Dynamic behavior of composite sandwich panel with CFRP outer layers

EVA KORMANÍKOVÁ, KAMILA KOTRASOVÁ

Institute of Structural Engineering and Transportation Structures, Faculty of Civil Engineering

Technical University of Kosice

Košice, SLOVAKIA

Abstract: — A sandwich panel with laminate faces is used for free vibration analysis. The periodic microstructure and Mori-

Tanaka model are used for homogenization of unidirectional fiber reinforced composite. The Shear Deformation Theory is

considered for analytical and numerical analysis. FEM in ANSYS is used for numerical analysis. The effect of sandwich design

parameters such as panel length, core thickness and fiber reinforced angle on vibration response is investigated. Natural

frequencies of sandwich panel versus sandwich design parameters are presented in graphical form. From the results can be

concluded that sandwich design parameters affect the natural frequencies of sandwich panels, and this effect is important for

designing of sandwich panels under dynamic load.

Keywords: —frequency analysis, sandwich plate, CRFP faces, FEM analysis

1. Introduction

Rapid growth in manufacturing industries has led to the

need for the betterment of materials in terms of strength,

stiffness, density, and lower cost with improved sustainability.

Composite materials have emerged as one of the materials

possessing such betterment in properties serving their potential

in a variety of applications [1].

The motivation for sandwich composites is two-fold [2]. If

a plate is bent, the maximum stresses occur at the top and the

bottom surface. So, it makes sense using high strength

materials only for the sheets and using low and lightweight

materials in the middle. The resistance to bending of a

rectangular cross-section is proportional to the cube of the

thickness. Increasing the thickness by adding a core in the

middle increases the resistance. The shear stresses have a

maximum in the middle of a sandwich beam requiring the core

to support the shear.

Sandwich panels are one of very important groups of

laminated composites. They consist of two thin faces with

thickness of h(1) and h(3) and a core with thickness of h(2). The

faces are made of high strength materials having good

properties under tension, such as fiber reinforced polymer

matrix laminates used in this paper, while the core is made of

lightweight materials such as foam. The advantage of weight

and bending stiffness makes sandwich composites more

attractive for some applications than other composite

or conventional materials [3],[4],[5],[6],[7],[8].

The analysis of simple sandwich structures may be

achieved by analytical methods, by adapting the classical tools

of structural analysis on anisotropic elastic structure face

elements [9]. For more complex structures, such as more

general boundary conditions or loading, numerical

methods such as Finite Element Method, Boundary

Element Method, etc. are used [10],[11].

The use of assumptions is necessary to

mathematical modeling of laminated composites. These

include an elastic behavior of fibers and matrices, a

perfect bonding between fibers and matrices, low

variation of the mechanical characteristics of the

individual fibers, uniform fiber diameters, their regular

arrangement in the matrix, etc.

Taking into account the different size scales of mechanical

modelling of structure elements composed of fiber

reinforced composites, the micro, macro and structural

modeling levels must be considered [12],[13],[14],[15],[16].

Free vibration analysis is a very important analysis.

The lowest natural frequency is often referred to as the

fundamental frequency, which is the most important

parameter for design engineers as many of the systems

are designed to operate below it [17],[18],[19],[20],[21],

[22],[23].

For obtaining the material characteristics of

composite laminated faces, the experimental tests were

performed [24],[25],[26],[27],[28].

2. Materials and Methods

The motivation for sandwich composites is two-fold: If

a beam is bent, the maximum stresses occur at the top and

the bottom surface. So, it makes sense using high

strength materials only for the sheets and using low and

lightweight

Received: January 15, 2022. Revised: November 17, 2022. Accepted: December 14, 2022. Published: December 31, 2022.

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materials in the middle. The resistance to bending of a

rectangular cross-sectional beam is proportional to the cube of

the thickness. Increasing the thickness by adding a core in the

middle increases the resistance. The shear stresses have a

maximum in the middle of a sandwich beam requiring the core

to support the shear. This advantage of weight and bending

stiffness makes sandwich composites more attractive for some

applications than other composite or conventional materials.

Composite materials consist of two or more constituents

and the modelling, analysis and design of structures composed

of composites are different from conventional materials such as

steel. There are three levels of modelling. At the micro-

mechanical level, the average properties of a single reinforced

layer (a lamina or a ply) have to be determined from the

individual properties of the constituents, the fibers and matrix.

The average characteristics include the elastic moduli, the

thermal and moisture expansion coefficients, etc. The micro-

mechanics of a lamina does not consider the internal structure

of the constituent elements but recognizes the heterogeneity of

the ply. The micro-mechanics is based on some simplifying

approximations. These concern the fiber geometry and packing

arrangement, so that the constituent characteristics together

with the volume fractions of the constituents yield the average

characteristics of the lamina. Note that the average properties

are derived by considering the lamina to be homogeneous. In

the frame of this paper only the micro-mechanics of

unidirectional reinforced laminates are considered. The

calculated values of the average properties of a lamina provide

the basis to predict the macrostructural properties. At the

macro-mechanical level, only the averaged properties of a

lamina are considered and the microstructure of the lamina is

ignored. The properties along and perpendicular to the fiber

direction, these are the principal directions of a lamina, are

recognized and the so-called on-axis stress strain relations for a

unidirectional lamina can be developed. A laminate is a stack

of laminae. Each layer of fiber reinforcement can have various

orientation and in principle each layer can be made of different

materials. Knowing the macro-mechanics of a lamina, one

develops the macro-mechanics of the laminate. Average

stiffness, flexibility, strength, etc. can be determined for the

whole laminate. The structure and orientation of the laminae in

prescribed sequences to a laminate lead to significant

advantages of composite materials when compared to a

conventional monolithic material. In general, the mechanical

response of laminates is anisotropic.

One very important group of laminated composites are

sandwich composites. They consist of two thin faces (the skins

or sheets) sandwiching a core. The faces are made of high

strength materials having good properties under tension such as

fiber reinforced laminates while the core is made of lightweight

materials such as foam, resins with special fillers, called

syntactic foam, having good properties under compression.

Sandwich composites combine lightness and flexural stiffness.

When the micro- and macro-mechanical analysis for

laminae and laminates are carried out, the global behavior of

laminated composite materials is known. The last step is the

modelling on the structure level and to analyze the global

behavior of a structure made of composite material. By

adapting the classical tools of structural analysis on anisotropic

elastic structure elements the analysis of simple structures as

beams or plates may be achieved by analytical methods, but for

more general boundary conditions and/or loading and for

complex structures, numerical methods are used.

3. First-Order Shear Deformation Theory

(FSDT)

FSDT considers that transverse normals do not remain

perpendicular to the midsurface after deformation. This theory

is used for thicker plates or sandwiches taking into account the

Reissner kinematics.

Transverse shear stresses are added to the state of plane

stress for this reason. Shear deformations are written following

the Figure 1:



yz,2

w/y

v12

v32

w/y

h(2)

h(1)

h(3)



xz,2

w/x

u12

u32

w/x

Fig. 1. Geometry of shear deformation.

(1)

where d is midplane distance of sandwich faces:

(2)

Internal forces are expressed by terms:

(3)

The constitutive equations for a sandwich are in the

form:

(4)

with stiffness coefficients:

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i, j = 4, 5,

(5)

where are coefficients of extension and bending

stiffness matrix, are coefficients of extension-bending

coupling stiffness matrix, are the transverse shear moduli of

the core.

4. Free Vibrations of Sandwich Plates

The equations to determine the natural frequencies of the

symmetric sandwich panel are used [1]:

(6)

where ks is the transverse shear deformation factor given by value

5/6.

(7)

where ρk is the mass density of the kth layer.

For the simply supported plate on each edge let:

(8)

where m, n are integers only, ωmn is natural angular velocity.

After substituting of Eqs. (8) into the Eqs. (6) we get

, (9)

where:

(10)

and

(11)

Natural angular velocity can we obtain from:

where

(12)

5. Results and Discussion

The parametric study of free vibration has been solved

for a simply supported panel on each edge. Length of panel L

is varied from 2 m to 3.8 m, with step of 0.2 m, width b = 1 m

and thickness h of the panel is varied from 0.04 m to 0.12 m,

with step of 0.02 m.

The thickness of outer layers h(1) = 0.0004 m and h(3) =

0.0006 m. Outer layers are made of carbon/epoxy laminates

with stacking sequence of layers [0/90/0], [0]3.

Carbon reinforced fiber polymer (CRFP) was considered

with the following characteristics:

Ef = 230 GPa; Em = 3 GPa;



f = 0.3;



m = 0.3;



f = 0.6.

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Mass density of the laminated composite is ρLc = 1508

kg/m3 and ρSc = 43 kg/m3 for the polyurethane foam as

sandwich core.

The material characteristics versus fiber volume fraction

are illustrated in Figure 2. The graphs show similar results for

longitudinal direction and differences in the transversal

directions for bigger fiber volume fraction.

Fig. 2. Material characteristics vs. fiber volume fraction.

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FEM analysis in ANSYS was used for free vibration.

The influence of the variable thickness and length of the panel

with CFRP [0]3 faces to the first ten natural frequencies is

presented in Figure 3. For illustration (Figure 3a), a length of

2 m was selected for analytical and numerical solution of

natural frequencies depending on changing thickness.

Fig. 3. a) Natural frequencies for sandwich with [0]3 CFRP faces,

b) Natural frequencies for sandwiches with [0]3 CFRP faces, numerical

solution.

Increasing panel thickness causes an increase in natural

frequencies (also seen in Figure 3a (L = 2 m) and differences

between analytical and numerical solution. To study

frequencies, depending on the changing length, a panel of

thickness 0.08 m is used (Figure 3b). Fundamental frequencies

are considerably higher for the panel supported on all sides

opposite the panel with two opposite supports in the width

direction. The influence of the variable face layout [0]3 and

[0/90/0] to natural frequencies is presented in Figure 4a. The

analytical and numerical analysis of the effect of fiber

orientation to the global coordinate system on the fundamental

frequency and considering the variable thickness and length is

shown graphically in Figure 4b and Figure 5. From Figures 4b

and 5 can be seen that for the laminate layout [0]3, the

frequencies are lower compared to laminate [0/90/0].

Differences between analytical and numerical solution are

greater for the [0/90/0] stacking sequence and the thicker

panel (Figure 5). It is also possible to observe the increase in

frequency with reducing the length of the panel.

Fig. 4. a) Natural frequencies for sandwich with [0]3 and [0/90/0] CFRP

faces, numerical solution, b) Fundamental frequency versus

thickness for sandwiches with [0]3 and [0/90/0] CFRP faces.

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Fig. 5. Fundamental frequencies versus length for sandwich with

[0]3 and [0/90/0] CFRP faces a) h = 0.04 m b) h = 0.12 m.

6. Conclusion

The periodic microstructure and Mori-Tanaka model were

used for homogenization of unidirectional fiber reinforced

composite. It is seen that both models provide similar results,

instead characteristics in the direction of the transverse axes.

The FSDT was considered for free vibration analysis. The

influence of the variable length, thickness, and fiber orientation

of CFRP faces with [0]3 and [0/90/0] layout to natural

frequencies was presented. It is also possible to observe the

increase in frequency with increasing the thickness of the panel

and by reducing the length of panel for both [0]3 and [0/90/0]

CFRP faces sandwich panels. The analytical and numerical

analysis of the effect of fiber orientation to the global

coordinate system on the fundamental frequency, taking into

account the variable thickness and length was shown

graphically.

From the results obtained by the presented work can be

concluded that sandwich design parameters affect the natural

frequencies of sandwich panels, and this effect has been taken

into consideration for designing of sandwich panels under

dynamic load.

Acknowledgment

This work was supported by the Scientific Grant Agency of

the Ministry of Education of Slovak Republic and the Slovak

Academy of Sciences under Projects VEGA 1/0307/23 and

VEGA 1/0363/21.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author(s) contributed in the present research,

at all stages from the formulation of the problem

to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

This work was supported by the Scientific Grant

Agency of the Ministry of Education of Slovak

Republic and the Slovak Academy of Sciences under

Projects VEGA 1/0307/23 and VEGA 1/0363/21.

Conflict of Interest

The author(s) declare no potential conflicts of interest

concerning the research, authorship, or publication of

this article.

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