Optimisation of sandwich panels with functionally graded core and faces

Composites Science and Technology 69 (2009) 575–585

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Optimisation of sandwich panels with functionally graded core and faces U. Icardi, L. Ferrero * Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino – Corso Duca degli Abruzzi 24, 10129 Torino, Italy

a r t i c l e

i n f o

Article history: Received 23 September 2008 Received in revised form 24 November 2008 Accepted 30 November 2008 Available online 11 December 2008 Keywords: A. Layered structures A. Sandwich C. Stress relaxation C. Computational mechanics C. Functionally graded materials

a b s t r a c t The distributions of properties across the thickness (core) and in the plane (face sheets) that minimise the interlaminar stresses at the interface with the core are determined solving the Euler–Lagrange equations of an optimisation problem in which the membrane and transverse shear energy contributions are made stationary. The bending stiffness is maximised, while the energy due to interlaminar stresses is minimised. As structural model, a refined zig-zag model with a high-order variation of displacements is employed. Simplified, sub-optimal distributions obtainable with current manufacturing processes appear effective for reducing the critical interfacial stress concentration, as shown by the numerical applications. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Sandwich structures are broadly used because they offer a high bending stiffness with the minimum mass, but also by virtue of their capability to be tailored in order to meet design requirements, their high damping properties and the great potential for impact protection, containment of explosions and projection of fragments [1] they offer. Unfortunately they suffer from strong stress concentrations at the interfaces among the face sheets, the weak adhesive layer and the core, as a consequence of the distinctly different properties of these materials in contact. It represents a critical design concern since it can have detrimental effects on structural performance and service life and can cause a premature failure at load levels much lower than the ultimate load. A number of sophisticate models has been recently published in order to accurately and efficiently predict these potentially deleterious effects, whose discussion is outside the purpose of this paper (see, e.g., the comprehensive review paper by Noor et al. [2] for details). It is just remarked that, in order to accurately predict the stress fields, the sandwich structural models have to account for the core deformability under compression and shear, as well as for the through-the-thickness shear and normal stresses continuity at the face–core interfaces necessary for keeping equilibrium. The recent fast growing interest to new technologies has provided an excellent opportunity for refining these models and, in general, the studies on sandwich composites. Some high-order models for

* Corresponding author. Tel.: +39 (0) 11 564 6872; fax: +39 (0) 11 564 6899. E-mail addresses: [email protected] (U. Icardi), [email protected] (L. Ferrero). 0266-3538/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2008.11.036

the analysis of sandwich composites can be found in Li et al. [3], Li and Kardomateas [4], Frostig [5] and Schwarts-Givli et al. [6]. New perspectives of solution for the stress concentration problem are represented by nano-technologies (see, e.g., Mahfuz et al. [7]) and functionally graded materials – FGM – (see, e.g., Suresh and Mortensen [8] and Fuchiyama and Noda [9]), because they enable either development of new high performance materials with improved damage tolerance, or a smooth property variation that can optimise some structural functions. Moreover, Aliaga and Reddy [10] proposed a study on FGM by a third-order plate theory. Although these new technologies are in their infancy, their potential advantages appear great. In the immediate future, the best chance for fully exploiting the lightweighting potential of sandwich composites appears to be the tailoring optimisation of the face sheets combined with a functionally graded core. To minimise the change in stiffness and then the interlaminar stress concentration at the interface, the core should have smoothly property variations across the thickness, that at this position are similar to those of the faces (see, e.g., Apetre et al. [11]). While the FGM do not complicate so much the simulation, or even make it much easier by virtue of their smooth property variation, they obviously represent manufacturing complications. The methods till now proposed have been developed for thermal coatings and thermal protection systems (see, e.g., [12–16]), but it is a common opinion that the technology is mature for the production of core foams with a desired variation of properties. Based on this consideration, recently, a series of papers has been presented by Sankar and Tzeng [17], Sankar, [18], Venkataraman and Sankar [19] and Apetre et al. [20] about FG laminated and sandwich beams with FG cores. The materials were assumed to be isotropic and with an exponential variation of the elastic

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stiffness coefficients across the thickness, in order to allow exact elasticity solution via Fourier transform methods. Polynomial distributions were considered by Apetre et al. [11,21] and Zhu and Sankar [22], who used the Galerkin method. These studies focusing on the capability of FGM to reduce the stress concentrations encourage both the development of new methods for the production of foam core materials with FG properties and further theoretical developments in order to understand which property distribution achieves the wanted properties. In the present paper, a numerical study of the potential advantages of combining tailoring optimisation and FGM is presented. A new optimisation technique recently developed by the authors [23] is employed for finding the orientation of the reinforcement fibres of the face sheet layers and the law of variation of the core properties across the thickness, able to minimise the interlaminar stress concentration. A refined zig-zag model is employed as structural model, in order to accurately and efficiently account for the interfacial stress continuity conditions. To make the structural model more efficient, a strain energy updating procedure is employed, like in Ref. [24]. The basic idea is to carry out the analysis with a classical C° parent finite element model based on the FirstOrder Shear Deformation Plate Theory-FSDPT and to update its energy to that of the zig-zag model in the post-processing phase, where the stresses are computed. 2. Structural model 2.1. Kinematics Assume the sandwich panel to consist of S layers with different thickness and material properties, the core being treated as a thick layer in a multilayer construction (the accuracy of this hypothesis will be assessed in Section 4). In order to accurately capture the stresses fields across the thickness, the whole set of displacement and stress contact conditions has to be fulfilled at the interfaces of the constituent layers. To this purpose, the displacement field is represented as

Uðx; y; zÞ ¼ uðx; y; zÞ þ Uðx; y; zÞ

ð1aÞ

Vðx; y; zÞ ¼ v ðx; y; zÞ þ Vðx; y; zÞ

ð1bÞ

Wðx; y; zÞ ¼ fw1 ðx; y; zÞ þ fw2 ðx; y; zÞ

ð1cÞ

according to Ref. [23]. Like for equivalent single-layer models, u and v give contributions to in-plane displacements that are continuous together with their first derivatives across the thickness

uðx; y; zÞ ¼ u

ð0Þ

ð0Þ x

þ zðc



wð0Þ ;x Þ

þ z Cx ðx; yÞ þ z Dx ðx; yÞ 2

3

ð0Þ 2 3 v ðx; y; zÞ ¼ v ð0Þ þ zðcð0Þ y  w;y Þ þ z C y ðx; yÞ þ z Dy ðx; yÞ

ð2aÞ ð2bÞ

while

Uðx; y; zÞ ¼

S1 X

ðkÞ

ðkÞ

/x ðx; yÞðz Z þ ÞHk

ð3aÞ

ðkÞ

/y ðx; yÞðz  ðkÞ Z þ ÞHk

ð3bÞ

k¼1

Vðx; y; zÞ ¼

S1 X k¼1

are continuous, but have discontinuous first derivatives at the interfaces. The two contributions to the transverse displacement have the following explicit expressions

fw1 ðx;y;zÞ ¼aðx; yÞ

ð4aÞ

a being assumed as the transverse displacement on the reference mid-plane w(o). Also in this case, a field with discontinuous derivatives is superposed to a polynomial representation. The continuity functions ðkÞ Ux , ðkÞ Uy ,ðkÞ wx and ðkÞ wy are determined enforcing the continuity of the transverse shear and normal stresses rxz, ryz, rzz and of the gradient rzz,z at the interfaces, as prescribed by the elasticity theory. The functionsC x; , C y , Dx , Dy , b, c, d, e are determined enforcing the fulfilment of the boundary conditions at the upper and lowers faces. Note that the functional degrees of freedom u(o), v(o), w(o), cx and cy coincide with those of conventional equivalent single-layer plate models. This will enable the possibility to update the strain energy of these models to that of the present model, with the purpose and the technique described hereafter. The zig-zag model is used in the optimisation process of Section 3, while the analysis of the optimised solutions is carried out as outlined in the following section. 2.2. Energy updating Unfortunately, a direct finite element formulation by the zig-zag model of Eqs. (1)–(4) involves second order derivatives of the nodal d.o.f. that make the element inefficient. Since, a finite element approach is required for treating the spatial variation of the mechanical properties set by the optimisation process, the problem is here overcame updating the strain energy of a C° parent finite element model based on the FSDPT, through the technique of Ref. [24]. This element is used for a preliminary analysis, then its energy is updated in the post-processing phase to that of the zig-zag model. As a result of this process, corrective terms for the d.o.f. of the FSDPT are computed and used for the stresses analysis. A broad summary of this technique is reported hereafter. The first operation is the interpolation and smoothening of the results by the FSDPT finite element analysis at discrete points, in order to compute the derivatives from the interpolation instead from the representation (with a lower order) by the shape functions. For this operation, spline functions are used. In this phase, the kinematics of the FSDPT model is made consistent with that ð0Þ ð0Þ ð0Þ ð0Þ of the zig-zag model, i.e. hx ¼ zðcx  w;x Þ, hy ¼ zðcy  w;y Þ. As(o) (o) (o) sume u , v , w , cx and cy as the functional d.o.f. of the FSDPT model, while the homologous terms of the zig-zag model be indicated by the superscript (). All the updating operations be carried out using corrective terms, i.e.

~ ð0Þ ¼ u ^ ð0Þ ^ ð0Þ þ Du u

ð5aÞ

v~ ð0Þ ¼ v^ ð0Þ þ Dv^ ð0Þ

ð5bÞ

^ ð0Þ þ Dw ~ ð0Þ ¼ w ^ ð0Þ w

ð5cÞ

~ð0Þ x ~ð0Þ y

ð5dÞ

^ð0Þ x ^ð0Þ y

D^ð0Þ x D^ð0Þ y

c ¼c þ c c ¼c þ c

ð5eÞ

It is remarked that u(o), v(o), w(o), cx and cy are computed by the FSDPT finite element model, while the corrective terms are unknown quantities to compute equating the energy contributions of interest by the penalty function method. The continuity functions ðkÞ Ux , ðkÞ Uy are preliminarily computed in an approximate way assuming w = w(o), i.e. fw2 ¼ 0, then they are made consistent with the present zig-zag model making the transverse shear strains exz,eyz least square compatible with their consistent counterparts. This is numerically more efficient than calculating them directly from the contact condition. The updating of the transverse shear energy is performed equating these two homologous quantities for the zig-zag and FSDPT models

fw2 ðx;y;zÞ ¼zbðx;yÞ þ z2 cðx;yÞ þ z3 dðx; yÞ þ z4 eðx;yÞ þ

S1 X

ðkÞ

k¼1

wx ðx;yÞðzðkÞ Z þ ÞHk þ

S1 X

ðkÞ

wy ðx;yÞðzðkÞ Z þ Þ2 H:k

k¼1

ð4bÞ

ðqe þ DqeK ÞT K fsdpt ðqe þ DqeK Þ ¼ qTe K zig-zag qe

ð6Þ

where qe represents the vector of nodal d.o.f., DqeK the corrective terms and K is the stiffness matrix (only the rows and columns rel-

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ative to the out-of-plane shears) of the two models. The left hand size member is based on the FEM model, the right hand side is an analytic expression representing the strain energy of the zig-zag model corresponding to the interpolation of the displacements qe . The strain energy due to transverse normal stress cannot be computed in a similar way, because the FSDPT model disregards this con tribution. A preliminary approximate expression for rzz is obtained by integrating the third differential equilibrium equation, while an  approximate expression of the transverse normal strain ezz is obtained by the stress–strain relations. An improved transverse normal   stress  rzz is computed substituting the last expression of ezz into  the stress–strain relations of linear elasticity. This stress rzz is then used to compute improved transverse shear stresses rxz, ryz (one at a time) by the third differential equilibrium equation. Once the new stresses rxz, ryz, rzz are computed, they are interpolated and used for improving the membrane energy with a procedure similar to Eq. (6). The entire process is repeated till convergence. This technique, that was successfully applied in Ref. [24], is effective because a limited number of iterations is required, the operations are carried out locally only and, furthermore, the zigzag and FSDPT models have a number of functional d.o.f. independent from the number of constituent layers. It achieves an equivalent accuracy level to that of the best models available to date (we mean all the models postulating separate stress and displacement fields for the constituent layers and impose interfacial constraint conditions) but with a lower computational effort. 3. Material properties optimisation

3.1. Stationary conditions for the strain energy contributions Since, the model of Eqs.(1)–(4) requires a too high mathematical effort, in the optimisation process the following third-order zig-zag model with a constant transverse displacement is used [25]. ð0Þ 2 3 Uðx; y; zÞ ¼ uð0Þ þ zðcð0Þ x  w;x Þ þ z C x ðx; yÞ þ z Dx ðx; yÞ

þ

ðkÞ

/x ðx; yÞðzðkÞ Z þ ÞH

ð7aÞ

ð0Þ 2 3 Vðx; y; zÞ ¼ v ð0Þ þ zðcð0Þ y  w;y Þ þ z C y ðx; yÞ þ z Dy ðx; yÞ S1 X

ðkÞ

/y ðx; yÞðzðkÞ Z þ ÞH

ð7bÞ

k¼1 ð0Þ

Wðx; y; zÞ ¼ w

The optimal stiffness property distribution in (x, y) is obtained solving the Euler–Lagrange equations resulting from the extremisation of the strain energy under variation of these properties. As examples, the stationary conditions for the bending energy

WR1 duð0Þ  WR2 dv ð0Þ  WR3 dwð0Þ   4 þ WR4  WR5 þ WR6 þ WR7 dcð0Þ x 3   4 þ WR8  WR9 þ WR10 þ WR11 dcð0Þ y 3

ð8Þ

 4 XRR1 þ XRaR1 þ XRdR1 þ XRR44 þ XRaR44 þ XRdS44  2 ðXRP1  XRP6 Þ 3h 1  ðXR26X2 þ XR31X2 þ XR36X2 þ XR41X2 Þ 2h  2  2 ðXR26X3 þ XR31X3 þ XR36X3 þ XR41X3 Þ duð0Þ 3h n 4 þ XRR2 þ XRaR2 þ XRdR2 þ XRR55 þ XRaR55 þXRdS55  2 ðXRP2  XRP7 Þ 3h 1  ðXR27X2 þ XR32X2 þ XR37X2 þ XR41X2 Þ 2h  2  2 ðXR27X3 þ XR32X3 þ XR37X3 þ XR42X3 Þ dv ð0Þ 3h  4 þ XRR3 þ XRaR3 þ XRdR3 þ XRR66 þ XRaR66 þ XRdS66  2 ðXRP3  XRP8 Þ 3h 1  ðXR28X2 þ XR33X2 þ XR38X2 þ XR43X2 Þ 2h  2  2 ðXR28X3 þ XR33X3 þ XR38X3 þ XR43X3 Þ dwð0Þ 3h n þ XRR4 þ XRaR4 þ XRdR4 þ XRR88 þ XRaR88 þ XRdS88  XRT88  XRaT88  XRdT88 

k¼1

þ

3.2. Face sheets

and for the transverse shear energy in the plane (x, z)

Suitably choosing the spatial property variation, certain strain energy contributions can be made stationary, as shown in Ref. [23], and used for ‘‘tuning” the energy absorption properties. In this way it is possible to minimise the energy absorbed by unwanted modes (e.g., that due to transverse shears) and maximise that involving acceptable modes (e.g., as membrane energy). In the preliminary applications of this technique to laminates [23], the magnitude of the interlaminar stresses was consistently reduced keeping unchanged, or even reducing deflections, by incorporating few optimised layers with spatially variable stiffness properties, i.e. the strength at the onset of delamination was improved without any stiffness loss1 In the present paper, this technique is used for finding the fibre orientation over the faces and the distribution of the core properties across the thickness that minimise the interlaminar stress concentration at the core interface.

S1 X

The contributions of the transverse deformation neglected by this model will be accounted for by updating its strain energy. In this case, a relation among the functional d.o.f. of the two zig-zag models is employed, that results by the comparison of their strain energy expressions. First, the strain energy stationary conditions for the model of Eq. (7) under variation of u(o), v(o), w(o), cx and cy (expression in Ref. [25]) are written, representing the equilibrium conditions to be fulfilled. Then, the stiffness coefficients are assumed to be functions of position and the stationary conditions for the membrane, bending, in-plane and out-of-plane energy contributions under spatial variation of these properties are written. A variation in (x, y) is assumed for the face sheets, while the properties of the core are assumed to vary in z.

2

ðXRP4  XRP9 Þ þ

þ XR39X2 þ XR44X2 Þ 

2

XRRR1 

1 ðXR29X2 þ XR34X2 2h

2

þ XRR5 þ XRaR5 þ XRdR5 þ XRR99 þ XRaR99  XRdS99  XRT99  XRaT99 4 3h

2

ðXRP5 þ XRP10 Þ þ

þXR40X2 þ XR45X2 Þ 1 Viscolelastic layers could be used for the same purpose, but they knock dawn the stiffness and in the case of sandwich panels they are ineffective, due to the thinness of the face sheets.

4 h

ðXR29X3  XR34X3 þXR39X3 þ XR44X3 Þ 2 3h  1 2 þ ðXR46X1 þ XR48X1 Þ þ 2 ðXR46X2 þ XR48X2 Þ dc0x h h n

XRdT99  ð7cÞ

4 3h

4 h

2

XRRR2 

1 ðXR30X2 þ XR35X2 2h

2

ðXR30X3 þ XR35X3 þ XR40X3 þ XR45X3 Þ  1 2 þ ðXR47X1 þ XR49X1 Þ þ 2 ðXR47X2 þ XR49X2 Þ dc0y h h 2

3h

ð9Þ

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U. Icardi, L. Ferrero / Composites Science and Technology 69 (2009) 575–585

under variation of the stiffness properties in (x, y) are reported. All the terms appearing in the former equations contains various orders partial derivatives of the stiffness quantities, as shown, for example, by the following term involving the bending stiffness components

WR3 ¼ D11;1111 þ 2D12;1122 þ 4D16;1112 þ 4D26;1222 þ 4D66;1122 þ D22;2222 ð10Þ An approximate solution of technical interest to the intricate system of coupled, partial differential equations that result from the energy stationary conditions are a second order polynomial approximation for the stiffness coefficients

Q 11 ¼ A1 þ A2 x þ A3 x2

ð11aÞ

Q 22 ¼ B1 þ B2 y þ B3 y2

ð11bÞ

Q 12 ¼ C 1 þ C 2 x þ C 3 y þ C 4 x2 þ C 5 y2 þ C 6 xy 2

ð11cÞ

2

Q 66 ¼ D1 þ D2 x þ D3 y þ D4 x þ D5 y þ D6 xy

ð11dÞ

Q 16 ¼ E1 þ E2 x þ E3 x2 þ E4 xy

ð11eÞ

2

Q 26 ¼ F 1 þ F 2 y þ F 3 y þ F 4 xy Q 44 ¼ G

ð11fÞ ð11gÞ

Q 55 ¼ L

ð11hÞ

Q 45 ¼ M

ð11iÞ

that is here referred to as the optimised ply. The coefficients appearing in the former expressions are determined enforcing the thermodynamic constraints (Lempriere, Lekhnitski and Chentsov’s relations, conservation of energy) and the mean vaule of the Qij in order to be physically consistent and to have the mean stiffness of the optimised ply coincident with that of the replaced ply. The enforcement of these constraints produces quite complex relations that preclude a closed form solution, as shown by the following expression of Q12, reported as an example

Q 12

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  C1Þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA1 þ A2 x þ A3 x2 ÞðB1 þ B2 yþ3 y2 Þ ¼ 1 þ C1 C

Q 11 ðz2 Gy;x  z2 W y;xx þ zU y;x Þ þ Q 12 ðzV  z2 W ;y þ z2 HÞþ

þ Q 16 ð2zU þ 2z2 G  3z2 W ;x  zV y;x þ z2 Hy;x Þ þ Q 26 ðzV x;y  z2 W x;yy ð13aÞ

where, as example

Z Z Z

u0;x dx ¼ U;

cy;x dx ¼ H; Gdy ¼ Gy ;

Z Z Z

v 0;x dx ¼ V; Udy ¼ U y ; Hdy ¼ Hy

Z Z

w0;x dx ¼ W; Vdy ¼ V y ;

Z Z

ð14Þ

In the case of the membrane energy contributions, F is expressed as (A + Bz), A and B being integrals in (x, y) of the spatial derivatives of displacements, as showed in (13b). In the case of the bending and transverse shear energy contributions, of interest for this paper, the extremal conditions are reached if F is expressed as

F ¼ ðAz2 þ BzÞ

ð15Þ

The former equation holds for Q11, Q12, Q22, Q16, Q26 and Q66, while in the case of Q44, Q45 and Q55 it as to be F=1. Therefore, in order to obtain the transverse shear energy and the bending energy of the core extremal, the stiffness properties Q11 to Q66 have to vary across the thickness according to ðAz2 þ BzÞ1 , while Q44, Q45 and Q55 have to be constant. If we assume the foam constituting the core as an isotropic material with properties E, G and t and represent the stiffness coefficients as Q 11 ¼ K 1 =ðz2 þ C 1 zÞ and Q 12 ¼ K 2 =ðz2 þ C 2 zÞ, once enforced the restrictions on the material constants due to thermodynamic and isotropy conditions, we find the following constraint relations for Ki and Ci to which symmetry consideration could be added

2K 1 < K 2 < K 1

ð16Þ

From the previous expression of Q11 and Q12, all the elastic coefficient are defined, by imposition of the same mean value of a constant case, considered as baseline and by considerations on the amount of realistic variation. In the present paper, the properties of the core are assumed to vary according to the former relations considering as upper and lower limits of the material data variation the minimum and maximum values of the ROHACELL foam. The xy-variable stiffness coefficient for the optimised faces (see Ref. [23]) are reported in Fig. 1a. In Fig. 1b and c the z-variable coefficient for optimised cores are depicted in two cases, as used in the numerical applications of Section 4. 4. Numerical applications

As an example of the operations involved, the strain energy contribution of the zig-zag model of Eq.(7a)–(7c) due to the transverse shear in the plane (x, z) is reported

þ z2 Hx;y ÞþQ 66 ðzV þ zU x;y  2z2 W ;y þ z2 H þ z2 Gx;y Þ  þQ 44 Gxy þ Q 45 Hxy dz

½Q ij ðzÞF dz ¼ Constant

ð12Þ

3.3. Core

Z h

Z

cx;x dx ¼ G;

On the contrary of Ref. [23], where the zig-zag model without updating was used for finding the optimised solutions (see. Eq. (7), while mixed solid elements were used for the analysis of the response, here all the computations will be carried out by the model of Section 2.2. The reason for this choice is that the improvement of accuracy brought by mixed solid elements is marginal, while the computational effort is larger, as appeared in a preparatory analysis. To justify the use of the zig-zag model, an assessment of its accuracy is premised to the analysis of the optimised solutions. To this purpose, the interlaminar stress fields of undamaged and damaged sandwich panels are compared to the exact elasticity solution [26]. To be self-contained, no results are reported for the transverse normal stress and for membrane stresses, but the model is accurate also for them. 4.1. Model accuracy assessment

Wdy ¼ W y ; ð13bÞ

Assuming a variation of the Qij across the thickness and considering the constraint relation that represents the effects of the energy updating, i.e. the relation among the functional d.o.f. obtained equating the strain energy of the model (1) to that of the model (7), the conditions that make extremal the strain energy of the core are obtained. These equations state that the extremal conditions under variation of the properties in z are reached when

The sample case considered is a sandwich plate with laminated faces and honeycomb core, simply-supported at the edges and cylindrically bent by a sinusoidal heap loading. The material properties for this case are reported in Table 1. The face sheets plies are made of materials 1–3, while the core is made of material 4. The sandwich panel is viewed as a multilayered composite with the following stacking sequence ([1]/[2]/[3]/[1]/[3]/[4])s. Material [1] is rather compliant in tension, compression and shear, while material [2] is stiff; material [3] is stiff in tension and compression, but rather compliant in shear. Material [4] is very compliant in tension

U. Icardi, L. Ferrero / Composites Science and Technology 69 (2009) 575–585

579

Fig. 1. Spatial variation of the stiffness properties of facesheet layers (a) and core (with maximum stiffness in the middle of core thickness (b) and with maximum stiffness at the ends of core thickness (c).

and compression and rather compliant in shear, as usual for the core. The ratio of the thickness of constituent plies with respect

the overall thickness is (0.01/0.025/0.015/0.02/0.03/0.4)S. Although it does not represent a realistic value, a length-to-thickness ratio of

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U. Icardi, L. Ferrero / Composites Science and Technology 69 (2009) 575–585

Table 1 Materials used for the model accuracy assessment and for the optimisation. Material

E11 (Gpa)

E22 (Gpa)

Sandwich beam for model accuracy assessment [1] 1 1 [2] 33 33 [3] 25 25 [4] 0.05 0.05 E1 E2 E3 (Gpa) Sandwich beam for optimisation Face sheets 142/9.8/9.8 Core 0.014

G13 (Gpa) 0.2 8 0.5 0.0217

m13 0.25 0.25 0.25 0.15

G12 G13 G23 (Gpa)

m12 m13 m23

q (Kg/m3)

7.1/7.1/3.3 0.005

0.3/0.3/0.25 0.3

1558.35 16.3136

4 is considered because it constitutes a very severe test for the present structural model. According to the ply discount theory, the damage is simulated reducing the elastic moduli of the damaged materials by the factor 102. In the event of face sheets failure the moduli E11, E22, G12, and m12 are reduced, while in the event of core failure they are G13 and G23. Fig. 2 reports the transverse shears across the thickness for  this case; they are normalised as rxz  ¼ rxz =P , P° being the magnitude of the sinusoidal loading, z/h the non-dimensional thickness coordinate and h the overall thickness. It appears that the structural model based on strain energy updating can accurately capture the stress with the same accuracy of a solid element, or a discrete-layer element, also in presence of damage, but with a low computational effort. In this case the number of iterations of the strain energy updating process has been limited to a maximum of 10; with this limitation, the present structural model required less than 20 s on a PC equipped with a 1800 Hz processor using a 30 FSDPT element subdivision in the spanwise direction. 4.2. Stress fields of optimised configurations Numerical applications to single and dual core, pinned sandwich beams subjected to a centred point loading are presented with the purpose to assess the advantages offered by the optimised

tailoring of the faces and by the optimal core property distribution of Section 3 in terms of interlaminar stress fields. This loading was chosen because it can give useful information about the stress fields induced by the impact of foreign objects [24] that represent the most common event generating damage in service, although the contact conditions are not accounted for and the computations are static, as in the present case. 4.2.1. Single core sandwich Consider a sandwich beam, pinned at both ends, 50 cm long and with an overall thickness of 2 cm, whose material properties are reported in Table 1. The constituent materials be stacked with the two sequences represented in Fig. 3, the symmetric stack-up ONE CORE and the asymmetric ONE CORE2. Suppose the face sheets to be made of six and four layers, respectively, with equal thickness and assume each to be 0.217 mm thick. For each case several results will be compare: the solution for a classical sandwich beam without optimised layers; the solution ot having a variable stiffness in the core, with maximum in the thickness middle point, according to Fig. 1b; the solution ot2 having a variable stiffness in the core, with maximum at the thickness endpoints, according to Fig. 1c. Moreover, for the first case ONE CORE, two additional comparisons have been proposed, one with the same lay-up having optimised faces and classical core, as depicted in Fig. 3c, called OTs2_45deg, and one having both optimised faces and optimised core, called sand_opt_all. Fig. 4 presents the transverse displacement of the beam at the loading point (averaged across the thickness) for: the reference configuration with constant stiffness properties sand_core_cl; a configuration named OTs2_45deg where the two 45 and 45 layers are substituted with minimum shear layers, as in Fig. 1c; the two cases when the core has variable properties as in Fig. 1a and b, sand_core_ot and sand_core_ot2, respectively; the case sand_opt_all with both optimised faces and core, according to the scheme of OTs2_45deg and sand_core_ot for the faces and the core, respectively. Fig. 5 provides the interlaminar shear stresses for the same configurations, with the effects of damage considered as outlined in Section 4.1. It appears by these results that the optimised property distributions of Fig. 1 are able to reduce the deleterious inter-

Fig. 2. Normalized transverse shear stress across the thickness for a sandwich beam with damage in the core, in the upper face and without damage.

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Fig. 3. Lay-up of the sandwich beam with functionally graded core and faces.

Fig. 4. Deflection of the beam with various configurations, sample case ONE_CORE.

laminar shear stress concentration at the interface with the core, increasing the amount of strain energy stored by in-plane modes and decreasing that stored by out-of-plane modes. In effects, the in-plane stresses, not reported in the paper not being critical, increase by 25% at the interfaces of the core since an amount of energy has been transferred in this way. The amount of the strain energy stored as bending energy is reduced for all the solutions represented in Fig. 1, i.e. the stiffness is increased, while the most critical stress concentration at the interface with the core is reduced. This result, that cannot be achieved using materials with

constant properties, justifies further researches in this field and on manufacturing of materials with the stiffness variations considered in this paper. Consider now the case represented in Fig. 2b with asymmetric faces. Dimensions of the beam and constituent materials are the same of the previous case. The transverse displacement for this case is represented in Fig. 6, the transverse shear and normal stress in Fig. 7. The observations about the effectiveness of variable stiffness properties still hold for this case.

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Fig. 5. Transverse shear stress fields of the beam with various configurations.

Fig. 6. Deflection of the beam with various configurations, sample case ONE_CORE2.

4.2.2. Dual core sandwich Consider now the case of the dual core sandwich represented in Fig. 3d. Dimensions of the beam and constituent materials are

the same of the previous case. Dual core panels are of interest because they can bear the stresses due to loading also when failed because the intermediate cores and faces inhibit the dele-

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Fig. 7. Transverse shear stress fields of the beam with asymmetric faces.

Fig. 8. Dual core sandwich beam: lay-up and deflections.

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Fig. 9. Stress fields across the dual core sandwich beam.

terious spreading of failure and damage. Although these intermediate components do not contribute to the bending stiffness, they do not consistently increase the weight, because the single-core need to be over-sized in order to tolerate the damage. Fig. 3d gives the lay-up for this case, while Fig. 8 gives deflections. The stress fields for this sample case are reported in Fig. 9. Two different configuration are possible, for both optimisation scheme ot and ot2, since the optimised distribution on the core can be applied on the two cores, as they were one on it can be applied on the two cores separately; in the first case, the hyperbole used for stiffness is divided in the upper and lower core (case A); in the second one, the hyperbole is applied once on the upper core, once on the lower one (case B). Both configuration have been analyzed for the distribution with maximum stiffness in the middle (Fig. 1a) and for the one with maximum at the endpoints (Fig. 1b), because they lead to meaningful results; however, the stress field are quite similar, so just the case B will be presented hereafter. It appears by these numerical results that also in the case of the dual core beam, the stiffness property distributions considered in this paper are effective in

order to reduce the deleterious transverse shear stress concentrations at the interfaces, without a remarkable loss of stiffness. Besides, the former results focus on the benefits of multi-core sandwich structures. 5. Concluding remarks A technique for tuning the energy absorbed by sandwich composites with laminated faces in the bending, in-plane, and out-ofplane shear modes has been presented, aimed to reduce the detrimental stress concentrations at the interfaces. The stiffness properties are varied in the plane for facesheets and across the thickness for core according to an extremization process of the energy contribution of interest. Therefore, the resulting Functionally Graded sandwich shows an energy transfer from bending to in-plane and out-of-plane shear energy modes, or vice versa. For the analysis, a shell element based on strain energy updating as been extensively used, since it allows higher accuracy with respect to classical 2D elements at a low computational effort, unlike solid elements. These preliminary results encourage future developments of the present technique.

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