Mechanical characterization of a woven multi-layered hyperelastic composite laminate under uniaxial loading

Abstract

We characterize the material properties of a woven, multi-layered, hyperelastic composite that is useful as an envelope material for high-altitude stratospheric airships and in the design of other large structures. The composite was fabricated by sandwiching a polyaramid Nomex® core, with good tensile strength, between polyimide Kapton® films with high dielectric constant, and cured with epoxy using a vacuum bagging technique. Uniaxial mechanical tests were used to stretch the individual materials and the composite to failure in the longitudinal and transverse directions respectively. The experimental data for Kapton® were fit to a five-parameter Yeoh form of nonlinear, hyperelastic and isotropic constitutive model. Image analysis of the Nomex® sheets, obtained using scanning electron microscopy, demonstrate two families of symmetrically oriented fibers at 69.3°± 7.4° and 129°± 5.3°. Stress-strain results for Nomex® were fit to a nonlinear and orthotropic Holzapfel-Gasser-Ogden (HGO) hyperelastic model with two fiber families. We used a linear decomposition of the strain energy function for the composite, based on the individual strain energy functions for Kapton® and Nomex®, obtained using experimental results. A rule of mixtures approach, using volume fractions of individual constituents present in the composite during specimen fabrication, was used to formulate the strain energy function for the composite. Model results for the composite were in good agreement with experimental stress-strain data. Constitutive properties for woven composite materials, combining nonlinear elastic properties within a composite materials framework, are required in the design of laminated pretensioned structures for civil engineering and in aerospace applications.

Keywords

Multi-layered woven composite orthotropy symmetry high altitude airship constitutive model rule of mixtures

Introduction

Woven composite materials, comprised of a laminated layer sandwiched between two functional coatings, have attractive attributes such as high strength, toughness, and are lightweight which permit wide usage in civil engineering and aerospace applications. Pretensioned membrane structures in building construction are dependent on the superior mechanical properties of composite materials in the design of long-span structures as canopy roofs. Laminated membrane materials are also employed in the hull or outer envelopes of high-altitude stratospheric airships (HAA) that help contain helium, serve as a protective coating from UV degradation, withstand challenging environments due to the varying temperatures and low pressure, and endure external aerodynamic loads during operation. The development of high strength and lightweight materials, combined with additional attributes of environmental resistance and low helium permeability, have garnered much attention in recent years in the design and fabrication of envelope materials for HAA.¹

Multi-layer flexible laminates in envelope construction are comprised of sandwich structures with tailored mechanical properties, desired gas retention ability, and environmental resistance. Woven single/multilayer synthetic fabrics are a popular choice for the hull materials due to their suitable strength to weight ratio. Polyamide (Nylon®), polyester (Dacron®/Kosa®), aramid (Kevlar®), liquid crystal (Vectran®), carbon (Thornel®), polyethylene (Spectra®), polyhydroquinone-di-imidazopyridine (M5®), and PBO (Zylon®) are some materials with superior strength, toughness, and weatherability that are suitable as envelope materials for HAA. Challenges with respect to UV resistance, difficulty in processing, low stretch, and moisture resistance continue to persist with some of these materials.¹ A combination of Nomex® as a core material, sandwiched between Kapton® HN films, is a promising recent direction in the development of HAA materials.² Nomex® is a polyaramid, consisting of orthotropic material symmetry, and has good flame resistance, thermal insulation and tensile strength. In contrast, Kapton® HN is an aromatic and isotropic polyimide with resistance to UV, has a high dielectric constant, and is resistant to helium leakage. These properties make Kapton® an attractive choice in the fabrication of the outer layers of the composite in HAA applications.

Kapton® films undergo large deformations and are described using hyperelastic models. Phenomenological constitutive models, relating the micro-scale to macro-scale responses in composites, are described using a strain energy density function (SEF) based on strain invariants that account for the existing material symmetries in the materials.^3,4 A constitutive theory for hyperelastic, fiber reinforced composites was first given assuming the fibers to be line elements and by applying kinematic constraints of fiber inextensibility and incompressibility.⁵ Several studies have applied the differential fiber layouts to model fiber reinforced composites based on transversely isotropic symmetry.⁶ Previous works by Polignone and Horgan,⁷ Qui and Pence,^8,9 Merodio and Pence,¹⁰ and Merodio and Ogden^11–14 modelled fiber reinforced composites with transverse isotropic symmetry using a form of strain energy function based on a sub-class of strain invariants, I₁ and I₄. Specifically, the strain energy function consisted of one term, corresponding to the matrix contribution and included as a neo-Hookean term with I₁ dependence, and a second that accounted for fiber reinforcement, in terms of invariant I₄, along a specific direction. Constitutive models, such as Holzapfel-Gasser-Ogden (HGO) for fiber reinforced tissues like arteries and the myocardium,^15,16 are also widely used and have been applied to glass fiber reinforced thermoplastic composites with transversely isotropic symmetries.¹⁷ These models have widespread application in modelling mechanical operations that warrant large deformations, such as hemispherical punch forming in glass fiber reinforced thermoplastic composites,¹⁸ double dome forming procedures,¹⁹ thermoforming processes using glass/polypropylene prepegs,²⁰ and the assembly process for non-rigid parts involving multilayered carbon fiber reinforced thermoplastics composites.²¹ Kinematic constraints of fiber inextensibility and incompressibility are useful to reduce the number of constants that are required to assess the form of the functional in fiber reinforced materials that undergo large deformations.²² Material incompressibility is applicable in rubber-like materials and tissues that demonstrate negligible volumetric changes under deformation.

We fabricated a multilayered composite using Kapton® films sandwiched between Nomex® calendared paper and tested the materials under uniaxial tensile loads. We used a five-parameter incompressible Yeoh model²³ to describe the constitutive properties of Kapton® based on the experimentally obtained data. Scanning electron microscopy images of Nomex® clearly show two symmetrically oriented fibers families with fiber angles 69.3°± 7.4° and 129°± 5.3° respectively. We used a modified form of the HGO model to describe the orthotropic material symmetry of Nomex®. Finally, we used the rule of mixtures to obtain the constitutive model for the Kapton®-Nomex®-Kapton® composite using the energy superposition principle. Unknown coefficients to the model were estimated using experimental data from uniaxial testing of the Kapton® films, Nomex® sheet, and the composite material. The methods established in this study are a first step to obtain the mechanical responses of composites undergoing large deformations and can also be extended to multifunctional materials and structures that show much potential in the areas of structural health monitoring and smart actuation of control surfaces.

Materials and methods

The materials used in this study are Kapton® HN polyimide film, Nomex® and a composite with Nomex® sandwiched as the core material between Kapton® films. Mechanical tests were conducted to characterize the stress – strain behaviors under uniaxial monotonic loading conditions.

Material symmetry characterization for Nomex®

Material characterization was performed using scanning electron microscopy (ULTRA 55, Zeiss, Germany) on a 0.5 mm thick Nomex® sheet cut using a high precision microtome. Figure 1(a) and (b) show representative micrographs, imaged under 324X and 744X magnification respectively, that clearly demonstrate two families of ∼20 µm diameter fibers with near symmetric layouts. The background matrix was excluded, the fiber images binarized, and the orientation angles were quantified using Hough transform (MATLAB R 2016 b, The Mathworks, Inc., Natik, MA). 30 binarized images of the sample at various locations were used to obtain the overall angular orientation vector, split into bins of size 5°, for Nomex®. The probability distribution of the fiber angles with orientation was plotted (Figure 1(c)) and the results fitted to a bimodal Gaussian distribution function (Figure 1(d)) for each of the two fiber families. We obtained an angular mean and standard error (SEM) to be 69.3°± 7.4° for one fiber family and 129°± 5.3° for the other family.

Fabrication of composites using Nomex® and Kapton®

Kapton® films were cut from a large sheet minimizing notches at the edges. Nomex® was cut in two orthogonal directions, as indicated by the supplier, corresponding to the longitudinal and transverse directions. Composites were fabricated by sandwiching Nomex® core between two sheets of Kapton® HN using epoxy (Araldite LY 5052: Epoxy Phenol Novolac resin mixed with a hardener HY 5052; Huntsman, Switzerland). The resin and hardener were mixed in 138:38 ratio using a glass rod in an acetone-cleaned container. Air bubbles were removed by applying vacuum for 5–10 minutes.

A smooth Teflon® surface, sprayed with mold release, was used to fabricate composites in this study. The Kapton® film was placed on the surface over which the epoxy solution was uniformly spread using a hand roller. The Nomex® film was gently placed using a hand lay-up technique and the same procedure repeated with a second Kapton® film. The setup was vacuum bagged to remove trapped air bubbles and the composite was cured by placing a 5 kg block for over 12 hours. The cured sandwich composite was cut along the longitudinal and transverse directions in Nomex® as described earlier. The volume fractions of Kapton®, $ν_{Kap},$ and Nomex®, $ν_{Nomx},$ were calculated using the specimen thickness that was measured using a micrometer. The volume fraction of glue was not considered in the estimation of mechanical properties of the composites.

ν_{Kap} = 0.0476; ν_{Nomx} = 0.9523

(1)

Aluminum end tabs were attached and used to grip samples to the stretcher (Figure 2). Sample dimensions and other relevant details are given in Table 1.

Table 1.

Sample test conditions used to test Kapton®, Nomex®, and composite specimens in the study.

	Kapton® HN	Nomex®	Composite
ASTM Standard	D882-12	D828-16	D828-16
Dimensions(mm³)	100 × 10 × 0.025	100 × 25.4 × 0.5	100 × 25.4 × 0.6
Clamped length (mm)	12	38.1	38.1
Arm speed(mm/sec)	0.8	0.23	0.23
Load cell capacity	25 lbf	25 kN	25. kN
(Make)	(Interface USA)	(BiSS)	(BiSS)
Dimensions of end tabs (in³)	Nil	1.5 × 1.0 × 0.0787 (Aluminum)	1.5 × 1.0 × 0.0787 (Aluminum)

Mechanical testing

The Nomex® and composite specimens were cut to specified dimensions in the longitudinal and transverse directions, as indicated by the supplier, for uniaxial tensile tests. Kapton® was tested using a planar biaxial testing machine (Bangalore Integrated System Solutions (P) Ltd, India) with custom made clamps and a 25 lbf load cell (Model: WMC-25, Interface Inc., Arizona, USA, 11340 gm). Nomex® and the composite were also tested on a uniaxial testing machine (Makron 25; Bangalore Integrated System Solutions (P) Ltd, India with 25 kN Model Bi-06-105 load cell). The test sample was clamped using a hydraulic grip system to apply a uniform pressure of 180–200 kPa and tested to failure (n = 3 for all groups) under displacement control using strain rates listed in Table 1.

The components of the first Piola-Kirchhoff stresses, $P_{11}$ , were computed using forces during extension, $F_{1},$ and the referential sample dimensions (thickness h, and width w).The corresponding Cauchy stresses, $σ_{11},$ were calculated using the applied stretches, $λ_{1},$ in the direction of uniaxial loading and are given by

P_{11} = \frac{F_{1}}{h * w} & σ_{11} = λ_{1} * P_{11}

(2)

Marker beads in the central gage region in the specimens were tracked and the Green-Lagrange strains were quantified using custom code based on iso-parametric mapping in MATLAB R2016b.^24,25 Poisson’s ratio was computed using the linear range (up to 4%) of the stress-strain curve for all the Nomex®, Kapton® and composite materials.

Constitutive model descriptions for finite elastic and incompressible materials in the study

The strain energy density function (SEF), $ψ,$ for a homogeneous, isotropic, hyperelastic material is given in terms of three strain invariants,

I_{1} (B) = t r (B); I_{2} (B) = \frac{1}{2} [{I_{1}}^{2} - t r {(B)}^{2}] {; I}_{3} (B) = \det (B)

(3)

where B is the left Cauchy-Green strain tensor (

B = {FF}^{T}

), F is the deformation gradient, tr is the trace, and det is the determinant of the tensor. The symbol ^T refers to the transpose of the tensor. The incompressibility constraint used to describe the constitutive properties of materials used in our study is given by

I_{3} (B) = 1

Various forms of $ψ$ have been used in the literature, such as the neo-Hookean, that consists of the term (I₁-3) alone.^26,27 The Yeoh model uses a truncated form of the polynomial in the neo-Hookean term; three or five terms are generally used to capture the moderate to large deformations in rubber-like materials.²³ We used a five-parameter incompressible Yeoh model, given below, to describe the material properties of Kapton® film in our study.

ψ_{Kap} = c_{1} (I_{1} - 3) + c_{2} {(I_{1} - 3)}^{2} + c_{3} {(I_{1} - 3)}^{3} + c_{4} {(I_{1} - 3)}^{4} + c_{5} {(I_{1} - 3)}^{5}

(4)

The constants, c_i, i = 1:5, were determined by fitting experimental results to the model.

We used an orthotropic material symmetry to describe the strain energy function for the Nomex® sheet based on the presence of two symmetrically placed fiber families along directions $a_{0}$ and $b_{0}$ . We reduced the functional dependence for orthotropic materials from nine invariants to a smaller subset that included I₁, I₂, I₄ and I₆. The pseudo-invariants, I₄ and I₆, are given by

I_{4} (B) = a_{0} . B a_{0}; I_{6} (B) = b_{0} . B b_{0}

(5)

We use an additive decomposition of the SEF based on an isotropic contribution from the matrix and an anisotropic component due to the presence of fibers.

ψ_{Nomx} = {\hat{ψ}}_{iso} + {\hat{ψ}}_{aniso}

(6)

Such a form of SEF is commonly used to describe arterial tissues, the myocardium, and other materials that have defined fibers in different orientations.^15,28

We used a modified HGO model to describe Nomex® given by

ψ_{Nom} = c_{6} (I_{1} - 3) + c_{7} (I_{2} - 3) + c_{8} (I_{1} - 3) (I_{2} - 3) + \frac{k_{1}}{2 k_{2}} \sum_{i = 4, 6} (\exp [k_{2} (I_{i} - 1)^{2}] - 1)

(7)

$c_{6}, c_{7}, c_{8},$ $k_{1}$ are material constants with dimensions of stress and $k_{2}$ is a non-dimensional constant. We have also included terms in I₂ and coupled terms in I₁ and I₂ that are essential to obtain a good fit to our experimental data; these terms are not included in the HGO models used in tissue mechanics.

We enforce polyconvexity of the SEF by imposing constraints on the unknown material constants in equations (4) and (7) by

c_{i} > 0; i = 1, 3, 5

c_{j} < 0; j = 2, 4

c_{6} + c_{7} > 0; c_{8} > 0; k_{1} > 0; k_{2} > 0

(8)

A right-handed Cartesian coordinate system was introduced along the principal material axes corresponding to the longitudinal and transverse directions. The unit vectors along two fiber directions in Nomex® are given by

a_{0} = \{\begin{matrix} \cos θ \\ \sin θ \\ 0 \end{matrix}\} b_{0} = \{\begin{matrix} \cos θ \\ - \sin θ \\ 0 \end{matrix}\}

(9)

Let $λ_{i}$ (i = 1,2,3) be the principal stretches for the materials. Cauchy stresses for the isotropic (Kapton® sheet) and orthotropic (Nomex®) materials along the uniaxial stretch direction are given by

σ_{Kap} = - p I + 2 \frac{\partial ψ_{Kap}}{\partial I_{1}} B - 2 \frac{\partial ψ_{Kap}}{\partial I_{2}} B^{- 1} σ_{Nomx}

(10a)

σ_{Kap} = - q I + 2 \frac{\partial ψ_{Nomx}}{\partial I_{1}} B + 2 \frac{\partial ψ_{Nomx}}{\partial I_{2}} (I_{1} B - B^{2}) + 2 \frac{\partial ψ_{Nomx}}{\partial I_{4}} a \otimes a + 2 \frac{\partial ψ_{Nomx}}{\partial I_{6}} b \otimes b

(10b)

$p$ and $q$ are Lagrange multipliers imposed due to material incompressibility constraint.

The constitutive equation for the composite, fabricated using a Nomex® core sandwiched between sheets of Kapton®, was given using the rule of mixtures. The SEF for the composite is given by

ψ_{Comp} = ν_{Kap} ψ_{Kap} + ν_{Nomx} ψ_{Nomx}

(11a)

ψ_{Comp} = ν_{Kap} (c_{1} (I_{1} - 3) + c_{2} {(I_{1} - 3)}^{2} + c_{3} {(I_{1} - 3)}^{3} + c_{4} {(I_{1} - 3)}^{4} + c_{5} {(I_{1} - 3)}^{5}) + ν_{Nomx} \times (c_{6} (I_{1} - 3) + c_{7} (I_{2} - 3) + c_{8} (I_{1} - 3) + \frac{k_{1}}{2 k_{2}} \sum_{i = 4, 6} (\exp [k_{2} (I_{i} - 1)^{2}] - 1))

(11b)

Cauchy stresses for the composite material under uniaxial stretch are calculated as

σ_{Comp} = ν_{Kap} σ_{Kap} + ν_{Nomx} σ_{Nomx}

(12)

Experimentally obtained stresses were fit to the computed stresses for Kapton® (equation (10a)), Nomex® (equation (10b)), and the composite (equation (12)) using a nonlinear Levenberg-Marquardt optimization algorithm implemented using the ‘fmincon’ function in MATLAB. The differences between experimentally obtained Cauchy stresses in ( $σ^{\exp}$ ) and the theoretically calculated stresses ( $σ^{calc}$ ) were used to compute the error given by

Error = \sum_{j = 1}^{N} {(σ_{j}^{calc} - σ_{j}^{\exp})}^{2}

(13)

At least five different initial values were used in the nonlinear optimization to ensure that the converged solution corresponded to the global minimum. The optimization was terminated when either a tolerance value of 1E-08 on the objective function was reached or a value of 1E^-06 applied on the estimated parameter values. The solution with a maximum $r^{2}$ value is reported as a best fit solution in this study. In addition to the unknown constants in the SEF, the fiber angle, $θ$ , was determined using fits to the data as discussed earlier. The two fiber families in Nomex® and the composite were assumed to be symmetrically oriented at $θ$ and $- θ$ based on the scanning electron microscope (SEM) micrographs for Nomex® (Figure 1(a) and (b)).

Results and discussion

Uniaxial test results

Figure 3 shows uniaxial stress-stretch plots in the longitudinal and transverse directions for Kapton® (isotropic), Nomex®, and the composite specimens (n = 3 for each group). Results for Kapton® films show nonlinear rubber-like large deformations in both the tested directions. These results confirm that Kapton may be modeled as an isotropic material. In contrast, there are clear differences in the mechanical responses of Nomex® in the longitudinal and transverse directions with the longitudinal direction stiffer than the transverse. The mechanical properties of composite are hence dominated by Nomex® in both directions. The composite has an intermediate response to that of Kapton® and Nomex® in the longitudinal direction. There are however no differences between the composite and the Nomex® in the transverse direction.

The composite and Nomex® samples have near linear initial response of∼4% stretch and are followed by non-linear response until failure. We used these results to compute the elastic moduli, Poisson’s ratio, and the maximum stretch to failure in the two directions for all the samples in the study (Figure 4). The elastic moduli, reported as Mean ± Std., were significantly different in the longitudinal (2.86 ± 0.36 GPa; p < 0.05) direction as compared to the transverse (1.70 ± 0.10 GPa) direction for Nomex® samples (Figure 4(a)). Modulus in the longitudinal direction was 1.68 times higher as compared to the transverse direction. The elastic modulus of Kapton® was 2.00 ± 0.08 GPa. Similar differences were seen in the moduli of the composite in the longitudinal (2.80 ± 0.08 GPa) and transverse (1.76 ± 0.04 GPa) directions as compared to the corresponding directions in the Nomex® samples. A higher contribution from Nomex® samples is expected because of a higher volume fraction used during the composite fabrication.

Poisson’s ratios for Nomex® samples were higher in the longitudinal (0.46 ± 0.01) as compared to the transverse (0.40 ± 0.02) direction (p < 0.05) (Figure 4(b)). There were no significant differences in the Poisson’s ratio values for composite samples in the longitudinal direction (0.43 ± 0.02) as compared to the transverse direction (0.40 ± 0.02). Finally, Kapton® specimens had values of 0.37 ± 0.01 which is significantly lower than that for rubber-like materials. Figure 4(c) shows the percentage elongation at rupture for all the materials. There was higher variability in the values reported for the composites that demonstrate the importance of the fabrication process in the overall mechanical behavior of the composite.

Uniaxial tensile tests are useful in quantifying properties, such as elastic modulus, Poisson’s ratio, elongation at break, etc., of the Nomex® and sandwich composites in our study. We show that the stress-strain results obtained from uniaxial tests on samples, oriented in the longitudinal and transverse directions respectively, may be used to estimate the role of fiber orientations in the overall material behaviors of composites. Kapton® is primarily important as a gas barrier in the fabrication of composites for HAA applications as compared to Nomex® that is useful in the structural properties. Combining both these materials in the sandwich construction hence offers several advantages that are not seen in the individual materials for the construction of HAA. Such materials, with high specific modulus and gas diffusion properties, make them suitable for HAA applications.

Verification of constitutive models with experimental data for materials in the study

A five-parameter isotropic and incompressible Yeoh model was used for Kapton® samples (equation (4)). In contrast, an incompressible HGO form of constitutive model (equation (7)) was used to fit the Nomex® data in either the longitudinal or the transverse directions based on the inclusion of two symmetrically orientated fiber families at $θ$ and $- θ$ . The model fit for a representative sample for Kapton® is shown in Figure 5; material constants are listed in Table 2 (n = 3). The r² values were > 0.99 that show good agreement of the model predictions with experimental data.

Table 2.

Unknown constants to material model for Kapton® were obtained by fitting a five parameter Yeoh model (equation (4)) to the experimentally obtained uniaxial data.

Kapton®	c₁ (GPa)	c₂ (GPa)	c₃ (GPa)	c₄ (GPa)	c₅ (GPa)	r²
Sample 1	0.39	–5.10	70.05	–572.18	1926.28	0.999
Sample 2	0.43	–7.01	124.67	–1295.39	5493.65	0.999
Sample 3	0.40	–6.06	93.09	–863.31	3422.64	0.999
Mean ± Std.	0.41 ± 0.02	–6.06 ± 0.8	95.94 ± 22.3	–910.3 ± 297.1	3614.2 ± 1462.7

Modified HGO model fits for a representative Nomex® sample are shown in Figure 6 and the constants from tested specimens in the longitudinal and transverse directions (n = 3) are listed in Table 3. In addition to the unknown material constants, values of fiber orientations (± $θ$ ) that were estimated by the optimization method are also listed. The mean fiber angular orientations (± $θ$ ) obtained from the model was 55.8°±0.6°. These values lie near the range of the experimentally quantified fiber orientation distributions using SEM micrographs calculated to be 69.3°± 7.4° for one fiber family and that for other family was 129°± 5.3°. Rule of mixtures (equation (12)) was used to quantify the properties of the composite material from the individual material properties of Kapton® and Nomex® as described earlier. The average material constant values for three samples from each material was used in this calculation. Data from experimental tests with the fiber orientation angles of ±55.4° were used to calculate the stress-strain results from the composite model. Figure 7(a) and (b) show model fits to the experimental results in longitudinal and transverse directions, respectively. Although the contribution of the glue was neglected in the mechanical response, we see that the $r^{2}$ value was 0.91 for both directions. The inclusion of glue contribution to quantify the properties of the composite using rule of mixtures may improve the goodness of fits obtained from the model to the experimental data and is a limitation of this study.

Table 3.

Modified HGO model constants (equation (7)) were estimated by fitting the uniaxial experimental results for Nomex® in the longitudinal (L) and transverse (T) directions.

Nomex®		c₁ (MPa)	c₂ (MPa)	c₃ (MPa)	k₁ (MPa)	k₂	θ°	r²
S 1	L	295.4	2.74	618.4	9747.4	666.3	55.7	0.996
S 1	T	162.0	1.12	322.7	5411.2	648.3	55.7^a	0.993
S 2	L	290.4	0.82	863.6	10630.8	692.5	55.5	0.990
S 2	T	258.3	1.18	694.7	9139.2	661.3	55.2^a	0.993
S 3	L	389.2	0.23	1086.4	14024.8	680.8	55.5	0.989
S 3	T	51.42	1.04	487.2	3972.1	3029.3	57.2^a	0.966
Mean± Std.		241.1 ± 107.9	1.19 ± 0.76	678.8 ± 247.5	8820.9 ± 3328.0	1063.1 ± 879.4	55.8 ± 0.6

L: longitudinal T: transverse.

^aWith reference to longitudinal direction.

Constitutive models for fiber reinforced composites are essential to predict the mechanical behavior under different loading conditions. Macro-scale models represent the composite material with average properties. In contrast, micromechanical models are based on the microstructure and constitutive properties of the elemental constituents like the matrix and fibers. Such approaches use homogenization techniques to estimate the effective mechanical properties of the composite us the using the individual contributions of the constituents and fiber-matrix interactions. Micromechanical models based on theories, such as the Hashin-Roshen model, Mori-Tanaka model, or the self-consistent models, are also useful in this context to estimate of the composite properties more accurately.^29–31

Constitutive models for fiber reinforced composites in the literature were obtained by superposing the stress-strain responses of the constituent fiber and matrix.^32–34 Aboshio and colleagues³² calculated the strain energy density by integrating the stress-strain curves and estimated the material parameters using the strain energy density versus invariant plots. Homogenization techniques to estimate the material properties of multi-layered, thick laminate composites undergoing small strains are based on force-deformation equivalence with strain equality at the interfaces.^35–37 Other micromechanical approaches include material heterogeneities at smaller scales and volume average the constitutive behavior of each phase to obtain the effective stress and strain potentials for the composite.^38,39 We used a linear superposition of the strain energy function for each individual material within a macroscopic rule of mixtures model and have neglected the interfacial stress discontinuities to quantify the overall mechanical properties of the composite. The strain energy of the composite is lower than the sum of the strain energies of the individual components associated with Kapton® and Nomex®. Because the strain energy function is continuous throughout the deformation regime, we show that the rule of mixtures provides an upper bound to the mechanical properties of composite, not only in the linear regime but also in the non-linear part. Earlier studies by Aboshio and co-workers ³² have used a similar approach based on the volume fractions of individual constituents in obtaining the mechanical properties of composite in the entire deformation regime. Such models are useful to obtain the envelope material in HAA which withstands the pressure differences between the interior and the exterior of the ship and maintains the desired aerodynamic configuration during lift. The constitutive model for the honey-comb structure of Nomex® is also useful in analyzing helicopter blades under different loading conditions. Methods used in the current study may be extended to equibiaxial and non-equibiaxial tensile tests to gain additional insights into the constitutive properties of sandwich composites that leverage direction dependent responses for specific applications. Determination of shear strength and tear resistance are other important criteria which will be required for such applications.

Conclusions

We fabricated a multi-layered composite using a Nomex® sheet sandwiched between Kapton® HN films with epoxy and vacuum bagging technique. The constitutive properties of the composite were obtained using a continuum mechanical framework and experimental results from uniaxial experiments. A five-parameter incompressible Yeoh model fit the stress - strain responses of Kapton® films. SEM images show two distinct families of symmetrically oriented fibers in the matrix of Nomex®. An incompressible, two fiber family, modified HGO model was used to quantify the material responses of Nomex®. The fiber orientation angles, estimated using the model for Nomex®, are in good agreement with data obtained from the analysis of SEM images. The material properties of Nomex® sheet dominate the stress-strain responses of the composite laminate. A constitutive model for the composite was obtained using a rule of mixtures model and a linear decomposition of the strain energy function based on individual components in the laminate structure. The theoretically obtained stress-strain results agree with experimental data. These results are useful in the design of composite laminates in engineering applications that undergo large deformation.

Footnotes

Acknowledgements

We thank Nimesh R. Chahare for help with mechanical tests. We also thank the MNCF lab staff members, CeNSE department, IISc for help with scanning electron microscopy.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Namrata Gundiah

References

Zhai

Euler

Material challenges for lighter-than-air system in high altitude applications. In: 5th Aviation, technology, integration, and operations conference (ATIO), Arlington, Virginia, 26--28 September 2005, AIAA 2005-7488.

Khoury

GA.

Airship technology. vol. 10. Cambridge: Cambridge University Press, 2012.

Rivlin

Saunders

DW.

Large elastic deformations of isotropic materials. VII. Experiments on the deformation of rubber. Philos Trans R Soc A Math Phys Eng Sci 1951; 243: 251–288.

Arruda

Boyce

MC.

A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys Solids 1993; 41: 389–412.

Spencer AJM. Continuum mechanics. New York: Dover Publications, 1972.

Polignone

Horgan

CO.

Cavitation for incompressible anisotropic nonlinearly elastic spheres. J Elast 1993; 33: 27–65.

Polignone

Morgan

CO.

Effects of material anisotropy and inhomogeneity on cavitation for composite incompressible anisotropic nonlinearly elastic spheres.

Int J Solids Struct 1993; 30: 3381–3416.

Qiu

Pence

TJ.

Remarks on the behavior of simple directionally reinforced incompressible nonlinearly elastic solids. J Elast 1997; 49: 1–30.

Qiu

Pence

TJ.

Loss of ellipticity in plane deformation of a simple directionally reinforced incompressible nonlinearly elastic solid. J Elast 1997; 49: 31–63.

10.

Merodio

Pence

TJ.

Kink surfaces in a directionally reinforced neo-Hookean material under plane deformation: I. Mechanical equilibrium.

J Elast Phys Sci Solids 2001; 62: 119–144.

11.

Merodio

Ogden

RW.

Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformation.

Arch Mech 2002; 54: 525–552.

12.

Merodio

Ogden

RW.

Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation. Int J Solids Struct 2003; 40: 4707–4727.

13.

Merodio

Ogden

RW.

Mechanical response of fiber-reinforced incompressible non-linearly elastic solids.

Int J Non Linear Mech 2005; 40: 213–227.

14.

Merodio

Ogden

RW.

On tensile instabilities and ellipticity loss in fiber-reinforced incompressible non-linearly elastic solids. Mech Res Commun 2005; 32: 290–299.

15.

Holzapfel

Gasser

Ogden

RW.

A new constitutive framework for arterial wall mechanics and a comparative study of material models.

J Elast Phys Sci Solids 2000; 61: 1–48.

16.

Holzapfel

Ogden

RW.

Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Phil Trans A Math Phys Eng Sci 2009; 367: 3445–3475.

17.

Pham

X-T

Bates

Chesney

Modeling of thermoforming of low-density glass mat thermoplastic.

J Reinf Plast Compos 2005; 24: 287–298.

18.

Aimène

Vidal-Sallé

Hagège

, et al. A hyperelastic approach for composite reinforcement large deformation analysis. J Compos Mater 2010; 44: 5–26.

19.

Peng

Guo

, et al. A simple anisotropic hyperelastic constitutive model for textile fabrics with application to forming simulation. Compos Part B Eng 2013; 52: 275–281.

20.

Gong

Peng

Yao

, et al. An anisotropic hyperelastic constitutive model for thermoplastic woven composite prepregs. Compos Sci Technol 2016; 128: 17–24.

21.

N-H

Pham

X-T

François

, et al. Characterization of multilayered carbon-fiber–reinforced thermoplastic composites for assembly process. J Thermoplast Compos Mater 2018; 32: 673--689.

22.

Chatterjee

Chahare

Kondaiah

, et al. Role of fiber orientations in the mechanics of bioinspired fiber reinforced elastomers. Soft Robotics (ahead of print). http://doi.org/10.1089/soro.2019.0191.

23.

Yeoh

Fleming

PD.

A new attempt to reconcile the statistical and phenomenological theories of rubber elasticity. J Polym Sci B Polym Phys 1997; 35: 1919–1931.

24.

Babu

Gundiah

Role of crosslinking and entanglements in the mechanics of silicone networks. Exp Mech 2014; 54: 1177–1187.

25.

Humphrey

JD.

Cardiovascular solid mechanics: cells, tissues, and organs. Berlin: Springer Science & Business Media, 2013.

26.

Boyce

Arruda

EM.

Constitutive models of rubber elasticity: a review. Rubber Chem Technol 2000; 73: 504–523.

27.

Yeoh

OH.

Some forms of the strain energy function for rubber. Rubber Chem Technol 1993; 66: 754–771.

28.

Gundiah

Ratcliffe

Pruitt

LA.

Determination of strain energy function for arterial elastin: experiments using histology and mechanical tests. J. Biomech 2007; 40: 586–594.

29.

Hashin

Rosen

BW.

The elastic moduli of fiber reinforced materials. J Appl Mech Trans ASME 1964; 31: 223–232.

30.

Mori

Tanaka

Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metal 1973; 21: 571–574.

31.

Hill

Theory of mechanical properties of fibre-strengthen materials-III. Self-consistent model. J Mech Phys Solids 1965; 13: 189–198.

32.

Aboshio

Green

JQ.

New constitutive model for anisotropic hyperelastic biased woven fibre reinforced composite. Plast Rubber Compos 2014; 43: 225–234.

33.

Luccioni

BM.

Constitutive model for fiber-reinforced composite laminates. J Appl Mech 2006; 73: 901–910.

34.

Cho

Fenner

Werner

, et al. A constitutive model for fiber-reinforced polymer composites. J Compos Mater 2010; 44: 3133–3151.

35.

Enie

Rizzo

RR.

Three-dimensional laminate moduli. J Compos Mater 1970; 4: 150–154.

36.

Sun

Three-dimensional effective elastic constants for thick laminates. J Compos Mater 1988; 28: 629–639.

37.

Sun

Chen

Tan

VBC

, et al. Homogenization and stress analysis of multilayered composite offshore production risers. J Appl Mech 2014; 81: 031003.

38.

Idiart

Danas

Castaneda

PP.

Second-order theory for nonlinear composites and application to isotropic constituents. Comp Rend Mechanique 2006; 334: 575–581.

39.

Furer

Castaneda

PP.

A symmetric fully optimized second-order method for nonlinear homogenization. Z Angew Math Mech 2018; 98: 222–254.