Stress intensity factor and energy release rate of externally pressurized thick cross-ply (very) long cylindrical shells with low-hardening transverse shear modulus nonlinearity

Engineering Fracture Mechanics 151 (2016) 138–160

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Stress intensity factor and energy release rate of externally pressurized thick cross-ply (very) long cylindrical shells with low-hardening transverse shear modulus nonlinearity Reaz A. Chaudhuri ⇑ Department of Materials Science and Engineering, 122 S. Central Campus Dr., Room 304, University of Utah, Salt lake City, Utah 84112-0560, United States

a r t i c l e

i n f o

Article history: Received 8 October 2015 Accepted 17 November 2015 Available online 2 December 2015 Keywords: Localization/delocalization Compression fracture Stress intensity factor Energy release rate Material nonlinearity

a b s t r a c t Combined effects of modal imperfections, transverse shear/normal deformation along with the nonlinear hypo-elastic transverse shear (GTT) material property on the emergence of interlaminar shear crippling type instability modes, related to the localization (onset of deformation softening), delocalization (onset of deformation hardening) and propagation of mode II compression fracture/damage, in thick imperfect cross-ply very long cylindrical shells under applied hydrostatic pressure, are investigated. The primary accomplishment is the (hitherto-unavailable) computation of the layer-wise mode II fracture toughness, fracture energy and kink crack band-width, under hydrostatic compression, from a nonlinear Finite Element Analysis (FEA), using Maxwell’s construction and Griffith’s energy balance approach. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Although localization was first discovered by Anderson [1] as a quantum mechanical phenomenon, localization and delocalization are multi-physics phenomena that transcend atomic scales. In Anderson localization model for opto-electronics [1] an electron is permitted to hop only to the nearest neighbor sites [1,2]. Almost all eigenstates are, according to this model (involving uncorrelated disorder), are known to be localized, and the quantum diffusion of an initially localized wavepacket is suppressed at the localization length [3]. de Moura and Lyra [3] have investigated one-dimensional Anderson models with long-range correlated diagonal disorder. Li et al. [4] have theoretically studied the behavior of electronic states in disordered systems subject to a strong nonresonant optical field, and have concluded that when the optical intensity reaches a critical value, which is dependent on the site-to-site coupling and the standard deviation of the random energy in the disordered system, a sharp transition from localized to extended electronic states takes place. Albrecht and Wimberger [5] have investigated two mechanisms that are responsible for delocalization, namely a correlated disorder potential and mutual interaction between two bosons. Stockman et al. [6] have shown that the eigenmodes (surface plasmons) of disordered nanosystems (modeled as random planar nano-composites) do not always undergo Anderson localization alone, but can display properties of both localized and delocalized states simultaneously. In the continuum scale, Hunt and Lucena Neto [7] have investigated the influence of localized imperfections on the elastic buckling of a long cylindrical shell (with large Batdorf parameter) under axial compression by using a double scale analysis

⇑ Tel.: +1 (801) 581 6282; fax: +1 (801) 581 4816. E-mail address: [email protected] http://dx.doi.org/10.1016/j.engfracmech.2015.11.008 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.

R.A. Chaudhuri / Engineering Fracture Mechanics 151 (2016) 138–160

139

Nomenclature df ELL, ETT,

mLT GLT, GTT h hj, hj+1 m N Ns n p pcr Ri rðhÞ SðkÞ ac f0 Vg W0 x, b, z ef , ee h qðkÞ

Rij f

fiber diameter longitudinal and transverse Young’s moduli, and major Poisson’s ratio, respectively, of a unidirectional lamina longitudinal and initial transverse shear moduli, respectively thickness of a laminated shell distance of the top and bottom surface of a sub-lamina from the bottom surface of the corresponding lamina ratio of reference yielding shear stress to transverse shear modulus, GTT, in the Ramberg–Osgood representation total number of laminas total number of sub-laminas inverse strain hardening parameter in the Ramberg–Osgood representation uniform hydrostatic pressure classical buckling pressure of a very long cylindrical shell (plane strain ring) inner radius of a long perfect cylindrical shell (plane strain ring) radial coordinate of the innermost (bottom) surface of an undeformed ring with modal imperfection elastic compliances of a lamina global displacement vector in FEA amplitude of modal imperfection of a plane strain ring coordinates of a point inside a layer force and energy convergence criteria, respectively angle measured from the global x3 axis radius of curvature of the inner surface of the kth layer of an imperfect plane strain ring component of the incremental compliance matrix of a lamina radial distance of a point inside a laminated cylindrical shell/ring measured from its innermost or bottom surface

including interaction modes. An investigation on the effect of localized imperfection must account for a large number of nearly coincident buckling modes and their nonlinear interactions, which can be accomplished by using amplitude equations of the Ginsburg–Landau type [8]. Localized buckling of a modally imperfect ring with bilinear hypoelastic material property has been analyzed by Kim and Chaudhuri [9]. The post-buckling behavior of a thin [90/0/90] cylindrical shell of finite length, and weakened by the presence of a localized imperfection, with surface-parallel shear modulus nonlinearity, is characterized by a limit or localization point beyond which the equilibrium path is unstable, although the same cylinder without localized imperfection and/or material nonlinearity displays stable post-buckling behaviors [10]. In what follows, occurrence of localization/delocalization in thick imperfect cross-ply very long cylindrical shells (plane strain rings), under applied hydrostatic pressure, is investigated. Most carbon fiber reinforced plastic (CFRP) thick cylindrical shells tested at Naval Surface Warfare Center (NSWC), Carderock, MD have failed at much lower pressures than expected [11–13]. Failure has usually been instantaneous and catastrophic, which has also been observed by Hahn and Sohi [14], Abdallah et al. [15], and others. Figs. 1 and 2 show typical failures observed in thick (Ri/h
6) CFRP cylinders tested at NSWC, Carderock [11–13]. Observations made from the fractured portion and detailed theoretical analysis indicate that the failure may have initiated at a stress concentration site such as initial fiber waviness or misalignment, shown in Figs. 3 and 4, and associated resin-rich area. As seen in Figs. 1 and 2, the failure creates a longitudinal crack through entire length of the cylin
in the range of 28–35° through the cross section. The fibers are broken into quite short lengths der at kink boundary angle, b, along almost the entire cylinder length within the kink band; failed cross-ply ring specimens, tested by Abdallah et al. [15] are shown in Figs. 9 and 10 of Chaudhuri [17]. The collapse pressure and apparent failure mode are about the same for all the carbon-composite thick cylindrical shell models tested at NSWC [11–13]. It has been concluded that initial fiber misalignment, ultimate shear strain of fibers and the two transverse shear moduli of a laminate are the key parameters limiting the compressive strength [16]. An analytical/experimental effort at improving the compressive strength/toughness of the CFRP material by using a hybrid carbon/glass commingling concept has been reported by Chaudhuri and Garala [17]. A Griffith type of fracture criterion for the stability of damage/crack growth, based on the principle of energy balance, is introduced to derive the then-unknown concept of kink toughness (i.e., resistance to kink band propagation) [17]. Propagation of kink band appears to be due to unstable (mode II) crack growth in the neighboring fibers along the kink
triggered by the energy released by the kinking or fracture of a defective (e.g., wavy or misaligned) fiber band angle, b, bundle. Chaudhuri et al. [18] and Chaudhuri [19] have investigated kink propagation in glass fiber reinforced unidirectional composites, utilizing eigenfunction expansion techniques [20–23] applied to fully bonded fiber–matrix kink-wedges. A three-dimensional eigenfunction expansion technique, based in part on separation of the thickness-variable and partly utilizing a modified Frobenius type series expansion [24–30] in conjunction with what is known as ‘‘Eshelby-Stroh formalism” has been used to compute the local stress singularity, in the vicinity of a kinked-fiber/matrix trimaterial junction front, representing a measure of the degree of inherent flaw sensitivity of unidirectional CFRP under compression. This process is further facilitated by the ‘‘yielding” and subsequent ‘‘plastic” deformation of the supporting matrix material, being investigated here.

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Fig. 1. End/external view of a failed carbon/epoxy cylinder tested at NSWC, Carderock with partially developed kink band and ensuing longitudinally propagated kink-crack [11,16].

Starnes and Williams [31] have reported a shear crippling type of failure to be prevalent in 0°-plies of a 48-ply laminate in the immediate vicinity of a hole as well as in a localized region of low velocity impact damage prior to catastrophic failure under compression. Various candidate sites for initiation of localized shear-crippling failures in laminated composite structural components, experimentally observed by Garala [11], Garala and Chaudhuri [12], Abdallah et al. [15], Starnes and Williams [31] and others are shown in Fig. 4 of Chaudhuri and Kim [32] and Fig. 4 of Chaudhuri [19]. The compressive response of a thin isotropic or laminated elastic ring or shell structure is known to be primarily characterized by the stability of equilibrium at the macro-structural level. The loss of stability of an isotropic or laminated thin or moderately thick elastic arch/ring/shell-type structure under external pressure has been studied extensively and is relatively well understood [33–37]. The classical postbuckling theory due to Koiter [33] can be used to compute, via the Lyapunov– Schmidt method, reductions of critical buckling pressures of imperfect thin cylindrical shells if the number of buckling modes is finite. A nonlinear resonance (eigenvalue) based semi-analytical solution technique for prediction of the elastic mode 2 collapse pressure (nonlinear eigenvalue) of a moderately thick cross-ply ring, weakened by a modal or harmonic type imperfection, has recently been presented by Chaudhuri et al. [38,39]. Compressive response of two extreme cases of cylindrical shell type structures weakened by the presence of localized imperfections has been investigated by Kim and Chaudhuri [40]. These two extreme cases are (i) thin metallic cylindrical shells, which are characterized by the absence of transverse shear deformation and (ii) thick laminated advanced fiber reinforced composite rings (very long cylindrical shells), whose response is dominated by the interlaminar or transverse (primarily shear) deformation. Kim and Chaudhuri [41] have investigated the role of lamination sequence, in conjunction with modal imperfection in lowering the load carrying capability of infinitely long thick laminated cross-ply cylindrical shells (plane strain rings) with linear material property. Results on symmetrically laminated [90/0/90] moderately thick (e.g., Ri/h = 12) and thick (e.g., Ri/h = 6) (plane strain) rings with modal imperfections and made of linear elastic materials show that a limit point appears on the post-buckling equilibrium path, due to the effect of interlaminar shear deformation, significantly reducing the load carrying capacity compared to their CLT (classical lamination theory) based linearized buckling loads [42]. A similar appearance of a limit point on the equilibrium path due to the effect of reduced transverse shear modulus of a lamina caused by the presence of uniformly distributed fiber misalignments has recently been presented by Chaudhuri [43]. The appearance of a limit point on the elastic post-buckling equilibrium path has been found to be a measure of the effect of interlaminar shear/normal deformation on localization. Tvergaard and Needleman [44] have shown stable postbucking behavior of an axially compressed elastic–plastic cylindrical panel made of a high hardening material, whereas

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141

Fig. 2. External view of a failed carbon/epoxy cylinder tested at NSWC, Carderock with longitudinally propagated kink-crack [11,16].

localization occurs in the otherwise same panel of a low hardening material. In what follows, the influence of modal imperfection in conjunction with thickness-based shear/normal deformation and low hardening coefficient on the emergence of localization and delocalization points on the equilibrium path is investigated. The intricacy of compression fracture of a CFRP cylindrical shell or ring arises out of the necessity of simulating the shear crippling or kink banding phenomena operating at multiple scales (ranging from angströms to cms and beyond) involving multiple physics [18]. Some of the uncertainties in the failure of thick CFRP cylindrical shells under compressive loading relate to very fundamental questions pertaining to (i) linkage between macro-structural instability, such as buckling/ post-buckling failure of a structural component, e.g., a ring or cylindrical shell at the geometric scale of at least several cms and larger [9,10,32,40–43], and micro-structural instability, such as kink band type failure at the fiber–matrix level at the geometric scale of about 10 lm [16–19,45,46]. A review of the above literature and the references therein reveals that although considerable efforts have been devoted to the issue of localization, not much analytical/computational attention has been directed toward either delocalization or the experimentally observed [11–13,15] compressive damage/fracture initiation and propagation till recently [47]. The combined effects of modal imperfections, transverse shear/normal deformation with the nonlinear hypo-elastic (wherein a separate unloading path has been ignored) transverse shear (GTT) material property with low hardening coefficient (e.g., inverse hardening parameter, n = 10) on the emergence of interlaminar shear crippling type instability modes, related to the localization (deformation softening), delocalization (deformation hardening) and propagation of mode II compression fracture/damage, are investigated. The important question that is begging to be asked and answered can be stated thus: what are the geometric and/or material parameters that would induce localized and delocalized states in cross-ply cylindrical shells under hydrostatic compression simultaneously, and what would be the consequences of such occurrences? The primary objective of the present investigation is to answer this question. The second and most important objective is the (hitherto unavailable) computation of the mode II stress intensity factor, energy release rate, and shear damage/crack band-width for a low hardening coefficient (i.e., n = 10), for a thick (Ri/h = 6) cross-ply CFRP plane strain ring (very long cylindrical shell) under external pressure, from a nonlinear FEA, using Griffith’s energy balance approach. Additionally, the shear crippling (kink band at the micro/nano scale) angle is determined using an analysis, pertaining to the elastic plane strain inextensional deformation of the thick hydrostatically compressed cross-ply long cylindrical shell (plane strain ring).

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Fig. 3. Localized fiber waviness in a thick composite cylindrical shell tested at NSWC, Carderock, MD [11,32].

2. Basic relations and solution strategy for nonlinear analysis of a long thick cylindrical shell 2.1. Three-dimensional kinematic relations for a thick shell Invoking the theory of parallel surfaces, the coefficients of the first fundamental differential quadratic form of a surface inside the kth layer of a laminated imperfect cylindrical shell can be written in terms of their bottom surface counterparts as follows [48]:

g ðkÞ x ðzÞ ¼ 1;

ðkÞ

ðkÞ

g b ðzÞ ¼ g b


 z 1 þ ðkÞ ;

g ðkÞ z ðzÞ ¼ 1:

q

ð1Þ

The components of the engineering nonlinear strain in terms of the physical components of the displacement vector at an arbitrary point inside the kth layer can be written as follows [48]:

"


@uðkÞ @x

2


ðkÞ 2
ðkÞ 2 # @v @w ; þ @x @x

eðkÞ 11 ðx; b; zÞ ¼

@uðkÞ 1 þ 2 @x

eðkÞ 22 ðx; b; zÞ ¼

"
2
ðkÞ
ðkÞ  2
ðkÞ 2 # @v 1 @uðkÞ @v @w ðkÞ ðkÞ ðkÞ þ ; þ þ v þ w þ w  ðkÞ ðkÞ 2 @b @b @b @b gb 2ðg Þ

ð2bÞ

eðkÞ 12 ðx; b; zÞ ¼



 @ v ðkÞ 1 @uðkÞ 1 @uðkÞ @uðkÞ @ v ðkÞ @ v ðkÞ @wðkÞ @wðkÞ þ ðkÞ þ ðkÞ þ þ wðkÞ þ  v ðkÞ ; @x @x @b @x @b @x @b g b @b gb

ð2cÞ

eðkÞ 33 ðx; b; zÞ ¼

@wðkÞ 1 þ 2 @z

þ

1

ð2aÞ

b

"
2
ðkÞ 2
ðkÞ 2 # @uðkÞ @v @w þ þ ; @z @z @z

ð2dÞ

R.A. Chaudhuri / Engineering Fracture Mechanics 151 (2016) 138–160

143

Fig. 4. Fiber misalignment of hoop layers (15) in a thick carbon/epoxy cylindrical specimen tested at NSWC, Carderock, MD [11,16].

eðkÞ 13 ðx; b; zÞ ¼

@wðkÞ @uðkÞ @uðkÞ @uðkÞ @ v ðkÞ @ v ðkÞ @wðkÞ @wðkÞ þ þ þ þ ; @x @z @x @z @x @z @x @z

ð2eÞ

eðkÞ 23 ðx; b; zÞ ¼


ðkÞ 
ðkÞ  ðkÞ
ðkÞ    ðkÞ @w @ v ðkÞ 1 @uðkÞ @uðkÞ @v @w ðkÞ ðkÞ @ v ðkÞ @w þ v v  þ þ þ w þ  : ðkÞ ðkÞ @b @z @b @z @b @z @b @z gb gb

ð2fÞ

1

2.2. Nonlinear Finite Element Analysis (FEA) The present study primarily focuses on the problem of nonlinear (both geometric and material) post-buckling behavior of a thick cross-ply (plane strain) ring, and includes, in contrast to the traditional von Karman approximation, all the nonlinear terms in the kinematic equations [48]. A fully nonlinear thick shell finite element is employed in order to obtain the discretized system equations [9,32,40–43,48]. The element utilizes the total Lagrangian formulation in the constitutive equations and incremental equilibrium equations. Hsia and Chaudhuri [40], Kim and Chaudhuri [48], and Chaudhuri and Hsia [49,50] have employed fully geometric nonlinear kinematic relations for perfect shallow homogeneous isotropic and cross-ply [0/90] panels. Also addressed by Kim and Chaudhuri [48] is the issue of accuracy of the von Karman nonlinear strain assumption which has been found to overestimate transverse displacements in the advanced nonlinear regime. A curved 16-node isoparametric layer-element, based on an assumed quadratic displacement field (in the surface-parallel coordinates), employs a layer-wise linear displacement distribution through thickness (LLDT). Chaudhuri and Hsia [49,50] have investigated the effect of thickness on the response of perfect isotropic and laminated cylindrical panels. After incorporating the boundary conditions, the principle of virtual displacement, in conjunction with the total Lagrangian formulation, is invoked to obtain the incremental equations of motion as follows [32,42]:

½KL f0 Vg þ ½KN f0 Vg ¼ ff L g  ff N g;

ð3Þ

in which [KL], [KN], [fL], [fN], and f0 Vg, are as defined in Chaudhuri and Kim [32,42]; see also Appendix A for details. The Newton–Raphson iteration scheme in conjunction with Aitken acceleration is employed to obtain the limit point (hydrostatic) pressure. Beyond this load, the post-buckling behavior is computed by an incremental displacement control scheme rather than the usual incremental force control scheme, when a limit point appears on the equilibrium path. The remaining equations and other details pertaining to the nonlinear finite element solution under consideration are available elsewhere [32,42], and will not be repeated here in the interest of brevity of presentation. A curved laminated shell element of triangular planform that assumes transverse inextensibility and layer-wise constant transverse shear-angle or

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cross-sectional warping (zigzag theory) as a reasonably accurate approximation to the nonlinear cross-sectional warping, obtainable from the three-dimensional nonlinear elasticity theory, developed by Chaudhuri [51], can also be employed for the present purpose. This will be explored as an alternative approach in the future. The ring geometry with modal imperfection is described in Fig. 5(a). The geometry of the inner surface of a laminated ring with a modal imperfection is described in Fig. 5(b). A modal or harmonic imperfection is given by

rðhÞ ¼ Ri  w0 cosð2hÞ

ð4Þ

where rðhÞ is the distance from the central axis of the perfect ring to the inner surface of the imperfect ring, with w0 denoting the amplitude of the modal imperfection that corresponds to mode 2 of classical buckling of a ring, shown in Fig. 5(a) and (b). The finite element model of a quarter of the plane strain (perfect) ring along with the prescribed boundary conditions is presented in Fig. 6. The corresponding imperfect one is similar and hence not shown here. Double symmetry conditions permit every model under consideration to be limited to only a quarter of the ring such that the corresponding surface-parallel displacements vanish along the center lines and the buckled shapes are assumed to be symmetric. Because the loading and geometric symmetries are assumed, boundary conditions on the surfaces in Fig. 6 can be prescribed as follows [42]: (a) Geometric Symmetry:

On the surfaces ABFE and CDHG : On the surface EHGF :

0 uðx; b; fÞ

0

v ðx; 0; fÞ ¼ 0 and 0 v ðx; p=2; fÞ ¼ 0

¼ 0;

ð5aÞ ð5bÞ

(b) Loading Symmetry: S

ðNþ1Þ

Traction force on the surface BCGF : f i ¼ pðx; b; qðNþ1Þ Þni

:

ð6Þ

The plane strain condition in the three-dimensional model is obtained by applying the displacement constraints as shown in Chaudhuri and Kim [32]:

(a)

(b)

Fig. 5. (a) Nomenclature and geometry and (b) Shape of the inner surface of a (plane strain) [90/0/90] ring with modal imperfection.

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145

Fig. 6. Finite element model of a [90/0/90] cylindrical (plane strain) ring.

3. Numerical results computed using the fully nonlinear FEA The inner radius, Ri, of the corresponding perfect ring is 8.89 cm (3.5 in.), while its thickness, h, is 1.48 cm (0.583 in.); see Fig. 5(a). Fiber orientation angles 90° and 0° represent the hoop and axial directions, respectively, of the cylinder/ring. The linear elastic properties of a Narmco 5605 graphite-epoxy orthotropic lamina under compression are as given below (see Appendix C of Jones [52]):

ELL ¼ 124:11 GPa ð18:0 MsiÞ; GLT ¼ 5:86 GPa ð0:85MsiÞ;

ETT ¼ 10:69 GPa ð1:55 MsiÞ; mLT ¼ 0:575;

GTT ¼ 3:10 GPa ð0:45 MsiÞ:

A convergence check is available in Table 1 of Kim and Chaudhuri [48]. A comparison of hydrostatic buckling pressures of perfect [90/0/90] plane strain rings computed using the present FEA with their Donnell theory counterparts is available in Table 2 of Kim and Chaudhuri [53]. The convergence criteria, given by Eqs. (47) of Ref. [42], are implemented in this investigation as follows: ef ¼ ee ¼ 5  103 . Chaudhuri and Abu-Arja [54] have demonstrated the importance of selection of proper step size of the loading in the nonlinear regime. One improperly chosen large step size has been shown to induce an artificial chaotic situation, even if subsequent load step sizes are properly selected. In what follows, the effect of a modal imperfection, in conjunction with hypoelastic (GTT only) material property, on the nonlinear (large displacement/strain) response of the plane strain ring is numerically investigated. The primary focus is on, as mentioned earlier, the geometric and/or material parameters that would induce localized and delocalized states in crossply cylindrical shells under hydrostatic compression simultaneously, and what would be the consequences of such occurrences. 3.1. Effect of hypoelastic (GTT only) material property with low hardening coefficient on localization and delocalization Fig. 7 provides the nonlinear elastic (hypoelastic) properties corresponding to transverse shear modulus, GTT. Since a fiber reinforced composite lamina is transversely isotropic, the plane of isotropy being the plane T–T (i.e., transverse to the fiber

Fig. 7. Nomenclature and property definitions of a nonlinear composite material: GTT curve for different n with m = SR44/GTT fixed.

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direction), this modulus is largely matrix dominated (along with the transverse Young’s modulus, ETT [55]), and is more likely to display nonlinear elastic (hypo-elastic) behavior [56]. This nonlinearity is described by setting the reference stress, Sr44 ¼ mGTT (refer to Eq. (8a) below and Fig. 7 for definition), for a unidirectional lamina. Taking into account nonlinear elastic (hypo-elastic) behavior (as opposed to plasticity) and neglecting the hygrothermal effects, the incremental strain–stress relations of the kth orthotropic lamina for the principal material directions (x, b, z) are given as follows [32,56]:

8 ðkÞ 9 > e > > > > 0 xx > > ðkÞ > > > > 0 ebb > > > > > > > > < eðkÞ > =

2

ðkÞ ðkÞ ðkÞ R11 R12 R13

0

0

ðkÞ ðkÞ R22 R23

0

0

6 ðkÞ 6 R12 6 6 ðkÞ 6 R13 0 zz 6 ¼ ðkÞ 6 > > > 6 0 0 ebz > > > > > 6 > > > > 6 ðkÞ > > > 4 0 0 exz > > > > : ðkÞ > ; 0 exb 0

ðkÞ 23

ðkÞ 33

0

0

0

0

RðkÞ 44

0

0

0

0

RðkÞ 55

0

0

0

0

R

R

38 ðkÞ 9 > 0 Sxx > > > > > ðkÞ > > 7> > > > > 0 Sbb > 0 7 > > > > 7> > > 7> ðkÞ < = 7 S 0 0 zz 7 7> ðkÞ >: S > 0 7> > 0 bz > > 7> > ðkÞ > > 7> > > 0 5> 0 Sxz > > > > > ðkÞ > > ; ðkÞ : R66 S 0 xb 0

ð7Þ

ðkÞ

In the above equation, for a unidirectional lamina (k = 1, N = 3), the compliance matrix component, R44 , nonlinearity can be approximated analytically by the method of Ramberg and Osgood [57], who have suggested that rising stress–strain curves with smooth knees be represented by the following type relation: ðkÞ 44

R

2 1 4 3 ¼ ðkÞ 1 þ 7 E44

RjjðkÞ ¼

1 ðkÞ

Ejj

;

t ðkÞ 0 S44 ðkÞ SR44

!n1 3 5;

ð8aÞ

j ¼ 1; 2; 3; 5; 6 ðno sum on jÞ

ð8bÞ

and ðkÞ ðkÞ ðkÞ R12 ¼ v 12 R11 ; ðkÞ

ðkÞ ðkÞ ðkÞ R13 ¼ v 13 R11 ;

ðkÞ ðkÞ RðkÞ 23 ¼ v 23 R22

ð9Þ

ðkÞ

where E44 (=GTT) and t0 S44 denote transverse shear modulus and transverse shear stress component at time t, respectively, of ðkÞ

the kth lamina material. Ejj , 1, 2, 3, 5, 6 (no sum on j) denote the three Young’s moduli as well as the other two shear moduli of a hoop layer, while v 12 ; v 13 and v 23 represent major Poisson’s ratios in the x–b, x–z and b–z planes (surfaces), respectively, of the same. Details of nomenclature and property definitions of the nonlinear (hypo-elastic) composite material are shown in Fig. 7 (see also Appendix B for further details). Swanson et al. [58] have experimentally verified the validity of the use of Ramberg–Osgood relationship for the surface-parallel shear modulus, GLT, of CFRP, in general, and AS4/3501-6 carbon/epoxy composite in particular. No such experimental data are available for the transverse shear modulus, GTT, however. Since the transverse shear modulus, GTT, is somewhat lower than its surface-parallel counterpart, n P 5 is regarded as a reasonable inverse hardening parameter for GTT. In addition, more compliant resin systems or hygro-thermal effects are expected to increase the inverse hardening parameter, n, from 4.1 to 10. Fig. 8 shows plots of the normalized pressure, p⁄ = p/pcr,Donnell, versus the normalized transverse displacement (deflection), w ¼ Riw at h = 0°, of modally imperfect [90/0/90] (plane strain) rings, with w⁄⁄ = w0/Ri = 0.005, for different inverse w0 ðkÞ

ðkÞ

ðkÞ

strain hardening parameter, n (yield parameter, m, being fixed at 0.03). Chaudhuri and Kim [55,56] have earlier studied the effects of nonlinear transverse Young’s modulus (ETT) and shear modulus (GTT) on localization. However, they have not computed beyond inverse hardening parameter n = 3. No delocalization is observed in the range of 1 6 n 6 3. Fig. 8 demonstrates that for n P 3.0, the ‘‘yield” point of the cross-ply ring coincides with the point where the localization (deformation softening) instability is triggered, and no distinct limit point appears on the post-‘‘yield” path, which represents an unstable equilibrium. For inverse strain hardening parameters, n > 1.5, a localization (deformation softening) type of behavior is expected to be triggered at/around this point, which corresponds to a much reduced load carrying capacity of the thick imperfect plane strain ring under investigation [56]. For n = 5, 10, the process of localization or deformation softening continues until the load drops by a factor of 2 or greater, followed by the appearance of a second limit point, called the forward tangent bifurcation, on the equilibrium path. The appearance of the second saddle-node or forward tangent bifurcation, which represents delocalization (extended or periodic orbit), is delayed as the inverse hardening parameter, n, increases. The combined effects of modal imperfection, transverse shear/normal deformation, geometric nonlinearity and transverse shear modulus nonlinearity are expected to trigger the emergence of interlaminar shear crippling (kink band) type propagating instability modes, caused by the localization (onset of deformation softening) and delocalization (onset of deformation hardening) phenomena, and leading to the compression damage/fracture at the propagation pressure [59]. This is investigated in the following sections.

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147

Fig. 8. Sensitivity of the post-buckling behavior of a thick [90/0/90] very long cylindrical shell (Ri/h = 6.0, w⁄⁄ = 0.005, nonlinear transverse shear modulus, GTT with m = 0.03) to the inverse hardening parameter, n.

3.2. Soliton (slightly disturbed integrable hamiltonian system) Analogy A qualitative understanding of the situation, specifically depicted in Fig. 8, can be gained by recalling the seminal numerical experiment by Fermi, Pasta and Ulam (FPU) [60], as narrated by Strogatz [61] (see also Chaudhuri [59]): ‘. . .Their test problem involved a deliberately simplified model of a vibrating atomic lattice, consisting of 64 identical particles (representing atoms) linked end to end by springs (representing the chemical bonds between them). . . . Fermi suspected that this nonlinear character of chemical bonds might be the key to the inevitable increase of entropy. . . . What the computer revealed was astonishing. Instead of a hiss, the string played an eerie tune, almost like music from an alien civilization. Starting from a pure tone, it progressively added a series of overtones, replacing one with another, gradually changing the timbre. Then it suddenly reversed direction, deleting overtones in the opposite sequence, before finally returning almost precisely to the original tone. . . . He had never guessed that nonlinear systems could harbor such a penchant for order.’ The KAM (Kolmogorov, Arnold and Moser) theory, which studies a slightly disturbed integrable Hamiltonian system, provides an understanding of the seemingly paradoxical FPU result [62]. In the present context, referring to Fig. 8, which represent the hydrostatic compression fracture counterpart of FPU [60], it can easily be seen that the first rising part of the curve up to the localization point is analogous to progressive addition of a series of overtones (starting from a pure tone – the linear elastic case), replacing one with another, gradually changing the timbre. An applied pressure above the localization pressure would represent the onset of Hamiltonian chaos, the musical counterpart of which would be a ‘‘meaningless hiss,” something akin to the static of a radio, which, however, does not happen here. The musical analog of the falling (softening) branch that follows the first saddle-node bifurcation is deletion of the afore-mentioned overtones in the opposite sequence, before finally returning almost precisely to the original tone, the forward tangent bifurcation representing the delocalization pressure (point D, Fig. 8), which brings back the original periodic buckling pattern. Below this pressure, the buckling pattern is assured to be periodic. If the experiment is continued, there is again a rising branch, reminiscent of the one before the localization pressure is reached, which is called recurrence in the soliton literature [59].

3.3. Maxwell’s construction, propagation pressure and localized shear crippling damage Chaudhuri and Kim [32] introduced a local or dimple shaped imperfection superimposed on a fixed modal one. With the increase of local imperfection amplitude, the limit load (hydrostatic pressure) decreases, and also the limit point appears at an increased normalized deflection. Additionally, the load–deflection curves tend to flatten (near-zero slope) to the lowest

Fig. 9. Idealization of an initial localized dent or shear damage.

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R.A. Chaudhuri / Engineering Fracture Mechanics 151 (2016) 138–160

pressure level, signaling the onset of ‘‘phase transition” in the localized region, and coexistence of two ‘‘phases”, i.e., a highly localized band of shear crippled (kinked) ‘‘phase” and its un-shear-crippled (unkinked) counterpart along the circumference of the long cylindrical shell. The Maxwell construction then implies the possibility that under favorable condition, such as the presence of a severe localized defect, in the form of, e.g., a dent (Fig. 9) or fiber misalignment region, the equilibrium configuration of the externally pressurized plane strain ring (very long cylindrical shell) would involve such a two‘‘phase” state. This numerical experiment by Chaudhuri and Kim [32] serves as a justification for employment of the artifice of Maxwell’s construction in the present investigation. Fig. 10 shows that the equilibrium path under the above conditions will follow the pathway, OACFG, instead of OABCDFG, computed using the nonlinear FEA. The straight equilibrium path, ACF, represents the normalized propagation pressure, pP , computed using Maxwell’s construction (see e.g., Chaudhuri and Kim [42] for a detailed discussion), for the inverse strain hardening coefficient, n = 10. The pathway, ACF, represents a mode II (or more likely, mixed mode with mode II dominance) crack growth under a fixed pressure, which can be used for computation of energy release rate, as in conventional fracture mechanics. The equilibrium path segment, FG. . ., represents a residual load carrying capacity of the long CFRP cross-ply cylindrical shell, provided the material is either intrinsically ductile, or its ductility is enhanced through commingling of carbon fibers with amorphous glass fibers at the tow level [17,19]. The CFRP material is generally considered brittle, however; therefore, the second option remains a possibility.

4. Analogy with anderson localization/delocalization (quantum opto-electronics) Anderson [1] and other quantum mechanics researchers, such as de Moura and Lyra [3] and Albrecht and Wimberger [5] have studied the effect of the degree of randomness or disorder on localization/delocalization. For example, Anderson [1] has presented the effect of W/V on localization, where W represents the width of the probability function for the random potential, Ei, and V denotes the magnitude of the hopping energy, Vji. W = 0 means Bloch (periodic) states. Figs. 18(a) and (b) of Anderson [1] show that when W/V is changed from 5.5 to 8.0, the eigenmodes change from delocalized to localized states. de Moura and Lyra [3] have found that the one-dimensional Anderson model with long-range correlated diagonal disorder exhibits a phase of extended electronic states if the on-site energy disorder distribution is governed by a power-law spectral density S(k) / 1/ka with a > 2 (i.e., whenever the energy sequency increments have a long-range persistent character). Here k is the wave number and a is a constant (parameter). a = 0 reduces the above relation to a white noise spectrum, and thus represents standard Anderson model [1] with d-correlated disorder. In the present situation, referring to Fig. 11 of Chaudhuri and Kim [42], it is seen that for a thin long cylindrical (modally imperfect) shell (Ri/h = 60) made of a linear elastic material, the pressure-deflection plot does not exhibit a limit point. The elastic post-buckling path is, in this case, stable, and the question of localization or limit point behavior does not arise. This is analogous to a Bloch state. With the decrease of Ri/h (i.e., with progressive increase of thickness induced shear) from 30 to 6, localization behavior dominates with concomitant reduction of the localization pressure. Referring back to Fig. 8 reveals transition from localization to localization–delocalization states on the equilibrium path for a thick long cylindrical (modally imperfect) shell (Ri/h = 6) with the increase of inverse hardening parameter, n, from 3 to 5 and finally to 10. Finally, Stockman et al. [6] state that all Anderson localized eigenmodes of surface plasmon are ‘‘dark”, and cannot be excited or observed from the far field. These eigenmodes ‘‘can, however, be excited and observed by near-field scanning optical microscopic probes in the near field region” [6]. In contrast, the delocalized (‘‘luminous”) eigenmodes ‘‘couple efficiently to the far zone fields” [6]. In the present case, referring back to Fig. 10, it is seen that localization–delocalization modes working in tandem give rise to the multi-scale and multi-physics phenomena of kink band at the nano/micro scale and shear crippling at the macro-scale (to be discussed below). Localization concentrates the energy in a ‘‘hot spot” [6], such as a region of misaligned fibers at the micron scale, and intracrystallite and intercrystallite disordered regions [35] at the nano-meter scale. The delocalized or extended states are responsible for kink-crack propagation at the propagation pressure.

5. Shear crippling type compression failure 5.1. Determination of the shear crippling angle Fig. 10 shows a steady state propagation of kink band or shear crippling damage at the propagating pressure, pP . It is assumed for the purpose of illustration that fiber misalignment defect is present at the bottom of each hoop or 90 lamina. This kind of defect may be restricted to a few fibers, which is considered to be the first sub-lamina (j = 1) of the kth lamina (k = 1, . . . , N). Each sub-lamina is assumed to be of equal thickness. The first step entails nucleation of a kink crack, of length, ðkÞ
ðkÞ Þ, and oriented at an angle, b
ðkÞ , with respect to the thickness (z-) direction, during which the long cylindrical a ¼ t k = cosðb j

j

j

shell deflects by an amount, Dc =N s ¼ ðwF  wF Þ=N s . The kink crack or the shear crippling damage then propagates at propagation pressure, pP , in a quasi-static manner, eventually covering the entire thickness of the kth lamina (k = 1, N = 3). Prediction of the microscopic failure mode and especially the shear crippling (or the kink band at the micro/nano-scale)
ðkÞ , j = 1, . . ., Ns, k = 1, . . ., N, would require an alternative analysis, angle of the jth sub-lamina belonging to the kth layer, b j

R.A. Chaudhuri / Engineering Fracture Mechanics 151 (2016) 138–160

149

Fig. 10. Maxwell diagram for computation of propagating pressure in a thick (very) long cylindrical shell (Ri/h = 6; w⁄⁄ = 0.005) with material nonlinearity (m = 0.03; n = 10).

pertaining to the elastic plane strain inextensional deformation of the compressed solid. What follows will consider a more realistic example of an initial localized dent or shear damage (see Fig. 9). This can be mathematically represented by [16]


ðkÞ w j ðy; zÞ ¼ 

f n ðkÞ cSj

   o ðkÞ ðkÞ yHðyÞ  y  cSj H y  cSj ;

ð10Þ

in which HðyÞ represents Heaviside function, while y in Eq. (10) is obtained from the relationship dy = gb db. Following the
ðkÞ can be obtained as follows: analysis of Chaudhuri [16], b j

31=2 2 ðkÞ ðkÞ   dy GLT  rj;ave ðkÞ
5 ; ¼ 4 tan b ¼ j ðkÞ dz E

ð11Þ

TT

as long as

y2 þ

  ðkÞ ðkÞ GLT  rj;ave ðkÞ

ETT

z2 – 0;

ð12Þ ðkÞ

which has to be true [16]. In Eq. (11), rj;ave is the average circumferential compressive stress through the thickness of the jth sub-lamina belonging to the kth layer (k = 1 or N = 3), and is given by

8 9 3 > > < = n gk  gk o Ns 6 1 1 7 ðkÞ ðkÞ ¼ 4Ak ð1  gk Þ   Ri þ h j gk   gk þ Bk ð1 þ gk Þ Ri þ hjþ1 5; ðkÞ > > tk : RðkÞ þ hjþ1 ; Ri þ hj i 2

ðkÞ rj;ave

ð13Þ

where

gk ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ffi 0ðkÞ 0ðkÞ s11 s22 ;

ðkÞ

0ðkÞ

ðkÞ

with scd ¼ scd 

sc3

ðkÞ

s33

;

c; d ¼ 1; 2

ð14Þ ðkÞ

and N s is the total number of sub-laminas in a lamina, while tk and Ri denote, respectively, the thickness and inner surface radius of the kth lamina (k = 1 or 3). Ak and Bk , k = 1 or 3, are as given in Appendix C. 5.2. Mode II fracture/damage A careful examination of Fig. 10 reveals that the above-discussed Maxwell’s construction corresponds to an assumed mode II (actually mode II dominated mixed mode) fracture at constant (propagating) pressure, pP, which corresponds to the steady state propagation of the kink band at the micro (fiber–matrix)-scale. The mode II energy release rate, GIIP, at fixed load (propagation pressure) is given by [47,59]

GIIP ¼

@U P @D ¼ ; @a 2L @a

ð15Þ

in which a is the kink-crack length, L is the length of a long cylinder (width of the plane-strain ring), and D represents deflection, while [47,59]

  ðkÞ P ¼ pP 2cSj L;

ð16Þ

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R.A. Chaudhuri / Engineering Fracture Mechanics 151 (2016) 138–160 ðkÞ

in which cSj denotes the half-band-width of the shear damage (kink crack) in the jth sub-lamina of the kth lamina. Since mode II energy release rate at propagation pressure, pP , in the jth sub-lamina of the kth lamina is given by [63] ðkÞ 2

K IIPj

ðkÞ

GIIPj ¼

EðkÞ

;

ð17Þ

in which

pffiffiffi ðkÞ 2ETT

EðkÞ ¼ rffiffiffiffiffiffi ðkÞ ETT

ðkÞ

ðkÞ

1=2 ;

ð18Þ

ðkÞ þ 1 þ mTT  mTL

ELL ðkÞ

and K IIPj , which denotes the mode II stress intensity factor of the kink-crack, when it has propagated up to the jth sub-lamina of the kth lamina, at propagation pressure, pP , can be approximated as ðkÞ pffiffiffiffiffiffiffiffi aj ;

ðkÞ

K IIPj
sPj

ð19Þ

p

ignoring the curvature and other geometric effects. Far-field applied shear stress, [17]:

sPjðkÞ


ðkÞ rj;ave

2

sðkÞ P , in Eq. (19) can be written as follows

 
ðkÞ sin 2b j;average ;

ð20Þ


in which rj;ave and b j;average represent average over first j sub-laminas. Substitution of Eqs. (13) and (16)–(20) into Eq. (15) and ðkÞ

ðkÞ

integration between a ¼ 0; D ¼ wA and a ¼ aj ; D ¼ wA þ jDc =N s for j = 1, . . ., Ns (see Fig. 10) yield the half-band-width of the shear damage (kink crack) as follows [47]:



pNs

ðkÞ

csj


2pP Dc E

ðkÞ

j

ðkÞ rj;ave

2

2

sin



  
ðkÞ cos2 b
ðkÞ a2 ; b j j j

j ¼ 1; . . . ; NS

ð21Þ

in which

 o1 t k Xn
ðkÞ cos b : j;ave Ns j1 j

Dc ¼ wF  wA ; and aj ¼

ð22a; bÞ

The corresponding (maximum or critical) mode II stress intensity factor and energy release rate in the kth lamina, as the mode II kink-crack propagates up to the jth sub-lamina, at propagation pressure are given as follows:

   pffiffiffiffiffiffiffiffi ðkÞ ðkÞ
ðkÞ cos b
ðkÞ K IIPjc
rj;ave sin b paj ; j;ave j;ave ðkÞ

GIIPjc


j ¼ 1; . . . ; Ns

   o2 paj n ðkÞ ðkÞ
ðkÞ rj;ave sin b
j;ave ; cos b j;ave ðkÞ

ð23Þ

j ¼ 1; . . . ; Ns

E

ð24Þ

6. Numerical results relating to shear crippling type compression failure and mode II fracture/damage The following normalized quantities are defined:

z ¼ z=t k ;

a ¼ a=t k ;

pP ¼

pP pcr;FEA

;

Dc ¼ 

ðwF  wA Þ ; Ri  w0

ðkÞ

ðkÞ

K IIP ¼

K IIPjc pffiffiffi ; pcr;FEA h

ðkÞ

ðkÞ

GIIP ¼

GIIPjc  ETT ðpcr;FEA Þ2 h

ðkÞ

;

ðkÞ

cS

¼

cSj

df

;

ð25Þ

where df = 8 lm (micro-meter). The critical buckling pressure, pcr,FEA of the corresponding perfect cross-ply (plane strain) ring, used here for normalization, have been computed using the linearized version of the same FEA (see Table 2, Kim and Chaudhuri [53]).
k , k = 1, 3, and the normalized kink-crack length, a , through Figs. 11 and 12 display variation of shear crippling angle, b thickness of a lamina (k = 1 or 3) of a thick [90/0/90] (very) long cylindrical shell, for the hypo-elastic (GTT only) material with the inverse hardening parameter, n = 10. These results are summarized in Table 1. The kink crack is assumed to grow simultaneously from a fiber misalignment defect located at the inner surface of each lamina. It should be noted that only the hoop or 90°-layers (i.e., k = 1 or 3) suffer from shear crippling type compression failure arising out of fiber misalignments at the micro-scale. Since the cylindrical shell is assumed to be infinitely long, the 0° layer behaves like a transversely isotropic solid, and is subjected to tensile stress in the circumferential direction.

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151


with normalized lamina thickness coordinate, z (Ri/h = 6, m = 0.03; n = 10). Fig. 11. Variation of b


k , through the thickness of each lamina is not substantial, Fig. 11 shows that the variation of the shear crippling angle, b
k;max value of 35.962° are justifying the employment of its average in Eqs. (22b), (23) and (24). Fig. 11 and Table 1 show that b in general agreement with experimentally observed values in the range of 28–35°. Figs. 13 and 14 display variation, with the normalized kink-crack length, aj , of the normalized stress intensity factor ðkÞ

ðkÞ

(fracture toughness), K II and normalized energy release rate (fracture energy), GII , respectively, all at the lamina-level (k = 1 or 3) of a thick [90/0/90] long cylindrical shell, for the afore-mentioned hypo-elastic (GTT only) material with the inverse hardening parameter, n = 10. Figs. 13 and 14 as well as Table 1 show that the normalized stress intensity factor and energy release rate of the outer hoop layer are somewhat higher than their inner hoop layer counterparts. Similar results for n = 5 are available in Chaudhuri [64]. A comparison between the two sets of results shows that these normalized quantities decrease with the increase of the inverse hardening parameter, n, from n = 5 to n = 10. Examination of Table 1 further reveals that the maximum computed mode II stress intensity factor and energy release pffiffiffiffiffi rate, at propagation pressure, for the hypo-elastic (GTT only) material (Ri/h = 6), are 16.6 MPa m and 25.5 kJ/m2, respecpffiffiffiffiffi tively, for n = 10. These values are comparable to the mode I fracture toughness of Al-alloy (23–45 MPa mÞ and fracture 2 energy (8–30 kJ/m ) [65]. Finally, Fig. 15 displays variation, with the normalized kink-crack length, aj , of normalized kink-crack half-band-width, ðkÞ

cs , at the lamina-level (k = 1 or 3) of a thick [90/0/90] long cylindrical shell, for the afore-mentioned hypoelastic (GTT only) material with the inverse hardening parameter, n = 10. Fig. 15 as well as Table 1 shows that the normalized kink-crack halfband-width of the outer hoop layer is slightly higher than its inner hoop layer counterpart. A comparison with Ref. [64] shows that this normalized quantity decreases with the increase of the inverse hardening parameter, n, from n = 5 to n = 10. Fig. 15 clearly exhibits the broadening of the kink-crack (damage) half band-width with the kink-crack growth starting from a fiber misalignment defect located at the inner surface of each lamina. This is in general agreement with the experimental results of Moran and Shih [45]. Fig. 1 of Moran and Shih [45] schematically depicts the steady ‘‘state kink band broadening in which edges propagate laterally into unhardened material. The material within the band is strain hardened and is highly resistant to further deformation”.

Fig. 12. Variation of a with normalized lamina thickness coordinate, z (Ri/h = 6, m = 0.03; n = 10).

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Table 1 Propagation pressure, shear crack width and maximum mode II fracture toughness/energy (under compression) of the hoop layers (k = 1, 3) of thick imperfect (w⁄⁄ = 0.005) carbon/epoxy [90/0/90] plane strain rings (Ri/h = 6.0; m = 0.03; n = 10). Layer # (k)

pP

Dc


1;max (deg.) b

K IIP;max

GIIP;max

cS;max

1 3

0.197 0.197

0.5126 0.5126

35.962 35.962

1.2505 1.3267

1.5514 1.7460

8.5901 9.6633

ð1Þ

ð1Þ

ð1Þ

Fig. 13. Variation of K II , with normalized kink-crack length, a (Ri/h = 6, m = 0.03; n = 10).

7. Multi-scale and multi-physics nature of the kink band/shear crippling phenomenon Shear crippling and kink band represent the same instability phenomenon viewed at three (or possibly more) geometrical scales: the former at the macro or meso-structural (lamina) level [32], while the latter at the micro or nano-structural (fiber–matrix) scale (see Fig. 12 of Chaudhuri [16]) as well as at the nano-meter (crystallite) scale, described in Chaudhuri [19]. This is because, defects are ubiquitous at all geometrical scales, in a fractal like formation, starting from modal imperfection, namely classical mode 2 (buckling) type, operating at the macro-scale (meter), to meso-scopic or lamina level operating at the scale of 130 lm, to fiber misalignment defects operating at the scale of 8–10 lm, to inter/intra-crystallite disorders operating at the scale of 5 nano-meters, to all the way down to the atomic scale (2–3 Å) at which a kink is nucleated by dislocation glide mechanism [19,47]. The classical buckling pattern has ultra-long range order of several centimeters to meters depending on the size of the structure, while at the other extreme, the Peierls–Nabarro stress associated with a dislocation has the shortest range order (<10 Å), with everything else in between these two vastly different scales, involving multiple physico-chemical mechanisms. Table 1 clearly shows that the above process of localization/delocalization helps nucleate and propagate a shear crack or damage of width of about 75–150 lm affecting about 5–10 fibers, depending on whether the initial defect is a fiber misalignment (asymmetric defect) or fiber waviness (symmetric defect as shown in Fig. 10). Fig. 9(a) of Chaudhuri [19] shows a group or bundle of misaligned carbon fibers, while the associated local rotational instability at the lamina scale (about 130 lm) is depicted in his Fig. 9(b). Chaudhuri [19] has recently investigated the stability or lack thereof of a trimaterial wedge like

Fig. 14. Variation of GII , with normalized kink-crack length, a (Ri/h = 6, m = 0.03; n = 10).

R.A. Chaudhuri / Engineering Fracture Mechanics 151 (2016) 138–160

153

Fig. 15. Variation of csI , with normalized kink-crack length, a (Ri/h = 6, m = 0.03; n = 10).

structure schematically shown in his Fig. 10(b), which appears to be ubiquitous in multiple scales: at macro/meso-scales, schematically depicted in Fig. 9 as well as at nano/micro-meter scales in a fractal-like manner. Micro-kinking (observed at the scale of a fiber diameter of about 8 lm) is caused by crystal boundaries and crystallite disorientations, as detected by the Raman and X-ray measurements, inside the fiber [66,67], as well as micro-notches and similar defects (during handling of fibers in the manufacturing of composites). A micro-kinked fiber, consisting of such crystallites and covalently bonded to the epoxy matrix, can also be idealized to form a wedge at a potential kink site. A nano-kinked graphite crystallite (54 Å for AS4 carbon fiber [67]) shown in Fig. 8 of Chaudhuri [19] is either located at the fiber surface and covalently bonded to the epoxy matrix, or is embedded inside the fiber surrounded by disordered carbon. Kink bands, thus formed at the nano-meter scale, can be considered to be atomically sharp. Displacement of carbon atoms at and around the kink boundary of a graphite crystallite structure during kink band formation is schematically illustrated in Figs. 8(a) and (b) of Chaudhuri [19]. Raman spectroscopy has revealed the presence of carbon atoms with sp3 hybridization in PAN/Rayon-based fibers with high intracrystallite disorder (Dc) [67,68]. Among experimental fibers listed in Table 1 of Chaudhuri et al. [67], those with R = I(A1g)/I(E2g) close to or greater than 0.8 and crystallite size of 5–6 nm or less are most likely to fail by kinking mechanism, where I represents the intensity of the Raman mode [19,67]. The commercial fibers such as AS4, Thornel 10 and Morganite II listed in Table 2 of Chaudhuri et al. [67] also belong to this category. Also included in this category are PAN-based fibers listed in Table 2 of Dobb et al. [68], such as C2, X, Y, Z and T1000. Once a kink-crack is nucleated at a nano/micro-scale by the above-mentioned localization mechanism, it will steadily propagate at the propagating pressure, pP , under the influence of the delocalization process. The active mechanism in simple kinking in a small graphite crystallite with intra/intercrystallite disorders, such as those
1} planes [19]. Formation of kink band would obtained in PAN-based AS4, Thornel 10 type fibers is dislocation glide on {1 0 1

(=20.15°), in the nano-scale (i.e., arrest further dislocation gliding on the {1 0 1 1} planes. Computed kink propagation angle, b crystallite) [19] is smaller than what has been experimentally observed (28–35°) in thick composite cylindrical shells tested at NSWC, Carderock. The difference can be attributed to presence of afore-mentioned intracrystallite and intercrystalite disorders, fiber tortuosity, crystallite disorientation, etc., at the nano-scale, and fiber misalignment, resin-rich areas, etc. at the micro-scale.

8. Conclusions Localization and delocalization are multi-scale and multi-physics phenomena that straddle the boundary between classical physics and quantum mechanics. The present study addresses the issue of localization/delocalization and compression fracture in thick imperfect cross-ply very long cylindrical shells (plane strain rings), under applied hydrostatic pressure. A fully nonlinear FEA for prediction of localization/delocalization and compression fracture of thick imperfect cross-ply very long cylindrical shells (plane strain rings), under hydrostatic pressure, is presented. The combined effects of modal imperfections, transverse shear/normal deformation with the nonlinear hypo-elastic (wherein a separate unloading path has been ignored) transverse shear (GTT) material property on the emergence of interlaminar shear crippling type instability modes, related to the localization (onset of deformation softening), delocalization (onset of deformation hardening) and propagation of mode II compression fracture/damage, are investigated. The important question that has long been begging to be asked and answered here can be stated thus: what are the geometric and/or material parameters that would induce localized and delocalized states in long cross-ply cylindrical shells under hydrostatic compression simultaneously, and what would be the consequences of such occurrences? An analogy to a soliton (slightly disturbed integrable Hamiltonian system) helps understanding the localization and delocalization phenomena leading to the compression damage/fracture at the propagation pressure. Hitherto unavailable computation of the layer-wise mode II fracture toughness (stress intensity factor),

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fracture energy (energy release rate) and shear damage/crack band-width, in a thick modally imperfect cross-ply long cylindrical shell under compression, from a fully nonlinear FEA, using Maxwell’s construction and Griffith’s energy balance approach fills a gap in the composites literature. Additionally, the shear crippling angles in the layers are determined using an analysis, pertaining to the elastic plane-strain inextensional deformation of the compressed (plane strain) ring. Numerical results primarily relate to the effects of hypoelastic (GTT only) material property, on localization and delocalization. What follows is a list of useful conclusions drawn from the present investigation: (i) Material nonlinearity with a relatively high inverse (Ramberg–Osgood) hardening parameter, such as n = 10, in conjunction with modal imperfection, can cause the appearance of localization (maximum pressure) and delocalization (minimum pressure) points on the postbuckling equilibrium path of a thick (Ri/h = 6) CFRP cross-ply long cylindrical shell, which is instrumental to the propagation of mode II fracture at the propagation pressure. (ii) Localization concentrates the energy in a hot spot, such as a region of misaligned fibers at the micron scale, and intracrystallite and intercrystallite disordered regions at the nano-meter scale. The delocalized or extended states are responsible for kink-crack propagation at the propagation pressure. (iii) Localization–delocalization modes working in tandem give rise to the multi-scale and multi-physics phenomena of kink band at the nano/micro scale and shear crippling at the macro-scale. Kink band formation is ubiquitous across all scales, starting from the nano-meter or crystallite level. The active mechanism in simple kinking in a graphite crys
1g planes. tallite is dislocation glide on f1 0 1 (iv) Maximum mode II stress intensity factor and energy release rate of a thick (Ri/h = 6) CFRP cross-ply long cylindrical shell with modal imperfection decrease with the increase of inverse hardening parameter, n. (v) Shear damage/crack band-width slightly decreases with the increase of inverse hardening parameter, n, from n = 5 to n = 10. (vi) The maximum computed mode II stress intensity factor and energy release rate, for the hypoelastic (GTT only) material pffiffiffiffiffi (Ri/h = 6), at propagation pressure, are 16.6 MPa m and 25.5 kJ/m2, respectively, for n = 10. These values are compapffiffiffiffiffi rable to the mode I fracture toughness of Al-alloy (23–45 MPa mÞ and fracture energy (8–30 kJ/m2). (vii) The computed broadening of the kink-crack (damage) half band-width with the kink-crack growth starting from a fiber misalignment defect located at the inner surface of each lamina, is in general agreement with the experimental results of Moran and Shih [45].
k;max ¼ 35:962 (to the loading direc(viii) The mode II shear crippling damage/fracture occurs in a narrow band at angle, b tion), in a thick (Ri/h = 6) cross-ply cylindrical hypoelastic (GTT only) material with low strain hardening parameter (n = 10), which is in general agreement with experimentally observed values in the range of 28–35°. (ix) Finally, the present investigation bridges a two to three orders of magnitude gap between the macro-mechanics and micro-mechanics by taking into account the effects of thickness shear, modal imperfections as wells material and geometric nonlinearities and combining them with the concepts of phase transition via Maxwell construction and Griffith–Irwin fracture mechanics.

Appendix A. Nonlinear finite element formulation A.1. The method of virtual work and incremental equations of motion The second Piola–Kirchhoff stress tensor and the Green–Lagrange strain tensor are conjugate quantities, because their properties are invariant under rigid body motions. The equilibrium of a deformable body at time t + Dt can be expressed using the principle of virtual displacements with tensor notation, using these two conjugate quantities in the total Lagrangian formulation, as follows:

Z

tþDt
0 tþDt 0 Sij d 0 ij dV

e

0V

¼ tþDt R

ðA1Þ

in which tþDt R; tþD0t Sij and tþD0t
eij denote the external virtual work, Cartesian components of the second Piola–Kirchhoff stress tensor and the total Green–Lagrange strain tensor, respectively, defined at time t + Dt, referred to the initial configuration. The latter two quantities can further be expressed as follows: tþDt 0 Sij

¼ t0 Sij þ 0 Sij ;

ðA2Þ

and tþDt
0 ij

e ¼ t0
eij þ 0
eij ;

t 0 Sij

0
ij

e ¼ 0
eij þ 0 g
ij ;

ðA3a; bÞ

where and 0 Sij denote components of the second Piola–Kirchhoff stress tensor defined at time t, and the incremental components of the same during the subsequent time step Dt, respectively, both referred to the initial configuration. The quan
ij in Eq. (A3b) represent the linear and the nonlinear incremental strains, respectively, that are referred to the tities 0
eij and 0 g initial configuration. The linear strain vector f0
eij g is here resolved into two parts, namely the pure linear part, f0
eLij g and the

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linearized part, f0
eNij g. The incremental constitutive relation, which relates the components of incremental stress and incremental strain both referred to the initial configuration, is given by 0 Sij

¼ 0 C ijrs 0
ers ;

ðA4Þ

in which 0 C ijrs is the incremental elastic stiffness (material property) tensor, referred to the initial configuration. Substitution of Eqs. (A2)–(A4) into the left hand side of Eq. (A1) finally yields the equations needed for the finite element formulation. The details are available in Chaudhuri and Kim [42]. The fact that the variation in the strain components is equivalent to the virtual strains permits the right hand side of Eq. (A1) to represent the virtual work done, when the body is subjected to a virtual displacement at time t + Dt. The corresponding virtual work is given by

Z

tþDt

R¼

tþDt s f j d0 usj tþDt ds;

tþDt s

ðA5Þ

s

where the tþDt f j is the surface force vector applied on the surface S at time t + Dt, and d0 usj is the ith component of the incremental virtual displacement vector evaluated on the loaded surface. When the hydrostatic pressure is applied, the loadingpath is always deformation-dependent. This requires that the load vector should be evaluated at the current configuration. The external virtual work can, however, be approximated to sufficient accuracy using the intensity of loading corresponding to time t + Dt, integrated over the surface area, tþDt sði1Þ calculated at the (i  1)th iteration (see Section 5 of Chaudhuri and Kim [32]) as follows:

Z

tþDt

R¼

tþDt s f j d0 usj tþDt ds:

tþDt sði1Þ

ðA6Þ

A.2. Isoparametric finite element discretization and iterative scheme On equating the left hand side of Eq. (A1) to the right hand side of Eq. (A6), followed by discretization using curved 16node isoparametric quadratic (in the surface-parallel coordinates) layer-elements and satisfaction of boundary conditions (see Chaudhuri and Kim [32,42] for details), the incremental equations of motion can be described by Eq. (3), in which ½KL  and ½KN  represent the linear global stiffness matrix, and nonlinear contribution to the global geometric stiffness matrix, respectively, while ff L g and ff N g denote the applied load vector and nonlinear internal force vector, respectively, with f0 Vg being the global nodal displacement vector. These are as given below:

½KL  ¼

NS Z NL X X SðmÞ

m¼1 k¼1

½KN  ¼ 2

Z

ff L g ¼

NS NL X X

SðmÞ

SðmÞ

Z

NL Z X

h

SðmÞ

m¼1 k¼1

hk

h

hk1

Z SðmÞ

h

hk1

ðN Þ BLL S

NS Z NL X X

h

hk

hk

Z

SðmÞ

 h ih i  T h ih ih i ðkÞ ðkÞ ðkÞ ðkÞ BLL T BT ½U Q ðkÞ BLL T BT ½UqðkÞ dz dS;

hk1

Z

m¼1 k¼1

m¼1

ff N g ¼

Z

NS Z NL X X m¼1 k¼1

þ

hk1

NS Z NL X X m¼1 k¼1

þ

hk

ih

hk

hk1

ðkÞ

BLL ðkÞ

BLN

ih

ih

ðA7Þ

i T h ih ih i ðkÞ ðkÞ ðkÞ T BT ½U Q ðkÞ BLN T BT ½UqðkÞ dz dS

i T

h ih i ðkÞ
ðkÞ BðkÞ T ðkÞ ½UqðkÞ dz dS T BT ½U Q LN BT

ih i T  ðkÞ h ih i ðkÞ ðkÞ ðkÞ ðkÞ tb BNN T BT ½UqðkÞ dz dS; BNN T BT ½U 0S

ðN Þ T BTS

h

i

ðkÞ

BLL

(

nðNS þ1Þ 0

) P r qðNS þ1Þ dS;

ðA9aÞ

i T n ðkÞ o ðkÞ t
qðkÞ dz dS; T BT ½U 0S

ðA9bÞ

½U ih

T

ðA8Þ

and

f0 VgT ¼

n

ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ 0 U b1 ; . . . ; 0 U b8 ; 0 V b1 ; . . . ; 0 V b8 ; 0 W b1 ; . . . ; 0 W b8 ; 0 U t1 ; . . . ; 0 U t8 ; 0 V t1 ; . . . ; 0 V t8 ; 0 W t1 ; . . . ; 0 W t8 ;

ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðNÞ 0 U b1 ; . . . ; 0 U b8 ; 0 V b1 ; . . . ; 0 V b8 ; 0 W b1 ; . . . ; 0 W b8 ; 0 U t1 ; . . . ; 0 U t8 ; 0 V t1 ; . . . ; 0 V t8 ; 0 W t1 ; . . . ; 0 W t8 ; 0 U b1 ; . . . ; ðNÞ ðNÞ ðNÞ ðNÞ ðNÞ ðNÞ ðNÞ ðNÞ ðNÞ ðNÞ ðNÞ 0 U b8 ; 0 V b1 ; . . . ; 0 V b8 ; 0 W b1 ; . . . ; 0 W b8 ; 0 U t1 ; . . . ; 0 U t8 ; 0 V t1 ; . . . ; 0 V t8 ; 0 W t1 ; . . . ; 0 W t8

o :

ðA10Þ

Here NL and NS denote the number of elements for each layer and number of layers, respectively. It may be noted that the total number of elements, N equals NLNS.

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R.A. Chaudhuri / Engineering Fracture Mechanics 151 (2016) 138–160

h i h i h i h i ðkÞ ðkÞ ðkÞ ðkÞ The differential operators, BLL ; BLN and BNN , are as presented below. The matrix BLL referred to in Eqs. (A7), (A8) and (A9a, b) is given as shown below:

2

@ @x

0

6 6 0 6 6 h i 6 6 0 ðkÞ BLL ¼ 6 6 0 6 6 6 @ 6 @z 4 1 @

1 ðkÞ gb

0 

1 ðkÞ gb

0 @ @x

@b

ðkÞ

gb

7 7 7 7 7 @ @z 7 1 @ 7 7; ðkÞ g b @b 7 7 7 @ @x 7 5 0

1 @ ðkÞ g b @b

@ @z

3

0

ðA11Þ

h i ðkÞ while the matrix BLN referred to in Eq. (A8) is given as follows:

2

ðkÞ

ðkÞ

gb

gb

3

ðkÞ @ @x

@ @ R11 @x R12 @x 6   ðkÞ 6 R21 @ ðkÞ @ ðkÞ 1 6 R22 @b  R23 ðkÞ ðkÞ 6 gb g b @b 6 6 ðkÞ @ ðkÞ @ R32 @z R31 @z h i 6 6 ðkÞ 6 ðkÞ ðkÞ ðkÞ BLN ¼ 6 R @ R32 @ R33 ðkÞ @ ðkÞ @ 31 þ R21 @z R22 @z þ ðkÞ  ðkÞ 6 gðkÞ @b @b gb gb 6 b 6 ðkÞ @ ðkÞ @ 6 RðkÞ @ þ RðkÞ @ R þ R 6 31 @x 11 @z 32 @x 12 @z 6 ðkÞ ðkÞ ðkÞ 4 R11 @ R12 @ R13 ðkÞ @ ðkÞ @ R22 @x þ ðkÞ @b  ðkÞ ðkÞ @b þ R21 @x

R13

7 7 7 7 gb 7 7 ðkÞ @ 7 R33 @z 7 7; ðkÞ ðkÞ R32 R33 @ ðkÞ @ 7 þ þ R 7 ðkÞ ðkÞ @b 23 @z gb gb 7 7 ðkÞ @ ðkÞ @ R33 @x þ R13 @z 7 7 7 ðkÞ ðkÞ R13 @ R12 5 ðkÞ @ R23 @x þ ðkÞ @b þ ðkÞ   ðkÞ @ ðkÞ R23 @b þ R22 ðkÞ

1

gb

gb

ðA12Þ

gb

in which

n o h i ðkÞ ðkÞ  ðkÞ
Rij ¼ BNN v ;

ðA13Þ

with

n 

ðkÞ

Rij

v
ðkÞ

o

¼ ¼

n

ðkÞ

ðkÞ

t

0u

ðkÞ

R12

R11

ðkÞ

t 0

ðkÞ

R13

v ðkÞ

ðkÞ

R21

t ðkÞ 0w

T

ðkÞ

R22

ðkÞ

R23

R31

ðkÞ

R32

ðkÞ

R33

oT

ðA14Þ

;

ðA15Þ

; h

i  ðkÞ ðkÞ
where the components of the vector v are known displacements at time t. The matrix, BNN , referred to in Eqs. (A8) and (A13), is given as follows:

2 h

@ @x

i 6 6 ðkÞ 0 BNN ¼ 6 6 4 0

0

0

1 @ ðkÞ g b @b

@ @x

0

0

0

@ @x

0 1

ðkÞ gb

0

0 @ @b



1

1

gb

gb

ðkÞ

1

ðkÞ gb

ðkÞ

@ @b

@ @z

0

0

@ @z

0

0

0

3T

7 7 07 : 7 5

ðA16Þ

@ @z

The layerwise linear distribution of displacement matrix, [TBT], referred to in Eqs. (A8) and (A9) can be written as follows Chaudhuri and Kim [42]:

2

h

i

6 ðkÞ T BT ðzÞ ¼ 6 4

1  hz

0

0 0

0

z hk

0

1  hz

0

0

z hk

0

1  hz

0

0

k

k

k

0

3

7 07 5:

ðA17Þ

z hk

The quadratic global interpolation function matrix, [U], referred to in Eqs. (A7), (A8) and (A9) is given by

2

fwg f0g f0g f0g f0g 6 f0g fwg f0g f0g f0g 6 6 6 f0g f0g fwg f0g f0g ½Uðr; sÞ ¼ 6 6 f0g f0g f0g fwg f0g 6 6 4 f0g f0g f0g f0g fwg f0g

f0g

f0g

f0g

f0g

f0g

3

f0g 7 7 7 f0g 7 7; f0g 7 7 7 f0g 5 fwg

ðA18Þ

R.A. Chaudhuri / Engineering Fracture Mechanics 151 (2016) 138–160

157

wherein

fwg ¼ fw1

w2

w3

w4

w5

w6

w8 g

w7

ðA19Þ

and {0} is 1  8 null matrix. wk ðr; sÞ; k ¼ 1; . . . ; 8, are the shape functions as used for displacements and coordinates. h i n o Finally, the stress matrix, t0 ^ Sij , and stress vector, t0
Sij , referred to in Eqs. (A8) and (A9b), respectively, are given as follows:

h

t^ 0 Sij

i

2 t

0S



6 ¼ 4 ½0

½0 t 

0S

½0 with

t

0S



½0

3

7 ½0 5; t 

0S

½0

ðA20Þ

2 t t t 3 0 S11 0 S12 0 S13 6 t t t 7 S S 5; ¼ 4 0 S12 t 0t 22 0t 23

0 S13 0 S23 0 S33

ðA21Þ

and

n

t
0 Sij

oT

¼

t

t 0 S22

0 S11

t 0 S33

t 0 S23

t 0 S13

t 0 S12



ðA22Þ

Appendix B. Constitutive relations for an orthotropic lamina  1 ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ¼ 0:7E44 , where ES44 is the secant The stress, SR44 , referred to in Eq. (8a), is defined as the stress, at which ES44 ¼ R44 shear modulus. The exponent n, known as the inverse hardening parameter, is found from the expression for the secant modulus thus defined: ðkÞ

E44 ðkÞ

ES44

ðkÞ

3 t0 S44 ¼1þ 7 SðkÞ R 44

!n1 ;

ðB1aÞ

with ðkÞ

ðkÞ

SR44 ¼ mE44 :

ðB1bÞ ðkÞ

ðkÞ

Evaluation of Eq. (B1a) at ES44 ¼ 0:85E44 gives

n¼

log10 ð0:441Þ  .  þ 1; ðkÞ ðkÞ log10 S2 SR44

ðkÞ

ðkÞ

ðB2Þ ðkÞ

where S2 is the stress at ES44 ¼ 0:85E44 . It may be noted that the linear elastic and the perfectly elastic–plastic cases can be ðkÞ

obtained by substituting SR44 ¼ 1 and n ¼ 1, respectively, into Eq. (8a). Appendix C. Definition of certain constants Chen et al. [69] have analyzed the problem of a pressurized thick cylindrically anisotropic infinitely long tube. For a thick cross-ply [90/0/90] plane strain ring (very long cylindrical shell), the constants, Ak and Bk, k = 1, 2, 3, referred to in Eq. (13), are given as follows:

A1 ¼

pP ; ða11 e1 þ a12 e2 Þ

B1 ¼ 

a1 A1 ; a2

A2 ¼ c 1 A1 ;

B2 ¼ c2 A1 ;

A3 ¼ e 1 A1 ;

B3 ¼ e2 A1 ;

in which

a1 ¼

ðg1 þ1Þ

Ri

;

ðg1 1Þ

a2 ¼ ð1 þ g1 ÞRi

;

a3 ¼

ð1  g1 Þ ðRi þ t 1 Þðg1 1Þ

;

a4 ¼ ð1 þ g1 ÞðRi þ t1 Þðg1 1Þ ;

1 1 ð1  g3 Þ ; a6 ¼ a8 ¼ 2; a7 ¼ ; a9 ¼ ; ðRi þ t 1 Þ2 ðRi þ 2t 1 Þ2 ðRi þ 2t1 Þðg3 þ1Þ ð1  g3 Þ ¼ ; a12 ¼ ð1 þ g3 ÞðRi þ 3t 1 Þðg3 1Þ ; ðRi þ 3t1 Þðg3 þ1Þ

a5 ¼ a11

ð1  g1 Þ

a10 ¼ ð1 þ g3 ÞðRi þ 2t 1 Þðg3 1Þ ;

ðC1Þ

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R.A. Chaudhuri / Engineering Fracture Mechanics 151 (2016) 138–160

a13

  0ð1Þ 0ð1Þ ð1  g1 Þ s11  s12 g1 ¼ ; g1 ðRi þ t1 Þg1

a15 ¼ 

a17 ¼ 

a19

  0ð2Þ 0ð2Þ s12  s22 ðRi þ t1 Þ   0ð2Þ 0ð2Þ s12  s22 ðRi þ 2t 1 Þ

c1 ¼

a1 a4 ; a2

g1

  0ð2Þ 0ð2Þ a16 ¼ 2 s12 þ s22 ðRi þ t 1 Þ;

;

  0ð2Þ 0ð2Þ a18 ¼ 2 s11 þ s12 ðRi þ 2t1 Þ;

c2 ¼

a20 ¼

 ð1 þ g3 Þ  0ð3Þ 0ð3Þ s11 þ s12 g3 ðRi þ 2t1 Þg3 ;

g3

ðC3Þ

ðb2 a5  b1 a15 Þ ; ða5 a16  a6 a15 Þ

ðC4Þ

d2 ¼ a17 c1 þ a18 c2 ;

e1 ¼

ðd1 a20  d2 a10 Þ ; ða9 a20  a10 a19 Þ

t1 ¼

h : 3

ðC2Þ

a1 a14 ; a2

b2 ¼ a13 

ðb1 a16  b2 a6 Þ ; ða5 a16  a6 a15 Þ

d1 ¼ a7 c1 þ a8 c2 ;

 ð1 þ g1 Þ  0ð1Þ 0ð1Þ s11 þ s12 g1 ðRi þ t 1 Þg1 ;

;

  0ð3Þ 0ð3Þ ð1  g3 Þ s11  s12 g3 ¼ ; g3 ðRi þ 2t 1 Þg3

b1 ¼ a3 

a14 ¼

e2 ¼

ðd2 a9  d1 a19 Þ ; ða9 a20  a10 a19 Þ

ðC5Þ ðC6Þ

ðC7Þ

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