Thermal buckling response of laminated and sandwich plates using refined 2-D models

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Thermal buckling response of laminated and sandwich plates using refined 2-D models

R. Vescovini, M. d’Ottavio, L. Dozio, O. Polit

To cite this version:

R. Vescovini, M. d’Ottavio, L. Dozio, O. Polit. Thermal buckling response of laminated and sandwich plates using refined 2-D models. Composite Structures, Elsevier, 2017, 176, pp.313-328.

�10.1016/j.compstruct.2017.05.021�. �hal-01981940�

Thermal buckling response of laminated and sandwich plates using refined 2-D models

R. Vescovini,M.D’Ottavio,L.Dozio,O. Polit

1. Introduction

Thermal loads play an important role in several fields of engi- neering. Aeronautical and space applications are a well-known example, where the structures commonly experience severe aero- dynamic heating and drastic changes of thermal environments. For instance, the launch and the re-entry of space vehicles is typically associated with severe temperature variations. Similarly, thermal loads are developed in supersonic and hypersonic aircrafts as the result of restrained thermal expansion[1].

These aspects, in conjunction with the typical thin-walled architecture of light-weight structures, have made thermal buckling the subject of several investigations in the past. Early works are based on classical lamination theory (CLT) and first order shear deformation theory (FSDT). Reviews of analytical solutions using classical CLT and FSDT are available on several textbooks [2,3]. A comprehensive overview of the topic is proposed by Jones [4]with regard to fiber-reinforced composite plates, where CLT is applied and the equivalent mechanical loading concept is

discussed. Within the context of CLT, Nemeth[5]investigated the thermal buckling response of infinitely long plates, introducing stiffness-weighted thermal-expansion parameters as a useful mean to obtain design charts, whose validity extends to a wide range of laminates. Recently, the effects of non-uniform tempera- ture distribution on thermal buckling have been assessed by Li et al.[6]using CLT in the context of an energy-based formulation.

The work was successively extended to account for random system properties[7]. Still in the context of CLT, variable stiffness plates have been optimized with regard to thermal buckling require- ments using finite elements in Ref.[8].

While CLT can be successfully applied to analyze a wide class of thin-plate problems, different strategies are needed to analyze more general plate configurations, where transverse shear defor- mation effects cannot be neglected. In this sense, a relatively vast amount of research efforts has regarded the development of more refined approaches, FSDT being the simplest one. An early work by Tauchert[9]discusses the derivation of closed-form solutions for anti-symmetric angle-ply plates based on FSDT, while mixed FSDT finite-elements are adopted by Noor and Peters[10]to address the sensitivity of thermal buckling loads to lamination and material properties. Recently, FSDT has been applied also to analyze the thermal buckling of FGM shells[11].

⇑ Corresponding author.

E-mail address:[email protected](M. D’Ottavio).

High-order shear deformation theories were employed in Refs.

[12,13]and closed-form solutions obtained for simply-supported plates subjected to a uniform temperature rise. Erroneous assump- tions on the plate constitutive law of Ref.[12]were successively pointed out by Rohwer[14], and results corrected accordingly.

Other refined techniques for the analysis of laminated plates included the high-order mixed formulation and the global/local higher order theory proposed in Refs.[15,16], respectively, while layerwise approaches have been carried out with finite elements in Refs.[17,18].

Despite the vast amount of research papers dealing with com- posite plates, less studies are available on the thermal buckling response of sandwich plates. A simplified analytical study is pro- posed in Ref.[19], based on the Ritz method and Libove-Batdorf sandwich theory[20]. A higher-order plate finite element is used in Refs.[21,22]to address the thermal buckling response of sand- wich panels, including skew configurations. An analytical approach is developed by Frostig[23] for the thermal buckling and post- buckling based on a high-order approach incorporating the effects of the core flexibility. Equivalent single layer models are discussed by Pradeep et al.[24], and Matsunaga with regard to plates with cross-ply and angle-ply face-sheets[25–27]. These latter studies include a simplified treatment of the pre-buckling deformations on the thermal buckling load, demonstrating their role on the crit- ical temperature prediction.

Three dimensional solutions were developed by Noor and co- workers[28–30]for both laminated and sandwich panels. In these pioneering works, the relevance of pre-buckling deformations was illustrated by providing results for temperature-independent[28]

and temperature-dependent[29]properties. Despite their useful- ness for deriving benchmark results, three dimensional solution procedures are usually too costly to be used for common design purposes, and more efficient approaches are needed.

In this sense, a powerful strategy for including several theories into one single formulation is given by the unified formulation pro- posed by Carrera, often denoted with the acronym CUF[31,32].

This kind of variable-kinematic approach has been adopted to ana- lyze the mechanical response of sandwich panels in several works [33–37], while the prediction of the critical temperatures is dis- cussed in Refs.[38,39]. Within the same framework, the thermal stability of functionally graded sandwich has been recently assessed[40].

The importance of retaining the transverse normal deformation in the presence of thermal loading and the influence of the actual temperature distribution across the plate thickness was pointed out by Carrera in Refs.[41,42].

In recent works by the authors, the Generalized Unified Formu- lation proposed by Demasi[43–45]was extended to include a sub- laminate approach[36,37] for the bending analysis of sandwich plates with the aim of optimizing the accuracy-to-cost ratio. The resulting approach was indicated as Sublaminate Generalized Uni- fied Formulation (S-GUF).

In the present paper, the S-GUF is developed in the context of a Ritz-based approach[46–48]and its formulation is extended to address the thermal buckling response of laminated plates and sandwich panels.

As opposed to the vast majority of previous studies, the initial stress produced by a uniform thermal loading is considered and the effect of the pre-buckling transverse normal deformation is assessed. This technique, suggested by Noor and Burton[28], intro- duces the deformed pre-buckling state into the classical eigenvalue problem for determining the critical temperature, which is then solved iteratively.

A more accurate identification of the initial stress state, that accounts for a refined representation of the thermal loading profile, the actual boundary conditions of the plate and, where necessary,

temperature-dependent material properties, would require the solution of a more complex non-linear problem and remains out of the scope of this paper.

The results illustrate the excellent accuracy of the predictions in relation with the relatively low number of degrees of freedom.

Novel results are finally proposed as a benchmark for future studies.

2. Formulation

The formulation is developed for the thermal buckling analysis of composite plates in the framework of the Sublaminate- Generalized Unified Formulation (S-GUF). Linear and nonlinear pre-buckling deformations are taken into account. The panels are flat and can be either monolithic or sandwich. A sketch of the panel is provided inFig. 1, where the reference system together with the typical dimensions of the panel are reported. In particular,aandb denote the length and the width of the panel, respectively, whileh is the total thickness. For generality purposes, the sketch refers to a sandwich configuration, where the quantityh_fdenotes the thick- ness of the single face-sheet. In the case of monolithic composite plates, the panel thickness is simply denoted with h, while hf

becomes meaningless.

The panel is subjected to a uniform temperature increase, although the formulation could be, in principle, extended to con- sider nonuniform distributions. The initial heating of the panels – which constitutes the pre-buckling condition – happens by assum- ing fully restrained boundary conditions, where immovable edges are considered. The buckling boundary conditions, as discussed next, can be any sort of combination involving the vanishing or non-vanishing of the three-inplane displacement components.

The formulation is thus characterized by two distinct steps, consisting in the solution of the pre-buckling and buckling condi- tion. When the nonlinear pre-buckling effects are accounted for, the equations are in the form of a nonlinear eigenvalue problem, which is solved iteratively. On the contrary, the classical approach based on a linear pre-buckling analysis leads to the solution of a standard eigenvalue problem.

The Sublaminate-Generalized Unified Formulation is an axio- matic approach for the analysis of composite structures based on the idealization of the structure as an assembly ofNksublaminates.

The fundamental ideas of the approach are here reviewed, while a deeper discussion is available in[36,37]. Each sublaminate is a cluster ofN^k_p plies, arbitrarily chosen by the user, where an arbi- trary kinematic theory is chosen for describing the displacement field. A fully displacement-based approach is adopted. By collect- ing the three displacement components of the generic ply p belonging to the sublaminatek, the 31 vectoru^p;kis defined as:

Fig. 1.Plate dimensions and reference system.

u^p;k¼nu^p_x^;k u^p_y^;k u^p_z^;koT

ð1Þ

whereu^p;k_r denotes the displacement component along the generic coordinate directionr¼x;y;z. The Generalized Unified Formulation (GUF) proposed by Demasi[43–45]is adopted as underlying kine- matic description for approximating the displacement components of Eq.(1). In particular, following the notation of GUF, the compo- nents of Eq.(1)are expressed as:

u^px^;kðx;y;zpÞ ¼FauxðzpÞu^px^;a^k_uxðx;yÞ aux ¼0;1;. . .;N^k_ux u^py^;kðx;y;zpÞ ¼FauyðzpÞu^py^;a^kuyðx;yÞ auy ¼0;1;. . .;N^k_uy u^pz^;kðx;y;zpÞ ¼Fa_uzðzpÞu^pza^;k_uzðx;yÞ auz¼0;1;. . .;N^k_uz 8>

><

>>

: ð2Þ

where separation of variables is assumed between the thickness- wise and in-plane directions. In particular, F_aur denotes the thickness function, while u^p;kraur are the generalized displacement

components associated with the kinematic assumptions. Note that N^k_ur denotes the order of the expansions, thus a different number of terms can be used for representing the displacement field along the three directions. This means that, for instance, the in-plane displacement components of a generic sublaminate could be described with an equivalent single layer theory of order 4, and the out-of-plane displacement with a similar theory of order 2. In this case, the sublaminate theory would be labeled as ED442. At lam- inate level, the kinematic theory is specified as a string reporting the approaches adopted for the various sublaminates. For instance, ED332/ED554/ED332denotes a three-sublaminate subdivision, where the first and the third sublaminates are modeled using ED332, while ED554is adopted for the second one. It is also remarked that each sublaminate can be modeled by assuming, independently, an equiv- alent single layer or a layerwise description for each displacement component.

2.1. Strain–displacement relations

As far as the formulation is developed within a displacement- based approach, it is necessary to introduce the relation between the strain and the displacement components. With the aim of treating the thermal buckling problem, the Green–Lagrange strain tensor is introduced as:

^ij¼eijþgij ð3Þ

where distinction is made between the linear and the nonlinear partseijandg_ij, whose components are:

eij¼1

2 ui;jþuj;i

ð4Þ and

g_ij¼1

2u_r;iu_r;j ð5Þ

Following the standard CUF formalism, the components of Eqs.

(4) and (5)are organized in vectors where the in-plane and the normal components are conveniently separated. Considering, for now, the linear part of the strain tensor, it is possible to write:

e^p;k_X ¼D_Xu^p;k e^pn^;k¼Dnu^p;kþDzu^p;k ð6Þ where the subscriptsXand n are introduced to denote the in-plane and the normal components, respectively. The two vectors of Eq.(6) read:

e^p_X^;k¼ⁿe^p_xx^;k e^p_yy^;k c^p_xy^;koT e^p_n^;k¼ⁿc^p_yz^;k c^p_xz^;k e^p_zz^;koT ð7Þ and the differential matrices of Eq.(6)are defined as:

D_X¼

@x@ 0 0

0 _@y^@ 0

@@y @

@x 0

2 64

3

75 Dn¼

0 0 _@x^@ 0 0 _@y^@ 0 0 0 2 64

3

75 Dz¼

@@z 0 0

0 _@z^@ 0 0 0 _@z^@ 2

64

3 75

ð8Þ In a similar fashion, the nonlinear terms of the strain tensorg_ij

expressed by Eq.(5)are organized as:

where, consistently with Eq.(6), distinction is made between the in-plane and the normal components.

2.2. Hooke’s law

The same partitioning into in-plane and normal components of Eq.(6)is adopted, and linear hyperelastic orthotropic behaviour is assumed. After rotating the components into a common laminate reference system, the 3D constitutive law of the generic plypof the sublaminatekcan be written as:

r^p;k_X ¼~C^p;k_XX^p;kX þ~C^p;k_Xn^pn^;k r^pn^;k¼~C^p;k_nX^p;kX þC~^p_nn^;kpn^;k ð10Þ where:

C~^p_XX^;k¼

eC^p₁₁^;k eC^p₁₂^;k eC^p₁₆^;k eC^p;k₁₂ eC^p;k₂₂ eC^p;k₂₆ eC^p₁₆^;k eC^p₂₆^;k eC^p₆₆^;k 2

664

3

775 C~^p_Xn^;k¼

0 0 eC^p₁₃^;k 0 0 eC^p;k₂₃ 0 0 eC^p₃₆^;k 2

664

3 775 C~^p_nn^;k¼

Ce^p₄₄^;k Ce^p₄₅^;k 0 Ce^p₄₅^;k Ce^p₅₅^;k 0 0 0 eC^p;k₃₃ 2

664

3 775

ð11Þ

Thermal deformations, which are of concern for the pre- buckling analysis, are introduced as[2,49]:

t^pX^;k¼ a^p_X^;kDT t^pn^;k¼ a^p_n^;kDT ð12Þ whereDTdenotes the temperature increase with respect to a refer- ence value, while the vectors collecting the coefficients of thermal expansion are defined as:

a^p_X^;k¼ⁿa^p_xx^;k a^p_yy^;k 2a^p_xy^;koT a^p_n^;k¼0 0 a^p_zz^;kT ð13Þ

2.3. Pre-buckling state

Following Noor and Burton[28], the pre-buckling analysis is per- formed by assuming uniform heating and boundary conditions of immovable edges. More specifically, the panel is constrained such that the normal in-plane displacement components are prevented along the four edges. On the other hand, the out-of-plane

g^p_X^;k¼ 1

2u^p_x;x^;k2þu^p_y;x^;k2þu^p_z;x^;k2 1

2u^p_x;y^;k2þu^p_y;y^;k2þu^p_z;y^;k2

u^p_x;x^;ku^p_x;y^;kþu^p_y;x^;ku^p_y;y^;kþu^p_z;x^;ku^p_z;y^;k

T

g^p_n^;k¼nu^p_x;y^;ku^p_x;z^;kþu^p_y;y^;ku^p_y;z^;kþu^p_z;y^;ku^p_z;z^;k u^p_x;x^;ku^p_x;z^;kþu^p_y;x^;ku^p_y;z^;kþu^p_z;x^;ku^p_z;z^;k u^p_x;z^;ku^p_x;z^;kþu^p_y;z^;ku^p_y;z^;kþu^p_z;z^;ku^p_z;z^;koT

ð9Þ

displacement is constrained in correspondence of the midsurface, so that the panel is free to expand or contract along the thickness-wise direction. It follows that the deformed pre-buckling deformed pat- tern is given by:

0uxðx;y;zÞ ¼0uyðx;y;zÞ ¼0

0uzðx;y;zÞ ¼0uzðzÞ

ð14Þ

where the subscript 0 is introduced to specify the pre-buckling con- dition. The internal stress distribution arising from the displace- ment field given by Eq.(14)is:

0ryzðx;y;zÞ ¼0rxzðx;y;zÞ ¼0

0rabðx;y;zÞ ¼0rabðzÞ witha;b¼x;y

ð15Þ

meaning that the only not-null pre-buckling stress components are those relative to the panel membrane behaviour. It is worth high- lighting that the terms0rab are functions ofzand are, in general, different from ply to ply.

For the pre-buckling condition specified by Eqs.(14) and (15), the evaluation of the normal displacement 0uz and the in-plane stress components can be performed in a closed-form manner.

After noticing that the only not null pre-buckling strain component is0^p;kzz, the internal stresses are available from the thermoelastic constitutive relation:

0r^p_X^;k¼ ~C^p_XX^;ka^p_X^;kD^Tþ~Cp_{Xn 0}^{;k pn;k}a^pn^;kD^T

0r^pn^;k¼ ~C^p_nX^;ka^p_X^;kD^{Tþ~Cpnn;k0pn;k}a^pn^;kD^{T ð16Þ} where:

0^pn^;k¼ 0 0 0^pzz^;k

T

ð17Þ The value of0^p;kzz can be determined after imposing the vanish- ing of the normal stressr^p;kzz, viz. considering plane stress condi- tions. More specifically, the conditionr^p;kzz ¼0 can be substituted in Eq.(16), and the pre-buckling strain is obtained as function of the temperature variation as:

0^pzz^;k¼0^p33^;k¼ Ce^p;k₁₃

Ce^p₃₃^;ka^p_xx^;kþeC^p;k₂₃

eC^p₃₃^;ka^p_yy^;kþ2Ce^p;k₃₆

Ce^p₃₃^;ka^p_xy^;kþa^p_zz^;k

!

D^{T ð18Þ}

and the corresponding in-plane stress components are retrieved by making use of Eq.(16), so:

0r^p;k_X ¼ ^C^p;ka^p;k_X D^T

0r^pn^;k¼0 ð19Þ

where C^^p;k¼

bC^p₁₁^;k Cb^p₁₂^;k Cb^p₁₆^;k bC^p₁₂^;k Cb^p₂₂^;k Cb^p₂₆^;k bC^p₁₆^;k Cb^p₂₆^;k Cb^p₆₆^;k 2

664

3

775 ð20Þ

and

bCij¼eC^p_ij^;keC^p;k_i3Ce^p;k_3j

eC^p₃₃^;k i;j¼1;2;6 ð21Þ If the effect of pre-buckling deformations is neglected – as it is done in standard linear buckling eigenvalue analysis – no addi- tional results are needed, and the solution given by Eq.(19)suffices for performing the subsequent buckling analysis.

On the other hand, an additional step is needed if the analysis is aimed at accounting for the effect of pre-buckling deformations. In particular, the additional quantity to be evaluated is the first derivative of the pre-buckling displacement0u_z;z, which is easily obtained after recalling the nonlinear expression of the Green–

Lagrange strain tensor of Eq. (3) and the solution of the pre- buckling problem of Eq.(14)as:

0u^p_z;z^;k¼ 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ20e^pzz^;k

q

ð22Þ Note that the solution of Eq. (22), despite its simplicity, accounts for the nonlinearity contained in the Green–Lagrange strain tensor. It is also worth observing the presence of the term

0e^p;kzz in Eq.(22), which makes it necessary the adoption of a theory of order higher than zero with respect to theuzcomponent.

2.4. Buckling condition

The buckling equations are derived in the context of a varia- tional framework, starting from the expression of the total poten- tial energy of the panel, which is:

P^¼1₂^Z

V^:rdV ð23Þ

where the potential of the external loads is, in the present investi- gation, identically null as no mechanical loads are prescribed. The strain tensor is intended as the contribution restricted to the mechanical deformations, as no strain energy is associated with the thermal deformations.

The first variation of the potential energy allows to derive the nonlinear pre-buckling equilibrium equations, whose closed-form solution is available from Eqs.(16) and (22). The buckling condi- tion can be found referring to the Trefftz criterion[3]:

dðd²P^{Þ ¼0 ð24Þ}

According to the stability criterion of Eq.(24), the buckling con- dition is readily found by recalling Eqs. (3) and (23), and is expressed as:

dðd²P^{Þ ¼} Z

V

de:rþkdg:0r

ð ÞdV¼0 ð25Þ

where0ris the pre-buckling stress tensor,kis the buckling multi- plier, whileeis:

e¼eþ^e ð26Þ

The components of the tensorseandgare available from Eqs.

(4) and (5), while the term^eis the linear part of the incremental Green–Lagrange strain tensor and its components read:

^eij¼1

2⁰u_r;iu_r;kþ0u_r;ku_r;i

ð27Þ It is highlighted that the components ^eij are identically null whenever pre-buckling deformations are null or negligible. In this case the variational statement of Eq. (25) reduces to the well- known form:

dðd²P^{Þ ¼Z}

V

de:rþkdg:0r

ð ÞdV¼0 ð28Þ

The more general case of non negligible pre-buckling deforma- tions is now considered. Introducing the S-GUF approximation, and recalling the pre-buckling condition expressed by Eq. (19), the buckling condition of Eq.(25)can be re-written as:

X^Nk

k¼1

X^Nkp

p¼1

Z

X

Zz^top_p z^bot_p

de^p;kTþd^e^p;kT

r^p;kþkdg^p_X^;kT0r^p_X^;k

h i

dzdX^{¼0 ð29Þ}

where, according to the pre-buckling solution given by Eq.(16), the components of^e^p;kare:

^e^p_X^;k¼f0 0 0g^T

^e^p_n^;k¼ ^c^p_yz^;k ^c^p_xz^;k ^^pzz^;k

n oT

¼n0u^p;k_z;zu^p;k_z;y 0u^p;k_z;zu^p;k_z;x 0u^p;k_z;zu^p;k_z;zoT

ð30Þ