A fracture-mechanics model of the microbond test with interface friction

0 2

…39†

Co …c† represents the compliance of the debonded ®ber given by Co …c† ˆ

c r2f Ef

…40†

Obviously,
ld is measured as relative displacement of the ®ber end with respect to the crack tip. Hence, in the case of an elastic matrix, an additional contribution to the total ®ber end displacement arises from the crack tip displacement (utip ) which is due to matrix deformation. The total ®ber end displacement consists of u ˆ
lfree ‡ u…z ˆ 0† ˆ
lfree ‡ utip ‡
ld

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knife edges remains constant during loading. This condition was enforced in the simulation. In practice it would require a small ¯at area at the tip of the droplet contacting the knife edges; the question remains open how good this condition can be satis®ed. Relation (42) holds for zero friction, but remains approximately valid for ®nite friction,  > 0. Thus the ®ber-end displacement can be written as   f C…c† 1 c2 u  1free ‡
f c ÿ …43† Ef Co …c† Ef rf where applied and crack-tip load have been eliminated by applied ®ber stress and friction stress according to Eq. (37). Fig. 8 gives an idea of the accuracy of this approximation.

…41†

as shown in Fig. 7. In the friction-free case,
ld and utip are related by Utip ˆ

C…c† ÿ Co …c†

ld Co …c†

…42†

where C…c† denotes the contribution of embedded ®ber and elastic droplet to the total compliance for the case of zero-friction. It can be computed by FE-analysis. Note that strictly speaking, the friction-free load/displacement relationship is linear only (with a load-independent compliance) if the contact area between droplet and Fig. 7. Contribution of debonded ®ber elongation (
ld ) and crack tip displacement (utip ) to the ®ber-end displacement. The free ®ber length is assumed to be zero (lfree ˆ 0).

Fig. 6. In¯uence of crack face friction on the energy release rate as function of crack length for di€erent applied ®ber stresses and an elastic matrix. The energy release rate is normalized by the friction-free ®ber contribution to the energy release rate, Gf … ˆ 0†: f, no friction; l, low ratio friction stress to applied ®ber stress (=f ˆ 3:8 MPa/1 GPa); h, high ratio friction stress to applied ®ber stress (=f ˆ 3:8 MPa/0.05 GPa). Solid curves show the exact FE-results. Dashed curves are obtained from the friction-free case, replacing according to Eq. (38) the ®ber load applied at the ®ber end by the reduced ®ber load acting at the crack tip.

Fig. 8. Fiber-end displacement on crack length during stable crack growth calculated by FE-analysis with interface friction (solid), and using approximation (43) with FE-data corresponding to zero friction.

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H. Kessler et al. / Composites Science and Technology 59 (1999) 2231±2242

During stable crack growth, energy conservation requires G ˆ Gc , and hence, from Eq. (38), we get s 4Ef Gc 2c ‡ f ˆ rf rf g…c; Ef =Em ; m †

…44†

Eqs. (43), (44), and P ˆ r2f f determine parametrically the load/displacement curve shown in Fig. 9.

During reloading we have 8 2 rf fc2 c > > < if fc <
r ‡ fc 1 C…c† rf 8 uc ˆ lfree ‡ 2 Ef Ef Co …c† > >  c ÿ c if  >
 : r fc fc rf

…47†

for short cracks or low friction behavior, and

3.3. Load/displacement hysteresis curve and dissipated energy

ÿ
  fc 1 C…c† c2 rf f2 ÿ fc2
f c ÿ ‡ ÿ uc ˆ lfree Ef Ef Co …c† rf 8

We assume that the ®ber-stress pro®le for a sti€ matrix, determined by Eqs. (5) and (9), applies without changes to the case of an elastic matrix. Then, using again approximation (42), we obtain the current ®berend displacement during unloading

in the opposite case. For low friction behavior, the crack length/compliance relationships, Eqs. (28) and (29) can be generalized as duc lfree 1 C…c†
c ˆ ‡ dfc Ef Ef Co …c†

8 c2 rf
…f ÿ fc †2 > >  c ÿ ÿ > f > > rf 8 > > > > if fc > f ÿ
r < fc 1 C…c† ‡ uc ˆ lfree Ef Ef Co …c† > > > c2 > > > fc c ‡ > > rf > : if fc < f ÿ
r In particular, the residual displacement at complete unloading is

ures

…49†

with fc ˆ 0 for the unloading branch and fc ˆ f for the reloading branch. Consider the hysteresis cycle for incomplete unloading with a minimum load min > f ÿ
r . Eq. (32) is generalized as …45†

8 2 c > > > < rf 1 C…c†
 ˆ  Ef Co …c† > c2 rf f2 > > ÿ : f c ÿ rf 8

…48†

for short cracks; c < cr for long cracks; c > cr …46†

uc ˆ lfree

  fc 1 C…c† rf …f ÿ fc †2
f
c ÿ 
c2 ÿ ‡ Ef Ef Co …c† 8

…unloading† …50† for unloading, and fc 1 C…c† ‡ uc ˆ lfree Ef Ef Co …c†   rf …f ÿ fc †…f ‡ fc ÿ 2min †
f
c ÿ 
c2 ÿ 8 …51† for reloading. For an incomplete unloading hysteresis cycle with a minimum load min > f ÿ
r , the load/ displacement slope at fc ˆ min (unloading) and fc ˆ f (reloading) follows as duc lfree rf C…c† f ÿ min ˆ ‡ dfc Ef Ef Co …c† 4

…52†

The energy dissipated during one such cycle is calculated as Fig. 9. Load/displacement curve calculated by FE-analysis with interface friction (solid), and using the approximation (43) with FEdata corresponding to zero friction.

Wdiss ˆ

C…c† r3f …f min †3 Co …c† 24Ef 

…53†

H. Kessler et al. / Composites Science and Technology 59 (1999) 2231±2242

4. Critical energy release rate and friction stress measurement Toughness evaluation from load/displacement curves is straightforward only for the friction-free case. In the presence of crack face-friction, load/displacement curves depend on at least one more parameter, the friction stress . The methods outlined below allow to determine both interface parameters, the interface toughness (Gc ) and interface friction stress (). 4.1. Extrapolation method of the load/displacement curve Consider the case of in-situ crack observation during load-controlled crack growth. Assume the crack has reached a length c ˆ ci  a=2 which at zero-friction corresponds to instability. Approximating according to Eq. (38) the energy release with interface friction rate by its friction-free equivalent, Gfriction  G…f ÿ 2c=rf ; c†, we have     @G @G df 2 ÿ ‡ …54† 0ˆ @c f ÿ2c=rf ˆconst @f dc rf As the friction-free instability crack length corresponds to @G=@c…c ˆ ci † ˆ 0, we obtain df …ci † 2 ˆ dc rf

…55†

Using Eq. (55), the friction force can be deduced by extrapolation as demonstrated in Fig. 10. Then, after calculating  from friction force, crack length and ®ber stress according to Eq. (37), the interface toughness can be determined from (38). Note that at c ˆ ci , the di€erence between Eq. (38) for the energy release rate in the case of elastic matrix and Eq. (14) for the energy release rate in the case of a sti€ matrix is small (Fig. 11), and either of them can be used.

Fig. 10. Extrapolation of the current friction force from the frictionfree instability crack length, c ˆ ci , to c ˆ 0.

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Besides in-situ monitoring, the crack length can be determined from unload or reload hysteresis slopes according to Eq. (49). This method requires low friction behavior (reversal of the friction stress at unloading over the total debond length) and a series of additional friction-free FE-analyses to obtain the compliance C…c†. It can be considered as a straightforward extension of standard procedures to estimate a crack length from compliance measurements in the friction-free case. 4.2. Residual displacement method In a di€erent approach, the crack length can be eliminated by the residual ®ber end displacement at complete unloading (ures ). From Eqs. (43) and (46) follows f ÿ

2c ˆ f
h…u ÿ
lfree ; ures † rf

…56†

where the function h is de®ned as 8u ÿ u 1 2 > for short cracks; c < cr > > for long cracks; c > cr : 1ÿ 2 u1 ÿ u2 …57† Alternatively, using the dissipated energy during incomplete unloading cycles (Wdiss ) according to Eq. (53), h can be expressed as s r2 …f ÿ min †3 …u ÿ
lfree † hˆ 1ÿ f 6f2 Wdiss

…58†

Inserting Eq. (56) into, (38) the crack length dependence of the energy release rate can be represented as rf
f2
h…u

4Ef g…c; Ef =Em ; m † gmin 1 ˆ 5  2 G G G ÿ
lfree ; ures † c c c …59†

Fig. 11. Crack length dependence of the ratio of energy release rates for elastic and sti€ matrix, respectively.

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H. Kessler et al. / Composites Science and Technology 59 (1999) 2231±2242

Evaluation of the minimum left hand side of Eq. (59) with high accuracy displacement measurements for repeated loading/unloading cycles gives an approximate lower bound for the interface toughness (Gc ). For a low or high friction stress, the ®rst or second form of the

function h…u1 ; u2 † should be used which correspond to short or long cracks, respectively. Alternatively, h can be evaluated from energy dissipation measurements during small hysteresis cycles with incomplete reversal of the friction stress over the total debond length. This should work in all cases. 5. Summary and conclusions

Fig. 12. In¯uence of interface friction on mixed mode angle (reference length r=1 mm). solid: no friction; small dashing: low ratio friction stress to applied ®ber stress (=f ˆ 3:8 MPa/1 GPa); large dashing: high ratio friction stress to applied ®ber stress (=f ˆ 3:8 MPa/0.05 GPa).

Fig. 13. FE-mesh showing the densely meshed crack tip in the droplet center. For symmetry reasons, meshing was limited to the right droplet half (®ber on the left-hand side).

In the preceeding paragraphs, the e€ect of a constant interface friction shear stress was considered with regard to relations between debond crack length, ®ber load and ®ber-end displacement. The analysis applies to conditions of a neglegible small plastic zone at the crack tip, initial defects large enough to facilitate stable growth of the debond crack, load-controlled crack growth, and a constant critical energy release rate (interface toughness) which does not depend on the mixed mode angle. The results are useful for interface toughness measurements and can be summarized in the following statements: 1. For conditions of zero friction and an elastic matrix, a debond crack becomes unstable when it reaches approximately half of the droplet size (Fig. 3). This crack length corresponds to the peak load of the load/displacement curve. By a FE-analysis, the interface toughness can be estimated from this peak load, the geometry and elastic constants of ®ber and droplet. However, even for a relatively compliant matrix, the explicit Eq. (14) gives (with  ˆ 0) a close lower estimate of the interface toughness. No droplet properties and no FE-analysis are needed in this approximation. 2. Interface friction e€ectively unloads the debond crack front. The energy release rate can be reduced to the friction free case by substituting the ®ber load at the crack tip for the applied ®ber load [Eqs. (36)± (38)]. For a sti€ matrix, there is no approximation involved. The friction debond force which determines the ®ber load at the crack tip can be estimated in two ways: by in-situ observation of the crack length and tangential extrapolation of the load/crack length curve to zero crack length (Fig. 10), or by taking advantage of the frictionally induced displacement hysteresis during repeated loading/unloading cycles [Eq. (59)]. In the latter case, an additional approximation assumes a friction-independent ratio of crack tip displacement to ®ber elongation [Eq. (42)]. For displacement hysteresis measurements, a good resolution, perhaps better than 1 mm, is required. For the case of low friction, the crack length can be determined from the slope of unloading/reloading hysteresis cycles by Eq. (49). 3. For the microbond test, the problem to determine a mixed mode dependent interface toughness

H. Kessler et al. / Composites Science and Technology 59 (1999) 2231±2242

remains unsolved. For not too small crack lengths, the mode mixity remains approximately constant and corresponds to a dominating interface shear load which is shifted towards mode-I loading by a higher ratio of friction load to applied load (Fig. 12). However, it appears impossible to adjust the mixed mode angle to a prede®ned value. Acknowledgements We appreciate useful discussions with E. Pisanova, V. Dutschk and S. Zhandarov, Institute of Polymer Science Dresden, and with H.-A. Bahr, Department of Mechanical Engineering at the Technical University Dresden. Appendix 1: FE-model The FE-model of a debond crack between isotropic, linear-elastic ®ber and droplet was veri®ed for the following particular combination of geometry and material parameters: . Young's modulus (E) and Poisson ratio () for ®ber (glass) and droplet (epoxy): Ef ˆ 64 GPa; Em ˆ 3 GPa; f ˆ 0:20; m ˆ 0:35 . Critical energy release rate: Gc ˆ 2 J=m2 . Constant interface friction shear stress:  ˆ 3:8 MPa . Fiber radius: rf ˆ 5mm . Axial and radial droplet radius: a=75 mm, b=60 mm The constraint imposed by the knife edges was simulated by suppression of the normal displacements of droplet surface nodes. The radial extension of this region (the difference between its outer and inner diameter) was kept equal to the ®ber diameter. Its inner radius was set equal to the ®ber radius. The details of crack initiation close to the knife edges are hardly accesible for detailed modeling. Moreover, for stable crack propagation, they do not in¯uence most of the load/displacement curve. Therefore, the FE-analysis of crack propagation between ®ber and elastic droplet was restricted to well formed debond cracks with crack lengths between half of the ®ber radius up to 90% of the axial droplet diameter. The in¯uence of friction was investigated for a given interface roughness and Coulomb friction law. Friction was incorporated by a special arrangement of regular contact elements described in Ref. [5], satisfying the following physical requirements: . Re-closure of an already opened debond crack is limited to the height of interface ledges. . The friction force between opposite crack faces is determined by the average slope of interface ledges and Coulomb friction.

2241

This friction model allows to explore the transition between dominating friction at small COD, applied loads and interface toughness to neglegible friction at large CODs, loads and toughness values, respectively. In the present paper, only the limiting case of dominating friction was considered, corresponding to a constant interface friction stress. The FE-mesh was generated from axisymmetric, isoparametric eight-node elements. The singular crack tip stress ®eld was captured by dense meshing of singular six-node crack tip elements (element size 1/1000 of the ®ber radius; see Fig. 13). Appendix 2: Energy release rate of a debond crack at the interface between ®ber and sti€ matrix droplet The external work done by the applied ®ber stress (f ) is dWex du ˆ r2f f dc dc

…A1†

From Eq. (6), employing Hooke's law and integrating the axial ®ber strain, we obtain the ®ber displacement along the debond crack (0 < z < c) as:  
1 ÿ 2 2 f …c ÿ z† ÿ c ÿz for 0 < z < c u… z† ˆ Ef rf …A2† with Ef denoting the ®ber modulus. The extension of the free ®ber end (
lfree ) is given by
lfree ˆ 1

f Ef

…A3†

The total ®ber end displacement consists of
lfree and u…z ˆ 0†:   f 1 c2 f c ÿ u ˆ
lfree ‡ u…z ˆ 0† ˆ lfree ‡ …A4† Ef Ef rf Hence, from Eq. (A1), the work done by the applied ®ber load is   dWex r2f 2f 2 ˆ f ÿ c …A5† dc Ef rf The strain energy is derived by integration of the strain energy density, f2 …z†=2Ef , over the debonded and free ®ber segments ( ÿ lfree < z < c). For load-controlled conditions, the free ®ber part is constant and has no e€ect on the energy release rate. Using Eq. (6) and taking the derivative, we obtain   dUel r2f 2c 2 ˆ f ÿ rf dc 2Ef

…A6†

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H. Kessler et al. / Composites Science and Technology 59 (1999) 2231±2242

During an incremental crack advance, the friction work done against the ®ber/matrix interfacial shear stress is …c …A7† Wfr ˆ 
u…z†
2rf dz 0

or, with Eq. (A2)   dWfr 2rf c 2c ˆ
f ÿ dc Ef rf

…A8†

Combining Eqs. (A5), (A6) and (A7), the energy release rate of the debond crack is derived as   rf 2c 2 f ÿ …A9† Gˆ 4Ef rf

References [1] Scheer RJ, Nairn JA. A comparison of several fracture mechanics methods for measuring interfacial toughness with microbond tests. J Adhesion 1995;53:45±68. [2] Pisanova E, Dutschk V, Lauke B. Work of adhesion and local adhesional strength in glass ®bre polymer systems. J Adhesion Sci Technol 1998;12(3):305±22. [3] Anderson TL. Fracture mechanics. London: CRC Press, 1995. [4] Gorbatkina Y. Adhesive strength of ®ber-polymer systems. New York/London: Ellis Horwood, 1992. [5] SchuÈller T, Bahr U, Beckert W, Lauke B. Fracture mechanics analysis of the microbond test. Composites Part A (special conference issue IPCM '97) 1998;29A:1083±9. [6] Outwater JO, Murphy MC. Fracture energy of unidirectional laminates. Modern Plastics 1970;47:160±9. [7] Beckert W. Modellierung des bruchmechanischen Verhaltens von Faserverbundwerksto€en mittels der Methode der ®niten Elemente. PhD thesis, University Kaiserslautern, 1994 (in German).