A powerful analytical methodology is discovered and is used to partition the total energy release rate of one dimensional (1D) interface fractures on brittle homogeneous or bi-material interfaces into its mode I and II components based on the classical and shear-deformable plate theories and 2D elasticity. Previously unsolvable problems are solved and confusions explained

Christopher Harvey (1256223)

Joseph D. Wood (7119710)

Simon Wang (1250952)

English

Loughborough University Institutional Repository

21ST International Conference on Composite Materials Xi’an, 20‐25th August 2017 THE MECHANICS OF INTERFACE FRACTURE IN LAYERED COMPOSITE MATERIALS: (1) BRITTLE HOMOGENEOUS AND BI-MATERIAL INTERFACES  Simon S. Wang1, 2, Christopher M. Harvey1 and Joseph D. Wood3  1 Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK E-mails: s.wang@lboro.ac.uk, c.m.harvey@lboro.ac.uk  2 School of Mechanical and Equipment Engineering, Hebei University of Engineering, Handan, China  3 Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK E-mail: joseph.wood@imperial.ac.uk  Keywords: Energy release rates, Interface fracture, Layered composite materials, Mixed mode parti-tion, Orthogonal pure modes  ABSTRACT A powerful analytical methodology is discovered and is used to partition the total energy release rate of one dimensional (1D) interface fractures on brittle homogeneous or bi-material interfaces into its mode I and II components based on the classical and shear-deformable plate theories and 2D elas-ticity. Previously unsolvable problems are solved and confusions explained. 1 INTRODUCTION Layered composite materials find many advanced applications in science, engineering and technol-ogy sectors. A major concern is the interface fracture during service, which leads to tremendous re-search effort on the mechanics of interface fracture in layered composite materials under various ser-vice conditions. Four well known traditional fundamental concepts on the topic are energy release rate (ERR), stress intensity factors (SIFs), virtual crack closure technique (VCCT) and J-integral, all of which are maturely understood. One less well known fundamental concept is the mixed mode partition (MMP). It, however, plays a key role in studying the mechanics of interface fracture since interface fracture toughness is not a purely intrinsic material property, but instead also depends on the mode mixity. There has been a great deal of controversies and confusions on MMP in last few decades until recently when a powerful orthogonal pure-mode partition theory (OPPT) [1-8] is discovered based on a fundamental understanding of the mechanics of interface fracture. This paper aims to give a brief summary on OPPT. 2 ORTHOGONAL PURE-MODE PARTITION THEORY (OPPT) Although interface fracture generally occurs as mixed-mode fracture with all three opening, shear-ing and tearing actions (i.e. mode I, II and III), 1D interface fractures have received much more atten-tion as it is simpler and still captures the essential fracture mechanics. The expression ‘1D interface fracture’ means that a fracture propagates in one direction with mode I and mode II actions only. Ex-amples of 1D interface fracture are shown in Fig. 1 including through-width delamination in double cantilever straight and curved beams and blisters in layered plates and shells. A central task in study-ing 1D interface fracture is to partition the total energy release rate (ERR) of a mixed-mode fracture into its individual mode I and II ERR components. The total ERR G  of a 1D mixed-mode fracture on a brittle homogeneous or bi-material interface can be expressed in the following form.     TBBBBBBBBBBBB PPNNMMCPPNNMMG 212121212121  (1)  Simon S. Wang, Christopher M. Harvey and Joseph D. Wood It is a quadratic form non-negative definite in terms of the crack tip bending moments per unit width BM1 , BM 2 , axial forces per unit width BN1 , BN2  and shear forces per unit width BP1 , BP2  as shown in Fig. 2 with subscripts 1 and 2 denoting the layers above and below the crack and with B denoting the crack tip. The coefficient matric  C  depends on material properties, layups of the layered materi-als and location of the interface fracture.   Figure 1: Examples of 1D interface fracture.   Figure 2: A representative 1D interface fracture model. (a) General description. (b) Details local to the crack tip.  21ST International Conference on Composite Materials Xi’an, 20‐25th August 2017 By using the VCCT, the individual mode I and II ERR components, that is, IG  and IIG  can be written in the following equations:  adFG nnaI   2lim 0  (2)  adFG ssaII   2lim 0  (3) where a  is the crack extension size, nF  and nd  are the crack tip opening force and relative opening displacement, respectively whilst sF  and sd  are the crack tip shearing force and relative shearing dis-placement, respectively. Obviously, the determination of nF , nd , sF  and sd  play the key role in the calculation of IG  and IIG . In the case of rigid interface fractures considered here both nF  and sF  have finite values due to the stress singularity at the crack tip. However, both the relative crack open-ing and shearing displacements right at the crack tip B are negligible. In the VCCT, i.e. Eqs. (2) and (3), the relative crack tip opening displacement nd  and shearing displacement sd  are measured at the location immediately behind the crack tip B by assuming crack extension similarity. The traditional analytical approach aims to obtain nF , nd , sF  and sd  by solving differential equations with a very limited capability leaving some important problems unsolved and confusions. A powerful orthogonal pure-mode partition theory (OPPT) is discovered recently. The OPPT ex-presses the IG  in Eq. (2) and IIG  in Eq. (3) in the following general equations:   5241322112152413221121BBBBBBBBBBBBIIPPNNMMPPNNMMcG (4)   5241322112152413221121BBBBBBBBBBBBIIIIPPNNMMPPNNMMcG (5) where Ic  and IIc  are two constants which are material property, layup and thickness ratio dependent. The first and second brackets of Eq. (4) represent nF  and nd  in Eq. (2), respectively. When either of them equals to zero, mode I ERR IG  becomes zero leading to a pure mode II fracture mode. The first bracket becomes zero in the following loading conditions:      00001 1212121 BBBBBB PPNNMM  (6)     00001 2212121 BBBBBB PPNNMM  (7)     00001 3212121 BBBBBB PPNNMM  (8)     00001 4212121 BBBBBB PPNNMM  (9)     5212121 00001 BBBBBB PPNNMM  (10) which are called force type pure mode II modes. The second bracket becomes zero in the following loading conditions:  Simon S. Wang, Christopher M. Harvey and Joseph D. Wood     00001 1212121 BBBBBB PPNNMM  (11)     00001 2212121 BBBBBB PPNNMM  (12)     00001 3212121  BBBBBB PPNNMM  (13)     00001 4212121 BBBBBB PPNNMM  (14)     5212121 00001  BBBBBB PPNNMM  (15) which are called displacement type pure mode II modes. Similarly, the first and second brackets of Eq. (5) represent sd  and sF  in Eq. (2), respectively. When either of them equals to zero, mode II ERR IIG  becomes zero leading to pure mode I. The first bracket becomes zero in the following loading condi-tions:      00001 1212121 BBBBBB PPNNMM  (16)     00001 2212121 BBBBBB PPNNMM  (17)     00001 3212121 BBBBBB PPNNMM  (18)     00001 4212121 BBBBBB PPNNMM  (19)     5212121 00001 BBBBBB PPNNMM  (20) which are called displacement type pure mode I modes. The second bracket becomes zero in the fol-lowing loading conditions:      00001 1212121 BBBBBB PPNNMM  (21)     00001 2212121 BBBBBB PPNNMM  (22)     00001 3212121  BBBBBB PPNNMM  (23)     00001 4212121 BBBBBB PPNNMM  (24)     5212121 00001  BBBBBB PPNNMM  (25) which are called force type pure mode I modes. All i , i  , i  and i  (with 5,4,3,2,1i ) are materi-al property, layup and thickness ratio dependent. It is extremely difficult if not impossible to find their analytical expressions using the traditional approach. This is particularly true to find the analytical ex-pressions for i  and i  as they are related to the interface stresses with singularity at the crack tip. In the powerful OPPT, the two most fundamental displacement type pure modes Eq. (11) and Eq. (16) , i.e. 1   pure mode II mode and 1  pure mode I mode are determined first either analytically or numer-ically or experimentally. Then, all other pure modes given in the above equations can be determined by using the powerful orthogonality principle. That is, for examples, using the following equations to find 1  and 2  when 1  is known.       00000100001 11 TC   (26)      00000100001 12 TC   (27) and then using the following equations to find 2  and 3 .      00000100001 21 TC   (28)      00000100001 31 TC   (29) 21ST International Conference on Composite Materials Xi’an, 20‐25th August 2017 Similarly, for examples, using the following equations to find 1  and 2  when 1   is known.       00000100001 11  TC   (30)      00000100001 12  TC   (31) and then using the following equations to find 2   and 3  .      00000100001 21  TC   (32)      00000100001 31  TC   (33)  ii  ,  and  ii  ,  (with 5,4,3,2,1i ) are called double set orthogonal pure modes. The orthogo-nality exists between any pairs in each set. The double set orthogonal pure modes are material proper-ty, layup and thickness ratio dependent. More importantly, they depend on the choice of mechanical theories.  When Euler beam or classical plate/shell theory are used Eqs. (4) and (5) become    EBEBEBBEBEBEBBIEIENNMMNNMMcG32211213221121   (34)    EBEBEBBEBEBEBBIIEIIENNMMNNMMcG32211213221121   (35) for both brittle homogeneous and bi-material interfaces where the subscript E denotes Euler beam or classical plate/shell theory. The two sets of orthogonal pure modes are different from each other and the through thickness shearing effects disappear. It was used to be claimed that mixed mode partition was unsolvable based on Euler beam or classical plate/shell theory. The powerful OPPT solves it. In the case of thin layer delamination Eqs. (34) and (35) reduce to    EBBEBBIEIENMNMcG211211   (36)    EBBEBBIIEIIENMNMcG211211   (37) When Timoshenko beam or first order shear deformable plate/shell theory are used Eqs. (4) and (5) become  252413221121   TBTBTBTBTBBITITPPNNMMcG   (38)  23221121   TBTBTBBIITIITNNMMcG   (39) for both brittle homogeneous and bi-material interfaces where the subscript T denotes Timoshenko beam or first order shear deformable plate/shell theory. The two sets of orthogonal pure modes coin-cide at the first set of orthogonal pure modes  ii  ,  and the through thickness shearing effects only produce mode I ERR. It was thought that Eqs. (34) and (35) were equal to Eqs. (38) and (39), respec-tively in the absence of thoroughness shear forces. The powerful OPPT clears the confusion. In the case of thin layer delamination Eqs. (38) and (39) reduce to  Simon S. Wang, Christopher M. Harvey and Joseph D. Wood  241211   TBTBBITITPNMcG   (40)  2211  TBBIITIITNMcG   (41) When 2D elasticity theory is used Eqs. (4) and (5) become  2252241232221212122   DBDBDBDBDBBDIDIPPNNMMcG   (42)  2252241232221212122   DBDBDBDBDBBDIIDIIPPNNMMcG   (43) for brittle homogeneous interfaces where the subscript 2D denotes 2D elasticity theory. The two sets of orthogonal pure modes coincide at each other and the through thickness shearing effects produce both mode I and II ERRs. In the case of thin layer delamination Eqs. (42) and (43) reduce to  2241221122   DBDBBDIDIPNMcG   (44)  2221122   DBBDIIDIINMcG   (45) Note that the through thickness shearing effects only produce mode I ERR for thin layer delamination. For brittle bi-material interfaces, Eqs. (4) and (5) remain the same, that is,   DBDBDBDBDBBDBDBDBDBDBBDIDIPPNNMMPPNNMMcG2522412322212121252241232221212122  (46)   DBDBDBDBDBBDBDBDBDBDBBDIIDIIPPNNMMPPNNMMcG2522412322212121252241232221212122  (47) The two sets of orthogonal pure modes are not only different from each other but are also crack exten-sion size-dependent. In the case of thin layer delamination Eqs. (46) and (47) reduce to    DBDBBDBDBBDIDIPNMPNMcG2412211241221122   (47)    DBBDBDBBDIIDIINMPNMcG2211241221122   (48) Finally, it is worth noting that the existence of double sets of orthogonal pure modes in the Euler beam or classical plate/shell partition theory is due to its global nature while the existence of double sets of orthogonal pure modes in 2D elasticity partition theory for brittle bi-material interfaces is due to the material mismatches resulting in phase difference in the variation of interface stresses and rela-tive separations. When the extension size a  is big enough, both the three partition theories will unify 21ST International Conference on Composite Materials Xi’an, 20‐25th August 2017 at the classical partition theory in the absence of the through thickness shear forces. That is why both the shear deformable and 2D elasticity partition theories are called local partition theory while the classical partition theory is called global partition theory. Validity of each theory has been assessed in a series of studies. Details of the orthogonal pure modes  ii  ,  and  ii  ,  in each theory can be found in the work [1-8]. 3 SOME COMPARISONS WITH FEM RESULTS  Fig. 3 shows the comparison between the present analytical predictions based on 2D elasticity par-tition theory with the marked FEM results for a bi-material interface. The details of the material prop-erties and FEM can be found in the work [6]. Excellent agreement is observed. Fig. 4 shows the comparison between the present analytical predictions based on 2D elasticity par-tition theory with FEM results for homogeneous interface fracture under general loading conditions. The details of the material properties and the FEM simulation can be found in the work [7]. Excellent agreement is observed again.  Figure 3: Comparison of the present analytical theory and FEM data for the ERR partition GGI  at crack extension size 05.0a  with 1012 BB MN  and various thickness ratios. 12 / hh  at a bi-material interface.  Simon S. Wang, Christopher M. Harvey and Joseph D. Wood  Figure 4: Comparison of the present analytical theory and the 2D FEM for the total ERR G  and the ERR partition GGI  for variable   and loading conditions BB PP 12 , BB MP 11  and BBe PN 11 . 21ST International Conference on Composite Materials Xi’an, 20‐25th August 2017 4 CONCLUSIONS The powerful OPPT reveals some fundamental mechanics of interface fractures, and solves previ-ously unsolvable problems and clears previous confusions. The analytical theories provide valuable means for studying interface fractures on macroscopic, microscopic and nano-scales. REFERENCES [1]  S. Wang S and C.M. Harvey, A theory of one-dimensional fracture. Composite Structures, 94, 2012, pp.758–767 (doi: 10.1016/j.compstruct.2011.09.011). Also a plenary lecture at the 16th International Conference on Composite Structures (ICCS16), 28–30 June 2011, Porto, Portugal. [2]  S. Wang and C.M. Harvey, Mixed mode partition theories for one dimensional fracture, Engi-neering Fracture Mechanics, 79, 2012, pp. 329-352 (doi: 10.1016/j.engfracmech.2011.11.013). [3]  C.M. Harvey and S. Wang, Mixed mode partition theories for one dimensional delamination in laminated composite beams, Engineering Fracture Mechanics, 96, 2012, pp. 737-759 (doi: 10.1016/j.engfracmech.2012.10.001). [4]  C.M. Harvey, J.D. Wood, S. Wang and A. Watson, A novel method for the partition of mixed-mode fractures in 2D elastic laminated unidirectional composite beams, Composite Structures 116, 2014, pp. 589-594 (doi: 10.1016/j.compstruct.2014.05.041). [5]  C.M. Harvey, J.D. Wood and S. Wang, Brittle interfacial cracking between two dissimilar elas-tic layers: Part 1- Analytical development, Composite Structures, 134, 2015, pp. 1076-1085 (doi: 10.1016/j.compstruct.2015.06.080). [6]  C.M. Harvey, J.D. Wood and S. Wang, Brittle interfacial cracking between two dissimilar elas-tic layers: Part 2 - Numerical verification, Composite Structures, 134, 2015, pp. 1087-1094 (doi: 10.1016/j.compstruct.2015.06.079). [7]  J.D. Wood, C.M. Harvey and S. Wang, Partition of mixed-mode fractures in 2D elastic ortho-tropic laminated beams under general loading, Composite Structures, 149, 2016, pp. 239-246 (doi: 10.1016/j.compstruct.2016.04.016). [8]  J.D. Wood, C.M. Harvey and S. Wang, Effect of Poisson’s ratio mismatch on brittle interfacial cracking between two dissimilar elastic layers, Composite Structures, 158, 2016, pp. 56-63 (doi: 10.1016/j.compstruct.2016.09.024). 

The mechanics of interface fracture in layered composite materials: (1)  brittle homogenous and bi-material interfaces

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The mechanics of interface fracture in layered composite materials: (1) brittle homogenous and bi-material interfaces

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