Buckling Analysis of Debonded Sandwich Panel Under Compressi

NASA Technical Memorandum 4701

Available electronically at the following URL address:632ea37a31b765ce05081447/ltrs/ltrs Printed copies available from the following:

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Introduction

Sandwich construction has been used in aircraft applications over the last 40 years because of its light weight, high bending rigidity, and good fatigue proper-ties (ref.1). However, debonding of the faceskin from the core, which can be by impact damage or defects in the manufacturing process, significantly degrades sand-wich construction performance. In particular, debonding can significantly affect the compression stability and strength of the sandwich structure. Moreover, the pres-ence of debonding is a major concern in sandwich struc-tures because the debonded region may grow under compression (ref.2).

There have been several studies on the buckling of debonded sandwich and delaminated composite struc-tures loaded under compression (refs.2–8). Webster investigated the buckling loads of composite laminates with circular debond defects using the Rayleigh-Ritz method(ref.3). Chai et al. (refs.4 and5) studied buck-ling and postbuckling of delaminated composite plates by modeling the failure in one and two dimensions. Hwu and Hu (ref.2) and Somers et al. (ref.6) derived closed-form solutions to study the buckling and postbuckling of delaminated sandwich beams. They applied elastic frac-ture mechanics (LEFM), using J-integral and Griffith cri-terion to compute strain energy release rates for predicting debond growth. Vizzini and Lagace (ref.7) studied sublaminate delamination in composite lami-nates. In their approach, a one-dimensional (1-D) model of a delaminated sublaminate was developed in which the sublaminate rested on an elastic foundation that mod-eled the resin layer. The buckling load of the sublaminate was predicted using the Rayleigh-Ritz energy method. Kim and Dharan (ref.8) used the approach developed by Vizzini and Lagace with LEFM to study faceskin deb-onding for composite sandwich panels.

This paper examines a sandwich panel with initial through-the-width debonds between the faceskin and sandwich foam core. (See fig. 1.) The buckling load of the panel is computed from a 1-D model, which models a sandwich panel strip as a beam (faceskin) on an elastic foundation (core material), and a two-dimensional (2-D) model, which models the cross section of the panel. The Rayleigh-Ritz and finite-difference methods are used to analyze the 1-D model, while the finite-element method using 4-node quadrilateral plane-strain elements is used to analyze the 2-D model.

Although many investigators used the 1-D beam-on-elastic-foundation model to study the sandwich panel debond problem, little research has been performed (to the authors’ knowledge) to assess the adequacy of mod-eling the foam core material as elastic springs by more accurate analysis methods, such as finite-element analy-sis. In the finite-element model, the foam core material is modeled as an isotropic material. Therefore, because the finite-element model properly includes the behavior of the faceskin and foam core, more accurate predictions of buckling will result.

This paper presents three different numerical meth-ods (Rayleigh-Ritz, finite-difference, and finite-element) for determining the buckling load of a sandwich panel with initial through-the-width debonds. Parametric stud-ies investigating the effects of sandwich foam stiffness and debond length on the buckling load of the debonded Figure 1. Debonded sandwich panel.

x

Abstract

A sandwich panel with initial through-the-width debonds is analyzed to study the buckling of its faceskin when subject to an in-plane compressive load. The debonded faceskin is modeled as a beam on a Winkler elastic foundation in which the springs of the elastic foundation represent the sandwich foam. The Rayleigh-Ritz and finite-difference methods are used to predict the critical buckling load for various debond lengths and stiffnesses of the sandwich foam. The accuracy of the methods is assessed with a plane-strain finite-element analysis. Results indicate that the elastic founda-tion approach underpredicts buckling loads for sandwich panels with isotropic foam cores.

2

faceskin are performed. Finally, the assumption of using the elastic foundation approach to model the foam core is evaluated by comparing the Rayleigh-Ritz and finite-difference results from the 1-D elastic beam foundation model to the results from the 2-D plane-strain finite-element model.

Numerical Methods for Determining Buckling Load

As previously mentioned, the Rayleigh-Ritz, finite-difference, and finite-element methods were used to ana-lyze the buckling of the debonded sandwich panel. In the Rayleigh-Ritz and finite-difference methods, a sandwich panel strip was modeled as a beam on an elastic founda-tion. In the finite-element method, 2-D plane-strain quad-rilateral elements were used to model the cross section of the debonded sandwich panel.

Beam-on-Elastic-Foundation Model

Figure 2(a) shows a strip of unit width b of the sand-wich panel loaded in compression with symmetric through-the-width debonds between the faceskins and core. The debonded region of length l d is located in the middle of the sandwich panel, and the applied compres-sive load is P , as shown in figure 2(b). The thickness of the faceskins is denoted by t b , and the thickness of the sandwich core is denoted by 2t f . Due to symmetry, only the upper half of the sandwich panel strip of length L is modeled, as shown in figure 3. The Young’s modulus and the moment of inertia of the faceskins are denoted by E b and , respectively. The bonded regions l f have foundation support. The faceskin, modeled as a beam, is partially supported by an elastic foundation with modulus k and has simple supported boundary conditions at its ends.

Rayleigh-Ritz method.The total potential energy of the beam-on-elastic-foundation system is expressed as

(1)

where U m and U b are membrane and bending energies,respectively,U f is the potential energy of the elastic foundation, and ? is the potential energy of the compres-sive load P . These energy terms are defined as

(2a)

(2b)

(2c)

(2d)

where A is the cross-sectional area of the faceskin and ,and are the longitudinal and out-of-plane deflections,respectively (ref.9). The extensional strain of the face-skin can be written as

(3)

Introduction into equation (1) gives

(4)

The elastic foundation modulus k (x ) in equation (2c) is

given by

(5)

(a) Sandwich beam with initial symmetric debonded regions between core and faceskin.

(b) Buckled shape of sandwich beam.

Figure 2. Buckling of debonded sandwich panel strip modeled as a beam on an elastic foundation.

I t b 3/12=ΠU m U b U f ?

+++=U m E b A 2

----------εb 2

x

d 0

L

∫

=U b E b I 2

--------w ″()2

dx

L

∫

=U f 1

2

--k x ()w 2dx

L

∫

=?P u ′dx

L

∫

=u w εb u ′1

2

--w ′()

2+=ΠE b A 2----------u ′12--w ′2+

2E b I 2--------w ″()21

2--kw 2Pu ′+++dx 0

L

∫

=

k x ()K ψx ()

=

3

where K is a constant and can be represented as E f b /t f and E f is the Young’s modulus of the sandwich foam core. The variable ψ(x ) is defined as

(6)

For stability of a system in equilibrium, the second variation of potential energy needs to be positive semidefinite,

(7)

When the second variation of potential energy is zero, a neutral equilibrium configuration exists and indicates the onset of buckling. This point is characterized as the bifurcation point (ref.10). The Trefftz criterion states that equation (7) is identically satisfied when the varia-tion of the second variation is zero (ref.11).

The expression for δ2Π is obtained by perturbing the system in equation (4) such that

(8)

where u 0 and w 0 represent the prebuckling configuration and the variations u (x ) and w (x ) are infinitesimally small increments. For the prebuckling configuration,

(9)Introducing equations (8) and (9) into equation (4),assuming inextensional buckling (i.e.,u = 0), and collect-ing the second-order terms gives the second variation of the potential energy:

(10)

In the Rayleigh-Ritz method, the out-of-plane defor-mation at the onset of buckling w of the panel faceskin is approximated as

(11)

where φm are selected functions that satisfy the geometric boundary conditions and A m represents their amplitudes.For simple support boundary conditions, a sinusoidal function is used:

(12)

The Trefftz criterion used with the Rayleigh-Ritz method to determine the buckling load can be written as

(13)

Dimensional parameters are nondimensionalized by

the length L as follows:

(14)

Figure 3. Elastic foundation model of debonded sandwich beam.

P

z , w

x, u

ψx ()1if 0x l f ≤≤0if l f x L l f

–≤≤1if L l f x L

≤≤–

=δ2

Πw []0

≥u u 0u +→w w 0w +→

u 0P

E b A ----------x –=w 00=

δ2

ΠE b I w ″()2K ψw 2P w ′()2

–+[]dx

L

∫

=

w x ()A m φm x ()

m 1

=M

∑

=

φm x ()m πx

L

----------sin =?δ2

Πw []

?A m

----------------------0=m 12…M ,,,=()ξx /L =ηl f /L =μl d /L 12η–==

4

The variable η is the foundation ratio, where a value of zero is equivalent to a faceskin (beam) with no foun-dation (unsupported) and a value of one-half is equiva-lent to a fully supported faceskin (beam). The variable μis the debond ratio and represents the fractional amount of the strip which is debonded. The functions φm in equa-tion (12) and ψ in equation (6) become and ,respectively, after the nondimensionalized parameters in equation (7) are used. Incorporating equation (13) gives the eigenvalue problem:

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