Transversely isotropic constitutive properties modeling of tubular sandwich composite structures

Abstract

Sandwich structures have been studied extensively for planar structures; however, the use of composite tubing, manufactured by pultrusion, in a bending situation where a core material can contribute to take shear stresses, can find many applications in modern structures made of composite materials. The objective of this article is to develop analytical solutions for axial effective modulus and major Poisson’s ratio of a pultruded unidirectional composite tubing filled with a core material. In this work, the unidirectional composite tubing and its core are considered to have transversely isotropic and isotropic properties, respectively. For the validation of the results, the obtained exact analytical solutions are reduced to a case where both the materials are isotropic and compared to the existing solutions for an isotropic material filling in an isotropic tube. Further validations of our exact analytical solutions for the transversely isotropic tubing and isotropic core are carried out employing a finite element analysis of the same structure, where the results show excellent agreements between the analytical solutions and the numerical results. Finally, a parametric study is conducted to investigate the variations in the effective properties of the two-phase composite cylinder based on the variations in the skin and/or core geometries and their material properties.

Introduction

Composite sandwich structures, as one of the commonly used composite systems and structures, have been among the best candidates for use in specific structural applications (such as bending) where ultralightweight and high-performance materials are needed. Typically, these composite structures comprise strong face sheets (e.g. thin sheets of metals, plastics, ceramics, and composite thin laminates) that are cocured to a very lightweight core material (e.g. low-to-high density foam as well as honeycomb structures). Sandwich structures provide an efficient method to increase the flexural rigidity without a significant increase in the structural weight. ¹ Thus, structures, based upon a minimum face sheet thickness adequate to carry the in-plane loads, can be made to carry out-of-plane loads and to be stable under compression without a significant weight penalty. ² Upon application specifications, composite sandwich structures have been used with different structural designs and geometries among which planar and tubular composite sandwich structures are the most commonly used geometries. Aerospace structures, in particular, modern launch vehicles, helicopter blades, optical benches for space applications, and nonferrous ship hulls are among some of the current applications. ^{1 –5} With increasing use of fiber-reinforced composite sandwich structures with different geometries for various applications, it has become necessary to investigate the mechanical behavior of such structures when they are subjected to external loads. Sarzynski and Ochoa ³ studied flexure responses of the composite sandwich beams with carbon-epoxy laminate face sheets and carbon foam core subjected to bending loads. In addition, failure modes of the composite sandwich beam structures are investigated by Daniel et al., ⁴ where the initiations of various failure modes depending on the material properties of the structural constituents (i.e. the face sheets and core materials), geometric dimensions, and the type of loading were observed. In their study, ⁴ the composite sandwich beams were made of unidirectional carbon/epoxy face sheets and aluminum honeycomb as well as polyvinylchloride closed cell foam cores.

The tubular sandwich composite structures can be effectively fabricated by means of pultrusion processing technology, where pultrusion is one of the fastest and most cost effective composites manufacturing methods. ⁶ Pultrusion is well suited to produce multilayer cylindrical composite structures, in particular, tubular sandwich structures, prestressing tendons, and reinforcing bars, because it can provide structures with a high degree of axial reinforcement. The basic operation of pultrusion is simple in concept but quite complex in detail because of the number of mechanical, chemical, and physical factors that are simultaneously involved in the process. ^{4 –8} A schematic of the pultrusion setup is shown in Figure 1.

Figure 1.

Schematic representation of the pultrusion setup. ⁹

By embedding fiber optic sensors in the composite part during pultrusion process, Kalamkarov et al. ⁹ have investigated the development of residual stresses in the composite part inside the pultrusion die. Kalamkarov et al. ^{10 , 11} developed smart pultruded fiber-reinforced polymer reinforcements with embedded Bragg grating and Fabry-Perot fiber optic sensors, where the pultruded smart reinforcements were evaluated and applied for structural strain monitoring and damage detection.

Many micromechanical models are available to predict the macroscopic behavior of fiber-reinforced composite materials, ^{12 , 13} where the fiber and the matrix materials were assumed to have isotropic properties. Basically, in these works, a representative geometric model of composite material is used to predict the effective properties of fiber-reinforced composite materials. One of the most used models is the composite cylindrical model introduced by Hashin and Rosen. ¹² To obtain transverse shear properties of a fiber composite system, Christensen ¹³ used a model that is very closely related to the composite cylinder model. This model assumes that the fiber, the matrix, and the interface are continuous materials and the constitutive relations for the bulk composite materials are formulated based on the assumptions of continuum mechanics. Effective characteristics of unidirectional fiber composites and composite shells and plates with rapidly varying thickness were derived by Kalamkarov ¹⁴ by means of application of the asymptotic homogenization techniques. In particular, the explicit analytical formulae for effective properties of the composite sandwich shells with honeycomb fillers and different types of rib- and wafer-reinforced shells and plates have been derived. ¹⁴ These analytical results have been applied for the design and optimization of the composite and reinforced shells and plates. ¹⁵ Most recently, these results have been generalized in the case of generally orthotropic smart (in particular, piezoelectric) honeycomb sandwich shells and smart wafer-reinforced composite shells and plates. ^{16 , 17}

Special classes of orthotropic materials are those that have the same properties in one plane and different properties in the direction normal to this plane. Such materials are called transversely isotropic, and they are described by five independent elastic constants (i.e. C₁₁, C₁₂, C₂₂, C₂₃, and C₆₆), instead of nine for fully orthotropic.

The objective of the present article is to develop analytical formulae for effective axial Young’s modulus and major Poisson’s ratio for a unidirectional pultruded composite tube filled with core material to give a tubular sandwich composite structure. Here, we consider the material of composite tube as transversely isotropic (with isotropy in the plane normal to the tube axis, i.e. 2–3 planes) and core material as isotopic. We have assumed that the bonding between the composite tube and the tubular core is perfect with no property variations and no discontinuity at the interface.

Analytical modeling of a two-phase tubular sandwich composite structure

In this work, a two-phase cylinder model is considered (see Figure 2). We assume that the inside core (i) is an isotropic and the outside tube (o) is a transversely isotropic material.

Figure 2.

Two-phase composite cylindrical model.

The outside radii of the core and the tube are denoted by b and c, respectively (see Figure 2). The problem of interest is axisymmetric with no dependence on polar angle θ. In addition, it is assumed that the inside core is perfectly bounded to the outside tube and an isostrain deformation takes place when cylinders are pulled in the x-direction (Figure 2). It means that the axial deformations are independent of radial coordinates at a given axial location. Therefore, the only equilibrium equation to be satisfied in this case is given as follows: ¹⁸

$\frac{d{\mathit{\sigma }}_r}{dr}+\frac{{\mathit{\sigma }}_r-{\mathit{\sigma }}_{\mathit{\theta }}}{r}=0$

1

Note that for this problem, all other equilibrium equations are automatically satisfied.

Determination of transversely isotropic effective Young’s modulus $\left(E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}\right)$

In general, the stress components in the cylindrical coordinate system can be formulated as

$\left\{\begin{array}{c}{\mathit{\sigma }}_x=C_{11}{\mathit{\epsilon }}_x+C_{12}{\mathit{\epsilon }}_{\mathit{\theta }}+C_{13}{\mathit{\epsilon }}_r\\ {\mathit{\sigma }}_{\mathit{\theta }}=C_{12}{\mathit{\epsilon }}_x+C_{22}{\mathit{\epsilon }}_{\mathit{\theta }}+C_{23}{\mathit{\epsilon }}_r\\ {\mathit{\sigma }}_r=C_{13}{\mathit{\epsilon }}_x+C_{23}{\mathit{\epsilon }}_{\mathit{\theta }}+C_{33}{\mathit{\epsilon }}_r\end{array}\right.$

2

where σ_x , σ_θ , and σ_r are normal stresses, and ε_x , ε_θ , and ε_r are normal strains. Then, the constitutive equations for the isotropic inside filler (i.e. material i) and the transversely isotropic outside tube (i.e. material o) can be written using superscripts (i) and (o) for σ, ε, and C terms in equation (2).

The strain–displacement relationships in the cylindrical coordinate system for this particular problem can be reduced to the following relations ¹⁹

${\mathit{\epsilon }}_r=\frac{\mathrm{\partial }w}{\mathrm{\partial }r}; {\mathit{\epsilon }}_{\mathit{\theta }}=\frac{w}{r}; {\mathit{\epsilon }}_x=\frac{\mathrm{\partial }u}{\mathrm{\partial }x}$

3

where the radial displacement is given by $w=w(r)$ , and u represents the displacement in the axial direction. By substituting equation (3) in equation (2) and then the obtained results in equation (1), we can obtain the following equation for the radial displacement, $w(r)$

$\frac{d^{2}w}{d{r}^2}+\frac{1}{r}\frac{dw}{dr}-\frac{w}{r^2}=0$

4

This is an Euler–Cauchy differential equation (a linear second order ordinary homogeneous differential equation), which has the following exact solution ²⁰

$w=Ar+\frac{B}{r}$

5

Therefore, the radial displacements in the outside tube and the inside core can be written as follows

$\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{u}\mathrm{b}\mathrm{e}\mathrm{:} w^{(\mathrm{o})}=A_{\mathrm{o}}r+\frac{B_{\mathrm{o}}}{r}; b\le r\le c$

6

$\mathrm{I}\mathrm{n}\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{:} w^{(\mathrm{i})}=A_{\mathrm{i}}r+\frac{B_{\mathrm{i}}}{r}; 0\le r<b$

7

In the problem under study, strain ${\mathit{\epsilon }}_x$ does not vary with axial coordinate x, hence it follows from equation (3) that

$u^{(\mathrm{o})}={\mathit{\epsilon }}^{(\mathrm{o})}x, b\le r\le c$

8

$u^{(\mathrm{i})}={\mathit{\epsilon }}^{(\mathrm{i})}x, 0\le r<b$

9

Since at the interface ${\left.u^{{}^{(\mathrm{i})}}\right|}_b={\left.u^{{}^{(\mathrm{o})}}\right|}_b$ , we obtain ${\mathit{\epsilon }}_x^{(\mathrm{i})}={\mathit{\epsilon }}_x^{(\mathrm{o})}=\mathit{\epsilon }=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$ , that is, an isostrain condition for Figure 2 when cylinder is pulled in x-direction. Note that in the core domain, radius reaches $r =0$ , therefore B _i in equation (7) should be zero, due to finiteness, and the radial displacement at the center of the tube is zero (i.e. $w^{(\mathrm{i})}(0)=0$ ). Hence, in the core region (i), equation (7) reduces to

$w^{(\mathrm{i})}=A_{\mathrm{i}}r 0\le r<b$

10

\Three constants A_i , A_o , and B_o entering equations (6) and (10) can be determined from the interface and outside boundary conditions. Substituting equations (6) and (10) into the strain–displacement equation (3), we obtain

${\mathit{\epsilon }}_{{}_r}^{(\mathrm{i})}=A_{\mathrm{i}}; {\mathit{\epsilon }}_{{}_{\mathit{\theta }}}^{(\mathrm{i})}=A_{\mathrm{i}} {\mathit{\epsilon }}_{{}_x}^{(\mathrm{i})}=\mathit{\epsilon }$

11

$\left\{\begin{array}{c}{\mathit{\epsilon }}_{{}_r}^{(\mathrm{o})}=A_{\mathrm{o}}-\frac{B_{\mathrm{o}}}{r^2}\\ {\mathit{\epsilon }}_{{}_{\mathit{\theta }}}^{(\mathrm{o})}=A_{\mathrm{o}}+\frac{B_{\mathrm{o}}}{r^2}\\ {\mathit{\epsilon }}_{{}_x}^{(\mathrm{o})}=\mathit{\epsilon }\end{array}\right.$

12

Substituting the strain components from equations (11) and (12) into constitutive equations for inner core (i) and outside tube (o) in equation (2) and after some mathematical manipulations, we obtain

$\left\{\begin{array}{c}{\mathit{\sigma }}_x^{(\mathrm{i})}=C_{11}^{(\mathrm{i})}{\mathit{\epsilon }}^{(\mathrm{i})}+2C_{12}^{(\mathrm{i})}{A}_{\mathrm{i}}\\ {\mathit{\sigma }}_{\mathit{\theta }}^{(\mathrm{i})}=C_{12}^{(\mathrm{i})}{\mathit{\epsilon }}^{(\mathrm{i})}+(C_{11}^{(\mathrm{i})}+C_{12}^{(\mathrm{i})})A_{\mathrm{i}}\\ {\mathit{\sigma }}_r^{(\mathrm{i})}=C_{12}^{(\mathrm{i})}{\mathit{\epsilon }}^{(\mathrm{i})}+(C_{11}^{(\mathrm{i})}+C_{12}^{(\mathrm{i})})A_{\mathrm{i}}\end{array}\right.$

13

$\left\{\begin{array}{c}{\mathit{\sigma }}_x^{(\mathrm{o})}=C_{11}^{(\mathrm{o})}{\mathit{\epsilon }}^{(\mathrm{o})}+2C_{12}^{(\mathrm{o})}{A}_{\mathrm{o}}\\ {\mathit{\sigma }}_{\mathit{\theta }}^{(\mathrm{o})}=C_{12}^{(\mathrm{o})}{\mathit{\epsilon }}^{(\mathrm{o})}+\left(C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}\right)A_{\mathrm{o}}+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)\frac{B_{\mathrm{o}}}{r^2}\\ {\mathit{\sigma }}_r^{(\mathrm{o})}=C_{12}^{(\mathrm{o})}{\mathit{\epsilon }}^{(\mathrm{o})}+\left(C_{22}^{(o)}+C_{23}^{(\mathrm{o})}\right)A_{\mathrm{o}}+\left(C_{23}^{(\mathrm{o})}-C_{22}^{(\mathrm{o})}\right)\frac{B_{\mathrm{o}}}{r^2}\end{array}\right.$

14

The interface conditions at r = b are

$w^{(\mathrm{o})}(b)=w^{(\mathrm{i})}(b)$

15a

$u^{(\mathrm{o})}(b)=u^{(\mathrm{i})}(b)$

15b

${\mathit{\sigma }}_r^{(\mathrm{o})}(b)={\mathit{\sigma }}_r^{(\mathrm{i})}(b)$

15c

The boundary condition at r = c is free and can be written as the following

${\mathit{\sigma }}_r^{(\mathrm{o})}(c)=0$

16

The condition given in equation (15b) has already been satisfied (i.e. ${\mathit{\epsilon }}^{(\mathrm{i})}={\mathit{\epsilon }}^{(\mathrm{o})}=\mathit{\epsilon }=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$ ). Replacing the radial displacement expressions of the inside and outside materials (i.e. equations (6) and (10)) for r = b into the interface condition given in equation (15a) yields

$A_{\mathrm{i}}b=A_{\mathrm{o}}b+\frac{B_{\mathrm{o}}}{b}$

17

By substituting the stresses from equations (13) and (14) into conditions (15c) and (16), a system of two equations that can be used along with equation (17) to obtain the three unknown constants A _o, B _o, and A _i (considering ${\mathit{\epsilon }}_x^{(\mathrm{i})}={\mathit{\epsilon }}_x^{(\mathrm{o})}=\mathit{\epsilon }$ ) can be yielded. The three unknown constants can now be determined from the equations in terms of ε (i.e. axial strain), b, and c, and the material constants of the inside core and outside tube as follows

$\begin{array}{cc}\frac{A_{\mathrm{i}}}{\mathit{\epsilon }}=& \left(\frac{C_{12}^{(\mathrm{o})}}{\left(C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}\right)+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)(b^2/c^2)}+\frac{C_{12}^{(\mathrm{i})}}{\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)\left(1-(b^2/c^2)\right)}\right)\\ & \times {\left(\frac{\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)(b^2/c^2)}{\left(C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}\right)+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)(b^2/c^2)}-\frac{\left(C_{11}^{(\mathrm{i})}+C_{12}^{(\mathrm{i})}\right)+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)\left(1-(b^2/c^2)\right)}{\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)\left(1-(b^2/c^2)\right)}\right)}^{-1}\end{array}$

18

$\begin{array}{cc}\frac{A_{\mathrm{o}}}{\mathit{\epsilon }}=& \left(\frac{\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)C_{12}^{(\mathrm{o})}(b^2/c^2)}{\left(C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}\right)+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)(b^2/c^2)}+\frac{C_{12}^{(\mathrm{i})}(b^2/c^2)}{\left(1-(b^2/c^2)\right)}\right)\\ & \times {\left(\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)\frac{b^2}{c^2}-\frac{\left(\left(C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}\right)+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)(b^2/c^2)\right)\left[\left(C_{11}^{(\mathrm{i})}+C_{12}^{(\mathrm{i})}\right)+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)\left(1-(b^2/c^2)\right)\right]}{\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)\left(1-(b^2/c^2)\right)}\right)}^{-1}\\ & -\frac{C_{12}^{(\mathrm{o})}}{\left(C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}\right)+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)\frac{b^2}{c^2}}\end{array}$

19

$\begin{array}{cc}\frac{B_{\mathrm{o}}}{\mathit{\epsilon }}=& b^2\left(\frac{\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)C_{12}^{(\mathrm{o})}}{\left(C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}\right)+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)(b^2/c^2)}+\frac{C_{12}^{(\mathrm{i})}}{\left(1-(b^2/c^2)\right)}\right)\left(C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}\right)\\ & \times {\left\{{\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)}^2\frac{b^2}{c^2}-\frac{\left(\left(C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}\right)+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)(b^2/c^2)\right)\left[\left(C_{11}^{(\mathrm{i})}+C_{12}^{(\mathrm{i})}\right)+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)\left(1-(b^2/c^2)\right)\right]}{\left(1-(b^2/c^2)\right)}\right\}}^{-1}\\ & +\left(\frac{c^2{C}_{12}^{(\mathrm{o})}}{C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}}\right)\left[1-\frac{\left(C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}\right)}{\left(C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}\right)+\left(C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}\right)(b^2/c^2)}\right]\end{array}\times$

20

The applied axial load F (see Figure 2) and the effective axial Young’s modulus, $E_{11}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ can be calculated from the following equations

$F=2\mathit{\pi }\left[\underset{0}{\overset{b}{\int }}{\mathit{\sigma }}_x^{(\mathrm{i})}rdr+\underset{b}{\overset{c}{\int }}{\mathit{\sigma }}_x^{(\mathrm{o})}rdr\right]$

21

$E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{{\mathit{\sigma }}_x}{{\mathit{\epsilon }}_x}=\frac{F/\mathit{\pi }{c}^2}{\mathit{\epsilon }}$

22

Substituting stresses ${\mathit{\sigma }}_x^{(\mathrm{i})}$ and ${\mathit{\sigma }}_x^{(\mathrm{o})}$ from equations (13) and (14) into equation (21) and then into equation (22) yields

$E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{b^2}{c^2}\left(C_{11}^{(\mathrm{i})}+2C_{12}^{(\mathrm{i})}\frac{A_{\mathrm{i}}}{\mathit{\epsilon }}\right)+\left(1-\frac{b^2}{c^2}\right)\left(C_{11}^{(\mathrm{o})}+2C_{12}^{(\mathrm{o})}\frac{A_{\mathrm{o}}}{\mathit{\epsilon }}\right)$

23

In order to obtain the final expression for $E_{11}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , one can substitute equations (18) and (19) into equation (23). Note that the volume fraction of the outside tube and the inside core appears explicitly in equation (23) as the follows

${\mathit{\gamma }}_{(\mathrm{o})}=1-\frac{b^2}{c^2}; {\mathit{\gamma }}_{(\mathrm{i})}=\frac{b^2}{c^2}$

24

Equation (23) can be written in the following form

$E_{11}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\gamma }}_{(\mathrm{i})}{E}^{(\mathrm{i})}\left(1+\mathit{\alpha }\right)+{\mathit{\gamma }}_{(\mathrm{o})}{E}_{11}^{(\mathrm{o})}\left(1+\mathit{\beta }\right)$

25

where parameters $\mathit{\alpha }$ and $\mathit{\beta }$ can be calculated with the use of equations (18) and (19). Note that for the isotropic material of the core, we have ²¹

$\left\{\begin{array}{c}{C}_{11}^{(\mathrm{i})}=\frac{E^{(\mathrm{i})}(1-{\mathit{\nu }}^{(\mathrm{i})})}{(1+{\mathit{\nu }}^{(\mathrm{i})})(1-2{\mathit{\nu }}^{(\mathrm{i})})}\\ C_{12}^{(\mathrm{i})}=\frac{{\mathit{\nu }}^{(\mathrm{i})}{E}^{(\mathrm{i})}}{(1+{\mathit{\nu }}^{(\mathrm{i})})(1-2{\mathit{\nu }}^{(\mathrm{i})})}\end{array}\right.$

26

For the transversely isotropic outside tube, we can write

$E_{11}=E_{11}^{(\mathrm{o})}; E_{22}=E_{22}^{(\mathrm{o})}; {\mathit{\nu }}_{12}={\mathit{\nu }}_{12}^{(\mathrm{o})}; {\mathit{\nu }}_{23}={\mathit{\nu }}_{23}^{(\mathrm{o})}; {\mathit{\nu }}_{12}={\mathit{\nu }}_{13}={\mathit{\nu }}_{12}^{(\mathrm{o})}={\mathit{\nu }}_{13}^{(\mathrm{o})}$

27

Coefficients $C_{11}^{(\mathrm{o})},C_{12}^{(\mathrm{o})},C_{22}^{(\mathrm{o})}, \mathrm{a}\mathrm{n}\mathrm{d} C_{23}^{(\mathrm{o})}$ can be expressed in terms of $E_{11},E_{22},{\mathit{\nu }}_{12},$ and ${\mathit{\nu }}_{23}$ as follows ²¹

$\left[\begin{array}{ccc}{C}_{11}^{(\mathrm{o})}& C_{12}^{(\mathrm{o})}& C_{12}^{(\mathrm{o})}\\ C_{12}^{(\mathrm{o})}& C_{22}^{(\mathrm{o})}& C_{23}^{(\mathrm{o})}\\ C_{12}^{(\mathrm{o})}& C_{23}^{(\mathrm{o})}& C_{22}^{(\mathrm{o})}\end{array}\right]={\left[\begin{array}{ccc}{S}_{11}& S_{12}& S_{12}\\ S_{12}& S_{22}& S_{23}\\ S_{12}& S_{23}& S_{22}\end{array}\right]}^{-1}$

28

where $S_{11}=1/E_{11},S_{22}=1/E_{22},S_{12}=-{\mathit{\nu }}_{12}/E_{11},$ and $S_{23}=-{\mathit{\nu }}_{23}/E_{22}$ .

Substituting formulae for $S_{11},S_{22},S_{12},$ and $S_{23}$ into equation (28), we obtain the following formulae for transversely isotropic materials

$\left\{\begin{array}{c}{C}_{11}^{(\mathrm{o})}=\frac{E_{11}^{(\mathrm{o})}\left(1-{\mathit{\nu }}_{23}^{(\mathrm{o})}\right)}{1-{\mathit{\nu }}_{23}^{(\mathrm{o})}-2{\mathit{\nu }}_{12}^{(\mathrm{o})}{\mathit{\nu }}_{21}^{(\mathrm{o})}}\\ C_{12}^{(\mathrm{o})}=\frac{{\mathit{\nu }}_{12}^{(\mathrm{o})}{E}_{22}^{(\mathrm{o})}}{1-{\mathit{\nu }}_{23}^{(\mathrm{o})}-2{\mathit{\nu }}_{12}^{(\mathrm{o})}{\mathit{\nu }}_{21}^{(\mathrm{o})}}\\ C_{22}^{(\mathrm{o})}=\frac{(1-{\mathit{\nu }}_{12}^{(\mathrm{o})}{\mathit{\nu }}_{21}^{(\mathrm{o})})E_{22}^{(\mathrm{o})}}{(1+{\mathit{\nu }}_{23}^{(\mathrm{o})})(1-{\mathit{\nu }}_{23}^{(\mathrm{o})}-2{\mathit{\nu }}_{12}^{(\mathrm{o})}{\mathit{\nu }}_{21}^{(\mathrm{o})})}\\ C_{23}^{(\mathrm{o})}=\frac{({\mathit{\nu }}_{12}^{(\mathrm{o})}{\mathit{\nu }}_{21}^{(\mathrm{o})}+{\mathit{\nu }}_{23}^{(\mathrm{o})})E_{22}^{(\mathrm{o})}}{(1+{\mathit{\nu }}_{23}^{(\mathrm{o})})(1-{\mathit{\nu }}_{23}^{(\mathrm{o})}-2{\mathit{\nu }}_{12}^{(\mathrm{o})}{\mathit{\nu }}_{21}^{(\mathrm{o})})}\end{array}\right.$

29

Also, note that ²¹

$\frac{{\mathit{\nu }}_{21}}{E_{22}}=\frac{{\mathit{\nu }}_{12}}{E_{11}}; {\mathit{\nu }}_{12}{\mathit{\nu }}_{21}=\frac{E_{22}}{E_{11}}{\mathit{\nu }}_{12}^2$

30

and

$\left\{\begin{array}{c}{C}_{11}^{(\mathrm{i})}=E^{(\mathrm{i})}+2{\mathit{\nu }}^{(\mathrm{i})}{C}_{12}^{(\mathrm{i})}; {\mathit{\nu }}^{(\mathrm{i})}=\frac{2C_{12}^{(\mathrm{i})}}{C_{11}^{(\mathrm{i})}+C_{12}^{(\mathrm{i})}}\\ C_{11}^{(\mathrm{o})}=E_{11}^{(\mathrm{o})}+2{\mathit{\nu }}_{12}^{(\mathrm{o})}{C}_{12}^{(\mathrm{o})}; {\mathit{\nu }}_{12}^{(\mathrm{o})}=\frac{C_{12}^{(\mathrm{o})}}{C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}}\end{array}\right.$

31

From equations (23), (25), (26), and (31), we obtain the following expressions for $\mathit{\alpha }$ and $\mathit{\beta }$ entering equation (25) (note that $E^{(\mathrm{i})} \mathrm{a}\mathrm{n}\mathrm{d} E_{11}^{(\mathrm{o})}$ are given in equation (31))

$\left\{\begin{array}{c}\mathit{\alpha }=\frac{2C_{12}^{(\mathrm{i})}}{E^{(\mathrm{i})}}\left({\mathit{\nu }}^{(\mathrm{i})}+\frac{A_{(\mathrm{i})}}{\mathit{ϵ}}\right)\\ \mathit{\beta }=\frac{2C_{12}^{(\mathrm{o})}}{E_{11}^{(\mathrm{o})}}\left({\mathit{\nu }}_{12}^{(\mathrm{o})}+\frac{A_{(\mathrm{o})}}{\mathit{\epsilon }}\right)\end{array}\right.$

32

where $A_{\mathrm{i}}/\mathit{ϵ} \mathrm{a}\mathrm{n}\mathrm{d} A_{\mathrm{o}}/\mathit{ϵ}$ are given in equations (18) and (19). For the inside isotropic core and outside transversely isotropic tube, we can write the following equations

$\left\{\begin{array}{c}{C}_{12}^{(\mathrm{i})}=2K^{(\mathrm{i})}{\mathit{\nu }}^{(\mathrm{i})}\\ C_{11}^{(\mathrm{i})}+C_{12}^{(\mathrm{i})}=2K^{(\mathrm{i})}\end{array}\right.$

33

$\left\{\begin{array}{c}{C}_{12}^{(\mathrm{o})}=2K_{23}^{(\mathrm{o})}{\mathit{\nu }}_{12}^{(\mathrm{o})}\\ C_{22}^{(\mathrm{o})}+C_{23}^{(\mathrm{o})}=2K_{23}^{(\mathrm{o})}\\ C_{22}^{(\mathrm{o})}-C_{23}^{(\mathrm{o})}=2G_{23}^{(\mathrm{o})}\\ C_{11}^{(\mathrm{o})}=E_{11}^{(\mathrm{o})}+4K_{23}^{(\mathrm{o})}{\left({\mathit{\nu }}_{12}^{(\mathrm{o})}\right)}^2\end{array}\right.$

34

where the definition of the new parameters in the above equations (in terms of engineering material properties, E and ν; plain strain bulk modulus, K; and shear modulus, G) are as follows

$\left\{\begin{array}{c}{K}^{(\mathrm{i})}=\frac{G^{(\mathrm{i})}}{(1-2{\mathit{\nu }}^{(\mathrm{i})})}=\frac{E^{(\mathrm{i})}}{2(1+{\mathit{\nu }}^{(\mathrm{i})})(1-2{\mathit{\nu }}^{(\mathrm{i})})}\\ G_{23}^{(\mathrm{o})}=\frac{E_{22}}{2(1+{\mathit{\nu }}_{23}^{(\mathrm{o})})}\\ K_{23}^{(\mathrm{o})}=\frac{E_{22}^{(\mathrm{o})}}{2(1-{\mathit{\nu }}_{23}^{(\mathrm{o})}-2{\mathit{\nu }}_{12}^{(\mathrm{o})}{\mathit{\nu }}_{21}^{(\mathrm{o})})}\end{array}\right.$

35

Substituting equations (33) and (34) into equations (18) and (19), we obtain the following expressions for $A_{\mathrm{i}}/\mathit{\epsilon } \mathrm{a}\mathrm{n}\mathrm{d} A_{\mathrm{o}}/\mathit{\epsilon }$ in terms of shear modulus, G, bulk modulus, K, Poisson’s ratio, ν, and volume fractions, ${\mathit{\gamma }}_{(\mathrm{i})}$ , and ${\mathit{\gamma }}_{(\mathrm{o})}$

$\left\{\begin{array}{c}{\displaystyle \frac{A_{\mathrm{o}}}{\mathit{\epsilon }}=\frac{-2K_{23}^{(\mathrm{o})}{G}_{23}^{(\mathrm{o})}{\mathit{\nu }}_{12}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{o})}-G_{23}^{(\mathrm{o})}{C}_{12}^{(\mathrm{i})}{\mathit{\gamma }}_{(\mathrm{i})}-2K_{23}^{(\mathrm{o})}{\mathit{\nu }}_{12}^{(\mathrm{o})}{K}^{(\mathrm{i})}}{2K_{23}^{(\mathrm{o})}{G}_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{o})}+2(K_{23}^{(\mathrm{o})}+G_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{i})})K^{(\mathrm{i})}}}\\ {\displaystyle \frac{A_{\mathrm{i}}}{\mathit{\epsilon }}=\frac{-2K_{23}^{(\mathrm{o})}{G}_{23}^{(\mathrm{o})}{\mathit{\nu }}_{12}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{o})}-(K_{23}^{(\mathrm{o})}+G_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{i})})C_{12}^{(\mathrm{i})}}{2K_{23}^{(\mathrm{o})}{G}_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{o})}+2(K_{23}^{(\mathrm{o})}+G_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{i})})K^{(\mathrm{i})}}}\end{array}\right.$

36

Next, using equations (32) and (36), we obtain the following explicit formulae for α and β

$\left\{\begin{array}{c}\mathit{\alpha }={\displaystyle \frac{4K^{(\mathrm{i})}{\mathit{\nu }}^{(\mathrm{i})}}{E^{(\mathrm{i})}}\left(\frac{K_{23}^{(\mathrm{o})}{G}_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{o})}({\mathit{\nu }}^{(\mathrm{i})}-{\mathit{\nu }}_{12}^{(\mathrm{o})})}{K_{23}^{(\mathrm{o})}{G}_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{o})}+(K_{23}^{(\mathrm{o})}+G_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{i})})K^{(i)}}\right)}\\ \mathit{\beta }={\displaystyle \frac{4K_{23}^{(\mathrm{o})}{\mathit{\nu }}_{12}^{(\mathrm{o})}}{E_{11}^{(\mathrm{o})}}\left(\frac{K_{23}^{(\mathrm{o})}{G}_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{o})}({\mathit{\nu }}_{12}^{(\mathrm{o})}-{\mathit{\nu }}^{(\mathrm{i})}){\mathit{\nu }}_{12}^{(\mathrm{o})}}{K_{23}^{(\mathrm{o})}{G}_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{o})}{\mathit{\nu }}^{(\mathrm{i})}+(K_{23}^{(\mathrm{o})}+G_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{i})})K^{(\mathrm{i})}{\mathit{\nu }}^{(\mathrm{i})}}\right)}\end{array}\right.$

37

Substituting the expressions for $\mathit{\alpha }\mathrm{a}\mathrm{n}\mathrm{d}\mathit{\beta }$ from equation (37) into equation (25) and after some mathematical manipulations, we obtain the explicit formula for $E_{11}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ as follows

$E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\gamma }}_{(\mathrm{o})}{E}_{11}^{(\mathrm{o})}+{\mathit{\gamma }}_{(\mathrm{i})}{E}^{(\mathrm{i})}+\frac{4G_{23}^{(\mathrm{o})}{\mathit{\gamma }}_{(\mathrm{i})}{\mathit{\gamma }}_{(\mathrm{o})}{\left({\mathit{\nu }}^{(\mathrm{i})}-{\mathit{\nu }}_{12}^{(\mathrm{o})}\right)}^2}{(G_{23}^{(\mathrm{o})}/K^{(\mathrm{i})}){\mathit{\gamma }}_{(\mathrm{o})}+(G_{23}^{(\mathrm{o})}/K_{23}^{(\mathrm{o})}){\mathit{\gamma }}_{(\mathrm{i})}+1}.$

38

which is the exact analytical solution for the effective axial Young’s modulus of a tubular sandwich composite structure for the case of transversely isotropic outside tubular skin and isotropic inside core.

Reduction in $\left(E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}\right)$ from transversely isotropic to fully isotropic case

We will use the following notation

${\mathit{\gamma }}_{\mathrm{i}}=\frac{b^2}{c^2}, {\mathit{\gamma }}_{\mathrm{o}}=1-\frac{b^2}{c^2}$

39

where ${\mathit{\gamma }}_{\mathrm{i}}$ is the volume content of the inside core and ${\mathit{\gamma }}_{\mathrm{o}}$ is the volume content of the outside tubular skin. As a check for the verification of the analytical solution for the effective axial Young’s modulus, given by equation (38), we reduce this formula for a particular case when both the outside tube material and the inside core are isotropic. In this limiting simplest case, we have the following relationships connecting various material parameters

$\left\{\begin{array}{c}{G}_{23}^{(\mathrm{o})}=G_{\mathrm{o}}:K_{23}^{(\mathrm{o})}=K_{\mathrm{o}}=k_{\mathrm{o}}+\frac{G_{\mathrm{o}}}{3}\\ G^{(\mathrm{i})}=G_{\mathrm{i}}:K^{(\mathrm{i})}=K_{\mathrm{i}}=k_{\mathrm{i}}+\frac{G_{\mathrm{i}}}{3}\end{array}\right.$

40

where G is the shear modulus, $K$ is the plain strain bulk modulus and k is the bulk modulus. From the above relationships, the stiffness constants can also be written in the following forms

$\left\{\begin{array}{c}{C}_{11}=C_{22}=C_{33}=E+4{\mathit{\nu }}^2\left(k+\frac{G}{3}\right)\\ C_{12}=C_{13}=2\mathit{\nu }\left(k+\frac{G}{3}\right)\end{array}\right.$

41

Substituting the parameters given by equation (40) into equation (38), we obtain the effective Young’s modulus for the isotropic case as

$E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}=E_{\mathrm{i}}{\mathit{\gamma }}_{\mathrm{i}}+E_{\mathrm{o}}{\mathit{\gamma }}_{\mathrm{o}}+\frac{4G_{\mathrm{o}}{\mathit{\gamma }}_{\mathrm{i}}{\mathit{\gamma }}_{\mathrm{o}}{\left({\mathit{\nu }}_{\mathrm{i}}-{\mathit{\nu }}_{\mathrm{o}}\right)}^2}{\left(\frac{G_{\mathrm{o}}}{K_{\mathrm{i}}}{\mathit{\gamma }}_{\mathrm{o}}+\frac{G_{\mathrm{o}}}{K_{\mathrm{o}}}{\mathit{\gamma }}_{\mathrm{i}}+1\right)}$

42

equation (42) coincides with a known formula that will be discussed later in this work.

Determination of effective major Poisson’s ratio $\left({\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}\right)$

From the above formulated problem, we can also determine the effective major Poisson’s ratio ${\mathit{\nu }}_{13}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ for the transversely isotropic case. By definition, the effective major Poisson’s ratio of our tubular sandwich composite structure (see Figure 2) can be determined from the following equation (consistent with the definition used in the study by Christensen ¹³ )

${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}=-{\left.\frac{{\mathit{\epsilon }}_{33}}{{\mathit{\epsilon }}_{11}}\right|}_c=-\frac{w^{(\mathrm{o})}(c)/c}{\mathit{\epsilon }}=-\frac{w^{(\mathrm{o})}(c)}{c\mathit{\epsilon }}$

43

Note that ${\mathit{\nu }}_{13}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\nu }}_{12}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ in our case, as seen in Figure 3.

Figure 3.

Effective composite cylinder model of radius “c”.

Using the relationships given by equations (6), (17), and (43), and coordinates shown in Figure 3, we can obtain

${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\nu }}_{\mathrm{12}}^{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{A_{\mathrm{o}}}{\mathit{\epsilon }}\left(\frac{b^2}{c^2}-1\right)-\frac{A_{\mathrm{i}}}{\mathit{\epsilon }}\frac{b^2}{c^2}$

44

$A_{\mathrm{i}}/\mathit{\epsilon } \mathrm{a}\mathrm{n}\mathrm{d} A_{\mathrm{o}}/\mathit{\epsilon }$ can be replaced into equation (44) from equations (36) and after some mathematical manipulation, we obtain the following explicit formula for the ${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$

${\mathit{\nu }}_{13}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\nu }}_{12}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\nu }}_{12}^{(\mathrm{o})}{\mathit{\gamma }}_{\mathrm{o}}+{\mathit{\nu }}_{12}^{(\mathrm{i})}{\mathit{\gamma }}_{\mathrm{i}}+\frac{{\mathit{\gamma }}_{\mathrm{i}}{\mathit{\gamma }}_{\mathrm{o}}\left({\mathit{\nu }}_{12}^{(\mathrm{i})}-{\mathit{\nu }}_{12}^{(\mathrm{o})}\right)\left((G_{23}^{(\mathrm{o})}/K_{23}^{(\mathrm{o})})-(G_{23}^{(\mathrm{o})}/K^{(\mathrm{i})})\right)}{\left((G_{23}^{(\mathrm{o})}/K^{(\mathrm{i})}){\mathit{\gamma }}_{\mathrm{o}}+(G_{23}^{(\mathrm{o})}/K_{23}^{(\mathrm{o})}){\mathit{\gamma }}_{\mathrm{i}}+1\right)}$

45

which is the exact analytical solution for the effective major Poisson’s ratio of a tubular sandwich composite structure, where the outside tube material is transversely isotropic in the plane normal to the tube axis and the inside core material is isotropic.

Reduction in $\left({\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}\right)$ from transversely isotropic to fully isotropic case

Consider equation (45) in the limiting simplest case when both the inside core and outside tube materials are isotropic. Formula (45) after applying the isotropic relationships given by equation (40) reduces to the following form

${\mathit{\nu }}_{13}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\nu }}_{\mathrm{12}}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\gamma }}_{\mathrm{i}}{\mathit{\nu }}_{\mathrm{i}}+{\mathit{\gamma }}_{\mathrm{o}}{\mathit{\nu }}_{\mathrm{o}}+\frac{{\mathit{\gamma }}_{\mathrm{i}}{\mathit{\gamma }}_{\mathrm{o}}\left({\mathit{\nu }}_{\mathrm{i}}-{\mathit{\nu }}_{\mathrm{o}}\right)\left((G_{\mathrm{o}}/K_{\mathrm{o}})-(G_{\mathrm{o}}/K_{\mathrm{i}})\right)}{\left(({\mathit{\gamma }}_{\mathrm{i}}{G}_{\mathrm{o}}/K_{\mathrm{o}})+({\mathit{\gamma }}_{\mathrm{o}}{G}_{\mathrm{o}}/K_{\mathrm{i}})+1\right)}$

46

Verification of the analytical solutions

The following formulae are known in the case when both the inside core material and the outside tube material are isotropic ¹³

$E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}=E_{\mathrm{f}}{\mathit{\gamma }}_{\mathrm{f}}+E_{\mathrm{m}}{\mathit{\gamma }}_{\mathrm{m}}+\frac{4{\mathit{\mu }}_{\mathrm{m}}{\mathit{\gamma }}_{\mathrm{f}}{\mathit{\gamma }}_{\mathrm{m}}{\left({\mathit{\nu }}_{\mathrm{f}}-{\mathit{\nu }}_{\mathrm{m}}\right)}^2}{\left(({\mathit{\mu }}_{\mathrm{m}}/(k_{\mathrm{f}}+({\mathit{\mu }}_{\mathrm{f}}/3))){\mathit{\gamma }}_{\mathrm{m}}+({\mathit{\mu }}_{\mathrm{m}}/(k_{\mathrm{m}}+({\mathit{\mu }}_{\mathrm{m}}/3))){\mathit{\gamma }}_{\mathrm{f}}+1\right)}$

47

${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\nu }}_{\mathrm{12}}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\gamma }}_{\mathrm{f}}{\mathit{\nu }}_{\mathrm{f}}+{\mathit{\gamma }}_{\mathrm{m}}{\mathit{\nu }}_{\mathrm{m}}+\frac{{\mathit{\gamma }}_{\mathrm{f}}{\mathit{\gamma }}_{\mathrm{m}}({\mathit{\nu }}_{\mathrm{f}}-{\mathit{\nu }}_{\mathrm{m}})({\mathit{\mu }}_{\mathrm{m}}/(k_{\mathrm{m}}+({\mathit{\mu }}_{\mathrm{m}}/3)))-({\mathit{\mu }}_{\mathrm{m}}/(k_{\mathrm{f}}+({\mathit{\mu }}_{\mathrm{f}}/3)))}{(({\mathit{\mu }}_{\mathrm{m}}/(k_{\mathrm{f}}+({\mathit{\mu }}_{\mathrm{f}}/3))){\mathit{\gamma }}_{\mathrm{m}}+({\mathit{\mu }}_{\mathrm{m}}/(k_{\mathrm{m}}+({\mathit{\mu }}_{\mathrm{m}}/3))){\mathit{\gamma }}_{\mathrm{f}}+1)}$

48

where m, f, and μ stand for the outside matrix, inside fiber, and shear modulus, respectively. The similarities between equations (42) and (47) as well as between equations (46) and (48) validate the correctness of the above derived much more general formulae for the effective axial properties in the case of a transversely isotropic outside composite tube having an isotropic core material inside.

Finite element analysis of the tubular sandwich composite structure

Finite element method (FEM) is employed to model and analyze the performance of the tubular sandwich composite structure. The goal is to compare the results obtained analytically (in the previous sections) with those obtained numerically (in this section) to verify the accuracy of the analytical method. ANSYS ²² finite element software has been used to perform the load–displacement, strain, and stress analyses of the tubular sandwich composite structure under study. A three-dimensional model (i.e. one eighth of the full model due to the symmetry, shown in Figure 4) of a unidirectional composite tube filled with polycarbonate (or high density foam) is generated and solved for the displacements, strains, and stresses. The following dimensions have been used for the geometry of the composite tube

Figure 4.

Three-dimensional solid model of one eighth of the sandwich composite tube with an end plate glued to the loading end (front end in red). Symmetric boundary conditions are applied to the cut surfaces (left, bottom, and rear end in gray) and uniform axial loading (normal to the front end in red) are applied to the tubular sandwich composite structure.

$L=250 \mathrm{m}\mathrm{m}; b=12.5 \mathrm{m}\mathrm{m}; c=13.5 \mathrm{m}\mathrm{m}$

49

where ‘L’, ‘b’, and ‘c’ are the length, inner radius, and outer radius of a carbon/poly(ether–ether–ketone) (PEEK) unidirectional composite tube, respectively (see Figure 2). SOLID45 element, as a suitable candidate for orthotropic materials, was chosen from ANSYS finite element analysis (FEA) element library, and appropriated isotropic and transversely isotropic material properties were assigned to the inner core material and outer composite tube, respectively. To simulate the gripping condition of the loading side of the tube, a thin stiff plate (as a rigid element) was modeled and glued to the loading end of the tube, where the surface load is to be applied (as shown in Figure 4).

Isotropic and transversely isotropic material properties, that is, modulus and Poisson’s ratios, were used for the inner core material (“i”, two different types: polycarbonate ²³ and dense polyurethane foam ²⁴ ), outside unidirectional composite tube (carbon/PEEK, “o”, ²⁵ ), and the thin end plate, “p”; their values are given as follows

$\left\{\begin{array}{c}{E}^{(\mathrm{i})}(\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{c})=2.7 \mathrm{G}\mathrm{P}\mathrm{a}\\ {\mathit{\nu }}^{(\mathrm{i})}(\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{c})=0.42\end{array}\right.$

50

$\left\{\begin{array}{c}{E}^{(\mathrm{i})}(\mathrm{D}\mathrm{P}\mathrm{F})=0.3 \mathrm{G}\mathrm{P}\mathrm{a}\\ {\mathit{\nu }}^{(\mathrm{i})}(\mathrm{D}\mathrm{P}\mathrm{F})=0.4\end{array}\right.$

51

$\left\{\begin{array}{c}{E}_{11}^{(\mathrm{o})}=138 \mathrm{G}\mathrm{P}\mathrm{a}\\ E_{22}^{(\mathrm{o})}=10 \mathrm{G}\mathrm{P}\mathrm{a}\\ E_{33}^{(\mathrm{o})}=10 \mathrm{G}\mathrm{P}\mathrm{a}\end{array}\right.; \left\{\begin{array}{c}{G}_{12}^{\mathrm{o}}=5.65 \mathrm{G}\mathrm{P}\mathrm{a}\\ G_{23}^{\mathrm{o}}=3.70 \mathrm{G}\mathrm{P}\mathrm{a}\\ G_{13}^{\mathrm{o}}=5.65 \mathrm{G}\mathrm{P}\mathrm{a}\end{array}\right.; \left\{\begin{array}{c}{\mathit{\nu }}_{13}^{(\mathrm{o})}=0.28\\ {\mathit{\nu }}_{23}^{(\mathrm{o})}=0.33\\ {\mathit{\nu }}_{13}^{(\mathrm{o})}=0.28\end{array}\right.$

52

$\left\{\begin{array}{c}{E}^{(\mathrm{p})}=1000 \mathrm{T}\mathrm{P}\mathrm{a}\\ {\mathit{\nu }}^{(\mathrm{p})}=0.358\end{array}\right.$

53

In the above equations, the terms E, G, and ν are modulus of elasticity, shear modulus, and Poisson’s ratio, respectively. The superscripts ‘i’, ‘o’, and ‘p’ represent the inner core, outer tube, and thin plate, respectively. In equations (50) and (51), the terms (DPF) and (Polyc) refers to polycarbonate and dense polyurethane foam, respectively, and the subscripts 1, 2, and 3 in equation (52) stand for axial, circumferential, and radial directions, respectively. Next, the created volumes (i.e. the inner core, outer tube, and thin plate) in the FEA were meshed by considering a proper mesh element size and type. It should be noted that due to the symmetry of the problem, only one eighth of the structure is modeled. Symmetric boundary conditions were applied at one end (i.e. rear end in gray) as well as the cut sides of the structure (i.e. left and bottom sides in gray) with fixed corner nodes. A uniform surface load was applied to the other end (i.e. front end in red) that was glued to the stiff end plate (see Figure 4).

Finally, the generated finite element model was solved for different cases (considering two different core materials, that is, polycarbonate and DPF), and the results obtained were to be used for the calculation of the effective axial constitutive properties of the sandwich composite tube. Mesh convergence studies (with a convergence criterion of less than 1% on displacement) were performed by increasing the number of the finite elements (i.e. decreasing the mesh size). Figure 5 shows the plots of the axial, circumferential, and radial displacement contours of the polycarbonate core sandwich composite structure subjected to an axial uniform load of 1000 N, respectively. These figures show that the axial displacements are uniform at any cross section of the tube along its longitudinal direction, as one would expect. Also, the radial displacements, away from the tube ends, are uniform. The circumferential displacements are zero or near zero, consistent with the assumption made in the analytical solution. It should be mentioned that as an alternative FEM approach, the end rigid plate was removed and the model was solved with all nodes at the loading side coupled to move together in the axial direction. These alternative approaches, away from the end loading points, produced similar results that will be discussed later in this work.

Figure 5.

The contour plots of axial, circumferential, and radial displacements of the polycarbonate core sandwich composite tube under axial loading.

The displacement results (for both types of the core material) were further used to calculate the effective axial and radial strains, ε ₁₁ and ε ₃₃, of the sandwich composite tube. The effective axial Young’s modulus, $E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , and the effective major Poisson’s ratio, ${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\nu }}_{\mathrm{12}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , were also determined and are explained in the following section.

Calculations of $E_{11}^{eff}$ and ${\mathit{\nu }}_{13}^{eff} ={\mathit{\nu }}_{12}^{eff}$ from FEA

The maximum axial displacement, at the loading end, and radial displacement, at the surface, from the FEA as well as the dimensions of the sandwich composite tube from equation (49) were substituted into equation (54) to find the values of the axial and radial strains, ε ₁₁ and ε ₃₃, respectively. Next, the averaged axial stress, σ ₁₁, is obtained from equation (55). Finally, the strain and stress values were used in equation (56) to calculate the effective axial Young’s modulus, $E_{11}^{eff}$ , and effective major Poisson’s ratio, ${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\nu }}_{\mathrm{12}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , of the sandwich composite tube

${\mathit{\epsilon }}_{11}=\frac{U_{\mathrm{x}}}{L/2}; {\mathit{\epsilon }}_{33}=\frac{U_{\mathrm{r}}}{c}$

54

${\mathit{\sigma }}_{11}=\frac{F}{A}; A=\mathit{\pi }{c}^2$

55

$E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{{\mathit{\sigma }}_{11}}{{\mathit{\epsilon }}_{11}}; {\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{-{\mathit{\epsilon }}_{33}}{{\mathit{\epsilon }}_{11}}={\mathit{\nu }}_{\mathrm{12}}^{\mathrm{e}\mathrm{f}\mathrm{f}}(\mathrm{s}\mathrm{e}\mathrm{e} \mathrm{F}\mathrm{i}\mathrm{g}\mathrm{.} \mathrm{2})$

56

Here, U_x is the maximum axial displacement, U_r is the radial displacement, and A is the total cross-sectional area of the sandwich composite tube.

The following are the numerical values obtained from these calculations for the case of polycarbonate core sandwich composite tube

$\left\{\begin{array}{c}{\left({\mathit{\epsilon }}_{11}\right)}_{\mathit{P}\mathit{o}\mathit{l}\mathit{y}\mathit{c}}=0.\mathrm{3168}\mathrm{e}-\mathrm{3}\\ {\left({\mathit{\epsilon }}_{33}\right)}_{\mathit{P}\mathit{o}\mathit{l}\mathit{y}\mathit{c}}=0.\mathrm{1267}\mathrm{e}-\mathrm{3}\\ {\left({\mathit{\sigma }}_{11}\right)}_{\mathit{P}\mathit{o}\mathit{l}\mathit{y}\mathit{c}}=\mathrm{6}\mathrm{.9862}\mathrm{M}\mathrm{P}\mathrm{a}\\ E_{\mathrm{11}}^{\mathit{e}\mathit{f}\mathit{f}}{\left(\mathrm{F}\mathrm{E}\mathrm{A}\right)}_{\mathit{P}\mathit{o}\mathit{l}\mathit{y}\mathit{c}}=\mathrm{22}\mathrm{.0524} \mathrm{G}\mathrm{P}\mathrm{a}\\ {\mathit{\nu }}_{\mathrm{13}}^{\mathit{e}\mathit{f}\mathit{f}}{\left(\mathrm{F}\mathrm{E}\mathrm{A}\right)}_{\mathit{P}\mathit{o}\mathit{l}\mathit{y}\mathit{c}}=0.\mathrm{3998}\end{array}\right.$

57

Similarly, the following results are obtained for the DPF core sandwich composite tube

$\left\{\begin{array}{c}{\left({\mathit{\epsilon }}_{11}\right)}_{\mathit{D}\mathit{P}\mathit{F}}=0.\mathrm{3504}\mathrm{e}-\mathrm{3}\\ {\left({\mathit{\epsilon }}_{33}\right)}_{\mathit{D}\mathit{P}\mathit{F}}=0.\mathrm{1248}\mathrm{e}-\mathrm{3}\\ {\left({\mathit{\sigma }}_{11}\right)}_{\mathit{D}\mathit{P}\mathit{F}}=\mathrm{6}\mathrm{.9862}\mathrm{M}\mathrm{P}\mathrm{a}\\ E_{11}^{\mathit{e}\mathit{f}\mathit{f}}{\left(\mathit{F}\mathit{E}\mathit{A}\right)}_{\mathit{D}\mathit{P}\mathit{F}}=\mathrm{19}\mathrm{.9378} \mathrm{G}\mathrm{P}\mathrm{a}\\ {\mathit{\nu }}_{\mathrm{13}}^{\mathit{e}\mathit{f}\mathit{f}}{\left(\mathit{F}\mathit{E}\mathit{A}\right)}_{\mathit{D}\mathit{P}\mathit{F}}=0.\mathrm{3562}\end{array}\right.$

58

The comparison of the FEA final results (i.e. the effective axial Young’s modulus, $E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , and effective major Poisson’s ratio, ${\mathit{\nu }}_{\mathrm{12}}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ ) for the sandwich composite tube with both core types (given by equations (57) and (58)) with the Young’s modulus and Poisson’s ratios of the inner cores and outer composite tube (given by equations (50) to (52)) reveals that the FEA results are within the acceptable range and are in between the values of the individual inner cores and outer composite tube (i.e. $E^{(\mathrm{i})}<E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}<E_{\mathrm{11}}^{\mathrm{(}\mathrm{o}\mathrm{)}}$ and ${\mathit{\nu }}^{(\mathrm{i})}<{\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}<{\mathit{\nu }}_{13}^{(\mathrm{o})}$ ).

Comparison of $E_{\mathrm{11}}^{ \mathrm{e}\mathrm{f}\mathrm{f}}$ and ${\mathit{\nu }}_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ values obtained from the analytical solutions and FEA

The exact analytical solutions for the effective axial Young’s modulus and effective major Poisson’s ratio of the sandwich composite tube for the transversely isotropic outer tube material and isotropic inner core material have been derived above and given in equations (38) and (45). It should be noted that the $G_{23}^{(\mathrm{o})}$ terms in the analytical solution given in equations (38) and (45) is different from the modulus $G_{23}^{\mathrm{o}}$ of the carbon/PEEK composite tube given in equation (52).

To calculate the effective axial Young’s modulus and effective major Poisson’s ratio for the sandwich composite tube, first the dimensions and material properties given by equations (49) to (52) are used to calculate the parameters given in equations (35) and (39) (i.e. ${\mathit{\gamma }}_{\mathrm{i}},{\mathit{\gamma }}_0,G_{23}^{(\mathrm{o})},K^{(\mathrm{i})},K_{23}^{(\mathrm{o})}$ ), which are then substituted into the final exact analytical solutions of $E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ and ${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ given by equations (38) and (45). MAPLE 7.0, a mathematical manipulation software, ²⁶ is used to handle these mathematical operations. Finally, the following results from the exact analytical solutions have been obtained

$\left\{\begin{array}{c}{E}_{\mathrm{11}}^{\mathit{e}\mathit{f}\mathit{f}}(\mathit{A}\mathit{n}\mathit{a}\mathit{l}\mathit{y}\mathit{t}\mathit{i}\mathit{c}\mathit{a}\mathit{l}{)}_{\mathit{P}\mathit{o}\mathit{l}\mathit{y}\mathit{c}}=\mathrm{22}\mathrm{.0265} \mathrm{G}\mathrm{P}\mathrm{a}\\ {\mathit{\nu }}_{\mathrm{13}}^{\mathit{e}\mathit{f}\mathit{f}}(\mathit{A}\mathit{n}\mathit{a}\mathit{l}\mathit{y}\mathit{t}\mathit{i}\mathit{c}\mathit{a}\mathit{l}{)}_{\mathit{P}\mathit{o}\mathit{l}\mathit{y}\mathit{c}}=0.\mathrm{4028}\\ E_{\mathrm{11}}^{\mathit{e}\mathit{f}\mathit{f}}(\mathit{A}\mathit{n}\mathit{a}\mathit{l}\mathit{y}\mathit{t}\mathit{i}\mathit{c}\mathit{a}\mathit{l}{)}_{\mathit{D}\mathit{P}\mathit{F}}=\mathrm{19}\mathrm{.9586} \mathrm{G}\mathrm{P}\mathrm{a}\\ {\mathit{\nu }}_{\mathrm{13}}^{\mathit{e}\mathit{f}\mathit{f}}(\mathit{A}\mathit{n}\mathit{a}\mathit{l}\mathit{y}\mathit{t}\mathit{i}\mathit{c}\mathit{a}\mathit{l}{)}_{\mathit{D}\mathit{P}\mathit{F}}=0.\mathrm{3632}\end{array}\right.$

59

It can be seen that these values are very close to those obtained from the FEA given by equations (57) and (58). This comparison shows that both the techniques are accurate and predict similar effective properties for the sandwich composite tube with relative errors of less than 0.12% for the effective Young’s modulus and less than 1.9% for the major Poisson’s ratio as shown below

$\left\{\begin{array}{c}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{f}{E}_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}(\mathrm{A}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{F}\mathrm{E}\mathrm{A}{)}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{c}}=\mathrm{0}\mathrm{.117}\mathrm{\%}\\ \mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{f}{\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}(\mathrm{A}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{F}\mathrm{E}\mathrm{A}{)}_{\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{c}}=0.\mathrm{730}\mathrm{\%}\\ \mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{f}{E}_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}(\mathrm{A}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{F}\mathrm{E}\mathrm{A}{)}_{\mathrm{D}\mathrm{P}\mathrm{F}}=\mathrm{0}\mathrm{.104}\mathrm{\%}\\ \mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{f}{\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}(\mathrm{A}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{F}\mathrm{E}\mathrm{A}{)}_{\mathrm{D}\mathrm{P}\mathrm{F}}=\mathrm{1}\mathrm{.914}\mathrm{\%}\end{array}\right.$

60

Therefore, an excellent agreement is achieved between the results obtained from the exact analytical solutions and the FEA introduced in this work. As mentioned earlier, a similar structure without the stiff end plate, but instead with coupled nodes at the loading end of the composite cylinder, was also modeled and solved in ANSYS FEA and produced similar results. The calculated effective longitudinal Young’s modulus, $E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , and effective major Poisson’s ratio, ${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , from the coupled end nodes FEA were more accurate and had less relative error percentage, that is, 0.02% and 0.07%, respectively, for the coupled-nodes case as opposed to 0.117% and 0.730% for the stiff plate, respectively, for the polycarbonate core, when the FEA results are compared with the analytical results.

Parametric study of the analytical solutions for the transversely isotropic tubular sandwich composite tube

The exact analytical solutions for $E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ and ${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ for the transversely isotropic tubular sandwich composite structures are derived in this work. They are verified by the comparison of the analytical and numerical results calculated for b (core radius) = 12.5 mm and c (tube outer radius) = 13.5 mm. In this section, the effects of the geometry (i.e. the volume fraction of the core, i.e. γ _i = b ² /c ² ) and material property variations on the effective properties are studied. Figure 6(a) shows the effect of the volume content, γ _i, of the core material (i.e. polycarbonate) on the effective Young’s modulus, $E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , of the sandwich composite tube. As one would expect, $E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ of the sandwich composite tube varies linearly with γ _i, and it represents the longitudinal Young’s modulus of the individual outside composite tube, that is, 1.38e + 11 Pa (see equation (52)), and that for the polycarbonate core, that is, 0.27e + 10 Pa (see equation (50)), when the volume content of the core material is 0 and 1, respectively. Note that for this parametric study, the sandwich composite tube with the polycarbonate core alone is considered, and hence the core material type throughout this section will be the polycarbonate material with properties given by equation (50).

Figure 6.

Effect of the volume content of the core material on (a) the effective longitudinal Young’s modulus and (b) the major Poisson’s ratio of the sandwich composite tube.

Similarly, Figure 6(b) gives the effect of the volume fraction of the core, γ _i, on the effective major Poisson’s ratio, ${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , of the sandwich composite tube. As seen in Figure 6(b), the relationship between the effective major Poisson’s ratio, ${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , and the volume content of the core material, γ _i, is linear. In addition, the values of the effective major Poisson’s ratio at core volume contents of 0 and 1 coincide with the individual Poisson’s ratios of the composite tube, that is 0.28 (see equation (52)), and the polycarbonate core, that is 0.42 (see equation (50)), respectively.

These graphs demonstrate that the analytical solutions derived in this work can be employed for design purposes of similar two-phase composite cylindrical structures. It is now readily possible to determine the required volume percentage of the transversely isotropic constituent materials to obtain a two-phase composite cylindrical structure with desired mechanical properties. It should be noted that these formulae are applicable to any similar structures with material anisotropy levels up to transversely isotropic for the outside tube and isotropic for the inside core and is a valuable tool for design purposes of composite tubular structures.

To investigate the influence of the outside material properties on the effective properties of the two-phase composite cylinder, parametric studies based on the variations in the material properties of the outside tube are carried out and discussed. For this purpose, all five independent transversely isotropic mechanical properties of the outside composite tube (i.e. $E_{11}^{(\mathrm{o})},E_{22}^{(\mathrm{o})}=E_{33}^{(\mathrm{o})},G_{12}^{\mathrm{o}}=G_{13}^{\mathrm{o}},{\mathit{\nu }}_{12}^{(\mathrm{o})}={\mathit{\nu }}_{13}^{(\mathrm{o})},{\mathit{\nu }}_{23}^{(\mathrm{o})},$ and $G_{23}^{\mathrm{o}}=E_{22}^{(\mathrm{o})}/2(1+{\mathit{\nu }}_{23}^{(\mathrm{o})})$ ) are simultaneously varied from their original values (see equation (52)) down to the properties of the core material (see equation (50)) and the effective longitudinal Young’s modulus, $E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , and major Poisson’s ratio, ${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ , are determined as the functions of the outside composite material properties variation. To carry out these parametric studies, the following matrix equation is introduced to simultaneously vary the properties of the outside composite tube, as described above

$\begin{array}{rl}& \left[P_{ij}^{(\mathrm{o})}\right]=\left[a_{ij}\right]+\left[b_{ij}\right]\mathit{\zeta }; \left(0\le \mathit{\zeta }\le 1\right)\\ & \left[P_{ij}^{(\mathrm{o})}\right]=\left[\begin{array}{ccc}{E}_{11}^{(\mathrm{o})}& G_{12}^{\mathrm{o}}& {\mathit{\nu }}_{12}^{(\mathrm{o})}\\ E_{22}^{(\mathrm{o})}& G_{13}^{\mathrm{o}}& {\mathit{\nu }}_{13}^{(\mathrm{o})}\\ E_{33}^{(o)}& G_{23}^o& {\mathit{\nu }}_{23}^{(o)}\end{array}\right]; \left[a_{ij}\right]=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]; \left[b_{ij}\right]=\left[\begin{array}{ccc}{b}_{11}& b_{12}& b_{13}\\ b_{21}& b_{22}& b_{23}\\ b_{31}& b_{32}& b_{33}\end{array}\right]\end{array}$

61

In equation (61), $\left[P_{ij}^{(\mathrm{o})}\right]$ denotes the properties of the outside composite tube material, ζ is the nondimensional variable factor, and [a_ij ] and [b_ij ] are constant coefficients that can be calculated from the following conditions

$\left[P_{ij}^{(\mathrm{o})}\right]=\left\{\begin{array}{c}\left[P_{ij}^{(\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{e})}\right];\mathit{\zeta }=0\\ \left[P_{ij}^{(\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e})}\right];\mathit{\zeta }=1\end{array}\right.$

62

Note that $\left[P_{ij}^{(\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{e})}\right]$ and $\left[P_{ij}^{(\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e})}\right]$ are mechanical properties matrices of the inside core material, that is equation (50), and outside composite tube, that is equation (52). Therefore, if one varies the nondimensional variable ζ from one to zero, all five independent transversely isotropic properties of outside composite tube will simultaneously vary from the carbon/PEEK composite skin properties to the polycarbonate core properties. Figure 7 shows the variations in the effective longitudinal Young’s modulus and major Poisson’s ratio of the two-phase sandwich composite tube as the functions of the variations in the mechanical properties of the outside composite tube.

Figure 7.

Variations in (a) the effective longitudinal Young’s modulus and (b) the major Poisson’s ratio of the sandwich composite tube versus the variation in the composite tube properties.

Both of these graphs demonstrate linear behavior. As one would expect, both graphs show that the effective properties of the sandwich composite tube are identical to the individual properties of the polycarbonate core, when properties of the core material is assigned to the outer composite tube, and identical to the effective properties of the sandwich composite tube, when the carbon/PEEK properties are assigned to the outer composite tube. The results presented in this section (i.e. Figures 6 and 7) can also be used for proper geometry and materials selection for the design of composite cylindrical structures with desired effective properties.

Conclusions

The exact analytical formulae for transversely isotropic effective longitudinal Young’s modulus and effective major Poisson’s ratio of tubular sandwich composite structure are derived and introduced here. In this work, a two-phase cylinder model is considered with a transversely isotropic outer composite tube material and an isotropic inner cylindrical core. These analytical solutions can be used to obtain the effective axial properties of any such tubes and fillers with known material properties. For the validation purposes, an FEA is performed to numerically investigate the responses of the sandwich composite tube under an axial loading where the displacement results are used to find the values of the effective axial Young’s modulus and effective major Poisson’s ratio. Next, the material properties of the individual inside core and outside composite tube, used in the FEA of the sandwich composite tube, are substituted into the exact analytical solutions to compute the effective material properties $E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ and ${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ . Finally, the results from the exact analytical solutions are compared with those obtained from the FEM. The comparison of the results reveals that both the techniques produce results that are in excellent agreement. Note that the exact analytical solutions are applicable to any such structure with a transversely isotropic outside tube and an isotropic inside core. These solutions can also be reduced to the limiting particular case of isotropic outside tubes with isotropic inside cores. In this case, the formulae converge to previously known results. To investigate the effects of the volume fraction and material properties of the constituents, parametric studies have also been conducted and the results are presented and discussed. The variations in these volume fractions and material properties primarily have linear effects on $E_{\mathrm{11}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ and ${\mathit{\nu }}_{\mathrm{13}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$ .