Composites: Part B 27B (1996) Printed in Great Britain. All rights ELSEVIER 13598368(9500035-6 13598368 Delaminations in composite structures: its origin, buckling, growth and stability Vladimir V. Bolotin Russian Academy of Sciences, Institute of Mechanica/ Engineering 101830 Moscow Centre, russia (Received 28 January 1995; accepted 14 June 1995) The most up-to-date research in the nations and related crack-like defects in laminate and fiber composites is discussed urface delaminations are considered In the latter case, local buckling of delamination between buckling, damage accumulation, crack growth and global buckling are cons of the evaluation of the residual load-carrying capacity of delaminated structural ssed, including the assessment of the fracture toughness with respect to impact loading. I INTRODUCTION structures. All these types of interlaminar defects will be discussed in this paper Delaminations. ie. interlaminar cracks and crack-like defects, are as typical for composite structures as, say, fatigue cracks are for common metal structures. 2 BRIEF HISTORICAL REMARKS Two kinds of delaminations are to be distinguished depending on their position in a structural member Although the intensive studies in the mechanics of Delaminations situated within the bulk of the mater delaminations were initiated in the 1980s the historic [Figure I(a)] are rather like the cracks studied in background goes back much further. In this context, the conventional fracture mechanics. The edge delamina- paper by Obreimoff ought to be cited. It was dedicated tions in thick members may also be partially attributed to to the assessment of the surface energy in splitting of this type of delamination. Delaminations situated near mica [Figure 2(a)]. Being not a composite, but an the surface of a structural member the surface delamina- extremely anisotropic natural laminate, mica is similar to tions[ Figure /(b)] are a special kind of crack-like defect. many modern composite materials. It seems appropriate The behavior of surface delaminations is, as a rule to mention that many years later, Stewart et al. analyzed accompanied by their buckling. Dealing with the surface the mechanical properties of body muscles which are also delaminations, we ought to take into account not only a kind of natural laminate material in a similar way their growth and interlaminar damage but also their With the direct reference to composite structures, the ability considered from the viewpoint of the theory of problem of delaminations was primarily considered by elastic stability. In addition, the local instability and Kulkarni and Frederick and Kachanov". In particular crack growth may produce the global instability of Kachanov considered a fiber-glass tube under compres structural components such as columns, plates and shells sion with a delamination situated near the internal under compression. Hence, the joint analysis of damage, surface [ Figure 2(b). The literature in this field consists fracture, local buckling and global stability is frequently of several dozen papers. A survey of Russian publica- required to predict the load-carrying capacity of tions could be found in book by Bolotinand a paper by composite structures with delaminations Grigolyuk et al. A survey of Western publications is Not only complete delaminations, but also the multiple presented in papers by Garg and Storakers. Many cracking without a separation of layers [Figure I(c) works had been performed earlier and at least in the typical for composite structures. These crack-like flaws USSR, were published with a significant delay. The also affect the load-carrying capacity and the safe life of presented survey is based mostly on the Russian iterature including the papers published in Mekhanika Kompozit- This paper was presented at the First International Conference nykh Materialov(Mechanics of Composite Materials) Composites Engineering (ICCE/I, New Orleans, 28-31 August edited in Riga, now the capital of Latvia
ELSEVIER 1359-8368(95)00035-6 Composites: Part B 27B (1996) 129-145 Copyright © 1996 Published by Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/96/$15.00 Delaminations in composite structures: its origin, buckling, growth and stability* Vladimir V. Bolotin Russian Academy of Sciences, Institute of Mechanical Engineering, 101830 Moscow Centre, Russia (Received 28 January 1995; accepted 14 June 1995) The most up-to-date research in the mechanics of delaminations and related crack-like defects in laminate and fiber composites is discussed. Both internal and near-surface delaminations are considered. In the latter case, local buckling of delaminations and the interaction between buckling, damage accumulation, crack growth and global buckling are considered. The problem of the evaluation of the residual load-carrying capacity of delaminated structural components is discussed, including the assessment of the fracture toughness with respect to impact loading. 1 INTRODUCTION Delaminations, i.e. interlaminar cracks and crack-like defects, are as typical for composite structures as, say, fatigue cracks are for common metal structures. Two kinds of delaminations are to be distinguished, depending on their position in a structural member. Delaminations situated within the bulk of the material [Figure 1 (a)] are rather like the cracks studied in conventional fracture mechanics. The edge delaminations in thick members may also be partially attributed to this type of delamination. Delaminations situated near the surface of a structural member, the surface delaminations [Figure 1 (b)] are a special kind of crack-like defect. The behavior of surface delaminations is, as a rule, accompanied by their buckling. Dealing with the surface delaminations, we ought to take into account not only their growth and interlaminar damage but also their stability considered from the viewpoint of the theory of elastic stability. In addition, the local instability and crack growth may produce the global instability of structural components such as columns, plates and shells under compression. Hence, the joint analysis of damage, fracture, local buckling and global stability is frequently required to predict the load-carrying capacity of composite structures with delaminations. Not only complete delaminations, but also the multiple cracking without a separation of layers [Figure 1 (c)] is typical for composite structures. These crack-like flaws also affect the load-carrying capacity and the safe life of * This paper was presented at the First International Conference on Composites Engineering (ICCE/1), New Orleans, 28-31 August 1994 structures. All these types of interlaminar defects will be discussed in this paper. 2 BRIEF HISTORICAL REMARKS Although the intensive studies in the mechanics of delaminations were initiated in the 1980s, the historical background goes back much further. In this context, the paper by Obreimoff 1 ought to be cited. It was dedicated to the assessment of the surface energy in splitting of mica [Figure 2 (a)]. Being not a composite, but an extremely anisotropic natural laminate, mica is similar to many modern composite materials. It seems appropriate to mention that many years later, Stewart et al. 2 analyzed the mechanical properties of body muscles which are also a kind of natural laminate material in a similar way. With the direct reference to composite structures, the problem of delaminations was primarily considered by Kulkarni and Frederick 3 and Kachanov 4. In particular, Kachanov 4 considered a fiber-glass tube under compression with a delamination situated near the internal surface [Figure 2 (b)]. The literature in this field consists of several dozen papers. A survey of Russian publications could be found in book by Bolotin 5 and a paper by Grigolyuk et al. 6. A survey of Western publications is presented in papers by Garg 7 and Storakers ~. Many works had been performed earlier and, at least in the USSR, were published with a significant delay. The presented survey is based mostly on the Russian literature including the papers published in Mekhanika Kompozitnykh Materialov (Mechanics of Composite Materials) edited in Riga, now the capital of Latvia. 129
Delaminations in composite structures: V. V. Bolotin Figure 1 Three types of delaminations: (a) internal,(b) near-surface and (c) multiple cracking khr (b Figure 2 Pioneering studies in mechanics of delaminations: (a) Obreimoff's study in splitting of mica, (b) Kachanov's problem of the compressed laminate tube 12 T-Tr bution of the residual specific fracture work along the 1-3 correspond to three increasing magnitudes of the Figure 3 Epox ies variation during the thermal treatment: 0 is shrinkage ratio, and E is the 10s Youngs modulus 3 DELAMINATIONS ORIGINATING IN THE MANUFACTURING PROCESS High strength of most laminated and fibrous composites in the direction of reinforcement is accompanied by low resistance against interlaminar shear and transverse ension. Therefore, the interlaminar cracks can originate both on the fabrication stage and on the stages of transportation, storage and service. Instabilities of the manufacturing process, imperfections of various natures, and thermal and chemical shrinkage of components may be the source of initial delaminations Figure 6 Distribution of summed cracked arca along the depth of a Delaminations in large-scale composite structures were specimen; coordinate is measured from the impacted surface met in the design of the deep underwater vehicle for ocean research. The vehicle was designed as a stiffened spheroidal that time which component of the shrinkage of the epoxy shell of glass/epoxy laminate. A set of multiple cracks was resin is more responsible for the occurrence of tensile stresses found in pilot specimens of these shells, and the stated thermal, chemical, or both. It depends, obviously, on how the objective was to avoid these defects. It was evident that these variation of shrinkage and compliance correlate in time 0, II cracks were produced when the transverse tensile stresses To study the shrinkage and compliance in situ, an occurred on the manufacturing stage, But it was not clear at amount of the liquid epoxy resin in a thin elastic shell was 130
Delaminations in composite structures. V. V. Bolotin (a) (b) (c) Figure 1 Three types of delaminations: (a) internal, (b) near-surface and (c) multiple cracking I Figure 2 / (a) (b) Pioneering studies in mechanics of delaminations: (a) ObreimofFs study in splitting of mica, (b) Kachanov's problem of the compressed laminate tube 0 Figure 3 Epoxy resin properties variation during the thermal treatment: 0 is shrinkage ratio, and E is the 10 s Youn'g's modulus 3 DELAMINATIONS ORIGINATING IN THE MANUFACTURING PROCESS High strength of most laminated and fibrous composites in the direction of reinforcement is accompanied by low resistance against interlaminar shear and transverse tension. Therefore, the interlaminar cracks can originate both on the fabrication stage and on the stages of transportation, storage and service. Instabilities of the manufacturing process, imperfections of various natures, and thermal and chemical shrinkage of components may be the source of initial delaminations 9. Delaminations in large-scale composite structures were met in the design of the deep underwater vehicle for ocean research. The vehicle was designed as a stiffened spheroidal shell of glass/epoxy laminate. A set of multiple cracks was found in pilot specimens of these shells, and the stated objective was to avoid these defects. It was evident that these cracks were produced when the transverse tensile stresses occurred on the manufacturing stage. But it was not clear at 7,k//m 2 (a) (b) (c) Figure 4 Impact damage tests of specimens: (a) spheroidal-head impactor; (b) flat-head impactor; (c) three-point impact testing 2.0 1.6 1.2 0.8 20 .................. _o ......... _o__0__ ~|~"~O ~ l 0 • • 0 • 0 [] -{3 [] A O //° I I I I -40 -20 0 20 ~mm Figure 5 Distribution of the residual specific fracture work along the specimen; lines 1-3 correspond to three increasing magnitudes of the impactor energy 5 I I IO z~mm Figure 6 Distribution of summed cracked area along the depth of a specimen; coordinate z is measured from the impacted surface that time which component of the shrinkage of the epoxy resin is more responsible for the occurrence of tensile stresses: thermal, chemical, or both. It depends, obviously, on how the variation of shrinkage and compliance correlate in timel°'ll. To study the shrinkage and compliance in situ, an amount of the liquid epoxy resin in a thin elastic shell was 130
Delaminations in composite structures: V. V. bolotin placed in glycerin and subjected to thermal treatment by Adams and Adams" 2, Bolotin et al. 3-15and imilar to that in the fabrication of composite structures gdanovich and Yarve. In the paper by bolotin et The density of the resin was measured continuously by al 3 experimental results are presented for three types of Archimed weighting. Visco-elastic compliance of the resin composites: organic fiber/epoxy, graphite/epoxy and was assessed with the use of an indentor and the known glass-textile/epoxy laminates. Specimens on the rigid analytical solution of Hertz 's problem for linear visco- foundation as well as beam specimens were tested (Figure elastic materials. The results are shown schematically in 4). Impactors with flat and spheroidal heads up to 15 kg Figure 3. There, the temperature difference T-T(T is of mass and up to 600 J of initial energy were applied the room temperature), shrinkage ratio 8=(Polp)-1(p To evaluate the level of multiple cracking, the residual is the mass density), and 10s Youngs modulus E are fracture work in ply-after-ply peeling was measured. An presented as a function of time t. It was found that the example is presented in Figure 5, where the longitudina hemical shrinkage of the epoxy resin, due to the distribution of the specific fracture work y is presented molecular linking, is more significant than was stated by Specimens were placed on the rigid foundation, and the ne manufacturers of the resin. Happily, the most flat-head impactor with the diameter of 20 mm was used intensive chemical shrinkage takes place when the After the impact no separation of layers was observed material compliance is high, therefore, due to filtration, however, the fracture work under the impact area stress relaxation, etc, the contribution of the chemical diminished significantly (see lines 1, 2 and 3 in Figure 5 shrinkage to the transverse tensile stresses is compara- corresponding to three increasing levels of impact tively small. As a result, a gentle thermal treatment and, in energy ). Figure 6 illustrates the damage distribution particular, a more prolonged cooling were recommended along the depth for a beam specimen subjected to a three to lessen nondesirable stresses. At the same time, it was point impact [ Figure 4(c)]. The specimens were dyed recommended that resins with the lesser shrinkage were after the impact, split in layers, and the summed cracked sed, and that the stresses in question are residual ones area S was measured. Both the multiple cracking under and in combination with other actions, could become a the impact area and the significant delamination near the source of delaminations in the later life of a structure middle surface were observed Some additional data are presented in Section 9 in the context of the residual load-carrying capacity of composites aft 4 LOW-ENERGY IMPACT AS A SOURCE OF An analytical-numerical study of cracking under the DELAMINATIONS low-energy impact was performed by Bolotin and Grishko. The composite was modelled as a multilayered There are a lot of causes of new-born delaminations after a solid with alternating elastic and elasto-plastic layers structure is manufactured. Among them are various addi- Very special properties were attributed to elasto-plastic tional ones not accounted for in design, loads and actions: layers which imitate the matrix and the interface. The local forces, thermal actions, low-energy surface impacts odel includes the change of the unloading modulus and etc. Without proper design and fabrication, holes, notches secondary moduli due to damage as well as the existence and connections also serve as sources of delamination of the ultimate tensile strain that corresponds to the quasi In recent years, a study of multiple cracking of lami- brittle inter-layer rupture(Figure 7). The wave propaga nates under surface impact was performed independently tion in a laminate plate under a rectangular-shaped Figure 7 Analytical model of multiple cracking under low-energy impact: (a) composite as a multilayered solid; (b)strain-stress relationship for ccount of damage and brittle
placed in glycerin and subjected to thermal treatment similar to that in the fabrication of composite structures. The density of the resin was measured continuously by Archimed weighting. Visco-elastic compliance of the resin was assessed with the use of an indentor and the known analytical solution of Hertz's problem for linear viscoelastic materials. The results are shown schematically in Figure 3. There, the temperature difference T - Tr (Tr is the room temperature), shrinkage ratio 0 = (Po/P) - 1 (p is the mass density), and 10s Young's modulus E are presented as a function of time t. It was found that the chemical shrinkage of the epoxy resin, due to the molecular linking, is more significant than was stated by the manufacturers of the resin. Happily, the most intensive chemical shrinka.ge takes place when the material compliance is high, therefore, due to filtration, stress relaxation, etc., the contribution of the chemical shrinkage to the transverse tensile stresses is comparatively small. As a result, a gentle thermal treatment and, in particular, a more prolonged cooling were recommended to lessen nondesirable stresses. At the same time, it was recommended that resins with the lesser shrinkage were used, and that the stresses in question are residual ones and, in combination with other actions, could become a source of delaminations in the later life of a structure. 4 LOW-ENERGY IMPACT AS A SOURCE OF DELAMINATIONS There are a lot of causes of new-born delaminations after a structure is manufactured. Among them are various additional ones not accounted for in design, loads and actions: local forces, thermal actions, low-energy surface impacts, etc. Without proper design and fabrication, holes, notches and connections also serve as sources of delamination. In recent years, a study of multiple cracking of laminates under surface impact was performed independently Delaminations in composite structures. V. V. Bolotin by Adams and Adams 12, Bolotin et al. 13 15 and Bogdanovich and Yarve 16. In the paper by Bo]otin et al.J3 experimental results are presented for three types of composites: organic fiber/epoxy, graphite/epoxy and glass-textile/epoxy laminates. Specimens on the rigid foundation as well as beam specimens were tested (Figure 4). Impactors with flat and spheroidal heads up to 15 kg of mass and up to 600 J of initial energy were applied. To evaluate the level of multiple cracking, the residual fracture work in ply-after-ply peeling was measured. An example is presented in Figure 5, where the longitudinal distribution of the specific fracture work 7 is presented. Specimens were placed on the rigid foundation, and the flat-head impactor with the diameter of 20 mm was used. After the impact no separation of layers was observed; however, the fracture work under the impact area diminished significantly (see lines 1, 2 and 3 in Figure 5 corresponding to three increasing levels of impact energy). Figure 6 illustrates the damage distribution along the depth for a beam specimen subjected to a threepoint impact [Figure 4 (c)]. The specimens were dyed after the impact, split in layers, and the summed cracked area S was measured. Both the multiple cracking under the impact area and the significant delamination near the middle surface were observed. Some additional data are presented in Section 9 in the context of the residual load-carrying capacity of composites after impact. An analytical-numerical study of cracking under the low-energy impact was performed by Bolotin and Grishko a4. The composite was modelled as a multilayered solid with alternating elastic and elasto-plastic layers 17'1s. Very special properties were attributed to elasto-plastic layers which imitate the matrix and the interface. The model includes the change of the unloading modulus and secondary moduli due to damage as well as the existence of the ultimate tensile strain that corresponds to the quasibrittle inter-layer rupture (Figure 7). The wave propagation in a laminate plate under a rectangular-shaped U f(a,0= 0 £/j ' , Eu ~0 (a) (b) Figure 7 Analytical model of multiple cracking under low-energy impact: (a) composite as a multilayered solid; (b) strain-stress relationship for interlayers with account of damage and brittle rupture 131
De/aminations in composite structures: VV Bolotin pressure impulse was studied with the numerical simula- Figure 9, plotted for a plate consisting of ten elastic layers ion to illustrate the consequent cracking of interlayers. In and nine elasto-plastic interlayers acquiring the damage Figure 8 the longitudinal strain e is plotted against the after each plastic straining. Here Po is the dynamic distance x from the surface for three following time pressure measured in MPa and k is the number of instants. There, eu denotes the ultimate strain interlayers, The occurrence of cracks is denoted with Various patterns of cracking were observed in the crosses. Delaminations situated near the face and near numerical simulation: cracking under the impacted the back side of the plate may be observed in Figure 9(a) surface; spalling near the back side produced from the and( b). They differ in the boundary conditions at the wave reflection; multiple cracking through almost the back surface Figure 9(c)is drawn for the matrix that is whole thickness of the specimen. It is illustrated in assumed to be elastic up to rupture due to the tensile strain. In the latter case, the resistance to multiple cracking occurs higher. It seems paradoxical, since the account of plasticity, generally, increases the impact 103 toughness. But the considered model also includes the tions. Omitting plasticity, we automatically exclude th damage [see Figure 7(b)]. The total number n of cracks as a function of the total impact impulse pot in Pa s is shown in Figure 10. Although the scatter of numerical data is large, able to draw E103 lower bound for the impact impulse that produces a given level of multiple cracking 5 STABILITY OF INTERNAL DELAMINATIONS Internal delaminations are rather similar to cracks in ordinary structural materials, and they are usually treated in terms of conventional fracture mechanics 103 This concerns, partially, the edge delaminations in thick laminated components. Conditions of stability with respect to the growth of delaminations may be for independent integrals as well as, in the case of linear elas of stress intensity factor particular, the interlaminar fracture energy per unit of the new surface is used widely to characterize the Figure 8 Longitudinal strain distribution for the three following time toughness of composites to the growth of delaminations 十+十十 十+++++十+十十十十十 十十十十十十++十+++十十+十 +十+++++十十十 ++十++十十 500100 300 500100 Figure9 Position of interlaminar cracks as a function of the impact pressure: (a)fixed back side;(b)free back side; (c)plastic deformation of matrix is 132
Delaminations in composite structures. V. V. Bolotin pressure impulse was studied with the numerical simulation to illustrate the consequent cracking of interlayers. In Figure 8 the longitudinal strain e is plotted against the distance x from the surface for three following time instants. There, eu denotes the ultimate strain. Various patterns of cracking were observed in the numerical simulation: cracking under the impacted surface; spalling near the back side produced from the wave reflection; multiple cracking through almost the whole thickness of the specimen. It is illustrated in E.11P 105 l" ° t -5 -10 e.10 ~ '° t 5 0 -5 -10 EU -~, hf .... hf .... hf .... hf .... hfj h.~f i t4- i -ti- ii- i- -ii- ii i i i 11 i !I II i I II II (a) r~ -- II II I II ',', ,,,, X (b) c'103 f Cu l; f ~ t, ,, ,, ,, (e) Figure 8 Longitudinal strain distribution for the three following time instants Figure 9, plotted for a plate consisting often elastic layers and nine elasto-plastic interlayers acquiring the damage after each plastic straining. Here Po is the dynamic pressure measured in MPa and k is the number of interlayers. The occurrence of cracks is denoted with crosses. Delaminations situated near the face and near the back side of the plate may be observed in Figure 9 (a) and (b). They differ in the boundary conditions at the back surface. Figure 9 (c) is drawn for the matrix that is assumed to be elastic up to rupture due to the tensile strain. In the latter case, the resistance to multiple cracking occurs higher. It seems paradoxical, since the account of plasticity, generally, increases the impact toughness. But the considered model also includes the damage of the interlayer attributed to plastic deformations. Omitting plasticity, we automatically exclude the damage [see Figure 7 (b)]. The total number n of cracks as a function of the total impact impulse po T in Pa s is shown in Figure 10. Although the scatter of numerical data is large, one might be able to draw an approximate lower bound for the impact impulse that produces a given level of multiple cracking. 5 STABILITY OF INTERNAL DELAMINATIONS Internal delaminations are rather similar to cracks in ordinary structural materials, and they are usually treated in terms of conventional fracture mechanics. This concerns, partially, the edge delaminations in thick laminated components 19. Conditions of stability with respect to the growth of delaminations may be formulated in terms of energy release rates and pathindependent integrals as well as, in the case of linear elasticity, in terms of stress intensity factors 2°-24. In particular, the interlaminar fracture energy per unit of the new surface is used widely to characterize the toughness of composites to the growth of delaminations. k 10 - 8 6 4 2 0 I00 Figure 9 neglected ÷+++++++++++++++ I0 +++ +++++ ++ ++++++ ++-F-t- ++ I I I I 200 300 400 500 (a) ++++++++++÷++ 4 - 2 - o I 100 200 +++++++++++ +++ +++÷++++ 10 "- +++÷+÷++++++++++ +++++++++++ + + I I I 0 I I I I 300 400 500 100 200 300 400 500 (b) (c) P0, MPa Position of interlaminar cracks as a function of the impact pressure: (a) fixed back side; (b) free back side; (c) plastic deformation of matrix is 132
Delaminations in composite structures: V, V. Bolotin +++++8 十 十+特普十普+6 计+十H++ +H什++ H計十什+十 20180240300360 PoTo MPa.s Figure 10 Total number of interlaminar cracks vs the impact impulse [(a)-(c)as in Figure 91 One distinguishes the energy release rates Gl, Gu and Gul for tensile, transverse shear and anti-plane shear respectively, as well he corresponding critical magnitudes GiC, Guc and Guc However, the patterns of interlaminar fracture are omplicated, and not only in the cas th strong anisotropy, interface friction, etc. To a larger degree, the complexity is connected with the mixed-mode type of fracture that is present in most of the practical cases When an interlaminar crack is situated between two layers with different properties, the fractographic picture becomes more complicate Some results concerning the'skew' interlaminar cracking are discussed by Bolotin et al.. In this section an analytical approach to the delaminations between two layers with similar properties is discussed, without going into the complications originating from dissimilarities of neighboring layers. tensile-shear interlaminar fracture mode of the glass-textile/epoxy ider the delaminations oriented aminate along the principal elastic axes. Then we may discuss the interlaminar fracture in terms of partial energy rates and their combinations quation()and similar relationships do not offer Although the strain energy release rates in elastic satisfaction from the academic viewpoint. It seems materials should be additive the condition G1+Gul+ more adequate to keep the summed energy release rate Gml=GC, due to the strong anisotropy, has no meaning: G as a generalized driving force, assuming that the the amount of fracture work depends significantly on th critical magnitude Gc (later, Gc= r)depends on the mode of fracture. Therefore, conditions are used such fracture mode. In fact, the fractographic picture of delaminations depends on which mode dominates in a mixed-mode fracture. In the general case, all three Gr modes contribute into the damage near the fronts of delaminations with empirical exponents mI, my and mll. Since equation Following the general approach based on the princi (1)is a kind of interpolation, it might fit experimental ple of virtual work for systems with unilateral con- data satisfactorilv24,27. An example taken from the straints, we may assume that the cracked body under paper by Shchugorev" is presented in Figure 11. There loading is in an equilibrium state if the virtual work the relationship between G and Gu is drawn fc satisfies condition mass-production glass-textile/epoxy laminate, The aver- 6A≤0 age critical magnitudes for 'pure modes are C 0.74 kJ/m and Guc=3.60kJ/m". This signifies a strong Denoting the crack sizes( Griffiths generalized coordi- anisotropy of fracture toughness a. and their variations with 133
Delaminations in composite structures. V, V. Bolotin I0 Ol 10 + + +++ ++ 8 ++ +l ~llll -H-t-H-tIIHIIIIII+--t#-'I#- +-it-+ 6 /JT : 2' "" + + + N., .N. I I I [ I I 0 60 121) 180 240 300 360 60 120 18 (a) 5 10 + / +++ 8 -I-HI-+ -I++ ++ 6 -t-I- +-t++ ++ llIll:l II I -I- 4 ............... ll:llll -I-+ + III I -tH-I- "l- 2 + + I I I ] o 180 240 300 360 (b) Figure 10 Total number of interlaminar cracks vs the impact impulse [(a) (c) as in Figure 9] ++ -I- I II II I I I ;i ',',', ',i I ',', II II III II t-t- It t t ................... II II~III;I Iiiiii ++ ',;:,++ , , , , 60 120 180 240 300 360 (c) PoXo 'MPa.s One distinguishes the energy release rates GI, Gu and GIII for tensile, transverse shear and anti-plane shear, respectively, as well as the corresponding critical magnitudes Gic, Gilt and GIIIC. However, the patterns of interlaminar fracture are complicated, and not only in the cases with strong anisotropy, interface friction, etc. To a larger degree, the complexity is connected with the mixed-mode type of fracture that is present in most of the practical cases. When an interlaminar crack is situated between two layers with different properties, the fractographic picture becomes more complicated 25. Some results concerning the 'skew' interlaminar cracking are discussed by Bolotin et al. 26. In this section an analytical approach to the delaminations between two layers with similar properties is discussed, without going into the complications originating from dissimilarities of neighboring layers. Moreover, we consider the delaminations oriented along the principal elastic axes. Then we may discuss the interlaminar fracture in terms of partial energy rates GI, Gu, Gill and their combinations. Although the Strain energy release rates in elastic materials should be additive, the condition GI + GII + Gil I = Gc, due to the strong anisotropy, has no meaning: the amount of fracture work depends significantly on the mode of fracture. Therefore, conditions are used such as: /G,~ "~|ml-p - GII -I- , //GIII "~ mln = 1 (1) with empirical exponents mi, mll and mii I. Since equation (1) is a kind of interpolation, it might fit experimental 24 27 data quite satisfactorily ' . An example taken from the paper by Shchugorev 28 is presented in Figure 11. There, the relationship between G I and Gn is drawn for the mass-production glass textile/epoxy laminate. The average critical magnitudes for 'pure' modes are Glc = 0.74 kJ/m 2 and GII C = 3.60 kJ/m 2. This signifies a strong anisotropy of fracture toughness. 0 1 2 3 GlbkJ/m 2 Figure 11 Relationship between the critical magnitudes for mixed tensile-shear interlaminar fracture mode of the glass-textile/epoxy laminate Equation (1) and similar relationships do not offer satisfaction from the academic viewpoint. It seems more adequate to keep the summed energy release rate G as a /generalized driving force, assuming that the critical magnitude Gc (later, Gc = F) depends on the fracture mode. In fact, the fractographic picture of delaminations depends on which mode dominates in a mixed-mode fracture. In the general case, all three modes contribute into the damage near the fronts of delaminations. Following the general approach based on the principle of virtual work for systems with unilateral constraints 5'29, we may assume that the cracked body under loading is in an equilibrium state if the virtual work satisfies condition ~SA ~< 0. (2) Denoting the crack sizes (Griffith's generalized coordinates) with al,...,am, and their variations with 133