Geometric modelling of kink banding in laminated structures
M. Ahmer Wadee, Christina Völlmecke, Joseph F. Haley, Stylianos Yiatros
aa r X i v : . [ n li n . PS ] M a y Geometric modelling of kink bandingin laminated structures
By M. Ahmer Wadee , ∗ , Christina V¨ollmecke , Joseph F. Haley and Stylianos Yiatros Department of Civil & Environmental Engineering, Imperial College London, London, UK LKM, Institut f¨ur Mechanik, Technische Universit¨at Berlin, Germany Department of Civil Engineering, Brunel University, Uxbridge, UK
Abstract
An analytical model founded on geometric and potential energy principles for kink band de-formation in laminated composite struts is presented. It is adapted from an earlier successfulstudy for confined layered structures which was formulated to model kink band formation inthe folding of geological layers. The principal aim is to explore the underlying mechanismsgoverning the kinking response of flat, laminated components comprising unidirectional com-posite laminae. A pilot parametric study suggests that the key features of the mechanicalresponse are captured well and that quantitative comparisons with experiments presented inthe literature are highly encouraging.
Keywords:
Kink banding; Laminated materials; Nonlinearity; Energy methods; Analyti-cal modelling
Kink banding is a phenomenon seen across many scales. It is a potential failure mode for anylayered, laminated or fibrous material, held together by external pressure or some form of internalmatrix, and subjected to compression parallel to the layers. Many examples can be found inthe literature concerning the deformation of geological strata [1, 2, 3], wood and fibre composites[4, 5, 6, 7, 8, 9, 10], and internally in wire and fibre ropes [11, 12]. There have been manyattempts to reproduce kink banding theoretically, from early mechanical models [13, 14], to moresophisticated formulations derived from both continuum mechanics [15], finite elasticity theory[16] and numerical perspectives for more complex loading arrangements [17].There has been much relevant work on composite materials with significant problems beingencountered as outlined thus. First, although two dimensional models are commonly employed[18, 19], modelling into the third dimension adds a significant extra component. It inevitablyinvolves a smeared approach in the modelling of material properties since there is a mix of laminaeand the matrix with the possibility of voids. Secondly, failure is likely to be governed by nonlinearmaterial effects in shearing the matrix material [20], and this is considerably less easy to measureor control than the combination of overburden pressure and friction considered in work on kinkbanding during geological folding [21, 22, 23].In the current paper, a pilot study is presented where the discrete model formulated for kinkbanding in geological layers is adapted such that it can be applied to unidirectional laminatedcomposite struts that are compressed in a direction parallel to the laminae. This is achieved byreleasing the assumption that voids are wholly penalized since, in the current case, no overburden ∗ Author for correspondence: [email protected] . A. Wadee et al. 2pressure actively compresses the layers in the transverse direction. Therefore, the rotation of thelaminae during the formation of the kink band causes a dilation, which is resisted by transversetensile forces generated within the interlamina region. The coincident shearing of this region alsogenerates an additional resisting force, but, as mentioned above, this can be subject to nonlinearity,in particular a reduced stiffness that may be either positive (hardening) or negative (softeningperhaps leading to fracture), which is currently formulated with a piecewise linear constitutivelaw. Work done from dilation and shearing are evaluated; additional features from the originalmodel: strain energies from bending and direct compression, and the work done from the externalload can be incorporated without significant alterations. An advantage of the presented model isthat the resulting equilibrium equations can be written and solved entirely in an analytical formwithout having to resort to complex continuum models or numerical solvers.The primary aim of the current work is to lay the foundations for future research. The geomet-ric approach has yielded excellent comparisons with experiments for the model for kink bandingin geological layers; the same is true currently with the present model being compared favourablyto previously published experiments [5]. Moreover, the relative importance of the parameters gov-erning the mechanical response are also identified in the current study. From this, conclusions aredrawn about the possible further studies that would extend the current model to give meaningfulcomparisons with the actual structural response for a variety of practically significant scenarios. A discrete formulation comprising springs, rigid links and Coulomb friction has been devised tomodel kink band deformation in geological layers that are held together by an overburden pressure[22]. It was formulated using energy principles and key parts of the model are shown in Figures 1and 2. It has been compared very favourably with simple laboratory experiments on layers of paper β qkPPPPP + H P + HP PPPq nP
Figure 1: Basic configuration of the discrete model for kink banding in n geological layers. Thetectonic load on each layer is P with a horizontal reaction force H , the axial stiffness of each layeris k , the overburden pressure on the layers is q and the kink band orientation angle is β .that were compressed transversely and then increasingly compressed axially to trigger the kinkband formation process. The testing rig used in that study is shown schematically in Figure 3(a)and a typical test photograph is shown in Figure 3(b). Assuming that the layers were transverselycompressible, the kink band orientation angle β was predicted theoretically for the first time; itbeing related purely to the initially applied transverse strain derived from the overburden pressure q . Figure 4 shows the characteristic sequence of deformation with (a) showing the undeformedstate with applied overburden pressure and the transverse pre-compression defining β ; (b) showingthe point where the interlayer friction is released when the internal transverse strain within thekink band is instantaneously zero and the band forming very quickly in the direction of β . It was Article submitted to The Royal Society
TEX Paper eometric modelling of kink bands in laminates b t cos( α − β ) k α t sin( α − β ) β µN c PN µN k f PPP cNk
Figure 2: Two internal layers of the geological model. The kink band width is b , the normal contactforce between layers is N , the coefficient of friction is µ , the stiffness of individual rotational springsmodelling bending is c , the elastic stiffness of surrounding medium per unit layer is k f and thekink band angle is α . A key assumption is the transverse compressibility of the layers.later demonstrated that beyond the condition shown in Figure 4(c), where all the layers have thesame thickness, whether internal or external to the kink band, lock-up begins to occur as shown inFigure 4(d). This marked the point where new kink bands formed and these could also be predictedby this approach after some modifications were made to the model [23]. An example comparisonbetween the theory and an experiment from that study is presented in Figure 5. Moreover, thismodel has also been demonstrated to be suitable for modelling internal kink band formation inindividual composite fibres under bending that are common in fibre ropes [12]. As discussed above, the system studied in [22] had layers that were bound together by the mecha-nisms of overburden pressure and interlayer friction. The deformation was in fact only admissiblegeometrically if the layers were transversely compressible; the relationship between the kink bandangle α , which could vary, and the orientation angle β , which was fixed, being such that interlayergaps, or voids, were not created. For a laminated strut, most experimental evidence from theliterature also suggests that the kink band orientation angle β is basically fixed for each laminateconfiguration [5, 7]. It is noted, however, in a recent study on laminates under combined compres-sion and shear that this angle can change as the kink propagates but the angle reaches a limit [24];in the current work, β is taken as a constant equivalent to this limiting value from the beginningof the kink band deformation process, which is a simplifying assumption.Kink band deformation in laminates involves different mechanisms that incorporate the inter-lamina region comprising the laminae and the matrix that binds the component together. Sincethe matrix is itself deformable and that there is no overburden pressure to close any voids, themodel needs significant modifications to account for the different characteristics of the laminatedstrut. It is worth noting that the assumption for the lay-up sequence of the composite in thepresent case is such that no twisting is generated from the applied compression. Figure 6 showsthe adapted 2-layer model which omits the following features that are not relevant in the currentcase: the foundation stiffness and the overburden pressure, i.e. q = k f = 0. The kink bandformation is thus intrinsically linked to the deformation of the interlamina region within the strut.Shearing within the interlamina region is the analogous process to sliding between the layersin the model for geological folding; the latter being modelled in the energy formulation as a workdone overcoming the friction force. A piecewise linear model is used to simulate the force versus Article submitted to The Royal Society
TEX Paper . A. Wadee et al. 4 ◦ . ◦ ◦ (a) (b)
120 mm30 mm320 mm
100 kNLoad Cell100 kNLoad Cell
Verti al on(cid:12)ning total overburden for e : Q n layers in axial ompressionkink bandHorizontal axial load : nP
Figure 3: (a) Schematic of the experimental rig used for testing the geological model. (b) A typicaldeformation profile in a physical experiment showing a sequence of kink bands with correspondingorientation angles β .displacement relationship in terms of the shear resistance (see Figure 7), where fracture modesthat are relevant for a linear–softening response, see Figure 7(a), are defined as in Figure 8.Tensile expansion, or dilation, of the interlamina region is modelled, however, with a purelylinear elastic constitutive law. In the model described in §
2, it was argued that when the interlayercontact force was released that not only would the friction be released but also that the overburdenpressure would inhibit the formation of subsequent voids within the layered structure. Since in thecurrent case there is no overburden or lateral pressure as such, potential dilation of the interlaminaregion needs to be included. As in the previous model, however, the lamina deformation is assumedto lock-up and potentially trigger a new band forming when α > β ; transverse compression inadjacent laminae would then be occurring and stiffening the response significantly. Hence, it isreasoned therefore that it would be energetically advantageous for the mechanical system to forma new kink band rather than continue to deform the current one [23]. The resistance to interlamina dilation while the kink bands deform is modelled with a linearconstitutive law; the dilation resisting force F I relating to the dilation displacement δ I , thus: F I ( α ) = C I δ I , δ I ( α ) = t (cid:20) cos( α − β )cos β − (cid:21) , (1)with C I being the transverse stiffness of the laminate, related to the transverse Young’s modulus,and t being the thickness of an individual lamina. Since the area over which the interlamina regiondilates depends directly on the kink band width b , the stiffness C I can be expressed as: C I = bdk I , (2)where k I is the transverse stiffness per unit area of the laminate and d is the breadth of the strut.However, with the lamina assumed to be transversely incompressible in the current model andthe dilation displacement being assigned purely to the softer interlamina matrix material, a cleardeparture from the geological model, the current lamina thickness is thus t rather than t cos( α − β ).This is shown in Figure 6 and is detailed in the highlighted area of that diagram. The relationshipin equation (1) for δ I is thus obtained from taking the length AB from Figure 6, where δ I = AB − t .Hence, there is a transverse tensile strain developed as the gap between the laminae grows as α Article submitted to The Royal Society
TEX Paper eometric modelling of kink bands in laminates (c) α = 2 β (d) α > ββ α (b) α = β (a) α = 0 ββ β αα Figure 4: Sequence of kink band deformation in the geological folding model. (a) initial state with β defined from applied q ; (b) instantaneous release of contact and hence friction within the kinkband when α = β ; (c) layers inside and outside of the kink band all have equal thickness when α = 2 β ; (d) lock-up occurs when α > β and a new band would form.increases from zero to β . The gap subsequently begins to reduce; when α = 2 β the gap returns tozero, marking the commencement of lock-up.The work done in the dilation process is therefore given by the expression: U D = Z δ I ( α )0 F I ( α ′ ) d (cid:26) t (cid:20) cos( α ′ − β )cos β − (cid:21)(cid:27) = k I bdt (cid:20) − cos( α − β )cos β (cid:21) . (3)It is assumed that the interlamina region would not be damaged in the process of dilation and thatthe only nonlinearity in the constitutive law would be under shear. This is because the dilationdisplacement is relatively smaller than the shearing displacement that is discussed next; this hasthe additional advantage of maintaining model simplicity such that any mixed mode fractureconsiderations can be left for future work. Interlamina shearing or the laminae sliding relative to one another is modelled with a piecewiselinear constitutive law with the force resisting shear F II relating to the shearing displacement δ II ,thus: F II ( α ) = C II δ II , δ II ( α ) = t cos β [sin( α − β ) + sin β ] , (4)with C II being the shearing stiffness of the combination of the matrix and laminae sliding relativeto one another. The relationship for δ II in terms of α and β in equation (4) is given by examiningthe length BC in Figure 6. However, since the band is basically assumed to form instantaneouslybefore any rotation occurs, it is implied that F II (0) = δ II (0) = 0. However, since β = 0, theexpression δ II = BC + t tan β is obtained, where the force and displacement conditions are satisfied.When the shearing displacement reaches the initial proportionality limit, i.e. when δ II = δ C (see Figure 7), the relationship between F II and δ II changes to: F II = C II δ C (cid:18) δ II − δ M δ C − δ M (cid:19) , (5) Article submitted to The Royal Society
TEX Paper . A. Wadee et al. 6 b (mm) (cid:1) (mm)
Load (kN) (cid:1) (mm)
Band 2 locks upExperiment Band 1 (cid:11) = 2(cid:12)1
Band 2 (cid:11) = 2(cid:12)2
Band 3 (cid:11) = 2(cid:12)3
Band 1 locks up Band 3 locks up
Figure 5: Graphs showing the total axial load P (left) and the kink band width b (right) bothversus the total end-displacement ∆, taken from [23]. The load versus displacement graph showsthe lockup of the current kink band as a circle on the theoretical curve.where δ M is the shearing displacement when the corresponding resistance force reduces to zero.Now, if δ II ( α C ) = δ C and δ II ( α M ) = δ M , the expressions for the resisting force can be written thus: F II = C II t [sin( α − β ) + sin β ] / cos β for α = [0 , α C ] ,C II t cos β " sin( α − β ) − sin( α M − β )sin( α C − β ) − sin( α M − β ) [sin( α C − β ) + sin β ] for α > α C and α = [ α C , α M ] if sgn( δ C ) = sgn( δ M ) > , α > α M and sgn( δ C ) = sgn( δ M ) > . (6)Moreover, since the shear area of contact depends on the kink band width b , the stiffness C II canbe expressed as: C II = bdk II , (7)where k II is the shear stiffness per unit area of the lamina and hence the effective shear stress τ isdefined: τ = k II δ II . (8)The work done in the shearing process is given by the expression: U S = Z δ II ( α )0 F II ( α ′ ) d (cid:26) t cos β [sin( α ′ − β ) + sin β ] (cid:27) = k II bdt (cid:20) sin( α − β ) + sin β cos β (cid:21) = k II bdt L ( α ) , (9) Article submitted to The Royal Society
TEX Paper eometric modelling of kink bands in laminates b tα δ II β δ I F II F II F I PnP PPP k k c c F I t C B A β α C B A t/ cos β t cos( α − β )cos βt sin( α − β )cos β ( α − β ) Figure 6: Two internal laminae of the laminated composite model, the shaded region shows theinterlamina region which is exaggerated in scale for clarity. Dilation and shearing forces with theircorresponding displacements are given by: F I and F II with δ I and δ II respectively. The highlightedsection shows the lengths AB and BC that directly relate to δ I and δ II respectively. δ C δ M Slope: C = F C /δ C Slope: Cδ C δ C − δ M − δ M − δ C F δ F C − F C δ C δδ M F C F (a) sgn( δ C ) = sgn( δ M ) (b) sgn( δ C ) = − sgn( δ M ) Figure 7: Piecewise linear force versus displacement model applied for interlamina shearing: (a) alinear–softening response which is more representative of a fracture model; (b) a linear–hardeningresponse which is more appropriate for materials that show post-yield strength.for α α C , or: U S = k II bdt (cid:26) L ( α C )+ Z αα C (cid:20) sin( α ′ − β ) − sin( α M − β )sin( α C − β ) − sin( α M − β ) (cid:21) (cid:20) sin( α C − β ) + sin β cos β (cid:21) cos( α ′ − β ) d α ′ (cid:27) = k II bdt β (cid:26) [sin( α C − β ) + sin β ] + (cid:20) sin( α C − β ) + sin β sin( α C − β ) − sin( α M − β ) (cid:21) (cid:20) sin ( α − β ) − sin ( α C − β ) + 2 sin( α M − β )[sin( α C − β ) − sin( α − β )] (cid:21)(cid:27) = k II bdt S ( α ) , (10)beyond the proportionality limit where α = [ α C , α M ]. However, if α > α M and sgn( δ C ) = Article submitted to The Royal Society
TEX Paper . A. Wadee et al. 8 age mechanisms leading to structural collapse is detachment of the skin and stiffener, typically occurring at a stiffener flangeedge. In co-cured stiffened panels, this detachment is caused by delamination growth at or near the skin-stiffener interface, andin secondary bonded panels it usually involves the adhesive disbonding between the skin and stiffener in addition todelamination. In order to include the effects of skin-stiffener separation in numerical analyses, it is necessary to capture boththe initiation and propagation of this type of damage. This paper is focused on the growth of an existing skin-stiffenerseparation, with the prediction of damage initiation from an intact structure to be the subject of future work.In structures manufactured from laminated composite materials, the phenomenon of skin-stiffener separation can beconsidered analogous to that of interlaminar cracking, for which the use of fracture mechanics to predict crack growth has be-come common practice over the past two decades [3-4]. This analogy is directly applicable between lamina in co-cured stiff-ened panels and is an approximation in the case of an adhesive layer in secondary bonded panels. In a fracture mechanics analy-sis, the strain energy released in crack growth is compared to a threshold maximum of strain energy release rate, called theinterlaminar fracture toughness . The strain energy release rate is typically split into three components according to theseparate mechanisms of crack growth: opening (I), sliding (II), and scissoring (III), as shown in Fig. 1. The strain energy re-lease rates and fracture toughnesses in all the three modes are usually used in single-mode criteria or combined in amixed-mode criterion to determine the onset of crack propagation, and these generally require curve-fitting parameters takenfrom experimental tests.The virtual crack closure technique (VCCT) is one of the most commonly applied methods for determining the components of the strain energy release rate along a crack front. The VCCT approach was proposed by Rybicki and Kanninen [5]and is based on two assumptions: 1) Irwin’s assumption that the energy released in crack growth is equal to the work required toclose the crack to its original length, and 2) that crack growth does not significantly alter the state at the crack tip. The use ofVCCT is advantageous as it allows the strain energy release rates to be determined with simple equations from a single finite-element (FE) analysis. Numerous researchers have applied the VCCT to analyse the crack growth properties of a pre-existing interlaminar damage in a range of structures, including fracture mechanics test specimens [6-7], bonded joints [8-9], andboth co-cured and secondary bonded skin-stiffener interfaces [10-12].Predicting the collapse of a structure with account of skin-stiffener separation also requires the disbonded area to begrown during the analysis. To date, the VCCT has been limited in this respect due to the requirement of a fine mesh of the orderof the ply thickness [13] and the need for complicated algorithms to monitor the shape of the crack front. An alternative approach for modelling the skin-stiffener separation is with the so-called cohesive elements, which are used to control the relationship between opening stresses and displacements in an interface [14-15]. Cohesive elements offer the advantages of incorporating both initiation and propagation of disbonding in such a way that the damage is initiated by using strength criteria andthe final separation is governed by fracture mechanics. However, like the VCCT approach, the cohesive elements require a finemesh to remain accurate and can become prohibitively inaccurate when larger mesh sizes are used, which makes their application to large structures problematic. Also, the standard cohesive-element formulation cannot account for an arbitrary crackfront shape; therefore, it does not differentiate between mode II and III directions, and in general, the exact location of the crack à Fig. 1. Crack growth modes: I-opening (a), II-sliding (b), and III-scissoring (c). (a) Mode I age mechanisms leading to structural collapse is detachment of the skin and stiffener, typically occurring at a stiffener flangeedge. In co-cured stiffened panels, this detachment is caused by delamination growth at or near the skin-stiffener interface, andin secondary bonded panels it usually involves the adhesive disbonding between the skin and stiffener in addition todelamination. In order to include the effects of skin-stiffener separation in numerical analyses, it is necessary to capture boththe initiation and propagation of this type of damage. This paper is focused on the growth of an existing skin-stiffenerseparation, with the prediction of damage initiation from an intact structure to be the subject of future work.In structures manufactured from laminated composite materials, the phenomenon of skin-stiffener separation can beconsidered analogous to that of interlaminar cracking, for which the use of fracture mechanics to predict crack growth has be-come common practice over the past two decades [3-4]. This analogy is directly applicable between lamina in co-cured stiff-ened panels and is an approximation in the case of an adhesive layer in secondary bonded panels. In a fracture mechanics analy-sis, the strain energy released in crack growth is compared to a threshold maximum of strain energy release rate, called theinterlaminar fracture toughness . The strain energy release rate is typically split into three components according to theseparate mechanisms of crack growth: opening (I), sliding (II), and scissoring (III), as shown in Fig. 1. The strain energy re-lease rates and fracture toughnesses in all the three modes are usually used in single-mode criteria or combined in amixed-mode criterion to determine the onset of crack propagation, and these generally require curve-fitting parameters takenfrom experimental tests.The virtual crack closure technique (VCCT) is one of the most commonly applied methods for determining the components of the strain energy release rate along a crack front. The VCCT approach was proposed by Rybicki and Kanninen [5]and is based on two assumptions: 1) Irwin’s assumption that the energy released in crack growth is equal to the work required toclose the crack to its original length, and 2) that crack growth does not significantly alter the state at the crack tip. The use ofVCCT is advantageous as it allows the strain energy release rates to be determined with simple equations from a single finite-element (FE) analysis. Numerous researchers have applied the VCCT to analyse the crack growth properties of a pre-existing interlaminar damage in a range of structures, including fracture mechanics test specimens [6-7], bonded joints [8-9], andboth co-cured and secondary bonded skin-stiffener interfaces [10-12].Predicting the collapse of a structure with account of skin-stiffener separation also requires the disbonded area to begrown during the analysis. To date, the VCCT has been limited in this respect due to the requirement of a fine mesh of the orderof the ply thickness [13] and the need for complicated algorithms to monitor the shape of the crack front. An alternative approach for modelling the skin-stiffener separation is with the so-called cohesive elements, which are used to control the relationship between opening stresses and displacements in an interface [14-15]. Cohesive elements offer the advantages of incorporating both initiation and propagation of disbonding in such a way that the damage is initiated by using strength criteria andthe final separation is governed by fracture mechanics. However, like the VCCT approach, the cohesive elements require a finemesh to remain accurate and can become prohibitively inaccurate when larger mesh sizes are used, which makes their application to large structures problematic. Also, the standard cohesive-element formulation cannot account for an arbitrary crackfront shape; therefore, it does not differentiate between mode II and III directions, and in general, the exact location of the crack b Fig. 1. Crack growth modes: I-opening (a), II-sliding (b), and III-scissoring (c). (b) Mode II age mechanisms leading to structural collapse is detachment of the skin and stiffener, typically occurring at a stiffener flangeedge. In co-cured stiffened panels, this detachment is caused by delamination growth at or near the skin-stiffener interface, andin secondary bonded panels it usually involves the adhesive disbonding between the skin and stiffener in addition todelamination. In order to include the effects of skin-stiffener separation in numerical analyses, it is necessary to capture boththe initiation and propagation of this type of damage. This paper is focused on the growth of an existing skin-stiffenerseparation, with the prediction of damage initiation from an intact structure to be the subject of future work.In structures manufactured from laminated composite materials, the phenomenon of skin-stiffener separation can beconsidered analogous to that of interlaminar cracking, for which the use of fracture mechanics to predict crack growth has be-come common practice over the past two decades [3-4]. This analogy is directly applicable between lamina in co-cured stiff-ened panels and is an approximation in the case of an adhesive layer in secondary bonded panels. In a fracture mechanics analy-sis, the strain energy released in crack growth is compared to a threshold maximum of strain energy release rate, called theinterlaminar fracture toughness . The strain energy release rate is typically split into three components according to theseparate mechanisms of crack growth: opening (I), sliding (II), and scissoring (III), as shown in Fig. 1. The strain energy re-lease rates and fracture toughnesses in all the three modes are usually used in single-mode criteria or combined in amixed-mode criterion to determine the onset of crack propagation, and these generally require curve-fitting parameters takenfrom experimental tests.The virtual crack closure technique (VCCT) is one of the most commonly applied methods for determining the components of the strain energy release rate along a crack front. The VCCT approach was proposed by Rybicki and Kanninen [5]and is based on two assumptions: 1) Irwin’s assumption that the energy released in crack growth is equal to the work required toclose the crack to its original length, and 2) that crack growth does not significantly alter the state at the crack tip. The use ofVCCT is advantageous as it allows the strain energy release rates to be determined with simple equations from a single finite-element (FE) analysis. Numerous researchers have applied the VCCT to analyse the crack growth properties of a pre-existing interlaminar damage in a range of structures, including fracture mechanics test specimens [6-7], bonded joints [8-9], andboth co-cured and secondary bonded skin-stiffener interfaces [10-12].Predicting the collapse of a structure with account of skin-stiffener separation also requires the disbonded area to begrown during the analysis. To date, the VCCT has been limited in this respect due to the requirement of a fine mesh of the orderof the ply thickness [13] and the need for complicated algorithms to monitor the shape of the crack front. An alternative approach for modelling the skin-stiffener separation is with the so-called cohesive elements, which are used to control the relationship between opening stresses and displacements in an interface [14-15]. Cohesive elements offer the advantages of incorporating both initiation and propagation of disbonding in such a way that the damage is initiated by using strength criteria andthe final separation is governed by fracture mechanics. However, like the VCCT approach, the cohesive elements require a finemesh to remain accurate and can become prohibitively inaccurate when larger mesh sizes are used, which makes their application to large structures problematic. Also, the standard cohesive-element formulation cannot account for an arbitrary crackfront shape; therefore, it does not differentiate between mode II and III directions, and in general, the exact location of the crack c Fig. 1. Crack growth modes: I-opening (a), II-sliding (b), and III-scissoring (c). (c) Mode III
Figure 8: Fracture modes. Mode I is tearing; Mode II is shearing and Mode III is scissoring. Inthe current model, only Mode II is relevant.sgn( δ M ) > U S becomes: U S = k II bdt (cid:26) L ( α C )+ Z α M α C (cid:20) sin( α ′ − β ) − sin( α M − β )sin( α C − β ) − sin( α M − β ) (cid:21) (cid:20) sin( α C − β ) + sin β cos β (cid:21) cos( α ′ − β ) d α ′ (cid:27) = k II bdt β (cid:26) sin β [sin β + sin( α C − β ) + sin( α M − β )] + sin( α C − β ) sin( α M − β ) (cid:27) = k II bdt S ( α M ) . (11)There would still be the potential for frictional forces to resist shear even though δ > δ M andthe interlamina region has lost all shear strength. However, this effect is currently neglected andhence the current model would tend to underestimate the true strength to some extent. As in the geological model, the strain energy stored in bending can be taken from a pair ofrotational springs of stiffness c : U b = cα . (12)The stiffness of the rotational springs is related differently from the geological model as the ex-pression for that model contained the overburden pressure q [22]. Since the bending energy shouldstrictly relate to curvature κ , where: U b = 2 Z b/ − b/ EIκ d x, (13)with x defining the domain of one bending corner and κ as the rate of change of the kink bandangle α over the kink band width b , as represented in Figure 9, thus: b/ b/ b/ b/ α α (c) b/ b/ b/ α α (b) b/ x = + b/ x = − b/ x (a) Figure 9: Bending of a lamina: (a) definition of x ; (b) idealized case; (b) actual case. Curvature κ is defined as the total angle change 2 α over the effective length of the band 2 b , hence κ ≈ α/b . Article submitted to The Royal Society
TEX Paper eometric modelling of kink bands in laminates κ ≈ αb ⇒ c ≈ EIb . (14)Hence, the rotational stiffness c is related to the flexural rigidity EI of a lamina with E being itsthe Young’s modulus in the axial direction and its second moment of area I = dt /
12. The strainenergy per layer associated with the in-line spring of stiffness k is hence given by: U m = 12 kδ a , (15)where δ a is the axial displacement of the springs. The in-line spring stiffness k = Edt/L for asingle lamina with L being the length of the strut. The work done by the external load can betaken simply as the sum of the displacement of the in-line springs δ a and from the band deformingmultiplied by the axial load P , which can be defined as the axial pressure p multiplied by thecross-sectional area of a lamina, dt : P ∆ = pdt [ δ a + b (1 − cos α )] . (16) The total potential energy V is given by the sum of the strain energies from bending U b , the in-linesprings U m , interlaminar dilation U D and shearing U S , minus the work done P ∆, thus: V = U b + U m + U D + U S − P ∆ . (17)Since the dilation terms are assumed to be linearly elastic throughout their loading history, thetotal potential energy per axially loaded lamina takes three forms:1. The case where α = [0 , α C ], so V = V L , i.e. linearly elastic in shear.2. The case where α > α C , so V = V S , i.e. the secondary shear stiffness is either a smallerpositive value than the primary shear stiffness or a negative value.3. The case where α > α M and sgn( δ C ) = sgn( δ M ) >
0, so V = V Z , i.e. no shear stiffness,which only occurs if the secondary shear stiffness is negative.These forms of the total potential energy are given by the expressions: V L = V I + k II bdt L ( α ) , V S = V I + k II bdt S ( α ) , V Z = V I + k II bdt S ( α M ) , (18)where V I is given by: V I = kδ a Edt α b + k I bdt (cid:20) − cos( α − β )cos β (cid:21) − pdt [ δ a + b (1 − cos α )] . (19)The total potential energy functions are nondimensionalized by dividing through by kt and canbe re-expressed in terms of rescaled parameters:˜ V L = ˜ V I + ˜ k II ˜ b L ( α ) , ˜ V S = ˜ V I + ˜ k II ˜ b S ( α ) , ˜ V Z = ˜ V I + ˜ k II ˜ b S ( α M ) , (20)where: ˜ V I = V I kt , ˜ V L = V L kt , ˜ V S = V S kt , ˜ V Z = V Z kt , ˜ δ = δ a t , ˜∆ = ∆ t , ˜ b = bt , ˜ p = pdk = pLEt , ˜ D = Ed k = L t , ˜ k I = k I dtk = k I LE , ˜ k II = k II dtk = k II LE . (21)
Article submitted to The Royal Society
TEX Paper . A. Wadee et al. 10
The equilibrium equations are defined by the condition of stationary potential energy with respectto the end-shortening δ a , the kink band angle α and the kink band width b ; these can be writtenin nondimensional terms, thus: ˜ p = ˜ δ, (22)˜ p = ˜ k I I α + ˜ k II J α + 2 ˜ Dα ˜ b sin α , (23)˜ p = ˜ k I I b + ˜ k II J b − ˜ Dα ˜ b (1 − cos α ) . (24)Equation (22) defines the pre-kinking fundamental equilibrium path that accounts for pure com-pression of the in-line springs of stiffness k . Equations (23)–(24) define the post-instability statesfor the non-trivial kink band deformations; equating them allows the kink band width b to beevaluated analytically: ˜ b = ( ˜ Dα [2 / sin α + α/ (1 − cos α )]˜ k I ( I b − I α ) + ˜ k II ( J b − J α ) ) / . (25)The expressions for I α and I b are given in detail thus: I α = (cid:20) − cos( α − β )cos β (cid:21) sin( α − β )sin α cos β , I b = 12 (1 − cos α ) (cid:20) − cos( α − β )cos β (cid:21) , (26)where these expressions apply for the entire range of α . However, the expressions for J α and J b change for each form of the total potential energy function; for ˜ V = ˜ V L , the expressions are: J α = cos( α − β ) [sin( α − β ) + sin β ]sin α cos β , J b = [sin( α − β ) + sin β ] − cos α ) cos β ; (27)for ˜ V = ˜ V S : J α = cos( α − β )sin α cos β (cid:20) sin( α C − β ) + sin β sin( α C − β ) − sin( α M − β ) (cid:21) [sin( α − β ) − sin( α M − β )] ,J b = (cid:20) sin( α C − β ) + sin β − cos α ) cos β (cid:21) (cid:26) sin( α C − β ) + sin β + sin ( α − β ) − sin ( α C − β ) + 2 sin( α M − β )[sin( α C − β ) − sin( α − β )]sin( α C − β ) − sin( α M − β ) (cid:27) ; (28)and for ˜ V = ˜ V Z : J α = 0 , J b = sin β [sin β + sin( α C − β ) + sin( α M − β )] + sin( α C − β ) sin( α M − β )2 cos β (1 − cos α ) . (29)The initial limiting case where α → b → ∞ and ˜ p → ˜ k I tan β + ˜ k II . The result for b suggesting that the kink band is initially prevalent throughout the structure and the result for p showing that the critical load depends primarily on the shear stiffness with a smaller contributionfrom the dilation stiffness that relates to β . This reproduces similar results from the literaturewhere the critical stress is related to the shear modulus [4, 5, 13]; it also reflects a significantdifference from the geological model which has an infinite critical load and where the kink bandwidth grows from zero length [22]. Article submitted to The Royal Society
TEX Paper eometric modelling of kink bands in laminates t = 7 × − mmOverall rod length L = 76 mmLongitudinal Young’s modulus: E = E = 130 .
76 kN / mm Transverse Young’s modulus: E = 10 .
40 kN / mm Shear modulus (initial to final): G = 6 . → .
68 kN / mm over 4% shear strainNondimensional stiffness quantities:Flexural rigidity: ˜ D = L/ (12 t )Dilation stiffness: ˜ k I = E L/ ( E t )Shearing stiffness: ˜ k II = G L/ ( E t )Table 1: Properties used in the validation study to compare the current model with experimentspresented in [5]. Results from the current model are initially compared with published experiments on circularcylindrical composite rods with confined ends that exhibited kink bands under axial compression[5]. Although the model is formulated for flat rectangular laminae, the lamina thickness t can beperceived to be equivalent to the diameter of an individual fibre rod. This aids the comparisonbetween the current model and the experiments such that both loading levels and the kink bandwidth can be compared; a similar approach was employed in [12].The dimensions of the overall sample had a diameter of 8 .
255 mm with the relevant propertiesgiven in Table 1. Note that the breadth d is not given since it cancels in all the relevant nondi-mensional quantities. The sample comprised ICI APC-2/AS4 composite fibres. Since the samplewas cylindrical, the system in [5] was presented in terms of a cylindrical polar coordinate systemwith x and x being the longitudinal and the radial coordinates respectively, as shown in Figure10. For the tests presented to measure the change in shear modulus G , there was no plateau x x
76 mm .
255 mm
Figure 10: Representation of the experimental sample in [5]. The rod comprised ICI APC-2/AS4composite fibres with properties as given in Table 1. The sample was confined such that there wasnegligible transverse compression but also that global buckling was not an issue.shown in the test data, see Figures A3 and A5 in [5]. In the current study, it is therefore assumedthat the piecewise linear model for the shear stiffness reflect the initial and final values foundin the experiments; hence sgn( δ M ) = − sgn( δ C ), i.e. a linear–hardening model is implemented asrepresented in Figure 7(b).The critical shear angle, γ C , which is effectively equal to the so-called engineering shear strain,for the piecewise linear idealization, is the angle beyond which the shear stiffness is replaced bya secondary smaller value; this is estimated from the aforementioned graphs in Figures A3 andA5 in [5] to be 0 .
012 rad (or 0 . ◦ ). The shear angle can be expressed in terms of the kink bandorientation angle β and the kink band angle, α , such that:tan γ C = δ II ( α C ) δ I ( α C ) + t = sin ( α C − β ) + sin β cos ( α C − β ) . (30)Given that β is assumed to remain constant during deformation, the critical kink band angle α C Article submitted to The Royal Society
TEX Paper . A. Wadee et al. 12can be found by rearranging (30), thus:tan γ C cos ( α C − β ) − sin ( α C − β ) − sin β = 0 , (31)and solving for α C . This is achieved by substituting the critical shear angle γ C from above andthe kink band orientation angle from [5], where β was reported to lie between 12 ◦ → ◦ to the x direction; for the specified values, α C is approximately equal to γ C ( α C = 0 . → . δ M /t = − .
094 is usedsuch that the ratios between the initial and final values of the shear stiffness reflect the reportedexperimental data.Figure 11 shows numerical results from the current model using the properties defined in
70 80 90 100707580859095100 0 0.2 0.4 0.6 0.801002003004005006000 0.2 0.4 0.605101520 0 0.01 0.02 0.03 0.04 0.05020406080100 τ ˜ p ˜ b α ˜∆ δ II /t ˜ pα (a) (b)(d)(c) Increasing β Increasing β Increasing β Figure 11: Nondimensional plots of load ˜ p versus (a) total end-shortening ˜∆ and (b) kink bandangle α (rad) ; (c) kink band width ˜ b versus kink band angle α (rad). Range of β = 12 ◦ → ◦ .(d) Piecewise linear–hardening relationship of the effective shear stress τ (N / mm ) versus thenormalized shearing displacement δ II /t for β = 16 ◦ . Properties of ICI APC-2/AS4 compositefibres and configuration and the range for β were taken from [5].Table 1 with β values as found in the published results. Note that the nondimensional totalend-shortening ˜∆ is defined thus: ˜∆ = ˜ δ + ˜ b (1 − cos α ) . (32)The actual kink band widths in the 5 tests were reported to range from 11 to 36 fibre diameters(directly corresponding to ˜ b in the current model) and the compressive strengths were found toaverage at 1 .
119 kN / mm with a standard deviation of 0 .
043 kN / mm (directly correspondingto p in the current model). The results from the current model show highly unstable snap-backand hence the critical load would never be reached realistically, see Figure 11(a) and (b); a wellestablished feature for systems of this type [4]. For comparison purposes, the pressure p is takenat the point at which the structure stabilizes and reaches a plateau; for the range of the β anglesconsidered, the nondimensional stabilization pressure ˜ p ranges from 80 . → . p ranging from 0 .
969 kN / mm → .
992 kN / mm : an error againstthe average from the experimental results of between 11% → Article submitted to The Royal Society
TEX Paper eometric modelling of kink bands in laminates β = 12 ◦ Case: β = 16 ◦ α = β ˜ b = 10 . b = 10 . α = 2 β ˜ b = 17 . b = 19 . α = β and α = 2 β are the points where the dilation within the bandare effectively maximized and minimized respectively; experiments in [5] reported ˜ b = 11 → α increases, initiallythe nondimensional kink band width ˜ b falls from a large value to a small value, approximately5 . α = 0 .
024 rad( ≈ . ◦ ). As α increases further, the kink band width begins to increaseslowly; see Table 2 for details of some key points. According to the sequence described in Figures4(b)–(d) the kink band itself maximizes dilation when α = β , minimizes it when α = 2 β and locksup when α > β . The results of the current model, particularly when α = 2 β , lie at the lower endof the range of observed values of the band widths from the published experiments. This seemssensible given that the lock-up condition used, where α = 2 β , represents a lower bound [22], whichimplies that the current model would also tend to predict lower bound kink band widths. Hence,the results from the comparisons between the current model and the published experiments [5] arehighly encouraging; they offer very good quantitative agreement for the loading and the geometricdeformation – key quantities that define the kink band phenomenon. The favourable comparisons between the current model with the published experiments in [5]imply that the fundamental physics of the system are captured by the current approach. Thestudy is therefore extended to present a series of model parametric variations to establish theirrelative effects. The basic geometric and material configuration is identical to that used in thevalidation study presented in Table 1. Material and geometric parameters are varied individually,while maintaining the remaining ones at their original values. The parameters that are variedare the kink band orientation angle β , the critical kink band angle, α C , the composite direct andshear moduli, E , E and G , and the shape of the piecewise linear relationship for shear. In the current model, the orientation angle β needs to be fixed a priori , hence the effects of differentstarting conditions for the model need to be established. Increasing β from 10 ◦ to 30 ◦ , a rangethat is representative of laminate experiments in the literature [5, 24], leads to a stiffer responsefor increasing α , as shown in Figure 12, with the pressure capacity for β = 30 ◦ being more thandouble the capacity for β = 10 ◦ for values of α < β . The graphs in Figure 13 raise an interestingpoint about the response particularly when β > . ◦ , which seems to define a boundary wherethe kink band width b loses its monotonically increasing property after it initially troughs for asmall value of α , which was identified as approximately 1 . ◦ in § α = 2 β temporarily peak when β = 22 . ◦ .For higher orientation angles the kink band width b in fact peaks beyond α = 1 . ◦ , then troughsand then resumes the monotonic rise as seen for β . ◦ . Moreover, this also explains thereason why the stabilization pressure increases for β > . ◦ , as shown in Figure 12(a), since thepressure has an inverse square relationship with the kink band width as shown in the equilibriumequations (23) and (24). The graphs presented in Figure 14 attribute this loss of monotonicity in b (beyond α = 1 . ◦ ) to the dominating influence of the dilation terms for larger β , particularlyin the region of maximum dilation where α ≈ β . In the first instance, it should be recalled thatwhen β is larger the potential maximum dilation displacement δ I is also larger relative to δ II when α = β . Figure 14(a) shows that the maximum of the dilation term ˜ k I ( I b − I α ) from the expression Article submitted to The Royal Society
TEX Paper . A. Wadee et al. 14
70 75 80 85 90 95 100707580859095100 ˜∆ ˜ p α ˜ p increasing β increasing β (a) (b) Figure 12: Equilibrium paths for different β = 10 ◦ → ◦ through nondimensional plots of load ˜ p versus (a) total end-shortening ˜∆ and (b) kink band angle α (rad). α ˜ b (a)
15 20 25 30727476788082848688 ˜ b ˜ p (b) β =12.5 ° β =15 ° β =17.5 ° β =20 ° β =22.5 ° β =25 ° β =27.5 ° β =30 ° β =10 ° increasing β Figure 13: (a) Nondimensional kink band widths ˜ b versus the kink band angle α (rad) for a rangeof orientation angles β = 10 ◦ → ◦ ; circles mark the the lower bound lock-up condition α = 2 β .(b) Values of nondimensional kink band widths ˜ b and applied axial pressure ˜ p at the lower boundlock-up condition.for b , i.e. equation (25), increases substantially with β whereas Figure 14(b) shows only verymarginal changes in the respective shear term ˜ k II ( J b − J α ). The numerator in equation (25), ˜ b num ,which represents the influence of bending, is independent of β , as shown in Figure 14(c) but therespective denominator, ˜ b den , shows that the dilation term influences the values significantly forthe higher β values, as shown in Figure 14(d). Once α gradually increases above β the dilationdisplacement progressively reduces and the shear term begins to dominate with the result that thekink band width resumes growth and lock-up occurs. This effect is similar to that found in thegeological model with the introduction of the foundation spring of stiffness k f [22], as shown inFigure 2; the kink band width was also found to plateau with higher foundation stiffnesses. It isworth noting that if destiffening in the constitutive law for dilation was introduced that the effectfound in the present case would be generally less pronounced. Increasing the critical kink band angle from α C = 0 . ◦ → . ◦ (with a fixed limiting displacement δ M /t = − .
094 as before) shows an increase in the critical shear stress before destiffening occurs –
Article submitted to The Royal Society
TEX Paper eometric modelling of kink bands in laminates α α α α ˜ k I ( I b − I α ) ˜ k II ( J b − J α ) ˜ b nu m ˜ b d e n Increasing β Increasing β Increasing β (a) (b)(d)(c) Figure 14: Graphs of various terms from the expression for the nondimensional kink band width˜ b , equation (25) for β = 10 ◦ → ◦ versus the kink band angle α (rad). (a)–(b) Plots of dilationand shear terms respectively. (c)–(d) Plots of the numerator and denominator of the ˜ b expression.see Figure 15(e) – and leads to a monotonic increase of the axial pressure p and the minimum kinkband width b – see Figures 15(a) and (c). A subtly different pattern is observed in Figures 15(b) and(d) where trends for increasing the initial shear modulus ( G init12 = 6 . / mm → . / mm ),lead to higher stabilization pressures but smaller minimum kink band widths. These are logicalresults since the effect of increasing the critical kink band angle will lead to a later destabilizationin shear and hence increase the load and band width; the increase in the initial shear modulusincreases the resistance against shearing – the process of kink banding therefore requires moreaxial pressure to overcome this increased stiffness. However, the increased shear stiffness reducesthe kink band width since there is a greater resistance to that type of deformation. The variation in the piecewise linear model for the shearing response is now discussed. Theconstitutive behaviour, F II versus δ II has been hitherto assumed to be a linear–hardening lawwhich corresponded with the data from the literature used in the validation exercise. Figure 16shows results for different secondary slopes while they remain positive (a linear–hardening law).Figure 17 shows results for reducing the secondary slope further such that they become negative(a linear–softening law). The results exhibit progressive behaviour; the reduced secondary slopesreduce the load carrying capacity but increase the kink band widths. This can be understood fromthe softening of the internal structure giving less resistance to the kink banding process, allowingfor larger rotations and gross deformations. For the cases presented in Figure 17, the negativesecondary slope mimics the behaviour of a fracture process where the shear stiffness and strengthhas vanished and mode II fracture and crack propagation would occur. However, as describedabove, a similar pattern remains with the strength reducing and the band widths increasing forweaker properties in shear, which appears to be entirely logical. The detailed effects of crackpropagation have been left for future work although recent work on buckling-driven delamination[25] has suggested that an analytical treatment of such effects may indeed be tractable. Article submitted to The Royal Society
TEX Paper . A. Wadee et al. 16
60 80 100 1206080100120 ˜∆ ˜ p (a)0.01 0.02 0.03 0.04 0.05 0.065.566.577.58 α ˜ b (c)0 0.02 0.04 0.06 0.08050100150 τ δ II /t (e) 60 80 100 1206080100120 ˜∆ ˜ p (b)0.01 0.02 0.03 0.04 0.05 0.06 0.075678 α ˜ b (d)0 0.02 0.04 0.06 0.08050100150 τ δ II /t (f) increasing α C increasing α C increasing G init increasing G init increasing G init increasing α C Figure 15: Comparison of the response of the kink band formation for different α C values in (a),(c) and (e) for the range of α C = 0 . ◦ → . ◦ and initial G values from 6 .
03 kN / mm → .
45 kN / mm (in (b), (d) and (f). Nondimensional plots of (a)–(b) load ˜ p versus total end-shortening ˜∆, (c)–(d) kink band width ˜ b versus kink band angle α (rad). (e)–(f) show the rela-tionship of the effective shear stress τ versus the normalized shearing displacement δ II /t . Notethat β = 12 ◦ throughout. Results for a two-fold increase in the axial modulus E suggest that this only has a marginaleffect on the stabilization pressure ( ≈ .
5% increase), whereas increasing the lateral modulus E results in a significantly stiffer response; the system stabilizing to a smaller kink band width (seeFigure 18). These, again, are logical results since the effect of increasing the lateral modulus E increases the resistance against dilation; the process of kink banding therefore requiring moreaxial pressure to overcome this. Increasing the axial modulus increases the axial stiffness k , whichin turn effectively reduces the relative dilation and shear stiffnesses without affecting the relativebending stiffness, see the scaling relationships in equation (21). Since bending is currently assumedto be purely linear, its relative effect becomes progressively more pronounced and then outweighsthe reduced dilation and shear effects at large rotations. Obviously, if the bending was assumedto plateau due to plasticity, this effect would be limited. Article submitted to The Royal Society
TEX Paper eometric modelling of kink bands in laminates
20 40 60 80 10020406080100 ˜∆ ˜ p (a) 0 0.1 0.2 0.3 0.4 0.5100200300400500600 α ˜ p (b)0 0.1 0.2 0.3 0.4 0.55101520 α ˜ b (c) 0 0.02 0.04 0.06 0.08050100 τ δ II /t (d) increasing δ M increasing δ M increasing δ M increasing δ M Figure 16: Nondimensional plots of load ˜ p versus (a) total end-shortening ˜∆ and (b) kink bandangle α (rad); (c) kink band width ˜ b versus kink band angle α (rad); (d) Piecewise linear–hardeningrelationship of the effective shear stress τ (N / mm ) versus the normalized shearing displacement δ II /t . Range of δ M /t = − . → − .
094 and β = 12 ◦ . An analytical, nonlinear, potential energy based model for kink banding in compressed unidirec-tional laminated composite panels has been presented. Comparisons of results with publishedexperiments from the literature suggest that very good agreement can be achieved from this rela-tively simple mechanical approach provided certain important characteristics are incorporated:1.
Interlamina dilation and shearing : the kink band rotation naturally causes shearing andchanges the gap between the laminae, the matrix within the composite needs to resist bothof these displacements for the laminate to have integrity and significant structural strength.2.
Bending energy : the resistance to rotation sets a length scale which in this case is the kinkband width b .Linear constitutive relationships for the mechanisms of bending and dilation, and piecewise linearfor the process of shearing, together with nonlinear geometric relationships have been applied. Theapproach has been successful such that the mechanical response captures the fundamental physicsof kink banding and agrees with the experiments from [5] in terms of kink band widths and loadinglevels without having to resort to sophisticated numerical or continuum formulations. Unlike thegeological model [22], where a relationship was derived for the band orientation β that was relatedto the overburden pressure q , in the current case the angle β has to be assumed a priori since, asfar as the authors are aware, no satisfactory procedure for predicting β for composite laminatesexists. For laminates, the magnitude of the orientation angle β has been largely attributed tothe manufacturing process [5, 15]. However, if the overburden pressure is considered to be thecontrolling parameter for the equivalent “manufacturing process” that keeps the geological layersbehaving together, then future work on modelling the process of manufacturing composite lam-inates may bear fruit; an indication of the parameters that govern the orientation angle for thecurrent case may be established. Although this is a shortcoming for the present model, the resultsfrom the parametric study are very encouraging with the trends appearing to be entirely logical.The current model can of course be used as a basis for further work. In the first instance, theformation of new kink bands can be investigated since work derived from similar approaches to Article submitted to The Royal Society
TEX Paper . A. Wadee et al. 18 ˜∆ ˜ p (a) 0 0.1 0.2 0.3 0.4 0.50200400600 α ˜ p (b)0 0.1 0.2 0.3 0.4 0.55101520 α ˜ b (c) 0 0.02 0.04 0.06 0.08020406080 τ δ II /t (d) increasing δ M increasing δ M increasing δ M increasing δ M Figure 17: Nondimensional plots of load ˜ p versus (a) total end-shortening ˜∆ and (b) kink bandangle α (rad); (c) kink band width ˜ b versus kink band angle α (rad); (d) Piecewise linear–softeningrelationship of the effective shear stress τ (N / mm ) versus the normalized shearing displacement δ II /t . Range of δ M /t = 0 . → .
482 and β = 12 ◦ .the current one exist for the geological model [23, 26]. In particular, in [23] the lock-up criterionwas employed as the condition to introduce a new kink band; although it was assumed that theoriginal kink band stops growing, the comparisons between theory and experiments were shown tobe very good. The piecewise linear formulation applied currently for shear could also be extendedto include dilation giving the possibility of mixed mode fracture [27] for the first kink band.Moreover, loading cases that are more complex than uniform compression could be investigated;for example, numerical approaches have been developed in [17, 28] with varying degrees of successto investigate the formation of kink bands where there is a combination of shear and compression.An additional complication in the combined loading case is that the kink band propagation acrossthe sample tends to occur more gradually in contrast to the present case where the formationprocess is fast. References [1] Anderson, T. B. Kink-bands and related geological structures.
Nature , 202:272–274, 1964.[2] Hobbs, B. E., Means, W. D., and Williams, P. F.
An Outline of Structural Geology . Wiley,New York, 1976.[3] Price, N. J. and Cosgrove, J. W.
Analysis of geological structures . Cambridge UniversityPress, Cambridge, 1990.[4] Budiansky, B. and Fleck, N. A. Compressive failure of fiber composites.
J. Mech. Phys.Solids , 41:183–211, 1993.[5] Kyriakides, S., Arseculeratine, R., Perry, E. J., and Liechti, K. M. On the compressive failureof fiber reinforced composites.
Int. J. Solids Struct. , 32:689–738, 1995.[6] Reid, S. R. and Peng, C. Dynamic uniaxial crushing of wood.
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Article submitted to The Royal Society
TEX Paper eometric modelling of kink bands in laminates
60 65 70 75 80 85657075 ˜∆ ˜ p (a)0 0.1 0.2 0.3 0.4510152025 α ˜ b (c) 70 80 90 1007075808590 ˜∆ ˜ p (b)0 0.1 0.2 0.3 0.405101520 α ˜ b (d) increasing E increasing E increasing E increasing E Figure 18: Nondimensional plots for the range of axial direct modulus E = 130 . / mm → . / mm in (a) and (c) and transverse direct modulus E = 10 . / mm →
52 kN / mm in (b) and (d). (a)–(b) Load ˜ p versus α ; (c)–(d) kink band width ˜ b versus the kink band angle α (rad). Note that β = 12 ◦ throughout.[7] Vogler, T. J. and Kyriakides, S. On the initiation and growth of kink bands in fiber composites.Part I: experiments. Int. J. Solids Struct. , 38:2639–2651, 2001.[8] Byskov, E., Christoffersen, J., Christensen, C. D., and Poulsen, J. S. Kinkband formation inwood and fiber composites—morphology and analysis.
Int. J. Solids Struct. , 39:3649–3673,2002.[9] Da Silva, A. and Kyriakides, S. Compressive response and failure of balsa wood.
Int. J. SolidsStruct. , 44(25–26):8685–8717, 2007.[10] Pimenta, S., Gutkin, R., Pinho, S. T., and Robinson, P. A micromechanical model for kink-band formation: Part I—experimental study and numerical modelling.
Compos. Sci. Technol. ,69:948–955, 2009.[11] Hobbs, R. E., Overington, M. S., Hearle, J. W. S., and Banfield, S. J. Buckling of fibres andyarns within ropes and other fibre assemblies.
J. Textile Inst. , 91(3):335–358, 2000.[12] Edmunds, R. and Wadee, M. A. On kink banding in individual PPTA fibres.
Compos. Sci.Technol. , 65(7–8):1284–1298, 2005.[13] Rosen, B. W. Mechanics of composite strengthening. In Bush, S. H., editor,
Fiber CompositeMaterials , pages 37–75. American Society of Metals, 1965.[14] Argon, A. S. Fracture of composites.
Treatise Mater. Sci. Technol. , 1:79–114, 1972.[15] Budiansky, B. Micromechanics.
Comput. & Struct. , 16:3–12, 1983.[16] Fu, Y. B. and Zhang, Y. T. Continuum-mechanical modelling of kink-band formation in fibrereinforced composites.
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Article submitted to The Royal Society
TEX Paper . A. Wadee et al. 20[17] Vogler, T. J., Hsu, S.-Y., and Kyriakides, S. On the initiation and growth of kink bands infiber composites. Part II: analysis.
Int. J. Solids Struct. , 38:2653–2682, 2001.[18] Budiansky, B., Fleck, N. A., and Amazigo, J. C. On kink-band propagation in fiber compos-ites.
J. Mech. Phys. Solids , 46:1637–1653, 1998.[19] Pimenta, S., Gutkin, R., Pinho, S. T., and Robinson, P. A micromechanical model for kink-band formation: Part II—analytical modelling.
Compos. Sci. Technol. , 69:956–964, 2009.[20] Fleck, N. A. Compressive failure of fiber composites.
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