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
Delaminations in composite structures: V. V. bolotin Ba1>0,., Sam>0, rewrite equation(2)as follows ∑(G-)a≤0 Here G are the generalized driving forces, and r are the corresponding generalized resistance forces. The unper- turbed state, as well as the perturbed ones, are considered here as equilibrium states with respect to Lagrangian generalized coordinates. Applied to composites, the idea appears to assume that the crack front is given with everal very close generalized Griffith's coordinates th the front of shear damage might be a little ahead of the opening(tensile)damage front, etc. Let'pure'modes be associated with variations da,, Saz, and Sa3(hereafter Arabic figures are used for the numbers of modes) When any two damage fronts coincide, we introduce variations Sa4, Sas and Saf. When all three modes contribute almost equally to damage, we label the variation Sa7. It ought to be stressed that all the Griffiths generalized coordinates a az correspond to the material points located in the process zone whose size is small compared with a,,..., ay. It means that the generalized coordinates take very similar tudes Supposing the variations to be independent and applying equation(3), we conclude that a delamination Figure 12 Schematic presentation of the limit surface for mixed begins to propagate if one of the following conditions is attained G G2=T2,G3=T3 We use case(a) for an open delamination. Its growth is +G2=12,G1+G3=13,G2+G3=2 accompanied by the formation of longitudinal cracks. In case(b)the delamination remains closed, ' pocket-like G1+G2+G3=F (4) Delaminations in the components under compression are shown in Figure 13(c)and (d). In case (c) the Here TI, T2 and T3 are generalized resistance forces in delamination is open and subjected to buckling. In case 'pure'modes;T12,T13, T23 and T123 are those in mixed (d)the delamination is also buckled but it is to be modes. The limit surface in the space of G1, G2, G3 is considered as closed. Edge delaminations are presented piece-planar (Fi ), but it is not so bad compare in Figure 13(e)and ( f). In the last case, a secondary crack say, with equation(1). Contrary to equation(1), which appears during the buckling of the delamination. One not more than a result of interpolation, equation(4)is in dimensional bending approximation is applicable in agreement with the principles of mechanics of solids cases (a)and(c). In the other cases we have to treat although it is based on the unconventional assumption delaminations with the use of the theory of plates and of the existence of several damage fronts) Finally, we may state that the limit surface for the shells. In cases(b)and (d), the delaminations may be considered as elliptical in the plane, and in cases(e) and mixed mode fracture can be developed in the framework () as half-elliptical of analytical mechanics Various versions of the energy approach are used to predict the stability of delaminations: the strain energy release rate approach, the path-invariant integral 6 BUCKLING AND STABILITY OF NEAR approach and the strain energy SURFACE DELAMINATIONS approach- In the case of a single-parameter nation in an elastic structural component 出 If a delamination is situated near the surface of a ways produce either identical or numerically close structural component, its behavior under loading is often rediction accompanied by buckling. This is The two simplest problems are depicted in Figure 14:a components under compression, under ce heating, beam-like delamination under compression and a d sometimes for components under ter ue to circular delamination with isotropic elastic properties Poisson's effect). Examples of delaminations are shown under the uniform two-dimensional compression. These in Figure 13. Cases (a)and(b)correspond to the problems were considered by a number of authors,31-39 delaminations propagating in a component under tension. A wide variety of simplifications might be used even in 134
Delaminations in composite structures: V. V. Bolotin 6a 1 ~ 0,..., 6a m >~ O, rewrite equation (2) as follows: m ~(G~ - rj)&j ~< O. (3) .j=l Here Gj are the generalized driving forces, and F/are the corresponding generalized resistance forces. The unperturbed state, as well as the perturbed ones, are considered here as equilibrium states with respect to Lagrangian generalized coordinates. Applied to composites, the idea appears 3° to assume that the crack front is given with several very close generalized Griffith's coordinates al,..., a m with independent variations ~al, (Sam. In fact, the front of shear damage might be a little ahead of the opening (tensile) damage front, etc. Let 'pure' modes be associated with variations 6al, 6a2, and 6a3 (hereafter Arabic figures are used for the numbers of modes). When any two damage fronts coincide, we introduce variations ~5a4, ~a5 and ~Sa 6. When all three modes contribute almost equally to damage, we label the variation 6a 7. It ought to be stressed that all the Griffith's generalized coordinates al, • • •, a7 correspond to the material points located in the process zone whose size is small compared with a~,..., a 7. It means that the generalized coordinates take very similar magnitudes. Supposing the variations to be independent and applying equation (3), we conclude that a delamination begins to propagate if one of the following conditions is attained: G1 = F1, G2 ~ 1"2, G3 = 1"3 G1 + G2 = 1"12, Gl + G3 = 1"13, G2 + G3 = 1"23 G1 + G2 + G3 = 1"123' (4) Here 1"1, 1"2 and 1"3 are generalized resistance forces in 'pure' modes; F12, I~13, F23 and 1"123 are those in mixed modes. The limit surface in the space of G l, G 2, G 3 is piece-planar (Figure 12), but it is not so bad compared, say, with equation (1). Contrary to equation (1), which is not more than a result of interpolation, equation (4) is in agreement with the principles of mechanics of solids (although it is based on the unconventional assumption of the existence of several damage fronts). Finally, we may state that the limit surface for the mixed mode fracture can be developed in the framework of analytical mechanics. 6 BUCKLING AND STABILITY OF NEARSURFACE DELAMINATIONS If a delamination is situated near the surface of a structural component, its behavior under loading is often accompanied by buckling. This is typical for the components under compression, under surface heating, and sometimes for components under tension (due to Poisson's effect). Examples of delaminations are shown in Figure 13. Cases (a) and (b) correspond to the delaminations propagating in a component under tension. Gm JIIC Gn Figure 12 Schematic presentation of the limit surface for mixed interlaminar fracture mode We use case (a) for an open delamination. Its growth is accompanied by the formation of longitudinal cracks. In case (b) the delamination remains closed, 'pocket-like'. Delaminations in the components under compression are shown in Figure 13 (c) and (d). In case (c) the delamination is open and subjected to buckling. In case (d) the delamination is also buckled but it is to be considered as closed. Edge delaminations are presented in Figure 13 (e) and (f). In the last case, a secondary crack appears during the buckling of the delamination. Onedimensional bending approximation is applicable in cases (a) and (c). In the other cases we have to treat delaminations with the use of the theory of plates and shells. In cases (b) and (d), the delaminations may be considered as elliptical in the plane, and in cases (e) and (f) as half-elliptical. Various versions of the energy approach are used to predict the stability of delaminations: the strain energy release rate approach 21, the path-invariant integral approach 22'23, and the strain energy density approach 2°. In the case of a single-parameter delamination in an elastic structural component, all these ways produce either identical or numerically close predictions. The two simplest problems are depicted in Figure 14: a beam-like delamination under compression and a circular delamination with isotropic elastic properties under the uniform two-dimensional compression. These problems were considered by a number of authors 29'31-39 A wide variety of simplifications might be used even in 134
Delaminations in composite structures: V. V. Bolotin Z Figure 13 Near-surface delaminations: (a)open delamination in tension; (b)closed one in tension; (c)open buckled delamination; (d)closed buckled one:() edge buckled delamination; (f the same with a Using a half-nonlinear'approach of the theory of elastic stability, we present the energy of the systems in the form U= const E、abh 2(1-vxyvy Here a, b and h are the dimensions shown in Figure 14 (a), Ex is Youngs modulus in the x-direction, vxy and vy are Poissons ratios. It is assumed that the general loading is strain-controlled with the applied strain Eoo, and the mem brane strain in the buckled delamination Figure 14 Elementary problems of buckled delaminations:(a)beam remains equal to the Euler's critical strain ike;(b)circular isotropic and isotropically strained delaminations (a) these comparatively transparent problems, especially concerning postbuckling behavior. In particular, assum- The energy in equation (7) is related to a half of the ing that the buckling mode is given with a single component, x>0. The generalized driving force, accord parameter, say, with the maximum lateral displacement, ing to equation(5),is one can calculate the potential energy of the system Ebh U=UC, a), where a is the size of the delamination G Figure 14). The generalized driving force is aU Equalizing, according to equation(6) the right-hand (5) side of equation(9) to the resistance force r=?b The generalized resistance force is r= yb for the beam come to the equation that connects the critical magi tudes of Em and a delamination(b is the width of the beam), and r= 2ray for the circular delamination. The growth of delamina- E2+2exe.(a)-32(a)=2 tions takes place under conditions similar to those in Here the notation is used 2y(1 As an example, assume that in the case depicted in It is easy to see that E, is the critical strain for an open Figure 14(a)the buckled mode is w(x)=fcos"(Tx/2a) delamination under tension [Figure 13(a)] if the work of 135
Delaminations in composite structures." V. V. Bolotin Z Z (a) Z 2' s½ (b) (c) Z Z 2b y Y~ X Y X S~ (d) (e) (0 Figure 13 Near-surface delaminations: (a) open delamination in tension; (b) closed one in tension; (c) open buckled delamination; (d) closed buckled one; (e) edge buckled delamination; (f) the same with a secondary crack Ca) (b) Figure 14 Elementary problems of buckled delaminations: (a) beamlike; (b) circular isotropic and isotropically strained delaminations these comparatively transparent problems, especially concerning postbuckling behavior. In particular, assuming that the buckling mode is given with a single parameter, say, with the maximum lateral displacement, one can calculate the potential energy of the system U = U(f,a), where a is the size of the delamination (Figure 14). The generalized driving force is OU a - Oa " (5) The generalized resistance force is P = 7b for the beam delamination (b is the width of the beam), and F = 2rra"/ for the circular delamination. The growth of delaminations takes place under conditions similar to those in equation (4): a = F. (6) As an example, assume that in the case depicted in Figure 14 (a) the buckled mode is w(x) =fcos2(Trx/2a). Using a 'half-nonlinear' approach of the theory of elastic stability, we present the energy of the systems in the form Exabh U = const (e 2 - 2e~e, + e2). (7) 2(1 - UxyUyx) Here a, b and h are the dimensions shown in Figure 14 (a), E~ is Young's modulus in the x-direction, Uxy and Yyx are Poisson's ratios. It is assumed that the general loading is strain-controlled with the applied strain e~, and the membrane strain in the buckled delamination remains equal to the Euler's critical strain. c,(a) = i5 (8) The energy in equation (7) is related to a half of the component, x >~ 0. The generalized driving force, according to equation (5), is a Exbh (e 2 - 2e~e, + e2,). (9) 2(1 - UxUy ) Equalizing, according to equation (6) the right-hand side of equation (9) to the resistance force P =-yb, we come to the equation that connects the critical magnitudes of e~ and a: e~ 2 + 2e~e,(a) 2 (10) - 3eZ(a) = et. Here the notation is used e 2 = 2"/(1 - UxyUyx) (11) Gh It is easy to see that el is the critical strain for an open delamination under tension [Figure 13 (a)] if the work of 135
Delaminations in composite structures: V.v. bolotin Figure 17 Near-surface delamination of arbitrary shape in the plan Figure 15 Stability chart for a beam-like delamination; region of buckling is light shaded, that of growth is dark shaded U= const-Ud(a1,…,am) 12 Here the constant represents the total energy in the depends on Griffiths coordinates am as well as on the loading parameters. The driving generalized forces ar aud ( Some obstacles may be met in the assessment of the generalized resistance forces. Apart from some difficulties of pure experimental character resulting, in particular, in a significant scatter of the specific fracture work(see, e.g Gutowski and Pankevicius), there are conceptual com- plications. It is evident that the energy required to create a Figure 16 Two-parameter elliptical delamination unit of the new fracture surface, being scalar, depends the direction of interlaminar cracking. Different amounts splitting In of energy, generally, will be needed to peel a ply along The stability diagram following from equations( 8) fibers or across them. Therefore, tensorial parameters are and (10)is shown in Figure 15. Line E is the boundary of to be introduced to describe this effect the Euler's buckling, and line G is the boundary of the The simplest way is to account for the anisotropy by initiation of fracture. The states of the system correspond- means of a second-rank tensor. It means that the specific ing to the descending part of line G are unstable, and those fracture work y in the equation to the ascending part are stable(in griffith's sense) Multiparameter delaminations were considered by 6Ar=- ?l ds x Sa l Bottelid2' Bolotin35 and Murzakhanov and Shchu- to be treated as a function of the unit normal vector n gorev142. The simplest among them is the elliptical to the front of the delamination(Figure 17) delamination that is assumed to remain elliptical during its growth. The semi-axes a and b here play the role of 7=7nk Griffith's generalized coordinates( Figure 16) Referring to the principal axes, equation(15) takes the In the case of a near-surface delamination whose form y(0)=2, cos 0+2 sin. Here y and n2 are the behavior does not affect the strain-stress field in the amounts of specific fracture work in xr-andx2" main bulk of the composite, one may present the strain directions, and 0 is the angle between the normal n to energy of a structure as the front and x-axis
Delaminations in composite structures." V. V. Bolotin E G Z= x 3 ff v X2 n 8a 0 0 E. I E~ Figure 15 Stability chart for a beam-like delamination; region of buckling is light shaded, that of growth is dark shaded 2a Figure 16 Two-parameter elliptical delamination ~a the longitudinal splitting is also included in '7. The stability diagram following from equations (8) and (10) is shown in Figure 15. Line E is the boundary of the Euler's buckling, and line G is the boundary of the initiation of fracture. The states of the system corresponding to the descending part of line G are unstable, and those to the ascending part are stable (in Griffith's sense). Multiparameter delaminations were considered by Bottega 4°, Bolotin 35 and Murzakhanov and Shchugorev 41'42. The simplest among them is the elliptical delamination that is assumed to remain elliptical during its growth. The semi-axes a and b here play the role of Griffith's generalized coordinates (Figure 16). In the case of a near-surface delamination whose behavior does not affect the strain-stress field in the main bulk of the composite, one may present the strain energy of a structure as Figure 17 Near-surface delamination of arbitrary shape in the plan U = const - Ud(al,..., am). (12) Here the constant represents the total energy in the absence of the delamination, and the term U d (al,..., am) depends on Griffith's coordinates al, ..., am as well as on the loading parameters. The driving generalized forces are 0ted. Gj = ~-a/(j = 1,...,m). (13) Some obstacles may be met in the assessment of the generalized resistance forces. Apart from some difficulties of pure experimental character resulting, in particular, in a significant scatter of the specific fracture work (see, e.g. Gutowski and Pankevicius2V), there are conceptual complications. It is evident that the energy required to create a unit of the new fracture surface, being scalar, depends on the direction of interlaminar cracking. Different amounts of energy, generally, will be needed to peel a ply along fibers or across them. Therefore, tensorial parameters are to be introduced to describe this effect. The simplest way is to account for the anisotropy by means of a second-rank tensor. It means that the specific fracture work '7 in the equation far = -.[s'7 [ ds × 6a I (14) is to be treated as a function of the unit normal vector n to the front of the delamination (Figure 17): 7 = "7jknjnk. (15) Referring to the principal axes, equation (15) takes the form 7(0) = "71 cos2 0 + '72 sin2 0. Here '71 and "72 are the amounts of specific fracture work in Xl- and x2- directions, and 0 is the angle between the normal n to the front and xl-axis. 136
Delaminations in composite structures: V.V. Bolotin D Longitudinal pressure: (a) scheme of loading: (6) boundary of bucklin 0.5 A number of more complicated problems were considered in the same manner, among them: initially ns previously subjected to the short-tit nal actional T=773 K buckled delaminations 3s, edge semi-elliptical delamina- and durations t=0. 15, 30, 60 and I 1-5, respectively) tions with secondary cracks", elliptical delaminations in shells*. etc. An illustrative elevated temperature, moisture and other environmental example is shown in Figure 18. A cylindrical laminated actions shell with an elliptical delamination situated near the Interlaminar fatigue of composites was studied internal surface is subjected to the longitudinal pressure. experimentally by a number of authors(see O'Brien The stability chart is drawn in the space of variables a, b Reifsnider"). The question arises how to present and E, where a and b are the semi-axes of th experimental results in the most rational and universal delamination, and Ex the nominal (membrane) strain form that allows the interpolation and extrapolation The surface ABC corresponds to the boundary of upon other loading levels, other initial crack sizes, etc. As buckling of the delamination that is assumed to be to the surface delaminations, the most sound approach is initially nonbuckled. The surface A'B'C corresponds to to use, as a controlling loading parameter, the range AG equations Ga=Ta and Gb=Tb, where indices a and b of the generalized driving force (in cyclic fatigue), or its relate to the semi-axes of the delamination both current magnitude G(in static fatigue) equations are satisfied on the line DB simultaneously Some results concerning graphite/epoxy, glass/epoxy, and organic fiber/epoxy composites subjected, before the cyclic loading, to a short-time-thermal action, were 7 GROWTH OF DELAMINATIONS DUE TO obtained by Bolotin et al.48-50. Figure 19 is obtained for FATIGUE the glass-textile/epoxy composite. There the interlami nar crack growth rate da/dN is plotted vs the range AG Delaminations in composite structures can grow both of the generalized driving force for specimens subjected under monotonous quasistatic loading and under cyclic to heating at 773 K Lines 1-5 correspond to increasing loading. They grow continuously, in a stable way when duration of thermal action up to 120 s The diagrams are the state of equilibrium(in the Griffith's sense) that is very similar to those for fatigue cracks in metals. The attained with the equality G;=r remains stable or at parameters of the lines, evidently, vary over a wide range least neutral during further growth. The growth will be A semi-theoretical equation jump-like, including that up to the complete splitting of a tructural component, when the attained state happens d4/4G-△G (16) to be unstable. The last situation is typical for most components under compression. was used to interpret these results. Material parameters The fatigue growth of delaminations is alike in many A, Gr, AGth, and r depend on the temperature and on the aspects to the fatigue crack growth in ordinary metal duration of thermal treatment. At gmax <<r the structures. Along with the proper (cyclic) fatigue, the exponent in equation(16) is m a 2. The remaining growth of delaminations also takes place under long- parameters can be estimated from the middle, Paris acting, sustained loading. This kind of damage, named Erdogan,'s part of the diagram. The estimates of Gr and here the static fatit typical for composites with AGuh in functions of T and t corresponding to Figure 19 polymer matrices as well as for composites subjected (b)are presented in Figure 137
Delaminations in composite structures: V. V. Bolotin P A -~2b (a) (b) C i b/h B Figure 18 Cylindrical shell with an elliptical delamination under longitudinal pressure: (a) scheme of loading; (b) boundary of buckling (ABC) and that of growth (A'B'C') A number of more complicated problems were considered in the same manner, among them: initially buckled delaminations 35, edge semi-elliptical delaminations with secondary cracks 41, elliptical delaminations in cylindrical 43'44 and spherical shells 45, etc. An illustrative example is shown in Figure 18. A cylindrical laminated shell with an elliptical delamination situated near the internal surface is subjected to the longitudinal pressure. The stability chart is drawn in the space of variables a, b and E~, where a and b are the semi-axes of the delamination, and e~ the nominal (membrane) strain. The surface ABC corresponds to the boundary of buckling of the delamination that is assumed to be initially nonbuckled. The surface A'B~C corresponds to equations G, = F, and Gb = Fb, where indices a and b relate to the semi-axes of the delamination. Both equations are satisfied on the line DB ~ simultaneously. 7 GROWTH OF DELAMINATIONS DUE TO FATIGUE Delaminations in composite structures can grow both under monotonous quasistatic loading and under cyclic loading. They grow continuously, in a stable way when the state of equilibrium (in the Griffith's sense) that is attained with the equality Gj = Fj remains stable or at 5 least neutral during further growth . The growth will be jump-like, including that up to the complete splitting of a structural component, when the attained state happens to be unstable. The last situation is typical for most components under compression. The fatigue growth of delaminations is alike in many aspects to the fatigue crack growth in ordinary metal structures. Along with the proper (cyclic) fatigue, the growth of delaminations also takes place under longacting, sustained loading. This kind of damage, named here the static fatigue, is typical for composites with polymer matrices as well as for composites subjected to dl mm (IN 'cycle 10-2 S ./.// 5 4 ./ o 10 -4 I I I I I 0.5 1 2 AG,~I2 Figure 19 Fatigue crack growth rate diagrams for composite specimens previously subjected to the short-time thermal action at T = 773 K and durations t = 0, 15, 30, 60 and 120s (lines 1-5, respectively) elevated temperature, moisture and other environmental actions. Interlaminar fatigue of composites was studied experimentally by a number of authors (see O'Brien 46, Reifsnider47). The question arises how to present experimental results in the most rational and universal form that allows the interpolation and extrapolation upon other loading levels, other initial crack sizes, etc. As to the surface delaminations, the most sound approach is to use, as a controlling loading parameter, the range AG of the generalized driving force (in cyclic fatigue), or its current magnitude G (in static fatigue). Some results concerning graphite/epoxy, glass/epoxy, and organic fiber/epoxy composites subjected, before the cyclic loading, to a short-time-thermal action, were obtained by Bolotin et al. 48 50. Figure 19 is obtained for the glass-textile/epoxy composite. There the interlaminar crack growth rate da/dN is plotted vs the range AG of the generalized driving force for specimens subjected to heating at 773 K. Lines 1-5 correspond to increasing duration of thermal action up to 120 s. The diagrams are very similar to those for fatigue cracks in metals. The parameters of the lines, evidently, vary over a wide range. A semi-theoretical equation 35 --dNd° ~ A (AG~-~Gth-~m(1-~9~) Gf ) \ (16) was used to interpret these results. Material parameters A, Gf, AGth, and £ depend on the temperature and on the duration of thermal treatment. At Gma x << 1 ~ the exponent in equation (16) is m ~ 2. The remaining parameters can be estimated from the middle, ParisErdogan's part of the diagram. The estimates of Gr and AGth in functions of T and t corresponding to Figure 19 (b) are presented in Figure 20. 137
Delaminations in composite structures: V. V Bolotin far field), and the global balance of forces and energy in system cracked body-loading or loading de without going into the general statement and analytical details, let us illustrate the theory with Gr example, an open beam-like delamination in a composite specimen subjected to cyclic loading(Figure 21) As in the quasistatic crack growth, fatigue crack growth governed by equation(6). Compared with the quasistatic case, the infuence of damage accumulated in the interlayer is be taken into account. Let us introduce the damage measure(x, I)in the interlayer, at xl> a. We assume that w=0 for the undamaged interlayer, and w= l when the adjacent layer is completely debonded. Let damage be I produced with the range AT of the tangential stress T(x, t) in the interlayer, and let the equation governing the damage be assumed to be of the form ON △=△m),△2 Here Tr is a material constant characterizing resistance of he interlayer to damage accumulation, and ATth is the threshold resistance range. Power exponent m is similar to exponents of fatigue curves in standard tests. The simplest equation for the stress r(x, t) holds in the membrane approximation. It has the form exp (18) with the shear modulus Gm, the effective interlayer 4 thickness hm, and the length parameter Ao. The latter may be interpreted as a characteristic length of the boundary effect, or an ineffective length in the vicinity of he ruptured layer, or as a characteristic size of the process zone. The loading is given in equation(18)with Figure 20 Parameters of the fatigue crack growth diagram as the applied(nominal) longitudinal strain Eoc = 0oc/E functions of temperature and duration of the initial short-time thermal Using equations(17)and(18)and the initial condition action: figures 1, 2,3,4 correspond to temperature levels T=573,673, w(x, 0)=wo(x), the damage measure (m)=wa(N), N 773and873K at the tip of the delamination can be evaluated For the further discussion, the relation T=r(o) between the resistance force T and the tip damage is to be specified. We assume that Tx, N T=I0(1-°) interlayer, and a>0, e.g. a=l. This assumption on more or less realistic. The cycle number N, of the termination of the initial stage is the first positive root of the equation G(N)=T(N) at a=a0=const. Using equation 9)at E,=0 and equations(17)-(19)we obtain Figure 21 A model of delamination growth due to fatigue an equation with respect to N. △r0(ao,N) To discuss the propagation of delaminations in an wo(ao)+ analytical way, the theory of fatigue crack growth5,29,1 may be applied in a whole scale. The leading idea of the E2(N) heory is that the fatigue crack growth is a result of the (20) interaction of two different mechanisms: the accumula- tion of microdamage(both near the crack tips and in the Here E, is the critical strain defined with equation(11)
Delaminations in composite structures." V. V. Bolotin ¢. t b 4~~mm 3 AGth 3 0 1 2 t, min Figure 20 Parameters of the fatigue crack growth diagram as functions of temperature and duration of the initial short-time thermal action: figures 1, 2, 3, 4 correspond to temperature levels T - 573, 673, 773 and 873 K It ,r(x,N) Figure 21 A model of delamination growth due to fatigue To discuss the propagation of delaminations in an analytical way, the theory of fatigue crack growth 5'29'51 may be applied in a whole scale. The leading idea of the theory is that the fatigue crack growth is a result of the interaction of two different mechanisms: the accumulation of microdamage (both near the crack tips and in the far field), and the global balance of forces and energy in the system cracked body-loading or loading device. Without going into the general statement and analytical details, let us illustrate the theory with the simplest example, an open beam-like delamination in a composite specimen subjected to cyclic loading (Figure 21). As in the quasistatic crack growth, fatigue crack growth is governed by equation (6). Compared with the quasistatic case, the influence of damage accumulated in the interlayer is be taken into account. Let us introduce the damage measure w(x, t) in the interlayer, at Ixl >~ a. We assume that = 0 for the undamaged interlayer, and ~ = 1 when the adjacent layer is completely debonded. Let damage be produced with the range A-r of the tangential stress T(x, t) in the interlayer, and let the equation governing the damage be assumed to be of the form 0~-N = AT[ -- ATth , [AT] >~Tth. (17) Tf Here zf is a material constant characterizing resistance of the interlayer to damage accumulation, and A~-th is the threshold resistance range. Power exponent m is similar to exponents of fatigue curves in standard tests. The simplest equation for the stress r(x, t) holds in the membrane approximation. It has the form IT I GmAOI¢~[ {Ixl - a~ -- hm exp ~k~ ) (18) with the shear modulus Gm, the effective interlayer thickness hm, and the length parameter A 0. The latter may be interpreted as a characteristic length of the boundary effect, or an ineffective length in the vicinity of the ruptured layer, or as a characteristic size of the process zone. The loading is given in equation (18) with the applied (nominal) longitudinal strain e~ = a~/E x. Using equations (17) and (18) and the initial condition w(x, 0) = w0(x), the damage measure 0(N) = w[a(N), N] at the tip of the delamination can be evaluated. For the further discussion, the relation F = F(q~) between the resistance force F and the tip damage ¢ is to be specified. We assume that V = V0(1 - ¢0) (19) where F0 is the resistance force for the undamaged interlayer, and a > 0, e.g. a = 1. This assumption is more or less realistic. The cycle number N, of the termination of the initial stage is the first positive root of the equation G(N)= F(N) at a = a0 = const. Using equation (9) at e, = 0 and equations (17)-(19) we obtain an equation with respect to N,: N, -- -- -- "] m 0/a0/ Jo CLLN,).] 1/& =1 ] (20) Here et is the critical strain defined with equation (11), 138
