Phil. Trans. R. Soc. A (2012) 370, 1827–1849doi:10.1098/rsta.2011.0380
Geometric modelling of kink banding inlaminated structures
BY M. AHMER WADEE1,*, CHRISTINA VÖLLMECKE2, JOSEPH F. HALEY1
AND STYLIANOS YIATROS3
1Department of Civil and Environmental Engineering, Imperial CollegeLondon, London SW7 2AZ, UK
2LKM, Institut für Mechanik, Technische Universität Berlin, Germany3Department of Civil Engineering, Brunel University, Uxbridge UB8 3PH, UK
An analytical model founded on geometric and potential energy principles for kink banddeformation in laminated composite struts is presented. It is adapted from an earliersuccessful study on confined layered structures that was formulated to model kink bandformation in the folding of geological layers. This study’s principal aim was to explorethe underlying mechanisms governing the kinking response of flat, laminated componentscomprising unidirectional composite laminae. A pilot parametric study indicates thatthe key features of the mechanical response are captured well and that quantitativecomparisons with experiments presented in the literature are highly encouraging.
Keywords: kink banding; laminated materials; nonlinearity; energy methods;analytical modelling
1. Introduction
Kink banding is a phenomenon seen across many scales. It is a potential failuremode for any layered, laminated or fibrous material, held together by externalpressure or some form of internal matrix, and subjected to compression parallelto the layers. Many examples can be found in the literature concerning thedeformation of geological strata [1–3], wood and fibre composites [4–10] andinternally in wire and fibre ropes [11,12]. There have been many attempts toreproduce kink banding theoretically, from early mechanical models [13,14] tomore sophisticated formulations derived from continuum mechanics [15], finiteelasticity theory [16] and numerical perspectives for more complex loadingarrangements [17].
There has been much relevant work on composite materials, with significantproblems being encountered as outlined thus. First, although two-dimensionalmodels are commonly employed [18,19], modelling into the third dimension addsa significant extra component. It inevitably involves a smeared approach in themodelling of material properties because there is a mix of laminae and the*Author for correspondence ([email protected]).
One contribution of 15 to a Theme Issue ‘Geometry and mechanics of layered structures andmaterials’.
matrix with the possibility of voids. Second, failure is likely to be governedby nonlinear material effects in shearing the matrix material [20], and this isconsiderably less easy to measure or control than the combination of overburdenpressure and friction considered in work on kink banding during geologicalfolding [21–23].
In the present paper, a pilot study is presented where the discrete modelformulated for kink banding in geological layers is adapted such that it can beapplied to unidirectional laminated composite struts that are compressed in adirection parallel to the laminae. This is achieved by releasing the assumption thatthe creation of voids between the layers is penalized by increasing the total systemenergy because, in the current case, no overburden pressure actively compressesthe layers in the lateral direction. It is worth noting that the lateral directionis defined throughout as orthogonal to the layers. Therefore, the rotation ofthe laminae during the formation of the kink band causes a dilation, which isresisted by lateral tensile forces generated within the interlaminar region. Thecoincident shearing of this region also generates an additional resisting force,but, as mentioned already, this can be subject to nonlinearity—in particular, areduced stiffness that may be either positive (hardening) or negative (softeningperhaps leading to fracture), which is currently formulated with a piecewiselinear constitutive law. Work done from dilation and shearing is evaluated;additional features from the original model (strain energies from bending anddirect compression) and the work done from the external load can be incorporatedwithout significant alterations. An advantage of the presented model is thatthe resulting equilibrium equations can be written and solved entirely in ananalytical form, without having to resort to complex continuum models ornumerical solvers.
The primary aim of the current work was to lay the foundations forfuture research. The geometric approach has yielded excellent comparisons withexperiments for the model for kink banding in confined layers of paper that wasused as an analogue for geological layers (termed the ‘geological model’ presently);the same is true currently with the present model being compared favourablywith previously published experiments [5]. Moreover, the relative importance ofthe parameters governing the mechanical response is also identified in the currentstudy. From this, conclusions are drawn about the possible further studies thatwould extend the current model to give meaningful comparisons with the actualstructural response for a variety of practically significant scenarios.
2. Review of model for geological layers
A discrete formulation comprising springs, rigid links and Coulomb friction hasbeen devised to model kink band deformation in geological layers that areheld together by an overburden pressure [22]. It was formulated using energyprinciples, and key parts of the model are shown in figure 1a,b. It has beencompared very favourably with simple laboratory experiments on layers of paperthat were compressed laterally and then increasingly compressed axially to triggerthe kink band formation process. The testing rig used in that study is shownschematically in figure 1c, and a typical test photograph is shown in figure 1d.Assuming that the layers were laterally compressible, a key assumption, the
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Figure 1. (a) Basic configuration of the discrete model for kink banding in n geological layers.Tectonic load on each layer is P with a horizontal reaction force H , the axial stiffness of eachlayer is k, the lateral overburden pressure is q and the kink band orientation angle is b. (b) Twointernal layers of the geological model. The kink band width and angle is b and a, respectively,normal contact force between layers is N , friction coefficient is m, individual rotational springsstiffness is c and stiffness of the surrounding elastic medium per unit layer is kf . (c) Schematic ofthe experimental rig used for testing the geological model. (d) A typical deformation profile in aphysical experiment showing a sequence of kink bands with corresponding orientation angles b.
kink band orientation angle b, was predicted theoretically for the first time,it being related purely to the initially applied lateral strain derived from theoverburden pressure q. Figure 2 shows the characteristic sequence of deformationwith figure 2a showing the undeformed state with the applied overburden pressureand the lateral pre-compression defining b, and figure 2b showing the point wherethe interlayer friction is released when the internal lateral strain within the kinkband is instantaneously zero and the band forms very quickly in the directionof b. It was later demonstrated that beyond the condition shown in figure 2c,where all the layers have the same thickness, whether internal or external to thekink band, lock-up begins to occur as shown in figure 2d, where the geometricconstraint forces the layers within the kink band to compress laterally, causingrestabilization. This marked the point where new kink bands formed and thesecould also be predicted by this approach after some modifications were made to
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Figure 2. Sequence of kink band deformation in the geological folding model. (a) Initial state withb defined from applied q; (b) instantaneous release of contact and hence friction within the kinkband when a = b; (c) layers inside and outside of the kink band all have equal thickness whena = 2b; (d) lock-up occurs when a > 2b and a new band would form.
the model [23]; a detailed discussion of the mechanical response and comparisonagainst experiments can be found in that article. Moreover, this model has alsobeen demonstrated to be suitable for modelling internal kink band formation inindividual composite fibres found commonly in fibre ropes under bending [12].
3. Pilot model for laminated composite struts
As discussed earlier, the system studied in Wadee et al. [22] had layers thatwere bound together by the mechanisms of overburden pressure and interlayerfriction. The deformation was in fact admissible geometrically only if the layerswere laterally compressible; the relationship between the kink band angle a, whichcould vary, and the orientation angle b, which was fixed, being such that interlayergaps, or voids, were not created. For a laminated strut in pure compression in thedirection parallel to the laminae, most experimental evidence from the literaturealso suggests that the kink band orientation angle b is basically fixed for eachlaminate configuration [5]. It is noted, however, in a recent study on laminatesunder combined compression and shear that this angle can change as the kinkpropagates but the angle reaches a limit [24]; in the current work, b is taken asa constant equivalent to this limiting value from the beginning of the kink banddeformation process, which is a simplifying assumption.
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Figure 3. Two internal laminae of the laminated composite model. The shaded region shows theinterlaminar region, which is exaggerated in scale for clarity. Dilation and shearing forces with theircorresponding displacements are given by FI and FII with dI and dII, respectively. The highlightedsection shows the lengths AB and BC , which directly relate to dI and dII, respectively.
Kink band deformation in laminates involves different mechanisms thatincorporate the interlaminar region comprising the laminae and the matrixthat binds the component together. Since the matrix is itself deformable andthere is no overburden pressure to close any voids, the model needs significantmodifications to account for the different characteristics of the laminated strut.It is worth noting that the assumption for the lay-up sequence of the compositein the present case is such that no twisting is generated from the appliedcompression. Figure 3 shows the adapted two-layer model, which omits thefollowing features that are not relevant in the current case: the foundation stiffnessand the overburden pressure, i.e. q = kf = 0. The kink band formation is thusintrinsically linked to the deformation of the interlaminar region within the strut.
Shearing within the interlaminar region is the analogous process to slidingbetween the layers in the model for geological folding, the latter being modelled inthe energy formulation as a work done overcoming the friction force. A piecewiselinear model is used to simulate the force versus displacement relationship interms of the shear resistance (figure 4), where fracture modes that are relevantfor a linear-softening response (figure 4a) are defined in figure 4c–e.
Tensile expansion, or dilation, of the interlaminar region is modelled, however,with a purely linear elastic constitutive law. In the model described in §2, it wasargued that when the interlayer contact force was released not only would thefriction be released but also the overburden pressure would inhibit the formationof subsequent voids within the layered structure. Since in the current case thereis no overburden or lateral pressure as such, potential dilation of the interlaminarregion needs to be included. As in the previous model, however, the laminadeformation is assumed to lock up and potentially trigger a new band formingwhen a > 2b; lateral compression in adjacent laminae would then be occurringand stiffening the response significantly. Hence, it is reasoned therefore that itcould be energetically advantageous for the mechanical system to form a newkink band rather than to continue to deform the current one [23,25].
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Figure 4. (a,b) Piecewise linear force versus displacement model for interlaminar shearing; (c–e)fracture modes. (a) Linear-softening response that is more representative of a fracture model,sgn(dC) = sgn(dM); (b) linear-hardening response that is more appropriate for materials that showpost-yield strength, sgn(dC) = −sgn(dM). (c) Mode I is tearing; (d) mode II is shearing and (e)mode III is scissoring. In the current model, only mode II is relevant.
(a) Potential energy formulation
(i) Interlaminar dilation
The resistance to interlaminar dilation while the kink bands deform is modelledwith a linear constitutive law with the dilation-resisting force FI relating to thedilation displacement dI, thus
FI(a) = CIdI, dI(a) = t[cos(a − b)
cos b− 1
], (3.1)
with CI being the lateral stiffness of the laminate, related to the lateral Young’smodulus, and t being the thickness of a single lamina. Since the area over whichthe interlaminar region dilates depends directly on the kink band width b, thestiffness CI = bdkI, where kI is the lateral stiffness per unit area of the laminateand d is the breadth of the strut. Moreover, kI can be related to the lateralYoung’s modulus E22, where E22 = kIt. However, with the lamina assumed tobe laterally incompressible in the current model and the dilation displacementbeing assigned purely to the softer interlaminar matrix material, a clear departurefrom the geological model, the current lamina thickness is thus t rather thant cos(a − b). This is shown in figure 3 and is detailed in the highlighted area ofthat diagram. The relationship in equation (3.1) for dI is thus obtained fromtaking the length AB from figure 3, where dI = AB − t. Hence, there is a lateral
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tensile strain developed since the gap between the laminae grows as a increasesfrom zero to b. The gap subsequently begins to reduce; when a = 2b the gapreturns to zero, marking the commencement of lock-up.
The work done in the dilation process is therefore given by
UD =∫ dI(a)
0FI(a′) d
{t[cos(a′ − b)
cos b− 1
]}= kIbdt2
2
[1 − cos(a − b)
cos b
]2
, (3.2)
where a′ is a dummy variable to facilitate evaluation of the definite integral. Itis assumed that the interlaminar region would not be damaged in the processof dilation and that the only nonlinearity in the constitutive law would beunder shear. This is because the dilation displacement is relatively smaller thanthe shearing displacement that is discussed next (see later text); this has theadditional advantage of maintaining model simplicity such that any mixed-modefracture considerations can be left for future work.
(ii) Interlaminar shearing
Interlaminar shearing or the laminae sliding relative to one another is modelledwith a piecewise linear constitutive law with the force-resisting shear FII relatingto the shearing displacement dII, thus
FII(a) = CIIdII, dII(a) = tcos b
[sin(a − b) + sin b], (3.3)
with CII being the shearing stiffness of the combination of the matrix and laminaesliding relative to one another. The relationship for dII in terms of a and b inequation (3.3) is given by examining the length BC in figure 3. However, sincethe band is basically assumed to form instantaneously before any rotation occurs,it is implied that FII(0) = dII(0) = 0. Moreover, as b �= 0, the expression dII = BC +t tan b is obtained, such that the force and displacement conditions are satisfied.Taking the limit a → 0 gives dI/dII → tan b, which shows that, for values of b <45◦, the shearing displacement dII is greater than the dilation displacement dI forpractical values of b that tend to be below 35◦ [5,24].
When the shearing displacement reaches the initial proportionality limit, i.e.when dII = dC (figure 4), the relationship between FII and dII changes to
FII = CIIdC
(dII − dM
dC − dM
), (3.4)
where dM is the shearing displacement when the corresponding resistance forcereduces to zero. Now, if dII(aC) = dC and dII(aM) = dM, the expressions for theresisting force can be written as
FII =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
CIIt[sin(a − b) + sin b]cos b
for a = [0, aC],CIItcos b
[sin(a − b) − sin(aM − b)sin(aC − b) − sin(aM − b)
][sin(aC − b) + sin b]
for a > aC and a = [aC, aM] if sgn(dC) = sgn(dM) > 0,
0 for a ≥ aM and sgn(dC) = sgn(dM) > 0.
(3.5)
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Moreover, since the contact area under shear depends on the kink band width b,the stiffness CII = bdkII, where kII is the shear stiffness per unit area of the lamina.Hence, the effective shear stress t = kIIdII and therefore kII can be related to thematerial shear modulus G12, where G12 = kIIt. The work done in the shearingprocess is given by
US =∫ dII(a)
0FII(a′) d
{t
cos b[sin(a′ − b) + sin b]
}= kIIbdt2
2L(a), (3.6)
where
L(a) =[sin(a − b) + sin b
cos b
]2
(3.7)
for a ≤ aC, or
US = kIIbdt2
2
{L(aC) +
∫a
aC
[sin(a′ − b) − sin(aM − b)sin(aC − b) − sin(aM − b)
beyond the proportionality limit where a = [aC, aM]. However, if a > aM andsgn(dC) = sgn(dM) > 0, the shear resistance force vanishes and the expression forUS becomes
US = kIIbdt2
2S(aM). (3.10)
There would still be the potential for frictional forces to resist shear even thoughd > dM, and the interlaminar region has lost all shear strength, particularly whena > 2b and the adjacent laminae are laterally compressed. However, when thelaminae are in lateral tension, it is assumed that the resistance to dilation isindependent of shear, such that the dilation resistance remains linearly elasticeven though the shear resistance may be zero owing to mode II fracture(figure 4d). This, again, is a simplifying assumption relying on the geometricconditions which dictate that dilation displacements are tending to reduce whenthose from shear in the matrix are approaching their maximum, where b ≤ a ≤ 2b.
(iii) Remaining energy contributions
As in the geological model, the strain energy stored in bending can be takenfrom a pair of rotational springs of stiffness c,
Ub = ca2. (3.11)
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Figure 5. Bending of a lamina within a kink band: (a) definition of x ; (b) idealized case; (c) actualcase. Curvature k is defined as the total angle change 2a over the assumed effective bending lengthof the band 2b, hence k ≈ a/b.
The stiffness of the rotational springs is related differently from the geologicalmodel as the expression for that model contained the overburden pressure q [22].The bending energy should strictly relate to curvature k, where
Ub = 2∫ b/2
−b/2
12EI k2 dx , (3.12)
with x defining the domain of one bending corner comprising two lengths ofb/2 that are external (x = −b/2) and internal (x = +b/2) to the kink band,respectively, as represented in figure 5. The quantity k is defined as the rate ofchange of the kink band angle a over the kink band width b, thus
k ≈ a
b⇒ c ≈ EI
b. (3.13)
The key point is that curvature changes sign at the midpoint of the kinkband. Hence, the rotational stiffness c is related to the flexural rigidity EI ofa lamina, with E being its Young’s modulus in the axial direction (denoted asE11 henceforth) and its second moment of area I = dt3/12. The strain energy perlayer associated with the in-line spring of stiffness k is hence given by
Um = 12kd2
a , (3.14)
where da is the axial displacement of the springs. The in-line spring stiffness isk = E11dt/L for a single lamina with L being the length of the strut. The workdone by the external load can be taken simply as the sum of the displacement ofthe in-line springs da and from the band deforming multiplied by the axial loadP, which can be defined as the axial pressure p multiplied by the cross-sectionalarea of a lamina dt,
PD = pdt[da + b(1 − cos a)]. (3.15)
(iv) Total potential energy functions
The total potential energy V is given by the sum of the strain energies frombending Ub, the in-line springs Um , interlaminar dilation UD and shearing US,minus the work done PD, thus
V = Ub + Um + UD + US − PD. (3.16)
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Since the dilation terms are assumed to be linearly elastic throughout theirloading history, the total potential energy per axially loaded lamina takesthree forms:
— The case where a = [0, aC]; so V = V L, i.e. linearly elastic in shear.— The case where a > aC, so V = V S, i.e. the secondary shear stiffness is
either a smaller positive value than the primary shear stiffness or a negativevalue.
— The case where a > aM and sgn(dC) = sgn(dM) > 0; so V = V Z, i.e. no shearstiffness, which occurs only if the secondary shear stiffness is negative.
These forms of the total potential energy are given by the expressions
V L = V I + kIIbdt2
2L(a), V S = V I + kIIbdt2
2S(a) and V Z = V I + kIIbdt2
2S(aM),
(3.17)where V I is given by
V I = kd2a
2+ E11dt3a2
12b+ kIbdt2
2
[1 − cos(a − b)
cos b
]2
− pdt[da + b(1 − cos a)].(3.18)
The total potential energy functions are non-dimensionalized by dividing throughby kt2 and can be re-expressed in terms of rescaled parameters,
V L = V I + k IIb2
L(a), V S = V I + k IIb2
S(a) and V Z = V I + k IIb2
S(aM),
(3.19)where
V I = V I
kt2, V L = V L
kt2, V S = V S
kt2, V Z = V Z
kt2, d = da
t, D = D
t, b = b
t,
p = pdk
= pLE11t
, D = E11d12k
= L12t
, k I = kIdtk
= E22LE11t
and k II = kIIdtk
= G12LE11t
.
(3.20)
(b) Equilibrium equations
The equilibrium equations are defined by the condition of stationary potentialenergy with respect to the end-shortening da , the kink band angle a and the kinkband width b; these can be written in non-dimensional terms, thus
p = d, (3.21)
p = k IIa + k IIJa + 2Da
b2 sin a(3.22)
and p = k IIb + k IIJb − Da2
b2(1 − cos a). (3.23)
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Equation (3.21) defines the pre-kinking fundamental equilibrium path thataccounts for pure compression of the in-line springs of stiffness k. Equations(3.22) and (3.23) define the post-instability states for the non-trivial kink banddeformations; equating them allows the kink band width b to be evaluatedanalytically,
b ={
Da[2/ sin a + a/(1 − cos a)]k I(Ib − Ia) + k II(Jb − Ja)
}1/2
. (3.24)
The expressions for Ia and Ib are given in detail as follows:
Ia =[1 − cos(a − b)
cos b
]sin(a − b)sin a cos b
and Ib = 12(1 − cos a)
[1 − cos(a − b)
cos b
]2
,
(3.25)
where these expressions apply for the entire range of a. However, the expressionsfor Ja and Jb change for each form of the total potential energy function; forV = V L, the expressions are
Ja = cos(a − b)[sin(a − b) + sin b]sin a cos2 b
and Jb = [sin(a − b) + sin b]22(1 − cos a) cos2 b
; (3.26)
for V = V S,
Ja = cos(a − b)sin a cos2 b
[sin(aC − b) + sin b
sin(aC − b) − sin(aM − b)
][sin(a − b) − sin(aM − b)] (3.27)
and
Jb =[sin(aC − b) + sin b
2(1 − cos a) cos2 b
] {sin(aC − b) + sin b
+ sin2(a − b) − sin2(aC − b) + 2 sin(aM − b)[sin(aC − b) − sin(a − b)]sin(aC − b) − sin(aM − b)
},
(3.28)
and for V = V Z, Ja = 0 and
Jb = sin b[sin b + sin(aC − b) + sin(aM − b)] + sin(aC − b) sin(aM − b)2 cos2 b(1 − cos a)
. (3.29)
The initial limiting case where a → 0 gives b → ∞ and p → k I tan2 b + k II. Theresult for b suggests that the kink band is initially prevalent throughout thestructure and the result for p shows that the critical load depends primarily on theshear stiffness with a smaller contribution from the dilation stiffness that relatesto b. This reproduces similar results from the literature where the critical stress isrelated to the shear modulus [4,5,13]; it also reflects a significant difference fromthe geological model, which has an infinite critical load and where the kink bandwidth grows from zero length [22].
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Figure 6. Representation of the experimental sample in Kyriakides et al. [5]. The rod comprisedICI APC-2/AS4 composite fibres, with properties as given in table 1. The sample was confinedsuch that there was negligible lateral compression but also that global buckling was not an issue.
Table 1. Properties used in the validation study to compare the current model with experimentspresented in Kyriakides et al. [5]. The shear modulus range is over 4% shear strain.
rod fibre diameter: t = 7 × 10−3 mmoverall rod length: L = 76 mmlongitudinal Young’s modulus: E = E11 = 130.76 kN mm−2
Results from the current model are initially compared with published experimentson circular cylindrical composite rods with confined ends that exhibited kinkbands under axial compression [5]. This aids the comparison between the currentmodel and the experiments such that both loading levels and the kink band widthcan be compared; a similar approach was used in Edmunds & Wadee [12].
The dimensions of the overall sample had a diameter of 8.255 mm with therelevant properties given in table 1. Note that the breadth d is not givenbecause it cancels in all the relevant non-dimensional quantities. The samplecomprised ICI APC-2/AS4 composite fibres. Since the sample was cylindrical,the system in Kyriakides et al. [5] was presented in terms of a cylindricalpolar coordinate system with x1 and x2 being the longitudinal and the radialcoordinates, respectively, as shown in figure 6. Although the current model isformulated for flat rectangular laminae, the lamina thickness t, which includes thecomposite action of the lamina and the matrix, can be perceived to be equivalentto the diameter of an individual fibre rod. This is due to the tight packing ofthe composite that has a relatively high volume fraction; any departures fromthis assumed value of t are likely therefore to be relatively small. For the testspresented to measure the change in shear modulus G12, there was no plateaushown in the test data (figs A3 and A5 in Kyriakides et al. [5]). In the currentstudy, it is therefore assumed that the piecewise linear model for the shearstiffness reflects the initial and final values found in the experiments; hencesgn(dM) = −sgn(dC), i.e. a linear-hardening model is implemented as representedin figure 4b.
The critical shear angle gC for the piecewise linear idealization is the anglebeyond which the shear stiffness is replaced by a secondary smaller value; thisis estimated from the earlier mentioned graphs in figs A3 and A5 in Kyriakides
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et al. [5] to be 0.012 rad (or 0.69◦). The shear angle can be expressed in terms ofthe kink band orientation angle b and the kink band angle a, such that
tan gC = dII(aC)dI(aC) + t
= sin(aC − b) + sin b
cos(aC − b). (4.1)
Given that b is assumed to remain constant during deformation, the critical kinkband angle aC can be found by rearranging (4.1), thus
tan gC cos(aC − b) − sin(aC − b) − sin b = 0, (4.2)
and solving for aC. This is achieved by substituting the critical shear angle gCfrom above and the kink band orientation angle from Kyriakides et al. [5], whereb was reported to lie between 12◦ and 16◦ (0.2094 and 0.2793 rad) to the x2direction; for the specified values, aC is approximately equal to gC (aC = 0.0120 −0.0121 rad). To obtain the correct final shear modulus (table 1), the value ofdM/t = −0.094 is used such that the ratios between the initial and final values ofthe shear stiffness reflect the reported experimental data (figure 4b).
Figure 7 shows numerical results from the current model, using the propertiesdefined in table 1 with b values as found in the published results. Note that thenon-dimensional total end-shortening D is defined as
D = d + b(1 − cos a). (4.3)
The actual kink band widths in the five tests were reported to range from 11to 36 fibre diameters (directly corresponding to b in the current model) and thecompressive strengths were found to average 1.119 kN mm−2, with a standarddeviation of 0.043 kN mm−2 (directly corresponding to p in the current model).The results from the current model show highly unstable snap-back and hence thecritical load would never be reached realistically (figure 7a,b), a well-establishedfeature for systems of this type [4]. For comparison purposes, the pressure p istaken at the point at which the structure stabilizes and reaches a plateau; inthe current model, this occurs at the geometric lock-up condition a = 2b. Forthe range of the b angles considered, the non-dimensional stabilization pressurep ranges from 80.5 to 82.3, which converts to an actual stabilization pressurep ranging from 0.969 to 0.992 kN mm−2: an error against the average from theexperimental results of 11–13%, which is sufficiently small to offer encouragement.
Of further interest is the comparison for the kink band width between the testsand the current model. Observing the graph shown in figure 7c, as the kink bandangle a increases, initially the non-dimensional kink band width b falls from alarge value to a small value, approximately 5.6 when a = 0.024 rad (≈1.4◦). As aincreases further, the kink band width begins to increase slowly; see table 2 fordetails of some key points. According to the sequence described in figure 2b–d,the kink band itself maximizes dilation when a = b, minimizes it when a = 2b andlocks up when a > 2b. The results of the current model, particularly when a = 2b,lie at the lower end of the range of observed values of the band widths from thepublished experiments. This seems sensible, given that the lock-up condition used,where a = 2b, represents a lower bound [22], which implies that the current model
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Figure 7. Non-dimensional plots of load p versus (a) total end-shortening D and (b) kink bandangle a (rad); (c) kink band width b versus kink band angle a (rad). Range of b = 12◦–16◦.(d) Piecewise linear-hardening relationship of the effective shear stress t (N mm−2) versus thenormalized shearing displacement dII/t for b = 16◦. Properties of ICI APC-2/AS4 composite fibresand configuration and the range for b were taken from Kyriakides et al. [5].
Table 2. Non-dimensional kink band width values from the current model at different stages ofdeformation. The conditions a = b and a = 2b are the points where the dilation within the bandare effectively maximized and minimized, respectively; experiments in Kyriakides et al. [5] reportedb = 11 − 36.
kink band angle case: b = 12◦ case: b = 16◦
a = b b = 10.4 b = 10.2a = 2b b = 17.0 b = 19.5
would also tend to predict lower bound kink band widths. Hence, the results fromthe comparisons between the current model and the published experiments [5]are highly encouraging; they offer very good quantitative agreement for theloading and the geometric deformation—key quantities that define the kink bandphenomenon.
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The favourable comparisons between the current model and the publishedexperiments in Kyriakides et al. [5] imply that the fundamental physics of thesystem are captured by the current approach. The study is therefore extendedto present a series of model parametric variations to establish their relativeeffects. The basic geometric and material configuration is identical to that usedin the validation study presented in table 1. Material and geometric parametersare varied individually, while maintaining the remaining ones at their originalvalues. The parameters that are varied are the kink band orientation angle b,the critical kink band angle aC, the composite direct and shear moduli, E11, E22and G12, and the shape of the piecewise linear relationship for shear. Finally, itis worth emphasizing that, where a range of a particular parameter is given, thegraphs show curves in equal increments that are inclusive of the limits stated ofthe parameter.
(i) Orientation angle
In the current model, the orientation angle b needs to be fixed a priori;hence the effects of different starting conditions for the model need to beestablished. Increasing b from 10◦ (0.1745 rad) to 30◦ (0.5236 rad), a range thatis representative of laminate experiments in the literature [5,24], leads to a stifferresponse for increasing a, as shown in figure 8a,b, with the pressure capacity forb = 30◦ being more than double the capacity for b = 10◦ for values of a < b. Thegraphs in figure 8c,d raise an interesting point about the response, particularlywhen b ≥ 22.5◦ (0.3927 rad), which seems to define a boundary where the kinkband width b loses its monotonically increasing property after it initially troughsfor a small value of a, which was identified as approximately 1.4◦ in §4a. Bothgraphs show that the kink band width at the lower bound lock-up conditiona = 2b temporarily peaks when b = 22.5◦. For higher orientation angles, the kinkband width b in fact peaks beyond a = 1.4◦, then troughs and then resumes themonotonic rise as seen for b ≤ 22.5◦. Moreover, this also explains the reason whythe stabilization pressure increases for b > 22.5◦, as shown in figure 8a, becausethe pressure has an inverse square relationship with the kink band width, asshown in the equilibrium equations (3.22) and (3.23). The graphs presentedin figure 9 attribute this loss of monotonicity in b (beyond a = 1.4◦) to thedominating influence of the dilation terms for larger b, particularly in the regionof maximum dilation where a ≈ b. In the first instance, it should be recalled thatwhen b is larger the potential maximum dilation displacement dI is also largerrelative to dII when a = b. Figure 9a shows that the maximum of the dilation termk I(Ib − Ia) from the expression for b, i.e. equation (3.24), increases substantiallywith b, whereas figure 9b shows only very marginal changes in the respectiveshear term k II(Jb − Ja). The numerator in equation (3.24), bnum, which representsthe influence of bending, is independent of b, as shown in figure 9c, but therespective denominator, bden, shows that the dilation term influences the valuessignificantly for the higher b values, as shown in figure 9d. Once a graduallyincreases above b, the dilation displacement progressively reduces and the shearterm begins to dominate with the result that the kink band width resumes growthand lock-up occurs. This effect is similar to that found in the geological model
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Figure 8. (a,b) Equilibrium paths for different b = 10–30◦ through non-dimensional plots of load pversus (a) total end-shortening D and (b) kink band angle a (rad). (c) Non-dimensional kink bandwidths b versus the kink band angle a (rad) for a range of orientation angles b = 10–30◦; circlesmark the lower bound lock-up condition a = 2b. (d) Values of non-dimensional kink band widthsb and applied axial pressure p at the lower bound lock-up condition.
with the introduction of the foundation spring of stiffness kf [22], as shown infigure 1b; the kink band width was also found to plateau with higher foundationstiffnesses. It is worth noting that if the stiffness loss in the constitutive lawfor dilation was introduced, then the effect found in the present case would begenerally less pronounced.
(ii) Critical shear angle and modulus
Increasing the critical kink band angle from aC = 0.69◦ to 0.96◦ (0.0120 to0.0168 rad), with a fixed limiting displacement dM/t = −0.094 as before, showsan increase in the critical shear stress before the loss in stiffness occurs (seefigure 10e) and leads to a monotonic increase in the axial pressure p andthe minimum kink band width b (figure 10a,c). A subtly different pattern isobserved in figure 10b,d, where trends for increasing the initial shear modulus
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Figure 9. Graphs of various terms from the expression for the non-dimensional kink band widthb, equation (3.24), for b = 10–30◦ versus the kink band angle a (rad). (a,b) Plots of dilation andshear terms, respectively. (c,d) Plots of the numerator and denominator of the b expression.
(G init12 = 6.03–8.45 kN mm−2) lead to higher stabilization pressures but smaller
minimum kink band widths. These are logical results because the effect ofincreasing the critical kink band angle will lead to a later destabilization inshear and hence increase the load and band width; the increase in the initialshear modulus increases the resistance against shearing—the process of kinkbanding therefore requires more axial pressure to overcome the increased stiffness.However, the increased shear stiffness reduces the kink band width because thereis a greater resistance to that type of deformation.
(iii) Hardening and softening in shear
The variation in the piecewise linear model for the shearing response is nowdiscussed. The constitutive behaviour FII versus dII has been hitherto assumedto be a linear-hardening law, which corresponded with the data from theliterature used in the validation exercise. Figure 11 shows the results for differentsecondary slopes while they remain positive (a linear-hardening law). The resultsexhibit fairly progressive behaviour; the reduced secondary slopes reduce theload-carrying capacity but make only marginal changes to the kink band widths.Figure 12 shows results for reducing the secondary slope further such that itbecomes negative (a linear-softening law). In this case, the negative secondaryslope mimics the behaviour of a fracture process in which the shear stiffnessand strength have vanished and mode II fracture and crack propagation would
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Figure 10. Comparison of the response of the kink band formation for different aC values in (a),(c) and (e) for the range of aC = 0.69–0.96◦ and initial G12 values from 6.03 to 8.45 kN mm−2 in(b), (d) and (f ). Non-dimensional plots of (a,b) load p versus total end-shortening D and (c,d)kink band width b versus kink band angle a (rad); (e,f ) show the relationship of the effective shearstress t versus the normalized shearing displacement dII/t. Note that b = 12◦ throughout.
occur. However, a pattern is seen with the strength reducing and the band widthsincreasing for weaker properties in shear, which appears to be entirely logical.The softening of the internal structure gives less resistance to the kink bandingprocess, allowing for larger rotations and gross deformations. The detailed effectsof crack propagation have been left for future work, although recent work onbuckling-driven delamination [26] has suggested that an analytical treatment ofsuch effects may indeed be tractable.
(iv) Young’s moduli
Results for a twofold increase in the axial modulus E11 suggest that thishas only a marginal effect on the normalized stabilization pressure p, withan approximately 1.5 per cent increase. However, this result does imply an
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Figure 11. Non-dimensional plots of load p versus (a) total end-shortening D and (b) kink bandangle a (rad); (c) kink band width b versus kink band angle a (rad); (d) piecewise linear-hardeningrelationship of the effective shear stress t (N mm−2) versus the normalized shearing displacementdII/t. Range of dM/t = −0.435 to −0.094 and b = 12◦.
approximate doubling of the actual stabilization pressure p; hence, the effectis quantitative without significantly affecting the kink band deformation nor theload-displacement response qualitatively. However, increasing the lateral modulusE22 results in a significantly stiffer response, the system stabilizing to a smallerkink band width (figure 13). These, again, are logical results because the effectof increasing the lateral modulus E22 increases the resistance against dilation;the process of kink banding therefore requires more axial pressure to overcomethis. Increasing the axial modulus increases the axial stiffness k, which in turneffectively reduces the relative dilation and shear stiffnesses without affectingthe relative bending stiffness—see the scaling relationships in equation (3.20).Since bending is currently assumed to be purely linear, its relative effect becomesprogressively more pronounced and then outweighs the reduced dilation and sheareffects at large rotations. Obviously, if the bending was assumed to plateau owingto plasticity, this effect would be limited.
(v) Summary
The parametric studies have shown that the orientation angle b of a laminateaffects the interplay between dilation and shear within the matrix. For smallervalues of b, shear tends to dominate throughout. For larger values of b,dilation dominates in the first part of the deformation a ≤ b with its effectdiminishing beyond this point and shear dominating once again. It has also
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Figure 12. Non-dimensional plots of load p versus (a) total end-shortening D and (b) kink bandangle a (rad); (c) kink band width b versus kink band angle a (rad); (d) piecewise linear-softeningrelationship of the effective shear stress t (N mm−2) versus the normalized shearing displacementdII/t. Range of dM/t = 0.096–0.482 and b = 12◦.
been shown that changes in the shear and lateral Young’s moduli alongside theconstitutive law used to model mode II fracture within the matrix have significanteffects. However, although varying the axial modulus has a significant effectquantitatively, it has only a minor effect qualitatively in terms of changing thenormalized axial pressure and kink band widths and angles.
5. Concluding remarks
An analytical, nonlinear, potential-energy-based model for kink banding incompressed unidirectional laminated composite panels has been presented.Comparisons of results with published experiments suggest that very goodagreement can be achieved from this relatively simple mechanical approach,provided that certain important characteristics are incorporated:
— Interlaminar dilation and shearing: the kink band process naturally causesshearing and changes the gap between the laminae. The matrix within thecomposite needs to resist both these displacements for the laminate tohave integrity and significant structural strength.
— Bending energy: the resistance to rotation sets a length scale, which—inthis case—is the kink band width b.
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Figure 13. Non-dimensional plots for the range of axial direct modulus E11 = 130.7–261.5 kN mm−2
in (a,c) and lateral direct modulus E22 = 10.4–52 kN mm−2 in (b,d). (a,b) Load p versusa; (c,d) kink band width b versus the kink band angle a (rad). Note that b = 12◦throughout.
Linear constitutive relationships for the mechanisms of bending and dilation,and a piecewise linear relationship for the process of shearing, together withgeometrically nonlinear relationships, have been implemented. The approach hasbeen successful such that the mechanical response captures the fundamentalphysics of kink banding and agrees with the experiments from Kyriakides et al.[5] in terms of kink band widths and loading levels, without having to resort tosophisticated numerical or continuum formulations. Unlike the geological model[22], where a relationship was derived for the band orientation b that wasrelated to the overburden pressure q, in the current case the angle b has tobe assumed a priori because, as far as the authors are aware, no satisfactoryprocedure for predicting b for composite laminates exists. For laminates, themagnitude of the orientation angle b has been largely attributed to the precisionand tolerances involved within the manufacturing process, where fibre wavinessand misalignments can be introduced [5,15]. However, if the overburden pressureis considered to be the controlling parameter for the equivalent ‘manufacturingprocess’ that keeps the geological layers behaving together, then future work onmodelling the process of manufacturing composite laminates may bear fruit; anindication of the parameters that govern the orientation angle for the current case
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may be established. Although this is a shortcoming for the present model, theresults from the parametric study are very encouraging, with the trends appearingto be entirely logical.
The current model can of course be used as a basis for further work. In thefirst instance, the piecewise linear formulation applied currently for shear couldalso be extended to include dilation, giving the possibility of mixed-mode fracture[27] in the kink band. It would appear from the current study that this wouldbe more prevalent in laminates that have a naturally larger orientation angleb, where the dilation process is relatively more significant. Moreover, loadingcases that are more complex than uniform compression could be investigated;for example, numerical approaches have been developed in Vogler et al. [17] andGutkin et al. [28] with varying degrees of success to investigate the formation ofkink bands where there is a combination of shear and compression. An additionalcomplication in the combined loading case is that the kink band propagationacross the sample tends to occur more gradually in contrast to the present case,where the formation process is found to be fast and has been assumed to beso currently.
References
1 Anderson, T. B. 1964 Kink-bands and related geological structures. Nature 202, 272–274.(doi:10.1038/202272a0)
2 Hobbs, B. E., Means, W. D. & Williams, P. F. 1976 An outline of structural geology. New York,NY: Wiley.
3 Price, N. J. & Cosgrove, J. W. 1990 Analysis of geological structures. Cambridge, UK:Cambridge University Press.
4 Budiansky, B. & Fleck, N. A. 1993 Compressive failure of fiber composites. J. Mech. Phys.Solids 41, 183–211. (doi:10.1016/0022-5096(93)90068-Q)
5 Kyriakides, S., Arseculeratne, R., Perry, E. J. & Liechti, K. M. 1995 On thecompressive failure of fiber reinforced composites. Int. J. Solids Struct. 32, 689–738.(doi:10.1016/0020-7683(94)00157-R)
6 Reid, S. R. & Peng, C. 1997 Dynamic uniaxial crushing of wood. Int. J. Impact Eng. 19,531–570. (doi:10.1016/S0734-743X(97)00016-X)
7 Vogler, T. J. & Kyriakides, S. 2001 On the initiation and growth of kink bands in fibercomposites. I. Experiments. Int. J. Solids Struct. 38, 2639–2651. (doi:10.1016/S0020-7683(00)00174-8)
8 Byskov, E., Christoffersen, J., Christensen, C. D. & Poulsen, J. S. 2002 Kinkband formationin wood and fiber composites—morphology and analysis. Int. J. Solids Struct. 39, 3649–3673.(doi:10.1016/S0020-7683(02)00174-9)
9 Da Silva, A. & Kyriakides, S. 2007 Compressive response and failure of balsa wood. Int. J.Solids Struct. 44, 8685–8717. (doi:10.1016/j.ijsolstr.2007.07.003)
10 Pimenta, S., Gutkin, R., Pinho, S. T. & Robinson, P. 2009 A micromechanical model for kink-band formation. I. Experimental study and numerical modelling. Compos. Sci. Technol. 69,948–955. (doi:10.1016/j.compscitech.2009.02.010)
11 Hobbs, R. E., Overington, M. S., Hearle, J. W. S. & Banfield, S. J. 2000 Buckling offibres and yarns within ropes and other fibre assemblies. J. Textile Inst. 91, 335–358.(doi:10.1080/00405000008659512)
12 Edmunds, R. & Wadee, M. A. 2005 On kink banding in individual PPTA fibres. Compos. Sci.Technol. 65, 1284–1298. (doi:10.1016/j.compscitech.2004.12.034)
13 Rosen, B. W. 1965 Mechanics of composite strengthening. In Fiber composite materials(ed. S. H. Bush), pp. 37–75. Metals Park, OH: American Society for Metals.
14 Argon, A. S. 1972 Fracture of composites. Treatise Mater. Sci. Technol. 1, 79–114.
Phil. Trans. R. Soc. A (2012)
on April 21, 2018http://rsta.royalsocietypublishing.org/Downloaded from
15 Budiansky, B. 1983 Micromechanics. Comput. Struct. 16, 3–12. (doi:10.1016/0045-7949(83)90141-4)
16 Fu, Y. B. & Zhang, Y. T. 2006 Continuum-mechanical modelling of kink-band formation in fibrereinforced composites. Int. J. Solids Struct. 43, 3306–3323. (doi:10.1016/j.ijsolstr.2005.05.031)
17 Vogler, T. J., Hsu, S.-Y. & Kyriakides, S. 2001 On the initiation and growth ofkink bands in fiber composites. II. Analysis. Int. J. Solids Struct. 38, 2653–2682.(doi:10.1016/S0020-7683(00)00175-X)
18 Budiansky, B., Fleck, N. A. & Amazigo, J. C. 1998 On kink-band propagation in fibercomposites. J. Mech. Phys. Solids 46, 1637–1653. (doi:10.1016/S0022-5096(97)00042-2)
19 Pimenta, S., Gutkin, R., Pinho, S. T. & Robinson, P. 2009 A micromechanical modelfor kink-band formation. II. Analytical modelling. Compos. Sci. Technol. 69, 956–964.(doi:10.1016/j.compscitech.2009.02.003)
20 Fleck, N. A. 1997 Compressive failure of fiber composites. Adv. Appl. Mech. 33, 43–117.(doi:10.1016/S0065-2156(08)70385-5)
21 Hunt, G. W., Peletier, M. A. & Wadee, M. A. 2000 The Maxwell stability criterionin pseudo-energy models of kink banding. J. Struct. Geol. 22, 669–681. (doi:10.1016/S0191-8141(99)00182-0)
22 Wadee, M. A., Hunt, G. W. & Peletier, M. A. 2004 Kink band instability in layered structures.J. Mech. Phys. Solids 52, 1071–1091. (doi:10.1016/j.jmps.2003.09.026)
23 Wadee, M. A. & Edmunds, R. 2005 Kink band propagation in layered structures. J. Mech.Phys. Solids 53, 2017–2035. (doi:10.1016/j.jmps.2005.04.005)
24 Gutkin, R., Pinho, S. T., Robinson, P. & Curtis, P. T. 2010 On the transition from shear-driven fibre compressive failure to fibre kinking in notched CFRP laminates under longitudinalcompression. Compos. Sci. Technol. 70, 1223–1231. (doi:10.1016/j.compscitech.2010.03.010)
25 Edmunds, R., Hicks, B. J. & Mullineux, G. 2010 Drawing parallels: modelling geologicalphenomena using constraint satisfaction. J. Struct. Geol. 32, 997–1008. (doi:10.1016/j.jsg.2010.06.005)
26 Wadee, M. A. & Völlmecke, C. 2011 Semi-analytical modelling of buckling driven delaminationin uniaxially compressed damaged plates. IMA J. Appl. Math. 76, 120–145. (doi:10.1093/imamat/hxq062)
27 Hutchinson, J. W. & Suo, Z. 1992 Mixed mode cracking in layered materials. Adv. Appl. Mech.29, 63–191. (doi:10.1016/S0065-2156(08)70164-9)
28 Gutkin, R., Pinho, S. T., Robinson, P. & Curtis, P. T. 2010 Micro-mechanical modellingof shear-driven fibre compressive failure and of fibre kinking for failure envelopegeneration in CFRP laminates. Compos. Sci. Technol. 70, 1214–1222. (doi:10.1016/j.compscitech.2010.03.009)
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