¼1
3zm ð2hi1 þ ti Þ where zm is given by zm ¼
m X
ð6Þ ,
Ei ti ð2hi1 þ ti Þ
i¼1
2
m X
Ei ti
ð7aÞ
i¼1
The corresponding radius of curvature, rm, in region II is m X rm ¼ b Ei ti 6h2i1 þ 6hi1 ti þ 2t2i i¼1
3zm ð2hi1 þ ti Þ=6M The change in the elastic strain energy, dU, is
ð7bÞ
C. H. Hsueh et al. / Scripta Materialia 60 (2009) 721–724
dU ¼ U II U I
ð8Þ
The increase in the interface energy, dG, due to crack extension is dG ¼ bDaG
ð9Þ
Also, because of the crack extension, region I, with a radius of curvature rn, is replaced by region II, with radius of curvature rm. This change in curvature would induce additional deflection of the beam under a constant moment, M, and work is done due to this deflection. For an elastic system, the work done, dW, is twice of the change in elastic strain energy [10,11], i.e. dW ¼ 2dU
ð10Þ
The critical moment required for steady-state interface cracking can be obtained from the energy balance relation, such that dW ¼ dU þ dG
ð11Þ
Substitutions of Eqs. (5), (6), (8)–(10) into Eq. (11) yield
( 3M 2 1 2 G ¼ 2 Pm 2 b E t 6h þ 6h t i i i1 i þ 2ti 3zm ð2hi1 þ ti Þ i1 i¼1 ) 1 2 Pn 2 i¼1 E i t i 6hi1 þ 6hi1 t i þ 2ti 3zn ð2hi1 þ t i Þ
ð12Þ
where zm and zn are given by Eqs. (7a) and (3a), respectively. The solution given by Eq. (12) is valid for multilayered beams with any number of layers and fracture at any interface. Also, each layer has its individual elastic properties and thickness. However, to elucidate the essential trends of the dependences of steady-state interface energy release rate on the layer thickness and modulus, the number of layers, and the location of the fracture interface, specific results are calculated for alternate layered systems in the present study. A layer arrangement of ABABA— is considered, with layer A being at the bottom of the system. All odd-number layers, A, have the same thickness and modulus, tA and EA. Similarly, all even-number layers, B, have the same thickness and modulus, tB and EB.
Figure 2. The normalized steady-state interface energy release rate, GEA b2 h3n =P 2 l2 , as a function of the thickness ratio, tB/tA, at different modulus ratios, EB/EA, for bilayered systems.
723
For bilayered systems, the normalized steady-state interface energy release rate, GEA b2 h3n =P 2 l2 , as a function of the thickness ratio, tB/tA, is shown in Figure 2 at different ratios of EB/EA. This corresponds to n = 2 and m = 1 in the closed-form solution, Eq. (12). The results shown in Figure 2 recover Charalambides et al.’s solutions (Fig. 3 in Ref. [3]). For a constant moment and fixed tA and EA, U1 decreases with increasing tB and EB while U2 remains constant. As a result, the normalized G increases with increasing thickness ratio, tB/tA, and modulus ratio, EB/EA, as shown in Figure 2. For trilayered systems (n = 3), the results are shown in Figure 3(a) and (b), respectively, for cracking at interface 2 (m = 2) and interface 1 (m = 1). It should be noted that the ratio of the total thickness of layer B to the total thickness of layers A is defined as the thickness ratio, tB/2tA, in plotting Figure 3(a) and (b). When tB = 0, the system cracks at the mid-plane of a monolithic material and the normalized G is 10.5, which can be observed from the solutions for (i) tB/tA = 1 and EB/EA = 1 in Figure 2 and (ii) tB = 0 in Figure 3(a) and (b). For n = 3 and m = 2, both U1 and U2 decrease with increasing tB when the other parameters are fixed. Depending upon the balance between these two decreasing rates, the normalized G can go through a maximum and then decrease with increasing thickness ratio, e.g. see the two lines for EB/EA = 0.1 and 0.2 in Figure 3(a). Also, both U1 and U2 decrease with increasing EB when other parameters are fixed; however, U2 has
Figure 3. The normalized steady-state interface energy release rate, GEA b2 h3n =P 2 l2 , as a function of the thickness ratio, tB/2tA, at different modulus ratios, EB/EA, for trilayered systems with cracking at (a) interface 2 and (b) interface 1.
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U2 is constant and, while U1 is insensitive to EB for small values of tB, it becomes much smaller than U2 for large values of tB. As a result, the normalized G becomes insensitive to the modulus ratio, and the two lines for EB/EA = 0.1 and 5 are hard to distinguish in Figure 3(b). For four-layered systems (n = 4), the results are shown in Figure 4(a)–(c), respectively, for cracking at interface 3 (m = 3), interface 2 (m = 2) and interface 1 (m = 1). For m = 3, the discussion for Figure 2 can be applied to Figure 4(a). For m = 2, the discussion for Figure 3(a) can be applied to Figure 4(b). Although local maxima are not shown in Figure 4(b), the line for EB/ EA < 1 will go through a maximum and then decrease with increasing tB/tA when tB/tA is sufficiently large (>1). Also, when EB/EA = 1, the system becomes cracking at the mid-plane of a monolithic material and the normalized G is a constant, 10.5. For m = 1, the discussion for Figure 3(b) can be applied to Figure 4(c). In conclusion, we derive the general closed-form solution for the determination of steady-state interface energy release rate of multilayered systems from fourpoint bending tests. Fracture can occur at any interface. We apply the solution to alternate bilayered, trilayered and four-layered systems, with the results shown, respectively, in Figures 2–4. The physical meanings of the trends of the dependences of the normalized steady-state interface energy release rate on the thickness ratio and modulus ratio are discussed for each location of the fractured interface. This research was sponsored by the National Science Council, Taiwan under Contract No. NSC962811-E-002-022.
Figure 4. The normalized steady-state interface energy release rate, GEA b2 h3n =P 2 l2 , as a function of the thickness ratio, tB/tA, at different modulus ratios, EB/EA, for four-layered systems with cracking at (a) interface 3, (b) interface 2, and (c) interface 1.
the higher decreasing rate. As a result, the normalized G decreases with increase in the modulus ratio, EB/EA. For n = 3 and m = 1 (see Fig. 3(b)), U2 is constant and U1 decreases with increasing tB when the other parameters are fixed. Hence, the normalized G increases with increasing thickness ratio. Also, for fixed tA and EA,
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