Failure indentation analysis of composite sandwich plates via hierarchical models

Abstract

In this study, an indentation failure analysis of sandwich composite plates via several higher-order theories is presented. The unified formulation by Carrera is adopted in order to derive displacement based or mixed, refined theories based on an equivalent single layer approach or a layerwise one. In this manner, out-of-plane normal and transverse shear stresses responsible for the indentation failure can be accurately described. A closed form, Navier-type solution is assumed. Simply supported plates made of fibre-reinforced polymeric skins and foam cores are investigated. The side-to-thickness ratio, a/h, and the indenter side dimension are considered as analysis parameters. The effect of skins' lamination and thickness is also investigated. Failure indentation is described in terms of failure loading value and failure position. The variation of displacement and stress fields near the indentation area are also presented. The accuracy of the proposed theories is assessed towards Pagano's three-dimensional exact solution. The obtained results show that higher-order models are required to accurately describe the indentation failure of sandwich plates.

Keywords

Failure indentation sandwich structures hierarchical models plates Navier solution

Introduction

Sandwich panels are structures made of two rigid skins separated by a low-density compact core. Besides non-conventional configurations, the faces are, usually, made of composite materials or lightweight metal alloys, whereas the core can be made of carbon or metallic foams, honeycomb made of metallic alloys or Nomex, balsa or plastic materials. In such a manner, a high stiffness-to-weight ratio is obtained and sandwich panels can be, therefore, employed in components that require low weight and high structural performances typical of engineering fields such as space, aeronautics and automotive. This article presents a failure analysis of sandwich composite plates subjected to static indentation. This study is motivated by the fact that sandwich structures, due to their applications, can undergo relevant localised compressive loadings or impacts that may lead to a significant damage, especially at the interface between skin and core and a consequent reduction of strength. An example is represented by sandwich floors in military and commercial air planes subjected to localised pressure loadings due to the payload. Static indentation is, therefore, an important research topic. However, the prediction of the stress and deformation fields in such a type of structure might be difficult. A correct and effective design of these structures calls for advanced models to describe their mechanics and failure as accurately as possible. To the best of the authors' knowledge, one of the first examples of sandwich structures used in civil engineering can be found in Fairbairn [1]. Allen [2] was one of the first to present a general tractation of sandwich structures. Burton and Noor [3] assessed several sandwich panel models based on two-dimensional (2D) shell theories, performing a parametric study in terms of material and geometric properties. Among the several reviews written over the past years, the works by Noor et al. [4] and Vinson [5] are worth to be mentioned. Ha [6] presented a review on finite element based on displacements or stress hybrid models for the analysis of sandwich structures. Hu et al. [7] presented a review of classical and high-order theories to model sandwich structures. Results were compared with finite element solutions. Failure in sandwich structure can occur in many different ways: face yielding, face wrinkling, intra-cell dimpling, core shear and local indentation. The type of failure depends on various parameters such as structure, geometry, loading conditions and mechanical properties of the core and the faces. Petras and Sutcliffe [8] presented a failure mode map accounting for the different failure modes in sandwich structures. They [9,10] investigated the indentation failure of sandwich beams, proposing an indentation failure criterion. The higher-order analytical theory by Frostig and Baruch [11], in which the classical thin-plate theory was used for the skins, whereas the core mechanics was described via a 3D elasticity solution was used. Numerical results were assessed towards experimental investigations. Lee and Tsotsis [12] formulated an analytical model considering the sandwich structure as a plate resting on elastic foundations to evaluate shear and compression failure indentation of honeycomb sandwich structures. Shuaeib and Soden [13] studied the indentation failure of polyvinyl chloride foam core sandwich beams. They modelled the top skin as a linear elastic beam resting on elasto-plastic foundations. The influence of geometrical parameters was investigated. Experimental tests were performed by Othman and Barton [14] on beams and panels made of carbon skins and Nomex core accounting for quasi-static and impact loadings. An analysis of the influence of core thickness on structure strength was carried out. A finite element model was developed by Castanié et al. [15] to study the compressive strength of sandwich composite plates after impact. They used non-linear springs to model the core behaviour and Reissner's first-order shear deformation theory (FSDT; see Reissner [16]) to model the linear elastic behaviour of the faces. Numerical results were compared with the experimental ones. Coté et al. [17] investigated the through-thickness compressive strength of square honeycomb sandwich panels via analytical models for buckling of the sandwich walls and elastic wrinkling and plastic micro-buckling of the skins. Results were validated towards experimental tests and 3D finite-element analyses. Rizov et al. [18] predicted the residual strength and damage dent after indentation of sandwich plates with a foam core via 3D finite element analyses in ABAQUS. Experimental tests were also presented. The unified formulation (UF) by Carrera [19] is used here. UF represents a systematic approach towards 2D modelling for deriving several higher-order theories via either the principle of virtual displacement (PVD; see Reddy [20]) or Reissner's mixed variational theorem (RMVT; see Reissner [21,22]) and adopting either an equivalent single layer (ESL) approach or a layerwise (LW) one. A Navier-type closed-form solution is assumed. Simply supported plates are, therefore, considered. This study represents a first approach to static failure indentation. A flat-faced square indenter is considered in order to describe the failure in terms of failure loading and location. In such a manner, a contact analysis is not required. The failure indentation criteria proposed by Lee and Tsotsis [12] and Petras and Sutcliffe [10] are used. Transverse displacement and out-of-plane stress components are also presented. The effect of the indenter size and the skin lay-up and thickness is investigated. The accuracy of UF models is assessed towards Pagano's [23] 3D exact solution.

Overview of the considered plate theories

Plate's geometry and reference system are shown in Figure 1. The through-the-thickness direction is z and Ω = {(x, y, z) : z = 0} is the reference plane of the plate. The reference system axes are bounded such that: 0 ≤ x ≤ a, 0 ≤ y ≤ b and −h/2 ≤ z ≤ h/2. The components of the displacement vector along x-, y- and z-axes are u_x, u_y and u_z, respectively. N_l represents the total number of layers, k counts the laminae and h_k stands for a k-layer thickness. A vast variety of 2D theories can be formulated on the basis of different kinematic assumptions.

Classical theories

Classical theories were conceived in the last two centuries in order to model the global membrane-bending mechanics.

Classical lamination theory

Classical lamination theory (CLT) is based on Cauchy [24], Poisson [25] and Kirchhoff's [26] kinematic assumptions

\begin{matrix} u_{q} (x, y, z) = u_{q 0} (x, y) - z \partial u_{z 0} \partial q with q = x, y \\ u_{z} (x, y, z) = u_{z 0} (x, y) \end{matrix}

(1)

They postulate that the normals to the reference surface Ω remain normal, straight and unstrained after deformation. Displacement components in correspondence to the reference plane are denoted via the subscript ‘0’. CLT does not account for shear and normal transverse deformations. Shear and normal out-of-plane stress components can be obtained a posteriori upon integration of the indefinite equilibrium equations.

First-order shear deformation theory

Reissner and Mindlin postulated a kinematic field that accounts for constant through-the-thickness transverse shear stress and strain components

\begin{matrix} u_{q} (x, y, z) = u_{q 0} (x, y) + z u_{q 1} (x, y) with q = x, y \\ u_{z} (x, y, z) = u_{z 0} (x, y) \end{matrix}

(2)

The model based on these kinematic assumptions is named FSDT. The normal deformation is neglected. More accurate values of the out-of-plane shear stresses can be obtained via integration of the indefinite equilibrium equations.

Higher-order theories

Refinements of CLT and FSDT can be introduced by including higher-order terms in the kinematic assumptions in equation (2)

\begin{matrix} u_{q} (x, y, z) = u_{q 0} (x, y) + z^{r} u_{q r} (x, y) with q = x, y, z r = 1, 2, \dots, N \end{matrix}

(3)

The summing convention for the repeated indexes has been adopted. The kinematic model in equation (3) accounts for both shear and normal transverse deformations. N is the through-the-thickness approximation order and, in this study, it is assumed to be as high as four. These theories are addressed as EDN. Letter ‘E’ denotes that the kinematics is based on the ESL approach, ‘D’ indicates that displacements are assumed as unknowns. Figure 2 presents the through-the-thickness first- and third-order ESL approximation.

LW theories

Multilayered plates' mechanics can be modelled by means of kinematic assumptions that are independent in each layer. According to Reddy [20], this approach is stated as LW. MacLaurin's expansion along the thickness, as for ESL models, is no longer convenient in this case. Displacement interlaminar congruency can be imposed more conveniently assuming interface values as unknown variables

\begin{matrix} u_{q}^{k} = F_{t} u_{q t}^{k} + F_{b} u_{q b}^{k} + F_{r} u_{q r}^{k}, with q = x, y, z r = 2, 3, \dots, N; k = 1, 2, \dots, N_{l} \end{matrix}

(4)

Subscripts ‘t’ and ‘b’ denote values related to a k-layer top and bottom surfaces. The thickness functions F_t, F_b and F_r depend upon a local, through-the-thickness dimensionless coordinate ζ_k

ζ_{k} = \frac{2}{z_{k}} h_{k}

(5)

where z_k is a local reference coordinate centred at k-layer mid-point. F_t, F_b and F_r are defined as follows

F_{t} = P_{0} + P_{1} 2, F_{b} = P_{0} - P_{1} 2, F_{r} = P_{r} - P_{r - 2}, r = 2, 3, \dots, N

(6)

where P_j = P_j(ζ_k) is the j-order Legendre's polynomial function. The first four Legendre's polynomials are

P_{0} = 1, P_{1} = ζ_{k}, P_{2} = 3 ζ_{k}^{2} - \frac{1}{2}, P_{3} = \frac{5}{ζ_{k}^{3}} 2 - \frac{3}{ζ_{k}} 2, P_{4} = 35 ζ_{k}^{4} 8 - \frac{1}{5} ζ_{k}^{2} 4 + \frac{3}{8}

(7)

Since the following properties hold

ζ_{k} = {\begin{matrix} 1 : & F_{t} = 1, F_{b} = 0, F_{r} = 0 \\ - 1 : & F_{t} = 0, F_{b} = 1, F_{r} = 0 \end{matrix}

(8)

the interlaminar displacement congruency can be easily imposed

\begin{matrix} u_{q t}^{k} = u_{q b}^{(k + 1)} with q = x, y, z; k = 1, 2, \dots, N_{l} - 1 \end{matrix}

(9)

The acronym for these theories is LDN, where ‘L’ stands for an LW approach. First- and third-order LW approximations along the plate thickness are reported in Figure 3.

Mixed theories

The kinematics described before does not ensure interlaminar continuity of shear and normal out-of-plane stresses at the interface between two adjacent layers. It can be fulfilled ‘a priori’ via RMVT. Within the RMVT framework, shear and normal transverse stresses are assumed as primary variables together with displacements. In an LW case, the following model is used to approximate the shear and normal out-of-plane stresses

\begin{matrix} σ_{q}^{k} = F_{t} σ_{q t}^{k} + F_{b} σ_{q b}^{k} + F_{r} σ_{q r}^{k}, with q = x, y, z r = 2, 3, \dots, N k = 1, 2, \dots, N_{l} \end{matrix}

(10)

Interlaminar continuity is imposed straightforwardly

\begin{matrix} σ_{q t}^{k} = σ_{q b}^{(k + 1)} with q = x, y, z k = 1, 2, \dots, N_{l} - 1 \end{matrix}

(11)

A model of this group is denoted as LMN, where ‘M’ means mixed models based on RMVT.

The UF

The considered theories are all unified considering that CLT and FSDT are a particular case of higher-order, ESL models. The latter can be regarded as a peculiar case of LW models in which the number of layers is equal to the unit and the polynomial approximation is obtained via the classical base {z^r : r = 0, 1, …, N}. Equations (1) to (4) can be unified into the following compact vectorial notation

\begin{matrix} u^{k} = F_{t} u_{t}^{k} + F_{b} u_{b}^{k} + F_{2} u_{2}^{k} + \dots + F_{N} u_{N}^{k} = F_{τ} u_{τ}^{k} τ = t, b, r; r = 2, 3, \dots, N; \\ σ^{k} = F_{t} σ_{t}^{k} + F_{b} σ_{b}^{k} + F_{2} σ_{2}^{k} + \dots + F_{N} σ_{N}^{k} = F_{τ} σ_{τ}^{k} k = 1, 2, \dots, N_{l} \end{matrix}

(12)

u^{k} = {\begin{matrix} u_{x}^{k} \\ u_{y}^{k} \\ u_{z}^{k} \end{matrix}}, σ^{k} = {\begin{matrix} σ_{xz}^{k} \\ σ_{yz}^{k} \\ σ_{zz}^{k} \end{matrix}}

(13)

The derivation of the governing equations according to the chosen variational statement becomes general, regardless the approximation approach (ESL or LW), the polynomial functions or the approximation order. The acronym system used to address each theory is summarised in Figure 4. In the case of displacement-based theories, the interlaminar stresses are obtained through integration of the indefinite equilibrium equations.

Governing equations

For the sake of brevity, governing equations and related mechanical and geometrical boundary conditions are addressed assuming the PVD only. Their derivation from RMVT is straightforward and it can be found in Carrera [19]. Considering a generic pressure loading $p^{kT} = {p_{x}^{k}, p_{y}^{k}, p_{z}^{k}}$ acting on the top, Σ ^t _k , and the bottom, Σ ^b _k , of each lamina, the variational statement based upon the PVD is

\sum {N_{l}}_{k = 1} \int Ω_{k} \int h_{k} (δ ε_{p}^{kT} σ_{p}^{k} + δ ε_{n}^{kT} σ_{n}^{k}) d z_{k} d Ω_{k} = \sum {N_{l}}_{k = 1} \int Σ_{k}^{b} ⋃ Σ_{k}^{t} δ u^{kT} p^{k} d Ω_{k}

(14)

in which

ε_{p} = {\begin{matrix} ε_{xx} \\ ε_{yy} \\ γ_{xy} \end{matrix}}, ε_{n} = {\begin{matrix} γ_{xz} \\ γ_{yz} \\ ε_{zz} \end{matrix}}, σ_{p} = {\begin{matrix} σ_{xx} \\ σ_{yy} \\ σ_{xy} \end{matrix}}, σ_{n} = {\begin{matrix} σ_{xz} \\ σ_{yz} \\ σ_{zz} \end{matrix}}

(15)

are the in- and out-of-plane strains and stresses in the structure reference system, see (Figure 1). Displacements-strains relations are

\begin{matrix} ε_{p}^{k} = D_{p} u^{k} \\ ε_{n}^{k} = D_{n Ω} u^{k} + D_{nz} u^{k} \end{matrix}

(16)

D_p, D_nΩ and D_nz are matrices of differential operators defined as follows

D_{p} = [\begin{matrix} \frac{\partial}{\partial x} & 0 & 0 \\ 0 & \frac{\partial}{\partial y} & 0 \\ \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \end{matrix}], D_{n Ω} = [\begin{matrix} 0 & 0 & \frac{\partial}{\partial x} \\ 0 & 0 & \frac{\partial}{\partial y} \\ 0 & 0 & 0 \end{matrix}], D_{nz} = [\begin{matrix} \partial \partial z & 0 & 0 \\ 0 & \partial \partial z & 0 \\ 0 & 0 & \partial \partial z \end{matrix}]

(17)

Stresses and strains of a k-layer are related via Hooke's law

\begin{matrix} σ_{p}^{k} = {\tilde{C}}_{pp}^{k} ε_{p}^{k} + {\tilde{C}}_{pn}^{k} ε_{n}^{k} \\ σ_{n}^{k} = {\tilde{C}}_{np}^{k} ε_{p}^{k} + {\tilde{C}}_{nn}^{k} ε_{n}^{k} \end{matrix}

(18)

where terms

{{\tilde{C}}_{ij}^{k} : i, j = p, n}

are the material stiffness matrices for a k-layer in the global reference system, see Carrera [19]. Replacing equations (16), (18) and the postulated, unified displacement field u = F_τ u _τ into equation (14) and after some algebraic manipulations and application of Gauss–Green's theorem, the variational statement becomes

\begin{matrix} \sum {N_{l}}_{k = 1} \int Ω_{k} δ u_{τ}^{kT} \int h_{k} {- F_{τ} D_{p}^{T} [{\tilde{C}}_{pp}^{k} F_{s} D_{p} + {\tilde{C}}_{pn}^{k} (F_{s} D_{n Ω} + F_{s, z})] \\ + (- F_{τ} D_{n Ω}^{T} + F_{τ, z}) [{\tilde{C}}_{np}^{k} F_{s} D_{p} + {\tilde{C}}_{nn}^{k} (F_{s} D_{n Ω} + F_{s, z})]} d z_{k} u_{s}^{k} d Ω_{k} \\ + \sum N_{l}_{k = 1} \int Γ_{k} δ u_{τ}^{kT} \int h_{k} {F_{τ} I_{p}^{T} [{\tilde{C}}_{pp}^{k} F_{s} D_{p} + {\tilde{C}}_{pn}^{k} (F_{s} D_{n Ω} + F_{s, z})] + F_{τ} I_{n Ω}^{T} [{\tilde{C}}_{np}^{k} F_{s} D_{p} \\ + {\tilde{C}}_{nn}^{k} (F_{s} D_{n Ω} + F_{s, z})]} d z_{k} u_{s}^{k} d Ω_{k} = \sum N_{l}_{k = 1} \int Σ_{k}^{b} ⋃ Σ_{k}^{t} δ u_{τ}^{kT} p_{τ}^{k} d Ω_{k} \end{matrix}

(19)

p_{τ}^{k}

is a variationally consistent load vector coming from the applied loadings p^k. Equation (19) yields the governing equations and the boundary conditions through an appropriate choice of the virtual displacements

\begin{matrix} K_{d}^{k τ s} u_{s}^{k} = p_{τ}^{k} \\ u_{τ}^{k} = {\bar{u}}_{τ}^{k} & (x, y, z) \in Γ_{k}^{g} \\ Π_{d}^{k τ s} u_{s}^{k} = Π_{d}^{k τ s} {\bar{u}}_{s}^{k} & (x, y, z) \in Γ_{k}^{m} \end{matrix}

(20)

being the k-layer geometrical boundary conditions (

{\bar{u}}_{τ}^{k}

) applied on Γ ^g _k and the mechanical ones (

Π_{d}^{k τ s} {\bar{u}}_{s}^{k}

) on Γ ^m _k . Differential matrices

K_{d}^{k τ s}

and

Π_{d}^{k τ s}

are

\begin{matrix} K_{d}^{k τ s} = \int h_{k} {- F_{τ} D_{p}^{T} [{\tilde{C}}_{pp}^{k} F_{s} D_{p} + {\tilde{C}}_{pn}^{k} (F_{s} D_{n Ω} + F_{s, z})] \\ + (- F_{τ} D_{n Ω}^{T} + F_{τ, z}) [{\tilde{C}}_{np}^{k} F_{s} D_{p} + {\tilde{C}}_{nn}^{k} (F_{s} D_{n Ω} + F_{s, z})]} d z_{k} \end{matrix}

(21)

\begin{matrix} Π_{d}^{k τ s} = \int h_{k} {F_{τ} I_{p}^{T} [{\tilde{C}}_{pp}^{k} F_{s} D_{p} + {\tilde{C}}_{pn}^{k} (F_{s} D_{n Ω} + F_{s, z})] \\ + F_{τ} I_{n Ω}^{T} [{\tilde{C}}_{np}^{k} F_{s} D_{p} + {\tilde{C}}_{nn}^{k} (F_{s} D_{n Ω} + F_{s, z})]} d z_{k} \end{matrix}

(22)

A Navier's solution of equation (20) can be obtained by assuming the following harmonic form for the applied loadings and unknown displacements

\begin{matrix} (u_{x}^{k}, p_{x z}^{k}) = \sum \bar{m}_{m = 1} \sum \bar{n}_{n = 1} (U_{x}^{k} (z), P_{x z}^{k} (z)) \cos (m π a x) \sin (n π b y) \\ (u_{y}^{k}, p_{y z}^{k}) = \sum \bar{m}_{m = 1} \sum \bar{n}_{n = 1} (U_{y}^{k} (z), P_{y z}^{k} (z)) \sin (m π a x) \cos (n π b y) \\ (u_{z}^{k}, p_{zz}^{k}) = \sum \bar{m}_{m = 1} \sum \bar{n}_{n = 1} (U_{z}^{k} (z), P_{zz}^{k} (z)) \sin (m π a x) \sin (n π b y) \end{matrix}

(23)

Indentation failure criteria

Among the different types of failure that can occur in a sandwich plate, failure indentation is characterised by core crushing due to out-of-plane normal and shear stress components {σ_iz : i = x, y, z} (see Petras and Sutcliffe [8]). Two criteria are considered here to predict this type of failure. In Lee and Tsotsis' [12] one (LTC), the out-of-plane stress components are accounted for separately. Indentation failure occurs when one of the following equalities is verified

σ_{zz} σ_{cc} = 1, σ_{xz} σ_{sc} = 1, σ_{yz} σ_{sc} = 1

(24)

where σ_cc and σ_sc are the normal compressive yield strength and the out-of-plane shear strength, respectively.

Indentation failure loading amplitude $p_{zz}^{(F)}$ is obtained, thanks to problem linearity

p_{zz}^{(F)} = p_{zz} min {σ_{cc} σ_{zz}, σ_{sc} σ_{xz}, σ_{sc} σ_{yz}}

(25)

where p_zz is a reference loading value. The second failure criterion is the one proposed by Petras and Sutcliffe [10] (PSC)

σ_{zz} σ_{cc} + (σ_{xz} + σ_{yz}) σ_{sc} = 1

(26)

The indentation failure loading amplitude is

p_{zz}^{(F)} = p_{zz} σ_{zz} σ_{cc} + (σ_{xz} + σ_{yz}) σ_{sc}

(27)

Results and discussion

Simply supported square plates are investigated. The thickness of each skins' ply is 0·125 · 10⁻³ m. Core's thickness is 9·4 · 10⁻³ m. The side-to-thickness ratio (a/h) is as high as 50 and as low as 5. Thin and thick plates are, therefore, investigated. The faces are all made of T300/BSL914C epoxy (see Soden et al. [27]) and its mechanical properties are: E_L = 138 · 10³ MPa, E_T = 11 · 10³ MPa, G_LT = 5·5 · 10³ MPa, G_TT =3· 4 · 10³ MPa, ν_LT = 0·28 and ν_TT = 0·4; ‘L’ stands for a direction parallel to the fibres, whereas ‘T’ represents a direction perpendicular to them. Strength parameters are: X_t = 1515 MPa, X_c = 900 MPa, Y_t = Z_t = 27 MPa, Y_c = Z_c = 200 MPa, S = T = 80 MPa and R = 67·6 MPa; ‘t’ stands for tension and ‘c’ for compression. The core is made of Rohacell 51A foam and its stiffness and strength properties are: E = 70 MPa, G = 19 MPa, ν = 0·4, σ_cc = 0·9 MPa and σ_sc = 0·8 MPa. Mechanical properties of the foam core are such that a linear-elastic behaviour can be assumed, as in Petras and Sutcliffe [9] and Lee and Tsotsis [12]. Unless differently stated, a symmetric [0/core/0] configuration is considered. The effect of the skins' staking sequence will be investigated in the final section. Plate's stacking sequence starts from plate top. Ply angles are measured towards x-axis. A flat-faced square indenter is simulated considering a uniform localised pressure p_zz applied as shown in Figure 5. The reference indenter side length Δ is equal to 12·7 · 10⁻³ m. Indentation loading is applied at centre of plate's top. The number of harmonic terms $\bar{m}$ and $\bar{n}$ in Fourier's expansion of the loading is equal to 73 such that a displacement convergence up to four significant digits is obtained. Indentation failure is predicted in terms of failure loading value and location. In order to ensure that failure is effectively due to core indentation, Tsai–Wu's and max-stress first-ply failure criteria are also considered to verify that skin failure load is higher than the indentation one. For the sake of brevity, these two criteria are not reported here. For their formulation, refer to Tsai and Wu [28] and Reddy [20].

Indentation loading

Indentation failure loading value and location via LTC and PSC are presented in Tables 1 and 2, respectively. According to the 3D exact solution, failure occurs at core top. For a/h = 50, the in-plane failure location is at x/a = y/b = 1/2 for both criteria. Because, in this point, the out-of-plane shear stress components are equal to zero for symmetry, the same failure loading value is obtained regardless the considered failure criterion. For a/h as high as 30, failure location depends upon the failure criterion. In the case of LTC, failure location depends upon the side-to-thickness ratio, whereas for PSC failure occurs always at

x / a = y / b = \frac{1}{2} \pm \frac{1}{92}

. It is worth to clarify that the failure location is accounted for since the minimum of the failure loading is investigated. The failure location should be regarded as a quantity to compare the results obtained via the several considered theories rather than a result by itself since it depends upon many analysis parameters such as the number of half-waves in the Fourier approximation and the number of points in which the solution is computed. Results have shown that failure occurs in a small area around the failure point, the failure load in that area being very close to the one of the failure point. Classical theories do not always predict correctly the failure location. In the case of correct failure location, they yield inaccurate results, the error being about 30% at best. They always overestimate the failure loading value. EDN models also do not yield accurate results. An accurate prediction of the failure indentation calls for LW theories. For LTC, a third-order approximation yields accurate results in value and location, whereas for a first order and in the case of PSC, the error is about 5% at worst. For a fixed expansion order and in the case of an LW approach, results can be improved via the assumption of mathematical interfaces, that is, a physical layer is ideally divided into more sub-layers. In the LM4^* theory, a mathematical interface is used in the core such that it is fictitiously split into two sub-layers of thickness, starting from the plate top, equal 0·4 · 10⁻³ and 9 · 10⁻³ m, respectively. The two sub-layers have different thickness due to the out-of-plane stress component gradient in the neighbourhood of the loading area. LM4^* model matches the reference 3D solution. The variation of the transverse displacement and stress components at the core top surface along x- or y-axes are also presented. The side-to-thickness ratio a/h is equal to 50 and 5. The transverse displacement is reported in Figure 6. It is put into the following dimensionless form

{\bar{u}}_{z} = u_{z} E_{L} h^{3} p_{zz} a^{4}

(28)

Classical theories totally fail to predict the actual variation of

{\bar{u}}_{z}

. ED4 model describes it qualitatively. LM2 matches Pagano's solution. Out-of-plane stress components are adimensionalised as follows

({\bar{σ}}_{xz}, {\bar{σ}}_{yz}, {\bar{σ}}_{zz}) = \frac{1}{p_{zz}} (σ_{xz}, σ_{yz}, σ_{zz})

(29)

σ_xz and σ_yz are presented in Figures 7 and 8, respectively. As shown in Lee and Tsotsis [12], shear stress components reach their maximum in the neighbourhood of the indenter edge. They present a high gradient in the indenter application area, since in a small area they vary from zero (at plate centre) to their maximum value. Fourth-order LW models match Pagano's exact solution for a/h = 50, whereas, in the case of thick plates, results are accurate except for the indenter edge where a steep gradient is present. In this case, a mathematical interface is required. LM4^* model matches the reference exact solution. The normal stress σ_zz is shown in Figure 9. For a/h = 50, its maximum absolute value is at the plate's centre, whereas for thick plates, it is almost constant in the neighbourhood of the plate centre. In the case of a/h = 50, LD1 model matches the reference solution, whereas an LM4 theory is required for thick plates since a high gradient is present in the neighbourhood of the indenter edge. As a remark, classical and ESL theories are not able to predict the mechanical response of a sandwich plate subjected to indentation loadings. LW theories are recommended to match Pagano's 3D reference solution. Moreover, the advantage of the proposed formulation is the manner the model are enriched to derive several theories, from the classical to the most refined ones. The drawback is represented by the computational costs that are, of course, affected by this enrichment. The computational time is a few seconds when using classical theories and about a minute for the most refined considered model. Since a Navier-type solution is used, the computational time is not very high when more refined models are used.

Table 1.

Minimum indentation failure loading value [10⁻¹ MPa] and location via LTC.

a/h	50	30	10	5
3D	6.638, $(\frac{1}{2}, \frac{1}{2})^{a}$	8.494, $(\frac{1}{2}, \frac{1}{2} \pm \frac{1}{92})$	8.784, $(\frac{1}{2}, \frac{1}{2})$	8.873, $(\frac{1}{2}, \frac{1}{2} \pm \frac{1}{92})$
CLT	11.696, 1.76^b	14.641	15.172, 1.73	15.709
FSDT	11.696, 1.76	14.641	15.172, 1.73	15.709
ED2	12.508, 1.88	14.521	14.494, 1.65	14.345
ED4	7.395, 1.11	8.942	9.017, 1.03	9.052
LD1	6.579, 0.99	8.387	8.779, 1.05	8.828
LD2	6.617, 1.00	8.487	8.769, 1.00	8.890, 1.00
LD3	6.634, 1.00	8.505, 1.00	8.782, 1.00	8.896, 1.00
LD4	6.638, 1.00	8.499, 1.00	8.775, 1.00	8.886, 1.00
LM1	6.551, 0.99	8.347	8.779	8.779
LM2	6.614, 1.00	8.490	8.753, 1.00	8.869, 1.00
LM3	6.632, 1.00	8.504, 1.00	8.780, 1.00	8.897, 1.00
LM4	6.638, 1.00	8.499, 1.00	8.776, 1.00	8.886, 1.00
LM4^*c	6.638, 1.00	8.495, 1.00	8.784, 1.00	8.874, 1.00

LTC: Lee and Tsotsis criterion; CLT: classical lamination theory; and FSDT: first-order shear deformation theory.

In-plane failure location (x/a, y/b).

2D–3D failure loading ratio in the case of coincident failure location.

The core has been split into two fictitious sub-layers via a mathematical interface.

Table 2.

Minimum indentation failure loading value [10⁻¹ MPa] and location via PSC.

a/h	50	30	10	5
3D	6.638	6.998	7.369	7.373
	$(\frac{1}{2}, \frac{1}{2})^{a}$	$(\frac{1}{2} \pm \frac{1}{92}, \frac{1}{2} \pm \frac{1}{92})$	$(\frac{1}{2} \pm \frac{1}{92}, \frac{1}{2} \pm \frac{1}{92})$	$(\frac{1}{2} \pm \frac{1}{92}, \frac{1}{2} \pm \frac{1}{92})$
CLT	9.976	9.263, 1.32^b	10.050, 1.36	9.826, 1.33
FSDT	9.976	9.263, 1.32	10.050, 1.36	9.826, 1.33
ED2	9.628	10.545	9.746	9.452
ED4	6.301	7.769	7.056	6.687
LD1	6.579, 0.99	7.339, 1.05	7.196, 0.98	7.377, 1.00
LD2	6.617, 1.00	7.049, 1.01	7.136, 0.97	7.197, 0.98
LD3	6.634, 1.00	6.884, 0.98	7.142, 0.97	7.141, 0.97
LD4	6.638, 1.00	6.944, 0.99	7.309, 0.99	7.289, 0.99
LM1	6.551, 0.99	7.364, 1.05	7.185, 0.98	7.402, 1.00
LM2	6.614, 1.00	7.045, 1.01	7.132, 0.97	7.184, 0.97
LM3	6.632, 1.00	6.883, 0.98	7.141, 0.97	7.141, 0.97
LM4	6.638, 1.00	6.945, 0.99	7.311, 0.99	7.291, 0.99
LM4^*c	6.638, 1.00	6.986, 1.00	7.344, 1.00	7.347, 1.00

PSC: Petras and Sutcliffe criterion; CLT: classical lamination theory; and FSDT: first-order shear deformation theory.

In-plane failure location (x/a, y/b).

2D–3D failure loading ratio in the case of coincident failure location.

The core has been split into two fictitious sub-layers via a mathematical interface.

Effect of the indenter size

Analyses are carried out for thin and thick plates subjected to an indentation loading with Δ as low as 2 · 10⁻³ m and as high as 25 · 10⁻³ m. Figure 10 presents the minimum failure loading for a thin plate (a/h = 50) obtained considering both LTC indentation and max-stress first-ply failure criteria. Max-stress first-ply failure criterion is also considered since, increasing the indenter size, failure is expected to change from core indentation to skins' first-ply failure. Failure transition effectively occurs for Δ equal to about 15 · 10⁻³ m and, in that point, a change in the slope of the minimum failure loading is present. Increasing Δ, CLT converges to the reference 3D solution since the loading is less and less localised and the problem becomes a thin plate under bending. In the case of indentation failure, CLT yields poor results. ED4 theory gives good results, whereas LD2/LW2 theories match Pagano's exact solution. Figure 11 presents the indentation failure analysis via PSC only, whereas PSC core indentation failure and skin failure loading obtained via Tsai–Wu's criterion are compared in Figure 12. Failure transition occurs for Δ equal to about 16 · 10⁻³ m. The change in slope of the core indentation failure loading is due to a change in failure location. As far as the theories accuracy is concerned, the same remark as in the previous case is still valid. The case of a thick plate is presented in Figures 13 to 15. In this case, LW models are mandatory for an accurate prediction of the indentation failure. Figure 13 presents a failure mode transition for Δ equal to about 18 · 10⁻³ m for LTC and max-stress criteria. This transition value is higher than the one of the previous case since the side-to-thickness ratio is smaller. On the contrary, Figure 15 shows that no failure mode transition occurs for the considered indenter sizes when PSC and Tsai–Wu's criteria are considered.

Effect of skins stacking sequence and thickness

Different symmetric and anti-symmetric cross-ply staking sequences, for top and bottom skins, are accounted for. The resulting sandwich structure is symmetric with respect to the core. Results are presented in Figures 16 to 17 for a/h = 50 and 5, respectively. The increase of indentation failure loading is mainly due to the increase of the skin thickness. In particular, the increment is higher for a skin ply number higher than two. An equivalent analysis considering a constant skin thickness for the same cross-ply stacking sequences has also been carried out and no considerable variation of the indentation failure loading has been found. These results are not here reported for the sake of brevity. In all cases, CLT yields inaccurate results, whereas ESL theories are qualitatively similar to the reference solution. An accurate prediction of the indentation failure loading calls for LW models. A second-order expansion yields accurate results for a/h = 50, whereas a third-order one is required for thick plates.

Conclusions

A study of indentation failure in sandwich composite plates via classical as well as several higher-order theories has been presented. Indentation failure has been addressed in terms of minimum failure loading and location. Petras and Sutcliffe and Lee and Tsotsis indentation failure criteria have been used. In order to verify that failure is effectively due to indentation, max-stress and Tsai–Wu's first-ply failure criteria have also been considered. Both thin and thick plates have been investigated. Plates' mechanics has been described via 2D models formulated via a UF, which allows for obtaining a vast variety of higher-order, displacement or mixed, ESL or LW theories. Classical theories are obtained as special cases. Governing equations have been solved via Navier's solution. Simply supported panels made of glass fibre reinforced polymer skins and foam core have been investigated. The side-to-thickness ratio, the indenter side dimension and the skin stacking sequence and thickness have been considered as analysis parameters. Accuracy of UF models has been verified through comparison with Pagano's 3D exact solution. Results underline that classical models do not yield accurate results, as the problem is governed by transverse shear and normal stresses and local high 3D stress gradients. Thanks to UF, 2D models that match the 3D solution can be obtained. In particular, it has been shown that higher-order ESL models yield good results, but only LW models match Pagano's exact solution. These last models, therefore, should be used in the case of indentation failure loading analysis.

Footnotes

Funding

First and Third authors are supported by the Fonds National de la Recherche (FNR) Luxembourg via the CORE project C09/MS/05 FUNCTIONALLY. Second author is supported by FNR through Aides à la Formation recherche grant PHD-09-184.

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