Free vibration analysis of a skew sandwich plate with bamboo biocomposite and polylactic acid core under temperature and moisture conditions

Abstract

This article investigates the impact of moisture and temperature on vibration characteristics of bio-composite skew-laminated composite sandwich (SLCS) plates. The bio-composite SLCS plates with bamboo face sheets and polylactic acid (PLA) cores are biodegradable, radiolucent, lightweight, high strength, and withstand vibrations. The coupled thermo-elastic and hygro-elastic finite element (FE) model of the SLCS plate is derived using the higher-order shear deformation theory (HSDT). An initial stress stiffness matrix is developed using the nonlinear strain-displacement relations to incorporate the effect of temperature and moisture in FE modeling. Temperature and moisture-dependent material properties of bamboo fiber-reinforced biocomposite (BFB) and PLA core are employed for the analysis. A comprehensive investigation has been carried out to study the impact of moisture and temperature situations with different geometrical parameters and various skew angles on the frequency of the SLCS plate. The results indicate that the biocomposite sandwich material has excellent potential for structural applications under different environmental conditions in various applications like food processing, and biomedical applications, including MRI and CT scan beds.

Keywords

Natural fibers bamboo fiber-reinforced biocomposite laminated bio-composite sandwich plate biodegradable materials impact of moisture and temperature

Introduction

The sandwich structures with a thin fiber-reinforced composite face sheet and a thick core are often widely used in numerous engineering fields, including aerospace, automobile, healthcare, and construction.¹ This wide range of usage is due to desirable mechanical characteristics, including a high strength-weight ratio, specific stiffness, vibration damping, and fatigue resistance.² The laminated composite (LC) face sheet in sandwich structures is fabricated using synthetic fibers such as carbon, glass, kevlar, and aramid. Similarly, synthetic core materials, especially polyvinyl chloride and polyethylene terephthalate, have been used in sandwich structures.³ The concerning impact on the environment and the high production cost of synthetic fibers accelerate the development of sustainable, atmosphere-friendly, and ecological materials. The composites with natural fiber reinforcement as face-sheet and biodegradable cores for sandwich structures are in trend due to less manufacturing cost and minimal environmental impact.

Many studies were carried out on various natural fibers for reinforcement in biocomposites, namely banana, bamboo, sisal, kenaf, jute, pineapple leaf, flax, and hemp.^4–12 Among all the different natural fibers, bamboo fibers are relatively inferior to glass fiber in mechanical properties but much cheaper,¹³ making bamboo fiber-reinforced biocomposite (BFB) with different matrix materials a great option to replace synthetic fiber composite. Several kinds of research on bamboo, bamboo fibers, and BFB have been conducted; chemical and physical treatments are also carried out to enhance the bonding between bamboo fiber reinforcement and several matrix materials, which improves BFB performance.^14–21 Literature reveals several studies on the mechanical characteristics of BFB with a thermosetting matrix such as polyester and epoxy and a biodegradable matrix such as polypropylene.^22–28 Jindal²⁵ experimentally investigated mechanical characteristics and properties such as tensile strength, impact strength, and modulus of elasticity. The results demonstrate that BFB has an ultimate tensile strength equivalent to mild steel, although its density is around one-eighth that of mild steel. The study also concluded that the behavior of BFB is similar to the commonly used composites like glass fiber reinforced composite (GFRC). Mohanavel et al.²⁶ conducted a similar experimental study with unidirectional bamboo fiber and bidirectional bamboo strip mats as reinforcement. They found out that reinforcement in the orthogonal direction possesses valuable biaxial strength. Verma and Chariar²⁷ studied the mechanical properties of BFB, including compressive, tensile, and flexural strength as well as screw-holding capability. The study focuses on identifying the mode of failure at a macroscopic level, the mechanism of failure, and the environmental impact of BFB. Khan et al.²⁸ studied the fracture behavior of BFB, i.e., fracture toughness for BFB and fractographic results, which shows matrix crack, fiber breakage, fiber pull-out, and fiber-matrix debonding. Ismail et al.²⁹ investigated the vibrational and acoustic behavior of composite with kenaf and bamboo fiber. The investigation shows that the kenaf bamboo composite is suitable for sound absorption and noise control structures.

Another vital constituent, along with LC face sheet in the sandwich structure, is the core. Numerous studies use different core materials in a sandwich structure, such as functionally graded material, polylactic acid (PLA), viscoelastic material, ceramic, and carbon nanotube-reinforced composite core.^30–32 Zenkour and Alghanmi³⁰ investigated the static behavior of sandwich plates comprised of a functionally graded material core and piezoelectric face sheets. The study investigates the stresses and center deflection produced in sandwich plates when subjected to sinusoidal thermo-electro-mechanical loading conditions. Katariya et al.³¹ investigated the vibrational response of skewed laminated composite sandwich (SLCS) plates with epoxy-filled softcore numerically as well as experimentally. Daikh et al.³² investigated the static behavior of ceramic core sandwich plates exposed to thermo-mechanical stresses using higher-order shear deformation theory (HSDT). Literature shows that PLA has essential features among the different core materials, such as biodegradability, high tensile strength, and stiffness.³³ The thermoplastic polymer PLA is produced from renewable sources such as sugar or starch-rich crops such as sugar cane fermentation. Also, it can be manufactured using standard processes such as injection extrusion, hot compression molding, and 3D printing. Due to ease of manufacturing and desired properties, the usage of PLA has increased significantly in the packaging and biomedical sector.^34,35 PLA can also be utilized as a matrix material in composites with natural fiber reinforcement.³⁶ Antony et al.³⁷ investigated the mechanical properties of a hemp/PLA 3D-printed core. The mechanical parameters of the honeycomb sandwich construction were also determined by analytical analysis. It is gaining importance in numerous systems consisting of composite sandwich structures like 3D-printed honeycomb cores and skin on various composite panels.³⁸ Lascano et al.³⁹ studied the fabrication and mechanical characterization of the laminated composite sandwich (LCS) plate with flax/PLA face-sheet and PLA as the core.

Though natural and biodegradable materials provide several advantages, such as low density, ease of availability, and good mechanical properties, they are vulnerable to higher moisture concentrations and temperatures. The composite’s severe degradation in mechanical characteristics can be seen under the temperature and moisture environment. Numerous studies have been carried out on sandwich structure considering environmental conditions.^40–48 Ding et al.^44,45 studied the influence of various adverse environmental effects on the mechanical properties of LCS experimentally. Garg et al.⁴⁶ applied the modified trigonometric zigzag theory to understand the influence of hygro-thermo-mechanical load on the performance of composite and sandwich plates. Kallannavar et al.⁴⁷ studied the effects of change in temperature as well as moisture conditions on the free vibration characteristics of the hybrid skew laminated composite (SLC) and the LCS plates using FSDT. There is inadequate literature on moisture and temperature affecting natural fiber composite face sheets.^22,49–53

The sandwich structure with natural fiber-reinforced composite (NFRC) and biodegradable core can also be advantageous in biomedical applications. Many parts of machines used in the medical field can be replaced by parts made of eco-friendly and sustainable materials. Magnetic Resonance Imaging (MRI) machines and Computerized Tomography (CT) scan machines are two examples of vital medical equipment that are widely employed in medical research and hospitals.⁵⁴ Material required for such application should be radiolucent, lightweight, high strength, and withstand vibrations.^55,56 One of the clinical requirements for a vital aspect of the material for MRI and CT scan machines is that it provides minimal attenuation to strong X-ray photon beams used in radiotherapy treatment. The MRI machine utilizes a rotating gradient coil to create the rapidly changing orientation of high magnetic fields inside the machine for imaging. This rotating coil is the primary source of acoustic noise and vibrations while scanning and reduces the quality of scans. It may be uncomfortable and even hazardous to the hearing of patients and workers.⁵⁶ Many studies show that the use of composite materials than conventional materials are more likely to be used for more transmission and good structural strength.^55,57,58 Bie De Ost et al.⁵⁵ studied the use of carbon fiber as an excellent choice for radiotherapy plates along conventional table tops for different types of photon beams and validated its radiolucent properties. But some studies show that carbon fiber is less compatible for the parts of large MRI machines due to its severe radio frequency (RF) shading and some material heating due to RF coil.^59,60 Morris et al.⁶¹ studied NFRC with polystyrene core under mentioned environment and outperformed carbon fiber reinforced composites (CFRC) in RF shading and distortion in MRI. NFRC performed well in megavoltage imaging homogeneity and was superior to glass fiber-reinforced composite (GFRC). Paley et al.⁶² developed an MRI-compatible carbon fiber incubator to carry newborns to the MRI machine in order to detect problems and make early diagnoses. Because of the conductive nature of carbon fiber, RF shadowing distortions were observed in the diagnostic scans. Morris et al.^61,63 used NFRC and claimed that NFRC outperformed carbon fiber in imaging capabilities and had no effect on the scans obtained, i.e., good radiolucency.

It is observed from a comprehensive literature survey that though synthetic fiber reinforcement for composites serves the purpose of good strength, they are costly and ecologically destructive. In comparison, natural fiber-reinforced composites are environment-friendly and cost-effective for manufacturing, with fewer carbon footprints and good mechanical characteristics. Additionally, the natural fiber composites, i.e., biocomposite materials for structural application in biomedical imaging systems such as MRI or CT scan machines, provide good radiolucency and give better imaging capabilities than CFRC, GFRC, and different conventional materials. As biomedical imaging systems are less prone to harsh loading conditions, the natural fiber biocomposite with sufficient structural strength, good radiolucency, and ecologically sustainable nature can be a suitable replacement for synthetic fiber composites. Hence this has motivated the authors to investigate the behavior of biocomposites in structural applications for biomedical imaging systems under environmental factors.

There is a scarcity of published research on biodegradable materials which are radiolucent and good in vibration response when used for sandwich plates considering environmental conditions. To the best of the authors’ knowledge, there is limited research on the influence of thermal and moisture conditions on the vibrational properties of sandwich structures with biocomposite face sheets and biodegradable cores in the open literature. Consequently, the present research focuses on studying free vibration characteristics of laminated biocomposite sandwich (LBCS) plate consisting of bamboo fiber as reinforcement in composite face-sheet and biodegradable PLA core under changing moisture concentration and temperature. The behavior of the plate is predicted by FE analysis considering HSDT and the dynamic version of virtual work principle. The initial stress stiffness using nonlinear strain-displacement relation is considered for the effect of thermal and hygroscopic conditions along with linear strain-displacement relation based on mechanical stiffness matrix. The impact of length-to-width ratio, length-to-thickness ratio, core thickness to face-sheet thickness ratio, skew angles, and different boundary conditions (BC) under various thermal and hygroscopic conditions on the vibrational behavior of the LBCS plate are studied.

Mathematical modeling

A sandwich composite plate with a bamboo fiber-reinforced biocomposite as a face sheet and PLA as a biodegradable core material is represented in Figure 1(a). The sandwich plate is made of N numbers of orthotropic laminae in the top face sheet and bottom face sheet, and a core is considered equivalent isotropic material. The sandwich plate’s thickness, length, and breadth are h, l, and w, respectively. h_t is the top face sheet thickness, 2h_c is the total thickness of the core, and the thickness of the bottom face sheet is h_c. Further, h_l is the thickness of the lamina of LC face sheet. Figure 1(b) shows the deformation kinematics of composite sandwich plate in different planes. The angles φ_x, α_x, and β_x denote the rotation of normal on the top face sheet, core, and bottom face sheet, respectively, in the xz plane. Similarly, angles φ_y, α_y, and β_y denote rotation in the yz plane. The displacement field obtained from the kinematics of deformation for LCS plate using HSDT is written as follows:

Figure 1.

(a) Schematic representation of the LBCS plate, (b) kinematics of deformation of the LBCS plate.

For face-sheet laminate at the top,

u_{t} = u_{0} + h_{c} φ_{x} - (\frac{4}{3 h^{2}}) h_{c}^{3} (φ_{x} + \frac{\partial w_{0}}{\partial x}) + (z - h_{c}) α_{x} - (\frac{4}{3 h^{2}}) {(z - h_{c})}^{3} (α_{x} + \frac{\partial w_{0}}{\partial x})

v_{t} = v_{0} + h_{c} φ_{y} - (\frac{4}{3 h^{2}}) h_{c}^{3} (φ_{y} + \frac{\partial w_{0}}{\partial y}) + (z - h_{c}) α_{y} - (\frac{4}{3 h^{2}}) {(z - h_{c})}^{3} (α_{y} + \frac{\partial w_{0}}{\partial y})

(1)

w_{t} = w_{0}

for core,

u_{c} = u_{0} + z φ_{x} - (\frac{4}{3 h^{2}}) z^{3} (φ_{x} + \frac{\partial w_{0}}{\partial x})

v_{c} = v_{0} + z φ_{y} - (\frac{4}{3 h^{2}}) z^{3} (φ_{y} + \frac{\partial w_{0}}{\partial y})

(2)

w_{c} = w_{0}

for face-sheet laminate at the bottom,

u_{b} = u_{0} - h_{c} φ_{x} - (\frac{- 4}{3 h^{2}}) h_{c}^{3} (φ_{x} + \frac{\partial w_{0}}{\partial x}) + (z + h_{c}) β_{x} - (\frac{4}{3 h^{2}}) {(z + h_{c})}^{3} (β_{x} + \frac{\partial w_{0}}{\partial x})

v_{b} = v_{0} - h_{c} φ_{y} - (\frac{- 4}{3 h^{2}}) h_{c}^{3} (φ_{y} + \frac{\partial w_{0}}{\partial y}) + (z + h_{c}) β_{y} - (\frac{4}{3 h^{2}}) {(z + h_{c})}^{3} (β_{y} + \frac{\partial w_{0}}{\partial y})

(3)

w_{b} = w_{0}

In equations (1)–(3), u, v, and w are displacements along x, y, and z axis, respectively. The axial displacements of the top face-sheet, bottom face-sheet, and core are noted by suffixes as t, b, and c, respectively. The translational displacements of any point on the center plane of an LCS plate are represented as u₀, v₀, and w₀ along the x, y, and z axes, respectively. The translational displacement and rotational variables can be separated for the easy solution of a problem and can be written as follow:

{d_{t_d i s p}} = {\begin{array}{c} u_{0} & v_{0} & w_{0} \end{array}}^{T}

{d_{r_d i s p}} = {\begin{array}{c} φ_{x} & φ_{y} & α_{x} & α_{y} & β_{x} & β_{y} & Κ_{x} & Κ_{y} \end{array}}^{T}

(4)

where K_x and K_y are higher-order rotational variables.

Linear strain-displacement relations

The condition of strains at every location in the plate is divided into two different strain vectors, normal strain ( $ε_{b}$ ), and transverse strain ( $ε_{s}$ ); for incorporating the selective integration rule, it can be expressed by equations (5) and (6),

{ε_{b}}_{t} = {\begin{array}{c} ε_{x} \\ ε_{y} \\ ε_{x y} \end{array}}_{t}, {ε_{b}}_{b} = {\begin{array}{c} ε_{x} \\ ε_{y} \\ ε_{x y} \end{array}}_{b}, {ε_{b}}_{c} = {\begin{array}{c} ε_{x} \\ ε_{y} \\ ε_{x y} \end{array}}_{c}

{ε_{b}}_{t} = {\begin{array}{c} \frac{\partial u_{0}}{\partial x} + (h_{c} + c_{1} h_{c}^{3}) \frac{\partial φ_{x}}{\partial x} + ((z - h_{c}) + c_{1} {(z - h_{c})}^{3}) \frac{\partial α_{x}}{\partial x} + (c_{1} h_{c}^{3} + c_{1} {(z - h_{c})}^{3}) \frac{\partial Κ_{x}}{\partial x} \\ \frac{\partial v_{0}}{\partial y} + (h_{c} + c_{1} h_{c}^{3}) \frac{\partial φ_{y}}{\partial y} + ((z - h_{c}) + c_{1} {(z - h_{c})}^{3}) \frac{\partial α_{y}}{\partial y} + (c_{1} h_{c}^{3} + c_{1} {(z - h_{c})}^{3}) \frac{\partial Κ_{y}}{\partial y} \\ (\frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x}) + (h_{c} + c_{1} h_{c}^{3}) (\frac{\partial φ_{x}}{\partial y} + \frac{\partial φ_{y}}{\partial x}) + ((z - h_{c}) + c_{1} {(z - h_{c})}^{3}) (\frac{\partial α_{x}}{\partial y} + \frac{\partial α_{y}}{\partial x}) + (c_{1} h_{c}^{3} + c_{1} {(z - h_{c})}^{3}) (\frac{\partial Κ_{x}}{\partial y} + \frac{\partial Κ_{y}}{\partial x}) \end{array}}

{ε_{b}}_{c} = {\begin{array}{c} \frac{\partial u_{0}}{\partial x} + (z + c_{1} z^{3}) \frac{\partial φ_{x}}{\partial x} + (c_{1} z^{3}) \frac{\partial Κ_{x}}{\partial x} \\ \frac{\partial v_{0}}{\partial y} + (z + c_{1} z^{3}) \frac{\partial φ_{y}}{\partial y} + (c_{1} z^{3}) \frac{\partial Κ_{y}}{\partial y} \\ (\frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x}) + (z + c_{1} z^{3}) (\frac{\partial φ_{x}}{\partial y} + \frac{\partial φ_{y}}{\partial x}) + (c_{1} z^{3}) (\frac{\partial Κ_{x}}{\partial y} + \frac{\partial Κ_{y}}{\partial x}) \end{array}}_{c}

(5)

{ε_{b}}_{b} = {\begin{array}{c} \frac{\partial u_{0}}{\partial x} - (h_{c} + c_{1} h_{c}^{3}) \frac{\partial φ_{x}}{\partial x} + ((z + h_{c}) + c_{1} {(z + h_{c})}^{3}) \frac{\partial β_{x}}{\partial x} + (- c_{1} h_{c}^{3} + c_{1} {(z + h_{c})}^{3}) \frac{\partial Κ_{x}}{\partial x} \\ \frac{\partial v_{0}}{\partial y} - (h_{c} + c_{1} h_{c}^{3}) \frac{\partial φ_{y}}{\partial y} + ((z + h_{c}) + c_{1} {(z + h_{c})}^{3}) \frac{\partial β_{y}}{\partial y} + (- c_{1} h_{c}^{3} + c_{1} {(z + h_{c})}^{3}) \frac{\partial Κ_{y}}{\partial y} \\ (\frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x}) - (h_{c} + c_{1} h_{c}^{3}) (\frac{\partial φ_{x}}{\partial y} + \frac{\partial φ_{y}}{\partial x}) + ((z + h_{c}) + c_{1} {(z + h_{c})}^{3}) (\frac{\partial β_{x}}{\partial y} + \frac{\partial β_{y}}{\partial x}) + (- c_{1} h_{c}^{3} + c_{1} {(z + h_{c})}^{3}) (\frac{\partial Κ_{x}}{\partial y} + \frac{\partial Κ_{y}}{\partial x}) \end{array}}_{b}

{ε_{s}}_{t} = {\begin{array}{c} ε_{x z} \\ ε_{y z} \end{array}}_{t} = {\begin{array}{c} \frac{\partial w_{0}}{\partial x} + (1 + 3 c_{1} {(z - h_{c})}^{2}) α_{x} + {(3 c_{1} (z - h_{c})}^{2}) Κ_{x} \\ \frac{\partial w_{0}}{\partial y} + (1 + 3 c_{1} {(z - h_{c})}^{2}) α_{y} + {(3 c_{1} (z - h_{c})}^{2}) Κ_{y} \end{array}}_{t}

{ε_{s}}_{c} = {\begin{array}{c} ε_{x z} \\ ε_{y z} \end{array}}_{c} = {\begin{array}{c} \frac{\partial w_{0}}{\partial x} + (1 + 3 c_{1} z^{2}) β_{x} + (3 c_{1} z^{2}) Κ_{x} \\ \frac{\partial w_{0}}{\partial y} + (1 + 3 c_{1} z^{2}) β_{y} + (3 c_{1} z^{2}) Κ_{y} \end{array}}_{c}

(6)

{ε_{s}}_{b} = {\begin{array}{c} ε_{x z} \\ ε_{y z} \end{array}}_{b} = {\begin{array}{c} \frac{\partial w_{0}}{\partial x} + (1 + 3 c_{1} {(z + h_{c})}^{2}) β_{x} + {(3 c_{1} (z + h_{c})}^{2}) Κ_{x} \\ \frac{\partial w_{0}}{\partial y} + (1 + 3 c_{1} {(z + h_{c})}^{2}) β_{y} + {(3 c_{1} (z + h_{c})}^{2}) Κ_{y} \end{array}}_{b}

The normal strains along the x-axis and y-axis, respectively, are ε_x and ε_y. The in-plane shear strain is $ε_{x y}$ , and transverse shear strains are $ε_{x z}$ , $ε_{y z}$ . The strain vectors ${ε_{b}}_{t}$ , ${ε_{b}}_{c}$ , and ${ε_{b}}_{b}$ show the state of normal strains at any different locations in the face-sheet at top, core and face-sheet at bottom by equation (5). Similarly, ${ε_{s}}_{t}$ , ${ε_{s}}_{c}$ , and ${ε_{s}}_{b}$ represent transverse shear strains as mentioned in equation (6).

Governing equation

The behavior of LC plate can be governed by the constitutive relation shown by equation (7) when exposed to temperature and moisture conditions,

{F} = [D] {ε} - {F^{H T}}

(7)

where,

{F} = {\begin{array}{c} {N_{x}}^{i} & {N_{y}}^{i} & {N_{x y}}^{i} & {M_{x}}^{i} & {M_{y}}^{i} & {M_{x y}}^{i} & {Q_{x}}^{i} & {Q_{y}}^{i} \end{array}}^{T}

{F^{H T}} = {\begin{array}{c} {N_{x}}^{H T} & {N_{y}}^{H T} & {N_{x y}}^{H T} & {M_{x}}^{H T} & {M_{y}}^{H T} & {M_{x y}}^{H T} & 0 & 0 \end{array}}^{T}

{ε} = {\begin{array}{c} ε_{x}^{0} & ε_{y}^{0} & ε_{x y}^{0} & Κ_{x} & Κ_{y} & Κ_{x y} & ϕ_{x} & ϕ_{y} \end{array}}^{T}

[D] = [\begin{array}{c} A & B & 0 \\ B & C & 0 \\ 0 & 0 & S s \end{array}]

A = [\begin{array}{c} A_{11} & A_{12} & A_{16} \\ A_{21} & A_{22} & A_{26} \\ A_{16} & A_{26} & A_{66} \end{array}], B = [\begin{array}{c} B_{11} & B_{12} & B_{16} \\ B_{21} & B_{22} & B_{26} \\ B_{16} & B_{26} & B_{66} \end{array}], C = [\begin{array}{c} C_{11} & C_{12} & C_{16} \\ C_{21} & C_{22} & C_{26} \\ C_{16} & C_{26} & C_{66} \end{array}], S s = [\begin{array}{c} S_{44} & S_{45} \\ S_{45} & S_{55} \end{array}]

where, in-plane internal stress resultants are N_xⁱ, N_yⁱ, N_xyⁱ, internal moment resultants are M_xⁱ, M_yⁱ, M_xyⁱ, and Q_xⁱ, Q_yⁱ are the transverse shear resultants. N_x^HT, N_y^HT, N_xy^HT are in-plane non-mechanical stress resultant, and M_x^HT, M_y^HT, M_xy^HT are non-mechanical moment resultant due to temperature and moisture conditions. The in-plane strains of the midplane are ε⁰_x, ε⁰_y, ε⁰_xy, curvatures of the plate are K_x, K_y, K_xy, and

ϕ_{x}

ϕ_{y}

are shear rotations.

(\begin{array}{c} A_{i j}, & B_{i j}, & C_{i j} \end{array}) = \sum_{k = 1}^{n} \int_{z_{k - 1}}^{z_{k}} {({\bar{Q}}_{i j}^{k})}_{k} (1, z, z^{2}) d z, . . . i, j = 1,2,6

(8)

(S_{i j}) = \sum_{k = 1}^{n} \int_{z_{k - 1}}^{z_{k}} {({\bar{Q}}_{i j}^{k})}_{k} d z, . . . i, j = 4,5

Finite element formulation

The iso-parametric quadratic elements with eight nodes are used to discretize the model. For each node, three translational and eight rotational degrees of freedom are considered. To generalize, the displacement vectors for any element can be stated as given in equation (9).

{d_{t i_d i s p}} = {\begin{array}{c} u_{0 i} & v_{0 i} & w_{0 i} \end{array}}^{T} a n d

{d_{r i_d i s p}} = {\begin{array}{c} φ_{x i} & φ_{y i} & α_{x i} & α_{y i} & β_{x i} & β_{y i} & Κ_{x i} & Κ_{y i} \end{array}}^{T}

(9)

The generalized nodal vectors are used to form generalized displacement vectors for elements as equation (10),

{d_{t_d i s p}} = [N_{t}] {d_{t_d i s p}^{e}}, a n d

{d_{r_d i s p}} = [N_{r}] {d_{r_d i s p}^{e}}

(10)

where,

[N_{t}] = {[\begin{array}{c} N_{t 1} & N_{t 2} & . . . & N_{t 8} \end{array}]}^{T}, a n d [N_{r}] = {[\begin{array}{c} N_{r 1} & N_{r 2} & . . . & N_{r 8} \end{array}]}^{T}

N_{t i} = η_{i} I_{t}; N_{r i} = η_{i} I_{r}

{d_{t_d i s p}^{e}} = {[\begin{array}{c} {d_{t 1_d i s p}^{e}}^{T} {d_{t 2_d i s p}^{e}}^{T} & . . . & {d_{t 8_d i s p}^{e}}^{T} \end{array}]}^{T}

{d_{r_d i s p}^{e}} = {[\begin{array}{c} {d_{r 1_d i s p}^{e}}^{T} {d_{r 2_d i s p}^{e}}^{T} & . . . & {d_{r 8_d i s p}^{e}}^{T} \end{array}]}^{T}

I_t = (3 × 3) Identity matrix, I_r = (8 × 8) Identity matrix, the natural coordinate’s shape functions are connected to the ith node is $η_{i}$ .

Elemental stiffness matrix

The linear strain can be written as follows in equation (11),

{ε} = [B] {δ_{e}}

(11)

where,

{δ_{e}} = {u_{1}, v_{1}, w, φ_{x 1}, φ_{y 1}, α_{x 1}, α_{y 1}, β_{x 1}, β_{y 1}, Κ_{x 1}, Κ_{y 1}, . . . . . ., u_{8}, v_{8}, w, φ_{x 8}, φ_{y 8}, α_{x 8}, α_{y 8}, β_{x 8}, β_{y 8}, Κ_{x 8}, Κ_{y 8}}^{T}

The relation between strain and displacement can be expressed as follows,

{ε_{b}}_{t o p} = [B_{t b}] {d_{t_d i s p}^{e}} + [Z_{1}] [B_{r b}] {d_{r_d i s p}^{e}}

{ε_{b}}_{c o r e} = [B_{t b}] {d_{t_d i s p}^{e}} + [Z_{2}] [B_{r b}] {d_{r_d i s p}^{e}}

{ε_{b}}_{b o t t o m} = [B_{t b}] {d_{t_d i s p}^{e}} + [Z_{3}] [B_{r b}] {d_{r_d i s p}^{e}}

(12)

{ε_{s}}_{t o p} = [B_{t s}] {d_{t_d i s p}^{e}} + [Z_{4}] [B_{r s}] {d_{r_d i s p}^{e}}

{ε_{s}}_{c o r e} = [B_{t s}] {d_{t_d i s p}^{e}} + [Z_{5}] [B_{r s}] {d_{r_d i s p}^{e}}

{ε_{s}}_{b o t t o m} = [B_{t s}] {d_{t_d i s p}^{e}} + [Z_{6}] [B_{r s}] {d_{r_d i s p}^{e}}

Various matrices used in equation (12) are elaborated in Appendix 1, The dynamic version of the virtual work principle is applied to obtain the equations of motion. Accordingly, the potential energy and kinetic energy can be represented by equation (13),

Τ_{p}^{e} = \frac{1}{2} [\begin{array}{l} {d_{t_d i s p}^{e}}^{T} [K t t e] {d_{t_d i s p}^{e}} + {d_{t_d i s p}^{e}}^{T} [K t r e] {d_{r_d i s p}^{e}} + \\ {d_{r_d i s p}^{e}}^{T} {[K t r e]}^{T} {d_{t_d i s p}^{e}} + {d_{r_d i s p}^{e}}^{T} [K r r e] {d_{r_d i s p}^{e}} - 2 {d_{t_d i s p}^{e}}^{T} {F^{e}} \end{array}]

Τ_{k}^{e} = \frac{1}{2} \int_{0}^{a^{e}} \int_{0}^{b^{e}} \bar{m} {{\dot{d}}_{t_d i s p}^{e}}^{T} {[N]}^{T} [N] {{\dot{d}}_{t_d i s p}^{e}} d x d y s

(13)

whereas,

[K t t e] = [K t t b e] + [K t t s e]

(14)

[K t r e] = [K t r b e] + [K t r s e]

(15)

[K r r e] = [K r r b e] + [K r r s e]

(16)

{F^{e}} = \int_{0}^{a^{e}} \int_{0}^{b^{e}} {[N]}^{T} {f} d x d y

(17)

Matrices used in equations (14)–(17) are defined in Appendix 2.

From solving the above equations (equations (11)–(13)), the equations of motion can be written as follow:

[Μ^{e}] {{\ddot{d}}_{t_d i s p}^{e}} + [K t t e] {d_{t_d i s p}^{e}} + [K t r e] {d_{r_d i s p}^{e}} = {F^{e}}

{[K t r e]}^{T} {d_{t_d i s p}^{e}} + [K r r e] {d_{r_d i s p}^{e}} = 0

(18)

The element stiffness matrix ([K_e]) can be obtained after solving the equations of motion.

Elemental initial stress stiffness matrix

The non-mechanical force and moment due to temperature and moisture are shown by equation (19),

{\begin{array}{c} {N_{x}}^{H T} & {N_{y}}^{H T} & {N_{x y}}^{H T} \end{array}} = \sum_{k = 1}^{n} {[{\bar{Q}}_{i j}]}_{k} (e_{k}) (z_{k} - z_{k - 1})

{\begin{array}{c} {M_{x}}^{H T} & {M_{y}}^{H T} & {M_{x y}}^{H T} \end{array}} = \frac{1}{2} \sum_{k = 1}^{n} {[{\bar{Q}}_{i j}]}_{k} (e_{k}) (z_{k}^{2} - z_{k - 1}^{2}), \dots i, j = 1,2,6

(19)

where,

{[{\bar{Q}}_{i j}]}_{k} = {[\begin{array}{c} c^{2} θ & s^{2} θ & s θ c θ \\ s^{2} θ & c^{2} θ & - s θ c θ \\ - 2 s θ c θ & 2 s θ c θ & c^{2} θ - s^{2} θ \end{array}]}^{- 1} [\begin{array}{c} Q_{11} & Q_{12} & Q_{16} \\ Q_{21} & Q_{22} & Q_{26} \\ Q_{12} & Q_{16} & Q_{66} \end{array}] [\begin{array}{c} c^{2} θ & s^{2} θ & s θ c θ \\ s^{2} θ & c^{2} θ & - s θ c θ \\ - 2 s θ c θ & 2 s θ c θ & c^{2} θ - s^{2} θ \end{array}] (i, j = 1, 2, 6)

{[\bar{Q}]}_{k} = {[\begin{array}{c} c θ & s θ \\ - s θ & c θ \end{array}]}^{- 1} [\begin{array}{c} Q_{44} & 0 \\ 0 & Q_{55} \end{array}] [\begin{array}{c} c θ & s θ \\ - s θ & c θ \end{array}], (i, j = 4, 5)

where θ is a fiber orientation in composite and c is a cosine, and s is a sine.

And,

{e}_{k} = {\begin{array}{c} e_{x}^{H T} \\ e_{y .}^{H T} \\ e_{x y}^{H T} \end{array}} = [\begin{array}{c} c^{2} θ & s^{2} θ \\ s^{2} θ & c^{2} θ \\ s 2 θ & c 2 θ \end{array}] {\begin{array}{c} β_{1} \\ β_{2} \end{array}}_{k} (C - C_{0}) + [\begin{array}{c} c^{2} θ & s^{2} θ \\ s^{2} θ & c^{2} θ \\ s 2 θ & c 2 θ \end{array}] {\begin{array}{c} α_{1} \\ α_{2} \end{array}}_{k} (T - T_{0})

(20)

e_x^HT, e_y^HT, e_xy^HT are non-mechanical strains because of hygroscopic and thermal conditions, α₁ and α₂ are thermal expansion coefficients, T₀ and T are room temperature and increased temperature. β₁ and β₂ moisture coefficient with C₀ as reference moisture and C as increased moisture.

The nonlinear strain can be stated as

{ε_{n}} = {ε_{x n}, ε_{y n}, ε_{x y n}} = [R] {d} / 2

(21)

where,

ε_{x n}

ε_{y n}

, and

ε_{x y n}

are nonlinear strains,

ε_{x n} = \frac{1}{2} [{\bar{u}}_{, x}^{2} + {\bar{v}}_{, x}^{2} + {\bar{w}}_{, x}^{2} + 2 z ({\bar{u}}_{, x} θ_{y, x} - {\bar{v}}_{, x} θ_{x, x}) + z^{2} (θ_{y, x}^{2} + θ_{x, x}^{2})]

ε_{y n} = \frac{1}{2} [{\bar{u}}_{, y}^{2} + {\bar{v}}_{, y}^{2} + {\bar{w}}_{, y}^{2} + 2 z ({\bar{u}}_{, y} θ_{y, y} - {\bar{v}}_{, y} θ_{x, y}) + z^{2} (θ_{x, y}^{2} + θ_{y, y}^{2})]

ε_{x y n} = [{\bar{u}}_{, x} {\bar{u}}_{, y} + {\bar{v}}_{, x} {\bar{v}}_{, y} + {\bar{w}}_{, x} {\bar{w}}_{, y} + z ({\bar{u}}_{, y} θ_{y, x} + {\bar{u}}_{, x} θ_{y, y}) - z ({\bar{v}}_{, y} θ_{x, x} + {\bar{v}}_{, x} θ_{x, y}) + z^{2} (θ_{y, x} θ_{y, y} + θ_{x, x} θ_{x, y})]

ε_{x z n} = [{\bar{u}}_{, x} θ_{, y} + {\bar{v}}_{, x} θ_{, x} + z (θ_{y} θ_{y, x} + θ_{x} θ_{x, x})]

ε_{y z n} = [{\bar{u}}_{, y} θ_{, y} + {\bar{v}}_{, y} θ_{, x} + z (θ_{y} θ_{y, y} + θ_{x} θ_{x, y})]

where u, v, and w are the x, y, and z components of structural displacement, and, and are total rotations of a plate member along the x and y direction.

{d} = {\begin{cases} \frac{\partial u_{0}}{\partial x}, \frac{\partial u_{0}}{\partial y}, \frac{\partial v_{0}}{\partial x}, \frac{\partial v_{0}}{\partial y}, \frac{\partial w_{0}}{\partial x}, \frac{\partial w_{0}}{\partial y}, \frac{\partial φ_{x}}{\partial x}, \frac{\partial φ_{x}}{\partial y}, \frac{\partial φ_{y}}{\partial x}, \frac{\partial φ_{y}}{\partial y}, \frac{\partial α_{x}}{\partial x}, \frac{\partial α_{x}}{\partial y}, \frac{\partial α_{y}}{\partial x}, \frac{\partial α_{y}}{\partial y}, . . . \\ . . ., \frac{\partial β_{x}}{\partial x}, \frac{\partial β_{x}}{\partial y}, \frac{\partial β_{y}}{\partial x}, \frac{\partial β_{y}}{\partial y}, \frac{\partial Κ_{x}}{\partial x}, \frac{\partial Κ_{x}}{\partial y}, \frac{\partial Κ_{y}}{\partial x}, \frac{\partial Κ_{y}}{\partial y}, θ_{x}, θ_{y}, α_{x}, α_{y}, β_{x}, β_{y}, Κ_{x}, Κ_{y} \end{cases}}^{T}

while {d} can also be represented as,

{d} = [G] {δ_{e}}

(22)

where,

[G] = [\begin{array}{c} {\hat{N}}_{1} & \tilde{O} \\ \hat{O} & {\hat{N}}_{2} \\ \bar{O} & \bar{I} \end{array}], {\hat{N}}_{1} = [\begin{array}{c} Ν_{i, x} & 0 & 0 \\ Ν_{i, y} & 0 & 0 \\ 0 & Ν_{i, x} & 0 \\ 0 & Ν_{i, y} & 0 \\ 0 & 0 & Ν_{i, x} \\ 0 & 0 & Ν_{i, y} \end{array}], {\hat{N}}_{2} = [\begin{array}{c} \bar{n} & \bar{o} & \bar{o} & \bar{o} \\ \bar{o} & \bar{n} & \bar{o} & \bar{o} \\ \bar{o} & \bar{o} & \bar{n} & \bar{o} \\ \bar{o} & \bar{o} & \bar{o} & \bar{n} \end{array}], \bar{n} = [\begin{array}{c} Ν_{i, x} & 0 \\ Ν_{i, y} & 0 \\ 0 & Ν_{i, x} \\ 0 & Ν_{i, y} \end{array}]

Õ, Ô, Ō, and ō are (6 × 8), (16 × 3), (8 × 3), and (4 × 2) null matrices, respectively. Ī is identity-matrix (8 × 8).

The initial stress stiffness matrix can be expressed as:

[K_{σ e}] = {\int \int [G]}^{T} [S] [G] d x d y

(23)

(S) matrix is elaborated in the Appendix 3.

Solution process

Gaussian integration rule is implemented to obtain bending and shear deformation at the element level. Then, the stiffness matrix [K_e], initial stress-stiffness matrix [K_σe], and mass matrix [M_e] for an element are computed. The calculated matrices at the elemental level are accumulated to the stiffness matrices [K], [K_σ], and [M] at the global level. The stiffness matrices determined at the global coordinate system are used to find the system’s natural frequency under free vibration by using the following equation.

([K] + [K_{σ}]) - ω^{2} [M] = 0

(24)

where the ω is the system’s free vibration natural frequency.

For skew sandwich plates, instead of rectangular coordinates (x, y, z), the skew coordinates (x', y', z') are used. The displacement vector, which is generalized for any point located on the skew edge of the plate, can be changed as given in equation (25),

{d_{t_d i s p}} = [L t] {d_{t_d i s p}^{'}}, {d_{r_d i s p}} = [L r] {d_{r_d i s p}^{'}}

(25)

For new coordinate system ${d_{t}^{'}}$ and ${d_{r}^{'}}$ are displacement vectors.

{d_{t_d i s p}^{'}} = {\begin{array}{c} u_{0}^{'} & v_{0}^{'} & w_{0}^{'} \end{array}}^{T}

{d_{r_d i s p}^{'}} = {\begin{array}{c} φ_{x}^{'} & φ_{y}^{'} & α_{x}^{'} & α_{y}^{'} & β_{x}^{'} & β_{y}^{'} & Κ_{x}^{'} & Κ_{y}^{'} \end{array}}^{T}

(26)

The transformation matrices for the above are given as,

[L_{t}] = [\begin{array}{c} c Ψ & s Ψ & 0 \\ - s Ψ & c Ψ & 0 \\ 0 & 0 & 1 \end{array}], [L_{r}] = [\begin{array}{c} L & 0 & 0 & 0 \\ 0 & L & 0 & 0 \\ 0 & 0 & L & 0 \\ 0 & 0 & 0 & L \end{array}] [L] = [\begin{array}{c} c Ψ & s Ψ \\ - s Ψ & c Ψ \end{array}]

(27)

where, [L_t] and [L_r], transformation matrices, are multiplied to global stiffness and mass matrices. In equation (24), those transformed global matrices are used in place of original global matrices. The computer code in MATLAB is developed to get the natural frequency of SLCS plate based on FE modeling for thermal and hygroscopic conditions.

Material properties

Numerous simulations were carried out to determine the free vibration characteristics of the LCS plate under the influence of moisture and thermal environment. Different materials were used to validate the developed code, such as Graphite-Epoxy, DYAD 606 viscoelastic material. The materials considered for the simulation is bi-directional bamboo fiber reinforced face sheet and biodegradable PLA for the core in the sandwich plate. The moisture and temperature-dependent material properties of graphite-epoxy are shown in Tables 1 and 2. Density is taken as 1600 kg/m³, Poisson’s ratio as 0.3, moisture expansion coefficient as β₁ = 0 and β₂ = 0.44, and thermal expansion coefficient as α₁ = −0.3 × 10⁻⁶/K, α₂ = 28.1 × 10⁻⁶/K. The mechanical properties of viscoelastic material DYAD 606 at different temperatures are obtained from Figure 2, with a density of 1200 kg/m³ and Poisson’s ratio of 0.49.

Table 1.

Temperature-dependent material properties of graphite reinforced epoxy composite, G₁₃ = G₁₂, G₂₃ = 0.5G₁₂.⁴²

Elastic moduli (GPa)	Temperature (K)
Elastic moduli (GPa)	300	325	350	375	400	425
E₁	130	130	130	130	130	130
E₂	9.5	8.5	8	7.5	7	6.75
G₁₂	6	6	5.5	5	4.75	4.5

Table 2.

Moisture-dependent material properties of graphite reinforced epoxy composite, G₁₃ = G₁₂, G₂₃ = 0.5G₁₂.⁴²

Elastic moduli (GPa)	Moisture content (%)
Elastic moduli (GPa)	0	0.25	0.5	0.75	1	1.25
E₁	130	130	130	130	130	130
E₂	9.5	9.25	9	8.75	8.5	8.5
G₁₂	6	6	6	6	6	6

Figure 2.

Mechanical properties of viscoelastic material dyad 606 (a) shear modulus, (b) loss factor.⁷¹

The moisture and temperature-dependent material properties for bamboo-epoxy composite²² and PLA⁶⁴ are shown in Figures 3 and 4, respectively, are not continuous for all temperature and moisture values. Therefore, to interpolate the required data, certain equations (equations (28)–(32)) are generated using different curve fitting techniques as follows:

Figure 3.

Degradation of mechanical properties of BFB with rising (a) temperature, (b) moisture concentration.²¹

Figure 4.

Modulus of elasticity of PLA varying with rising temperature.⁶³

For BFB,

{E_{1}}_{t e m p} = (\frac{1}{a_{1} + b_{1} T + c_{1} T^{2}}) G P a, a_{1} = 0.051257, b_{1} = 0.000293, c_{1} = 0.000006 .

(28)

{G_{12}}_{t e m p} = (a_{2} {(1 + \frac{c_{2} T}{b_{2}})}^{(\frac{- 1}{c_{2}})}) G P a, a_{2} = 2.771837, b_{2} = 646.470829, c_{2} = - 4.962168 .

(29)

{E_{1}}_{m o i s t} = (a_{3} (b_{3} - e^{- c_{3} M})) G P a, a_{3} = 0.653099, b_{3} = 32.974983, c_{3} = - 1.685657 .

(30)

{G_{12}}_{m o i s t} = (\frac{a_{4}}{1 + e^{(b_{4} - c_{4} M)}}) G P a, a_{4} = 2.889512, b_{4} = - 7.947971, c_{4} = - 4.422342 .

(31)

For PLA,

{E_{1}}_{t e m p} = (\frac{a_{5}}{1 + e^{(b_{5} - c_{5} T)}}) G P a, a_{5} = 2.802262, b_{5} = - 18.965464, c_{5} = - 0.376534 .

(32)

The density of BFB and PLA are taken as 1830 kg/m³ and 1240 kg/m³, respectively.

Boundary conditions

Four different cases (boundary conditions) are considered to mimic the actual supports in the radiotherapy bed or patient positioning system in the MRIsystem. Thus, the boundary conditions of the plate are all sides are clamped (CCCC), two opposite sides are clamped, and the other two are simply supported (CSCS), one side is clamped, and the other three sides are simply supported (CSSS). Finally, all four sides are simply supported (SSSS), and are given as follows:

C l a m p e d e d g e (C) : u_{x} = u_{y} = u_{z} = ϕ = ψ = 0

(33.a)

S i m p l y s u p p o r t e d e d g e (S) : u_{x} \neq 0; u_{y} = u_{z} = ϕ = ψ = 0 a t x = 0, a

u_{y} \neq 0; u_{x} = u_{z} = ϕ = ψ = 0 a t y = 0, b

(33.b)

Results and discussions

The developed mathematical formulation is used to investigate the influence of hygroscopic and thermal conditions on the natural frequency of the LC and LCS plates. While performing FE simulation, a uniform rise in temperature and moisture concentration is considered.

Convergence study

The fundamental non-dimensional natural frequencies of SSSS square LC plate with four-layer symmetric cross-ply laminae are determined using a convergence study at 300 K and 325 K temperature and 0.1% moisture concentration for different mesh divisions. The length-to-thickness (l/h) ratio is set to 100. The following material properties of graphite-epoxy are used for the convergence study.

E₁ = 130 GPa, E₂ = 9.50 GPa, G₁₂ = G₁₃ = 6.00 GPa, G₂₃ = 0.50G₁₂, μ₁₂ = 0.30, α₁ = −0.30×10⁻⁶/K, α₂ = 28.10×10⁻⁶/K, β₁ = 0, β₂ = 0.44, ΔT = 25, ΔC = 0.1%.

The computed natural frequency is converted to non-dimensional form as:

\bar{ω} = \frac{ω a^{2}}{H} \sqrt{\frac{ρ}{E_{2}}}

(34)

The non-dimensional frequencies of composite plates under hygrothermal conditions converge very well with mesh refinement, as seen in Table 3. It is also noticed that 10 × 10 mesh division is sufficient for reliably predicting higher-order behavior and saves computation time compared to 12 × 12 mesh divisions. Thus 10 × 10 mesh is adopted in the current numerical study.

Table 3.

Convergence study of non-dimensional fundamental frequencies for SSSS LC plate under hygrothermal conditions.

Mode	C = 0.1%			T = 300 K			T = 325 K
Mode	1	2	3	1	2	3	1	2	3
4 × 4	9.4290	20.5392	39.5042	12.0651	23.6590	41.2018	8.0922	19.0988	38.7802
6 × 6	9.3917	19.8929	39.2810	12.0416	23.1767	41.0390	8.0456	18.3648	38.5311
8 × 8	9.3902	19.8703	39.2763	12.0413	23.1667	41.0413	8.0435	18.3358	38.5232
10 × 10	9.3901	19.8685	39.2773	12.0414	23.1676	41.0443	8.0431	18.3326	38.5235
12 × 12	9.3900	19.8683	39.2783	12.0415	23.1684	41.0459	8.0431	18.3320	38.5241

Validation with previous studies

Under the elevated temperature and moisture conditions, the derived formulation has been validated for free vibration analysis of the SSSS composite plate. Tables 4 and 5 present the validation of non-dimensional fundamental natural frequency for elevated temperature and moisture conditions, respectively. Graphite reinforcement with epoxy resin is used as lamina for the LC. The l/h ratio is taken as 40, and a four-layered symmetric cross-ply is considered. The non-dimensional fundamental natural frequency is determined for different temperatures and moisture conditions, and varying length-to-width (l/w) ratios are compared with the existing research. The results obtained from the developed FE formulation in the current study are in good agreement with the results stated in the literature.^42,47 However, it may laso be obsereved that the results obtained using the FE formulation derived in the present study are lower valves as compared with Kallannavar et al.⁴⁷ It may be due to implementation of reduced order integegration. Nevertheless, very good agreement with Ram et al.⁴² can be noticed.

Table 4.

The non-dimensional fundamental frequency at different temperatures for (0°/90°/90°/0°) LC plate with SSSS boundary condition.

Temp	l/w = 0.5			l/w = 1.0			l/w = 2.0
Temp	Kallannavar et al.⁴⁷	Ram et al.⁴²	Present	Kallannavar et al.⁴⁷	Ram et al.⁴²	Present	Kallannavar et al.⁴⁷	Ram et al.⁴²	Present
300	10.2953	10.1714	10.2281	12.1095	11.8934	11.9558	23.4569	23.0044	22.9771
325	10.4077	9.9067	9.8279	12.1041	11.3951	11.4059	23.6263	22.1934	22.2675
350	10.2584	9.6410	9.4107	11.7558	10.8916	10.8281	23.219	21.3008	21.5345
375	10.1553	9.2964	8.9741	11.4469	10.5476	10.2177	22.8721	20.4877	20.7757
400	10.1337	8.9498	8.5151	11.2521	9.8887	9.5684	22.7194	19.7508	19.9880
425	9.9080	8.7633	8.0298	10.8171	9.3896	8.8717	22.2143	19.0919	19.1680

Table 5.

The non-dimensional fundamental frequency at various moisture concentrations for (0°/90°/90°/0°) LC plate with SSSS boundary condition.

Moisture (%)	l/w = 0.5			l/w = 1.0			l/w = 2.0
Moisture (%)	Kallannavar et al.⁴⁷	Ram et al.⁴²	Present	Kallannavar et al.⁴⁷	Ram et al.⁴²	Present	Kallannavar et al.⁴⁷	Ram et al.⁴²	Present
0	10.2953	10.2654	10.2281	12.1095	11.8169	11.9558	23.4569	22.8951	22.9771
0.25	9.587	9.6505	9.5087	11.1004	10.982	10.9643	22.217	21.6891	21.7061
0.5	8.8313	8.8126	8.7301	10.0005	9.9216	9.8737	20.9189	20.3351	20.3557
0.75	8.0156	8.2725	7.8749	8.7768	8.8612	8.6466	19.5524	18.9823	18.9091
1	7.1202	7.3613	6.9146	7.3689	7.6567	7.2136	18.1034	17.7763	17.3422
1.25	5.2376	6.5249	5.4139	5.2374	6.5258	5.4136	16.1422	16.4238	15.6187

Skew laminated composite sandwich plate

The study of an LCS plate consisting of DYAD 606 viscoelastic core as well as graphite-epoxy face-sheet is also validated for a square sandwich plate, and l/h ratio is considered 50 and l/w ratio 2. The temperature-dependent material properties used for the graphite-reinforced epoxy composite are shown in Tables 1 and 2, and the DYAD 606 viscoelastic material is shown in Figure 2. The complex modulus method is used to model the viscoelastic material. Accordingly, the temperature-dependent shear modulus is expressed in equation (35) as follows:

G = G^{'} (1 + i η)

(35)

where the temperature-dependent storage modulus is G', and the temperature-dependent loss factor is η. As temperature changes, these value changes, which can be taken from Figure 2 mentioned earlier. The results for the simulation of an LCS plate in different BCs, for different temperatures and skew angels, are compared with the results published in Kallannavar et al.⁴⁷ Table 6 shows that the results are nearly matching and follow the same trend while increasing temperature, as seen in the literature.

Table 6.

Effect of temperature on a SLCS plate with graphite-epoxy as a face-sheet and dyad 606 as a core.

BC.	Temperature (K)	Skew angle = 0°		15°		30°		45°
BC.	Temperature (K)	Kallannavar et al.⁴⁷	Present	Kallannavar et al.⁴⁷	Present	Kallannavar et al.⁴⁷	Present	Kallannavar et al.⁴⁷	Present
CCCC	300	23.4859	24.2969	23.7027	24.5468	24.4527	25.4411	26.1390	27.5555
	325	13.2707	13.5570	13.4053	13.7041	13.8563	14.2113	14.8004	15.3147
	350	10.0561	10.2478	10.1523	10.3542	10.4714	10.7223	11.1327	11.5636
	375	8.7084	9.3432	8.7624	9.4397	8.9750	9.8186	9.5497	10.8057
SSSS	300	21.0898	22.1264	21.3488	22.3264	22.2307	23.2509	24.0882	25.3970
	325	12.4751	12.9584	12.6105	12.9901	13.0852	13.4547	14.0754	14.5201
	350	9.3303	9.6233	9.4307	9.6411	9.7822	10.0015	10.5065	10.8289

Effect of initial stress stiffness matrix due to temperature and moisture on the natural frequency

Several studies show that the mechanical properties of composite materials change when subjected to various thermal and moisture environments. The stiffness of composite materials deteriorates under high-temperature and high-moisture environments, resulting in natural frequency deterioration. This behavior is due to the changing volume of the composite because of thermal expansion of the polymer matrix; a similar phenomenon happens with moisture absorption also. As a result, the geometry of the structure or a structural component deforms and loses stability. It causes initial strain, which results in severe geometric nonlinearity, which leads to a change in stiffness. This change can be incorporated by considering the initial stress stiffness (K_σ) owing to temperature and moisture. Thus, the temperature and moisture-dependent properties of materials can be considered. The simulation has been carried out to evaluate the impact of environmental conditions on the LBCS plate consisting of PLA as a core and BFB as a face sheet. The temperature and moisture-dependent material properties are determined from equations (27)–(31) are used for the analysis. The LBCS plate with l/h ratio as 40, l/w ratio as 1, and core thickness to face sheet thickness (h_c/h_t) ratio of 2 is considered for the computation. Figure 5(a) and (b) show that non-dimensionalized natural frequency (NDNF) decreases with increasing temperature and moisture. Figure 5 depicts that material with moisture and temperature-independent properties with or without K_σ underpredicts the NDNF of the LBCS plate. Whereas the consideration of temperature and moisture-dependent material properties and K_σ gives results in close agreement with experimental results, as shown in a study presented by Sit and Ray.²² Subsequently, these dependent material properties, and initial stress stiffness are used for further investigations. In Figure 5(a), we see a significant drop in NDNF from 50°C. This behavior is noticed because the glass transition temperature (T_g) of the PLA used as the core ranges from 50°C to 70°C,^65–67 which reduces the strength of the LBCS plate. No such behavior of the LBCS plate can be seen in the moisture environment in Figure 5(b). It is due to the material properties of only bamboo face sheets degrading with an increase in moisture, but there is a negligible change in the properties of PLA.⁶⁸

Figure 5.

Comparison of effect of initial stress stiffness matrix and temperature and moisture dependent material properties NDNF.

Effect of length-to-width ratio on the natural frequency of the plate in different boundary conditions under thermal environment

The influence of l/w ratio on the NDNF of the LBCS plate under various BCs using material properties of the BFB and PLA at room temperature and dry conditions are presented in Figure 6. For the computation, the l/h ratio of 40, as well as the hc/ht ratio of five is employed. According to the results, NDNF increases when the length-to-width ratio increase. This behavior is because of the transition from a thick to a thin plate as l/w ratio increases. As a result, the stiffness matrix in relation to the mass matrix increases, resulting in a rise in NDNF value. Several investigators^1,40,46,69 have found similar conclusions for different composite material structures. The plate with CCCC boundary condition possesses the highest NDNF for all l/w ratios considering other boundary conditions due to an increase in the plate’s stiffness due to all sides being in the clamped condition. But the increase in NDNF with l/w ratio changing from 0.5 to 2.0 is the most minor, i.e., increment of 284% compared to 594% in CSCS, 439% in CSSS, and 296% in SSSS boundary conditions.

Figure 6.

Effect of l/w ratio on NDNF for various boundary conditions at room temperature and dry condition.

The behavior of the LBCS plate under the thermal environment with varying l/w ratios for various boundary conditions is represented in Figure 7. Temperature-dependent material properties are utilized with an l/h ratio of 40, h_c/h_t ratio of 10, and four different boundary conditions: CCCC, SSSS, CSCS, and CSSS. As expected from previous results, the NDNF starts to drop rapidly after 50°C. It can be seen in all cases of various boundary conditions. The highest drop of 48.57% in NDNF from room temperature to 60°C can be seen in l/w ratio of 2.0 in CCCC boundary conditions. Similarly, for the same l/w ratio of 2, a rise in temperature causes a maximum reduction of 34.1%, 48.9%, and 41.8% in SSSS, CSCS, and CSSS boundary conditions, indicating that the drop in NDNF is more evident in higher l/w ratios as temperature increases.

Figure 7.

Effect of thermal environment on NDNF with different l/w ratio (a) CCCC, (b) SSSS, (c) CSCS, (d) CSSS.

Effect of ratio of length-to-thickness of LBCS plate on the natural frequency at different temperature

The effect of l/h ratio is studied on the BFB face-sheet/PLA core sandwich plate with l/w = 2 and h_c/h_t = 5 for different boundary conditions at room temperature and dry conditions. The results presented in Figure 8 show an increase in NDNF with an increase in l/h ratio. Further, the behavior of the LBCS plate with increasing l/h ratio as for the behavior under varying l/w ratio. Results show that the LBCS plate with CCCC has the maximum natural frequency while the SSSS exhibits a minimum natural frequency value in all the boundary conditions. The increment in non-dimensional frequency for CCCC, CSCS, CSSS, and SSSS are 61.6%, 62.4%, 38.2%, and 18.3%, respectively, for l/h value changing from 10 to 80.

Figure 8.

Effect of l/h ratio on NDNF with various boundary conditions.

Further, the effect of increasing temperature on NDNF for l/h as 2, 10, 20, and 40 is investigated for different boundary conditions depicted in Figure 9. The non-dimensional natural frequency decreases as the temperature value increases. For all the boundary conditions, as l/h increases, the increment in the NDNF decreases for the considered temperatures. This decrease in the increment of NDNF is substantial as the number of simply-supported sides increases. A similar conclusion can be observed in the study of some investigators.⁶⁸ The maximum reduction of 75.8%, 67.9%, 75.5%, and 72.7% in non-dimensional fundamental NDNF can be seen for CCCC, SSSS, CSCS, and CSSS boundary conditions, respectively.

Figure 9.

Effect of temperature on NDNF for different l/h ratio with (a) CCCC, (b) SSSS, (c) CSCS, (d) CSSS.

Effect of skew angle with changing temperature and moisture concentration on LBCS plate

The study of the influence of skew angle on NDNF in BFB face-sheet/PLA core sandwich plate with varying l/w ratios, l/h ratios, and boundary conditions under temperature and moisture environments is continued. Firstly, the influence of skewness is studied for different boundary conditions with h_c/h_t ratio of 5, l/w ratio of 1, and l/h ratio of 40. The results are illustrated in Figure 10, showing that the NDNF increases with an increasing skew angle for other parameters that remain unchanged. Many researchers studied the influence of skew angle and reported identical results.^70,71 The impact of BC increases as there is an increase in the skew angle, and the affected area of the plate reduces, increasing NDNF. Again, it is clearly seen that CCCC and SSSS have the maximum and minimum NDNF, respectively.

Figure 10.

Effect of skew angle on NDNF for a different boundary.

The current research is also conducted for several l/w and l/h ratios and the influence of skew angle on non-dimensional fundamental natural frequency. Figure 11(a) depicts the study for four different l/w ratios for l/h ratio of 40, h_c/h_t ratio of 5, and at CCCC boundary conditions in room temperature and dry conditions. From the figure, we can say that as the skew angle increases, the NDNF increase for every l/w ratio. The maximum gain of 55.8% from 0° to 45° skew angle is observed for l/w ratio of 2. A similar study has been done for varying l/h ratios to study its effect on NDNF for increasing skew angle. Figure 11(b) shows similar results of increasing NDNF for a rise in skew angle with l/w ratio of one and keeping everything else constant. A maximum of 51.9% increment can be observed in NDNF between two extreme skew angles. As we see in Figure 11(a) and (b), the change in NDNF between 0° to 15° is significantly less, but from 30° onward, the NDNF is altered substantially.

Figure 11.

Effect of skew angle on NDNF for different (a) l/w ratios, (b) l/h ratios.

The experiment is being carried out to assess the effect of temperature on NDNF for different skew angles. The l/h ratio of 40, l/w ratio of 1, and h_c/h_t ratio of five are considered for the simulation. As shown in Figure 12, increased temperature decreases the NDNF for all skew angles and boundary conditions. Further, as the skew angle increases, the difference in all the curves increases for every boundary condition. The maximum reduction in the NDNF for CCCC, SSSS, CSCS, and CSSS is 44.8%, 42.7%, 42.2%, and 42.3%, respectively, with varying temperatures from 20°C to 60°C.

Figure 12.

Effect of temperature on NDNF for different skew angle with (a) CCCC, (b) SSSS, (c) CSCS, (d) CSSS.

Further investigation is carried out to study the effect of moisture on NDNF for different skew angles keeping all the parameters the same; results are shown in Figure 13. There is no sudden drop in NDNF in case of changing moisture environment. It may be because of degradation in the mechanical properties of face sheets only, which gradually decreases over moisture exposure. Therefore, the decrease in NDNF is due to a reduction in the mechanical properties of the bamboo composite face sheet. The maximum reduction in NDNF for CCCC, SSSS, CSCS, and CSSS are 16.7%, 87.6%, 28.1%, and 48.5%, respectively.

Figure 13.

Effect of moisture on NDNF for different skew angle with boundary conditions: (a) CCCC, (b) SSSS, (c) CSCS, (d) CSSS.

Effect of change in core thickness to face-sheet thickness ratio with temperature and moisture concentration on the LBCS plate

The current study has continued to consider the effect of changing the core-to-face sheet thickness ratios on NDNF under different temperature and moisture conditions. Figure 14 shows the effect on NDNF due to change in h_c/h_t with different boundary conditions for l/h ratio of 40, l/w ratio of 1, and skew angle of 0°. As the h_c/h_t ratio increases, the NDNF decreases. While changing h_c/h_t, keeping the sandwich plate’s cross-section area constant, only the core and face sheet thickness varies. As h_c/h_t increases, the core thickness increases, and reduction in the thickness of the face sheet, which in turn reduces the stiffness and NDNF as well. The CCCC boundary condition possesses higher NDNF than the other boundary conditions for all h_c/h_t ratios. The maximum reduction of 39% can be seen in the CCCC boundary condition over h_c/h_t ratio of 1 to 20.

Figure 14.

Effect of h_c/h_t on NDNF for different boundary conditions.

Similar simulation is continued for different l/w ratios, l/h ratios, and skew angles, which are shown in Figure 15(a)–(c), respectively. The results in Figure 15(a) are computed at l/h ratio of 40 and skew angle of 0°, and in Figure 15(b), l/w ratio is 1, and the skew angle is 0°. Similarly, for Figure 15(c), the l/h and l/w ratios are 40 and 1. In all cases, NDNF decreases with an increase in h_c/h_t ratio. The maximum reduction in NDNF can be seen as 39.9%, 39.7%, and 39.0% for each case in Figure 15(a)–(c), respectively. The reduction can be seen as prominent after h_c/h_t ratio of 2, as a significant contribution of core increases in the strength of the BFB face-sheet/PLA core sandwich plate.

Figure 15.

Effect of h_c/h_t ratio on NDNF for different (a) l/w ratios (b) l/h ratios (c) skew angle.

The simulation is done to study the effect of h_c/h_t ratio on NDNF in thermal and moisture environments for different l/w ratios and skew angles. The results are presented for l/h ratio of 40, and all side-clamped conditions under varying temperatures in Table 7. The result shows decreasing NDNF for all temperatures with increasing h_c/h_t ratio but increasing NDNF with increasing l/w ratio and skew angle. Table 8 shows the results under varying moisture concentrations for l/h ratio 40, and all side-clamped conditions. The table clearly shows the decrease in NDNF with increasing moisture concentrations.

Table 7.

The effect of h_c/h_t ratio on the NDNF at different temperatures with varying l/w ratios and skew angles.

Skew angle	Temperature	h _c /h _t
Skew angle	Temperature	1	2	5	10	20
0°	20°	27.2246	25.8111	22.4138	19.4288	16.9070
	30°	27.0182	25.6405	22.2758	19.3039	16.7878
	40°	26.8355	25.4911	22.1584	19.2014	16.6940
	50°	25.5338	24.0517	20.6872	17.5415	14.6937
	60°	12.8996	11.6346	10.6982	9.9922	8.8910
15°	20°	28.4290	26.9577	23.4366	20.3321	17.7018
	30°	28.2250	26.7895	23.3008	20.2093	17.5847
	40°	28.0467	26.6440	23.1868	20.1101	17.4944
	50°	26.6382	25.0889	21.6161	18.3569	15.3941
	60°	13.2547	11.9170	10.9573	10.2656	9.1768
30°	20°	32.5656	30.8988	26.9719	23.4673	20.4678
	30°	32.3683	30.7377	26.8429	23.3511	20.3573
	40°	32.2042	30.6049	26.7400	23.2626	20.2780
	50°	30.3918	28.6139	24.7988	21.1700	17.8227
	60°	14.4451	12.8472	11.7964	11.1498	10.1088
45°	20°	41.7144	39.6316	34.9211	30.5990	26.8095
	30°	41.5262	39.4819	34.8039	30.4944	26.7107
	40°	41.3904	39.3752	34.7236	30.4276	26.6537
	50°	38.4933	36.2175	31.7980	27.4703	23.3374
	60°	17.0039	14.7664	13.4537	12.8774	11.9484

Table 8.

The effect of h_c/h_t ratio on the NDNF at different moisture concentrations (%) with varying l/w ratios and skew angles.

Skew angle	Moisture concentration (%)	h _c /h _t
Skew angle	Moisture concentration (%)	1	2	5	10	20
0°	0	27.2281	25.8153	22.4189	19.4351	16.9149
	0.25	26.3179	25.1473	22.0262	19.2054	16.8095
	0.50	25.3908	24.4819	21.6538	19.0076	16.7461
	0.75	24.4560	23.8332	21.3214	18.8662	16.7547
	1.00	23.5297	23.2262	21.0646	18.8276	16.8912
15°	0	28.4329	26.9623	23.4420	20.3388	17.7101
	0.25	27.5390	26.3073	23.0585	20.1162	17.6107
	0.50	26.6330	25.6584	22.6977	19.9276	17.5555
	0.75	25.7258	25.0310	22.3803	19.7988	17.5757
	1.00	24.8357	24.4520	22.1439	19.7778	17.7296
30°	0	32.5706	30.9047	26.9786	23.4753	20.4775
	0.25	31.7288	30.2914	26.6250	23.2763	20.3981
	0.50	30.8896	29.6957	26.3021	23.1185	20.3704
	0.75	30.0693	29.1374	26.0346	23.0316	20.4300
	1.00	29.2936	28.6504	25.8667	23.0711	20.6435
45°	0	41.7229	39.6412	34.9310	30.6100	26.8224
	0.25	40.9802	39.1098	34.6379	30.4599	26.7857
	0.50	40.2698	38.6207	34.3943	30.3676	26.8178
	0.75	39.6195	38.2051	34.2345	30.3728	26.9589
	1.00	39.0730	37.9150	34.2197	30.5485	27.3179

Conclusion

This article focuses on studying the influence of temperature and moisture conditions on the natural frequency of the LBCS plate with bamboo fiber-reinforced biocomposite face sheets and biodegradable PLA cores. The effect of temperature and moisture concentrations with different geometric parameters and various skew angles for the free vibration response of sandwich plates is thoroughly studied. The current analysis reveals that considering the initial stress stiffness matrix and the temperature and moisture-dependent material properties anticipate the proper behavior of biocomposite sandwich plates. Bamboo fiber-reinforced biocomposite and PLA core for sandwich plate follow a similar trend as a synthetic fiber-reinforced face-sheet sandwich plate in fundamental frequency when temperature or moisture concentration increases. A severe decrement in frequency is observed as the temperature reaches the T_g of PLA. A gradual degradation of natural frequency can be seen for increasing moisture concentration because of the deteriorating material properties of BFB.

The length-to-width and length-to-total thickness ratios significantly affect the natural frequency, and the frequency increases with an increase in the l/w and l/h ratio. The reduction in frequency for increasing temperature is more significant in thick plates. The stiffness of the sandwich structure increases as the skew angle in the BFB face-sheet/PLA core sandwich plate increases, which elevates the value of the fundamental frequency of the structure. The increment is observed in fundamental frequency with an increment in skew angle. The reduction in frequency is more compelling at a smaller skew angle for increasing moisture concentration. The ratio of core thickness to face sheet thickness significantly impacts the behavior of the sandwich plate. The frequency reduces with an increase in h_c/h_t ratio as there is an increase in the influence of the core in the sandwich plate system. For the change in h_c/h_t ratio, the decline in frequency is more prominent in a higher h_c/h_t ratio for increasing temperature or moisture concentration. Considering the vibration response under the effect of varying temperature and moisture concentration, the bamboo fiber-reinforced biocomposite and biodegradable PLA core for the sandwich plate with optimum geometric parameters can be a suitable replacement for conventional materials for various applications.

Footnotes

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would really like to take this opportunity to express their gratitude to the Science and Engineering Research Board (SERB) ASEAN-India S & T Collaborative (AISTDF) Project No. IMRC/AISTDF/CRD/2019/000128, Govt. of India, towards providing the grants and resources for the research.

ORCID iDs

Subhaschandra Kattimani

Hesam Kamyab

Saeed Althamer

Data availability statement

The used to support the findings in the study is mentioned within the article ⁴⁷.

Appendix

References

Kallannavar

Kattimani

Soudagar

MEM

Mujtaba

Alshahrani

Imran

. Neural network-based prediction model to investigate the influence of temperature and moisture on vibration characteristics of skew laminated composite sandwich plates. Materials 2021; 14: 3170. DOI: 10.3390/ma14123170.

Mahesh

. Nonlinear pyrocoupled deflection of viscoelastic sandwich shell with CNT reinforced magneto-electro-elastic facing subjected to electromagnetic loads in thermal environment. Eur Phys J Plus 2021; 136: 796.

Fahim

Chbib

Mahmoud

. The synthesis, production & economic feasibility of manufacturing PLA from agricultural waste. Sustain Chem and Pharm 2019; 12: 100142.

Ramesh

Atreya

TSA

Aswin

Eashwar

Deepa

. Processing and mechanical property evaluation of banana fiber reinforced polymer composites. Procedia Eng 2014; 97: 563–572.

Murali Mohan Rao

Mohana Rao

Ratna Prasad

. Fabrication and testing of natural fibre composites: Vakka, sisal, bamboo and banana. Mater Des 2010; 31: 508–513.

Azammi

Sapuan

Ishak

Sultan

. Physical and damping properties of kenaf fibre filled natural rubber/thermoplastic polyurethane composites. Def Technol 2020; 16: 29–34.

Garkhail

Heijenrath

RWH

Peijs

. Mechanical properties of natural-fibre-mat-reinforced thermoplastics based on flax fibres and polypropylene. Appl Compos Mater 2000; 7: 351–372.

Holbery

Houston

. Natural-fiber-reinforced polymer composites in automotive applications. JOM 2006; 58: 80–86.

Chao

Maneengam

Margabandu

Jindal

Patil

Zheng

Selvaraj

. Mechanical and damping property evaluation of jute fiber reinforced composite periodic core sandwich panels. J Nat Fibers 2022; 19: 11424–11434.

10.

Oksman

. Mechanical properties of natural fibre mat reinforced thermoplastic. Appl Compos 2000; 7: 403–414.

11.

Kumar

Saha

. Effects of stacking sequence of pineapple leaf-flax reinforced hybrid composite laminates on mechanical characterization and moisture resistant properties. Proc Inst Mech Eng C J Mech Eng Sci 2022; 236: 1733–1750.

12.

Ramraji

Rajkumar

Sabarinathan

. Mechanical and free vibration properties of skin and core designed basalt woven intertwined with flax layered polymeric laminates. Proc Inst Mech Eng C J Mech Eng Sci 2020; 234: 4505–4519.

13.

Huda

Reddy

Yang

. Ultra-light-weight composites from bamboo strips and polypropylene web with exceptional flexural properties. Compos B: Eng 2012; 43: 1658–1664.

14.

Lee

Wang

. Biodegradable polymers/bamboo fiber biocomposite with bio-based coupling agent. Compos Part A Appl Sci Manuf 2006; 37: 80–91.

15.

Huang

. A novel process to improve yield and mechanical performance of bamboo fiber reinforced composite via mechanical treatments. Compos B: Eng 2014; 56: 48–53.

16.

Gawande

Mukka Ramachanra

Kamyab

Trung

Kattimani

. Effect of radiofrequency coil and primary magnetic field on radiolucent composite plates. Noise Vib 2022; 54: 44–58. DOI: 10.1177/09574565221150195.

17.

Trujillo

Moesen

Osorio

Van Vuure

Ivens

Verpoest

. Bamboo fibres for reinforcement in composite materials: strength Weibull analysis. Compos Part A Appl Sci Manuf 2014; 61: 115–125.

18.

Manalo

Wani

Zukarnain

Karunasena

Lau

. Effects of alkali treatment and elevated temperature on the mechanical properties of bamboo fibre-polyester composites. Compos B: Eng 2015; 80: 73–83.

19.

Amada

Ichikawa

Munekata

Nagase

Shimizu

. Fiber texture and mechanical graded structure of bamboo. Compos B: Eng 1997; 28: 13–20.

20.

Sukmawan

Takagi

Nakagaito

. Strength evaluation of cross-ply green composite laminates reinforced by bamboo fiber. Compos B: Eng 2016; 84: 9–16.

21.

Essabir

Raji

Laaziz

Rodrique

Bouhfid

Qaiss

AEK

. Thermo-mechanical performances of polypropylene biocomposites based on untreated, treated and compatibilized spent coffee grounds. Compos B: Eng 2018; 149: 1–11.

22.

Sit

Ray

. Free vibration characteristics of glass and bamboo epoxy laminates under hygrothermal effect: A comparative approach. Compos B: Eng 2019; 176: 107333. DOI: 10.1016/j.compositesb.2019.107333.

23.

Fang

Dai

. Study on the properties of the recombinant bamboo by finite element method. Compos B: Eng 2017; 115: 151–159.

24.

Thwe

Liao

. Effects of environmental aging on the mechanical properties of bamboo–glass fiber reinforced polymer matrix hybrid composites. Compos Part A: Appl Sci Manuf 2002; 33: 43–52.

25.

Jindal

. Development and testing of bamboo-fibres reinforced plastic composites. J Compos Mater 1986; 20: 19–29.

26.

Mohanavel

Suresh Kumar

Vairamuthu

Ganeshan

NagarajaGanesh

. Influence of stacking sequence and fiber content on the mechanical properties of natural and synthetic fibers reinforced penta-layered hybrid composites. J Nat Fibers 2022; 19(13): 5258–5270. DOI: 10.1080/15440478.2021.1875368.

27.

Verma

Chariar

. Development of layered laminate bamboo composite and their mechanical properties. Compos B: Eng 2012; 43: 1063–1069.

28.

Khan

Yousif

Islam

. Fracture behaviour of bamboo fiber reinforced epoxy composites. Compos B: Eng 2017; 116: 186–199.

29.

Ismail

Jawaid

Naveen

. Void content, tensile, vibration and acoustic properties of kenaf/bamboo fiber reinforced epoxy hybrid composites. Materials 2019; 12: 2094.

30.

Zenkour

Alghanmi

. Hygro-thermo-electro-mechanical bending analysis of sandwich plates with FG core and piezoelectric faces. Mech Adv Mater 2021; 28: 282–294.

31.

KatariyaPanda

PSK

Mehar

. Theoretical modelling and experimental verification of modal responses of skewed laminated sandwich structure with epoxy-filled softcore. Eng Struct 2021; 228: 111509. DOI: 10.1016/j.engstruct.2020.111509.

32.

Daikh

Bensaid

Zenkour

. Temperature dependent thermomechanical bending response of functionally graded sandwich plates. Eng Res Express 2020; 2: 015006. DOI: 10.1088/2631-8695/ab638c.

33.

Lascano

Moraga

Ivorra-Martinez

et al. Development of injection-molded polylactide pieces with high toughness by the addition of lactic acid oligomer and characterization of their shape memory behavior. Polymers 2019; 11: 2099. DOI: 10.3390/polym11122099.

34.

Arrieta

García

López

, et al. Antioxidant bilayers based on PHBV and plasticized electrospun PLA-PHB fibers encapsulating catechin. Nanomaterials 2019; 9: 346. DOI: 10.3390/nano9030346.

35.

Villegas

Arrieta

Rojas

Torres

Faba

Toledo

Gutierrez

Zavalla

Romero

Galotto

Valenzuela

. PLA/organoclay bionanocomposites impregnated with thymol and cinnamaldehyde by supercritical impregnation for active and sustainable food packaging. Compos B: Eng 2019; 176: 107336. DOI: 10.1016/j.compositesb.2019.107336.

36.

Quiles-Carrillo

Montanes

Garcia-Garcia

Carbonell-Verdu

Balart

Torres-Giner

. Effect of different compatibilizers on injection-molded green composite pieces based on polylactide filled with almond shell flour. Compos B: Eng 2018; 147: 76–85.

37.

Antony

Cherouat

Montay

. Fabrication and characterization of hemp fibre based 3D printed honeycomb sandwich structure by FDM process. Appl Mater 2020; 27: 935–953.

38.

Azzouz

Chen

Zarrelli

Pearce

Mitchell

Ren

Grasso

. Mechanical properties of 3-D printed truss-like lattice biopolymer non-stochastic structures for sandwich panels with natural fibre composite skins. Compos Struct 2019; 213: 220–230.

39.

Lascano

Guillen-Pineda

Quiles-Carrillo

Ivorra-Martínez

Balart

Montanes

Boronat

. Manufacturing and characterization of highly environmentally friendly sandwich composites from polylactide cores and flax-polylactide faces. Polymers 2021; 13: 342–414.

40.

Rath

Dash

. Parametric instability of woven fiber laminated composite plates in adverse hygrothermal environment. Am J Mech 2014; 2: 70–81.

41.

Mukherjee

Das

, et al. Dynamic analysis of laminated composite plate structure with square cut-out under hygrothermal load. Int J Eng Res Technol 2014; 3: 2417.

42.

Ram

KSS

Sinha

. Hygrothermal effects on the free vibration of laminated composite plates. J Sound Vib 1992; 158: 133–148.

43.

Patel

Ganapathi

Makhecha

. Hygrothermal effects on the structural behaviour of thick composite laminates using higher-order theory. Compos Struct 2002; 56: 25–34.

44.

Ding

Wang

. Hygroscopic ageing of nonstandard size sandwich composites with vinylester-based composite faces and PVC foam core. Compos Struct 2018; 206: 194–201.

45.

Ding

Wang

. Assessment on the ageing of sandwich composites with vinylester-based composite faces and PVC foam core in various harsh environments. Compos Struct 2019; 213: 71–81.

46.

Garg

Karkhanis

Sahoo

Maiti

Singh

. Trigonometric zigzag theory for static analysis of laminated composite and sandwich plates under hygro-thermo-mechanical loading. Compos Struct 2019; 209: 460–471.

47.

Kallannavar

Kumaran

Kattimani

. Effect of temperature and moisture on free vibration characteristics of skew laminated hybrid composite and sandwich plates. Thin-Walled Struct 2020; 157: 107113. DOI: 10.1016/j.tws.2020.107113.

48.

Kallannavar

Kattimani

. Effect of temperature on the performance of active constrained layer damping of skew sandwich plate with CNT reinforced composite core. Mech Adv Mater 2022; 29: 5423–5442. DOI: 10.1080/15376494.2021.1955315.

49.

Thwe

Liao

. Environmental effects on bamboo-glass/polypropylene hybrid composites. J Mater Sci 2003; 38: 363–376. DOI: 10.1023/A:1021130019435.

50.

Samal

Mohanty

Nayak

. Polypropylene—bamboo/glass fiber hybrid composites: fabrication and analysis of mechanical, morphological, thermal, and dynamic mechanical behavior. J Reinf Plast 2009; 28: 2729–2747.

51.

Chen

Miao

Ding

. Influence of moisture absorption on the interfacial strength of bamboo/vinyl ester composites. Compos Part A: Appl 2009; 40: 2013–2019.

52.

Qian

Zhang

Yao

Sheng

. Effects of bamboo cellulose nanowhisker content on the morphology, crystallization, mechanical, and thermal properties of PLA matrix biocomposites. Compos Part B: Eng 2018; 133: 203–209.

53.

Godbole

Lakkad

. Effect of water absorption on the mechanical properties of bamboo. J Mater Sci 1986; 5: 303–304.

54.

Oveisi

Gudarzi

Mohammadi

, et al. Modeling, identification and active vibration control of a funnel-shaped structure used in MRI throat. J Vibroengineering 2013; 15: 1392–8716.

55.

De Ost

Vanregemorter

Schaeken

Van den Weyngaert

. The effect of carbon fibre inserts on the build-up and attenuation of high energy photon beams. Radiother Oncol 1997; 45: 275–277.

56.

Rodrigue

Mechefske

. Modal analysis of magnetic resonance imaging (MRI) scanner support structure. In: Volume 8: 30th conference on mechanical vibration and noise, Hamilton, Canada, 2018. American Society of Mechanical Engineers. DOI: 10.1115/DETC2018-85651.

57.

Meara

SJP

Langmack

. An investigation into the use of carbon fibre for megavoltage radiotherapy applications. Phys Med Biol 1998; 43: 1359–1366.

58.

Langmack

. The use of an advanced composite material as an alternative to carbon fibre in radiotherapy. Radiography 2012; 18: 74–77.

59.

Jafar

Reeves

Ruthven

Dean

MacDougall

Tucker

Miquel

. Assessment of a carbon fibre MRI flatbed insert for radiotherapy treatment planning. Brit J Radiol 2016; 89: 20160108. DOI: 10.1259/bjr.20160108.

60.

Silveira

Gomes

Rezende

Botelho

. Electromagnetic properties of multifunctional composites based on glass fiber prepreg and ni/carbon fiber veil. J Aerosp Technol Manag 2017; 9: 231–240.

61.

Morris

Geraldi

Stafford

, et al. Woven natural fibre reinforced composite materials for medical imaging. Materials 2020; 13: 1684. DOI: 10.3390/ma13071684.

62.

Paley

MNJ

Hart

Lait

Griffiths

. An MR-compatible neonatal incubator. Brit J Radiol 2012; 85: 952–958.

63.

Morris

Trabi

Spicer

Langmack

Boersma

Weightman

Newton

. A natural fibre reinforced composite material for multi-modal medical imaging and radiotherapy treatment. Mater Lett 2019; 252: 289–292.

64.

Grasso

Azzouz

Ruiz-Hincapie

Zarrelli

Ren

. Effect of temperature on the mechanical properties of 3D-printed PLA tensile specimens. Rapid Prototyp J 2018; 24: 1337–1346.

65.

Feng

Bian

Chen

. Compatibility, mechanical properties and stability of blends of polylactide and polyurethane based on poly(ethylene glycol)-b-polylactide copolymers by chain extension with diisocyanate. Polym Degrad Stab 2016; 125: 148–155.

66.

Farah

Anderson

Langer

. Physical and mechanical properties of PLA, and their functions in widespread applications — a comprehensive review. Adv Drug Deliv Rev 2016; 107: 367–392.

67.

Vouyiouka

Papaspyrides

. Mechanistic aspects of solid-state polycondensation In: Polymer science: a comprehensive reference 10 volume set 2012; Elsevier, Amsterdam, Netherlands, pp. 857–874.

68.

Algarni

. The influence of raster angle and moisture content on the mechanical properties of pla parts produced by fused deposition modeling. Polymers 2021; 13: 237.

69.

Padhi

Pandit

. Bending and free vibration response of sandwich laminate under hygrothermal load using improved zigzag theory. J Strain Anal Eng 2017; 52: 288–297.

70.

Park

Lee

Seo

Voyiadjis

. Structural dynamic behavior of skew sandwich plates with laminated composite faces. Compos Part B: Eng 2008; 39: 316–326.

71.

Neamah

Shukur

Al-Ansari

LS,

et al. Studying the effect of fiber orientation and skew angle on the fundamental frequencies of simply supported composite plates using finite element method. J Mech Eng Res Dev 2020; 43: 199–212.