Impact of blast and mechanical loads on the shear deformable stiffened sandwich plate with an auxetic core layer in thermal environment

Abstract

By using the first order shear deformation theory (FSTD), this paper presents the results of the nonlinear dynamic behavior and natural frequencies of sandwich plate supported by elastic foundations in thermal environment and subjected to mechanical load and blast loading. This work takes advantage of the sandwich plate configuration with three layers: graphene platelet –reinforced composite (GPL) layer – auxetic layer – FGM layer, to analyze the dynamic and vibration problem, in which the auxetic core layer has a negative Poisson's ratios and the FGM layer is reinforced by stiffeners made of full metal or full ceramic depending on a situation of stiffeners at the metal-rich or ceramic-rich side of the plate respectively. Corresponding to the combination of material layers, the mechanical quantities of the problem are processed and calculated to suit the structure and reinforcement conditions. Numerical results are provided to explore the influences of geometrical parameters, elastic foundation parameters, GPL volume fraction, blast and mechanical loads on the nonlinear dynamic behavior and vibration of sandwich plate resting on elastic foundation and in thermal environment. In addition, the study is not only assumed that the material properties depend on environment temperature variation, but also considered the thermal stresses in the stiffeners, as well as considered the effect of imperfections in the original shape of the structure.

Keywords

FSTD stiffened sandwich plate negative Poisson’s ratios GPL FGM blast loading thermal environment

Introduction

With the development of science and manufacturing technology, a great deal of advanced materials has been invented. As a result, the field of research on these materials is increasingly attracting the attention of many material scientists. However, it is a fact that the new material may be very expensive or easily damaged by external conditions, such as high temperatures, blast loading or complex loads. From that fact, it was thought of ways to reinforce the structure in possible cases to protect expensive or perishable layers of material. Two of these ways are to reinforce the stiffened or add layers of another protective material to form a reinforced stiffened structure or sandwich structure. Or, where possible, combine both of the above creating a new stiffened sandwich structure. Because that the newly created structure can take full advantage of above ways, optimizing both the hardness, price and weight of the structure. In fact, in recent years, studies of this structure are increasing, because of its application in conditions of load and temperature such as blast loading.

From open source, there are some research on this field such as Goel et al. in [1] proved that the provision of the stiffeners and foam core considerably improves the blast resistance as compared to both, the unstiffened panels with foam core and without using syntactic foam core, through examining the response of the aluminum cenosphere syntactic foam core stiffened and unstiffened structures subjected to blast load. The nonlinear dynamic behavior of a laminated composite plate with sandwich stiffener (with four layers carbon/epoxy and the stiffeners composed of carbon/epoxy face sheets and polypropylene honeycomb core) subjected to the non-uniform blast load was investigated by Balkan et al. in [2]. From comparison results in [3] between the bare hull and the equivalent model which increases the thickness of the thin or pressure hulls under the same mass conditions indicate that the method of using a sacrificial coating is more effective in improving the blast resistance of the structure. Study [4] presented experimental and numerical investigations into dynamic responses of aluminum foam core sandwich panels subjected to localize air blast loading by Chen et al. The effect of through-the-thickness stitching on the blast resistance of sandwich panels was studied both experimentally and numerically by Guan et al. in [5], with the blast resistance of the sandwich panels was modelled using finite element techniques. Problems related with the modeling and dynamic response to blast loading of doubly-curved anisotropic sandwich panels with laminated face sheets were developed by Hause and Librescu in [6]. In addition, there were other studies on this field in [7 –13].

As is well known, auxetics are new materials with a negative Poisson’s ratio. Thanks to this feature, the auxetic is expected to have mechanical properties such as high energy absorption and fracture resistance, and may be useful in applications such as body armor, robust shock absorbing material and especially for structures subjected to blast loading. However, controlling the behavior of the auxetic structures is still a big challenge. The study of structures made of auxetic materials has received a lot of attention from the authors, but the study of sandwich structure with auxetic materials is still limited, while sandwich structures make most of the advantages of each component material, so the study of fabrication and theoretical study of sandwich structures with the auxetic layer will create new structures with more additional advantages than inherent advantages of the auxetic. The amount of research in this area is still limited. The dynamic and static behavior of eccentrically stiffened circular cylindrical shells with the auxetic core layer subjected to blast loading; and of sandwich composite cylindrical panels with an auxetic honeycomb core layer by using Reddy’s order shear deformation shell theory were investigated by Duc and Cong in [14] and Duc et al. in [15,16]. Madke and Chowdhury’s research [17] has shown that, compared to non-auxetic foam, proposed auxetic sandwich structures with braided face sheets and 3D re-entrant cores were lightweight, had better energy absorption and high stiffness values in the in plane along with out-of-plane directions. Based on testing results of Smardzewski in [18], it was shown that the wooden sandwich panels with an auxetic core and oval cells exhibited auxetic properties and a strong orthotropy. Through the study of sandwich plates with functionally graded (FG) auxetic 3D lattice core in [19,20], Li et al. have proved that FG configurations have distinct effect on the natural frequencies, nonlinear to linear frequency ratios of sandwich plates, which will become stable when the vibration amplitude was sufficiently large, and compressive buckling and thermal post-buckling loads of sandwich plates with an auxetic core were remarkably higher than those of their counterparts having the 3D lattice core with positive Poisson’s ratios. The addition of carbon nanotube-reinforced composites (CNTRC) material layer to the auxetic structure also results in remarkable behavior of the structure. For example, Hajmohammad et al.’s research results in [21] revealed that by reinforcing the structure with CNTs, the dynamic deflection induced by blast load decreases about 59%. In addition, Shen and Reddy also have much research on FG-CNTRC laminated sandwich with negative Poisson’s ratio in [22 –24].

Recently, scientists have been studying auxetic graphene structures, for example, an effective turnable way is proposed for customizable fabrication for auxetic graphene assembled macro films with high conductivity and flexibility was studied by Li et al. in [25]. Anisotropic mechanical behavior and auxeticity of penta-graphene was examined by Winczewski and Rybicki in [26]. Study about a highly efficient sound and shock absorber based on three-dimensional (3D) auxetic foam with two-dimensional (2D) wrinkled graphene oxide presented by Oh et al. in [27]. With these auxetic graphene structures overview, it can be seen that the research in this regard is still a completely new field and has not received much attention. There are still a lot of issues that need further research, especially in cases where the structure has stiffened reinforcement.

By using FSDT, the object of the present investigation is to give analytical solutions to the problem of the nonlinear dynamic and vibration of sandwich plate supported by elastic foundations in thermal environment and subjected to mechanical loads and blast loading. The sandwich plate configuration consists of three layers: graphene platelet –reinforced composite (GPL-RC) layer – auxetic layer and FGM layer, in which the FGM layer is reinforced by stiffeners made of full metal or full ceramic depending on a situation of stiffeners at the metal-rich or ceramic-rich side of the plate respectively. In addition, the study is not only assumed that the material properties depend on temperature variation, but also considered the thermal stresses in the stiffeners.

Formulation of the problem

The configuration of sandwich plate with auxetic core layer

Consider stiffened sandwich plate with graphene platelet –reinforced composite (GPL)layer, auxetic layer and FGM layer, in which the FGM layer is reinforced by stiffeners made of full metal or full ceramic depending on a situation of stiffeners at the metal-rich or ceramic-rich side of the plate respectively (Figure 1). With such a combination of layers, the total plate thickness is $h_{P} = h_{GRA} + h_{A U} + h_{FGM}$ .

The stiffened sandwich plate with an auxetic core layer resting on an elastic medium modeled as shown in Figure 1(d). The reaction–deflection relation is given by [14,28]

q_{e} = K_{W} w - K_{P} \nabla^{2} w

(1)

with

\nabla

is Laplace operator,

w

is the deflection,

K_{W}

and

K_{P}

are Winkler foundation stiffness and shear layer stiffness of Pasternak foundation, respectively.

Graphene platelet-reinforced sandwich plate

The plate with the face sheet GPL layer is introduced in this paper. The elastic modulus of the GPL layer are estimated through the Halpin–Tsai model and the mass density and Poisson’s ratio of the GPL layer are calculated by applying the rule of mixture follows as [27]:

\begin{array}{l} E_{}^{GRA} = \frac{3}{8} (\frac{1 + ξ_{L} η_{L} V_{GPL}}{1 - η_{L} V_{GPL}}) E_{m} + \frac{5}{8} (\frac{1 + ξ_{T} η_{T} V_{GPL}}{1 - η_{T} V_{GPL}}) E_{m}, \\ η_{L} = \frac{E_{GPL} / E_{m} - 1}{E_{GPL} / E_{m} + ξ_{L}}, η_{T} = \frac{E_{GPL} / E_{m} - 1}{E_{GPL} / E_{m} + ξ_{T}}, \\ ξ_{L} = 2 (a_{GPL} / h_{GPL}), ξ_{T} = 2 (b_{GPL} / h_{GPL}) \\ ρ^{GRA} = ρ_{GPL} V_{GPL} + ρ_{m} V_{m}, \\ ν_{12}^{GRA} = ν_{21}^{GRA} = ν_{GPL} V_{GPL} + ν_{m} V_{m} \\ α_{1}^{GRA} = α_{2}^{GRA} = \frac{V_{GPL} E_{GPL} α_{GPL} + V_{m} E_{m} α_{m}}{V_{GPL} E_{GPL} + V_{m} E_{m}} \\ V_{m} = 1 - V_{GPL} \end{array}

(2)

The functions of non-dimensional thickness coordinate $Q_{i j}^{GRA}$ $(i j = 11, 12, 22, 66)$ are expressed as

\begin{array}{l} Q_{11}^{GRA} = \frac{E_{}^{GRA}}{1 - v_{12}^{GRA} v_{21}^{GRA}}, Q_{12}^{GRA} = \frac{v_{12}^{GRA} E_{}^{GRA}}{1 - v_{12}^{GRA} v_{21}^{GRA}}, Q_{22}^{GRA} = \frac{E_{}^{GRA}}{1 - v_{12}^{GRA} v_{21}^{GRA}}, \\ Q_{66}^{GRA} = G_{12}^{GRA}, Q_{44}^{GRA} = G_{23}^{GRA}, Q_{55}^{GRA} = G_{13}^{GRA} \end{array}

(3)

Honeycomb core materials

The auxetic layer with negative Poisson’s ratio (honeycomb core material) with unit cells of coremodeled as shown in Figure 1(c). Base on the geometric of the cell, the important parameters of core property are $t_{a u}, l_{a u}, h_{a u}$ .

Corresponding to auxetic material, the functions of non – dimensional thickness coordinate are given by the formula [16]

\begin{array}{l} Q_{11}^{A U} = \frac{E_{1}^{A U}}{1 - v_{12}^{A U} v_{21}^{A U}}, Q_{12}^{A U} = \frac{v_{12}^{A U} E_{2}^{A U}}{1 - v_{12}^{A U} v_{21}^{A U}}, Q_{22}^{A U} = \frac{E_{2}^{A U}}{1 - v_{12}^{C} v_{21}^{C}}, \\ Q_{66}^{A U} = G_{12}^{A U}, Q_{44}^{A U} = G_{23}^{A U}, Q_{55}^{A U} = G_{13}^{A U} \end{array}

(4)

With the elastic modulus of the auxetic layer are expressed as [29,30]

\begin{array}{l} E_{1}^{A U} = E_{m} \frac{n_{3}^{3} (η_{1} - \sin θ)}{\cos^{3} θ [1 + (\tan^{2} θ + η_{1} \sec^{2} θ) η_{3}^{2}]}; E_{2}^{A U} = E_{m} \frac{n_{3}^{3}}{\cos θ (η_{1} - \sin θ) (\tan^{2} θ + η_{3}^{2})}, \\ G_{12}^{A U} = E_{m} \frac{η_{3}^{3}}{η_{1} (1 + 2 η_{1}) \cos θ}; G_{13}^{A U} = G_{m} \frac{η_{3}}{2 \cos θ} [\frac{η_{1} - \sin θ}{1 + 2 η_{1}} + \frac{η_{1} + 2 \sin^{2} θ}{2 (η_{1} - \sin θ)}], \\ G_{23}^{A U} = G_{m} \frac{η_{3} \cos θ}{η_{1} - \sin θ}; v_{12}^{A U} = - \frac{\sin θ (1 - n_{3}^{2}) (η_{1} - \sin θ)}{\cos^{2} θ [1 + (\tan^{2} θ + \sec^{2} θ η_{1}) η_{3}^{2}]}; \\ v_{21}^{A U} = \frac{- \sin θ (1 - η_{3}^{2})}{(\tan^{2} θ + η_{3}^{2}) (η_{1} - \sin θ)}; ρ^{A U} = ρ_{m} \frac{η_{3} (η_{1} + 2)}{2 \cos θ (η_{1} - \sin θ)}; \\ α_{1}^{A U} = α_{m} \times \frac{η_{3} cos θ}{\sin θ + η_{1}}; α_{2}^{A U} = α_{m} \frac{η_{3} (η_{1} + \sin θ)}{(2 η_{1} + 1) cos θ} \end{array}

(5)

in which

η_{1} = h_{a u} / l_{a u}, η_{3} = t_{a u} / l_{a u} .

Eccentrically stiffened functionally graded material layer

The FGM layer is reinforced by stiffeners made of full metal or full ceramic depending on a situation of stiffeners at the metal-rich or ceramic-rich side of the plate respectively. The elasticity modulus $E^{FGM}$ , the coefficient of thermal expansion $α^{FGM}$ of this layer are [10,31]

[\begin{matrix} E^{FGM} (z) \\ α^{FGM} (z) \end{matrix}] = [\begin{matrix} E_{m} \\ α_{m} \end{matrix}] + [\begin{matrix} E_{c m} \\ α_{c m} \end{matrix}] \times {(\frac{2 z + h}{2 h})}^{N}

(6)

The functions of non-dimensional thickness coordinate of this layer are [10,31]

Q_{11}^{FGM} = Q_{22}^{FGM} = \frac{E^{FGM}}{1 - {(v^{FGM})}^{2}}, Q_{12}^{FGM} = \frac{v^{FGM} E^{FGM}}{1 - {(v^{FGM})}^{2}}, Q_{44}^{FGM} = Q_{55}^{FGM} = Q_{66}^{FGM} = \frac{E^{FGM}}{2 (1 + v^{FGM})}

(7)

with

v^{FGM}

– the Poison ratio.

Theoretical formulations

The FSDT are used in the present report establishing the governing equations and determine the mechanical and blast loads of the sandwich stiffened plate with negative Poisson’s ratio in auxetic honeycombs.

The strain-displacement relations taking into account the von Karman nonlinear terms [14]:

[\begin{matrix} ε_{x} \\ ε_{y} \\ γ_{x y} \end{matrix}] = [\begin{matrix} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \end{matrix}] + z [\begin{matrix} χ_{x} \\ χ_{y} \\ χ_{x y} \end{matrix}] = [\begin{matrix} \frac{\partial u}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2} \\ \frac{\partial v}{\partial y} + \frac{1}{2} {(\frac{\partial w}{\partial y})}^{2} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} + \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} \end{matrix}] + z [\begin{matrix} ϕ_{x, x} \\ ϕ_{y, y} \\ ϕ_{x, y} + ϕ_{y, x} \end{matrix}]; [\begin{matrix} γ_{x z} \\ γ_{y z} \end{matrix}] = [\begin{matrix} \frac{\partial w}{\partial x} + ϕ_{x} \\ \frac{\partial w}{\partial y} + ϕ_{y} \end{matrix}]

(8)

The combination of material layers to create a sandwich plate structure changes the stress and strain, but these components still follow Hooke’s law:

{\begin{matrix} σ_{x}^{(k)} \\ σ_{y}^{(k)} \\ σ_{x y}^{(k)} \\ σ_{x z}^{(k)} \\ σ_{y z}^{(k)} \end{matrix}} = {\begin{matrix} Q_{11}^{(k)} & Q_{12}^{(k)} & 0 & 0 & 0 \\ Q_{12}^{(k)} & Q_{22}^{(k)} & 0 & 0 & 0 \\ 0 & 0 & Q_{66}^{(k)} & 0 & 0 \\ 0 & 0 & 0 & Q_{55}^{(k)} & 0 \\ 0 & 0 & 0 & 0 & Q_{44}^{(k)} \end{matrix}} [{\begin{matrix} ε_{x} \\ ε_{y} \\ γ_{x y} \\ γ_{x z} \\ γ_{y z} \end{matrix}} - {\begin{matrix} α_{1}^{(k)} \\ α_{2}^{(k)} \\ 0 \\ 0 \\ 0 \end{matrix}} Δ T]

(9)

in which k = “^GRA”, “^AU”, “^FGM”.

The stress component for stiffened is described by the formula [32]:

[\begin{matrix} σ_{x}^{S} \\ σ_{y}^{S} \end{matrix}] = E^{S} [\begin{matrix} ε_{x} \\ ε_{y} \end{matrix}] - \frac{E^{S}}{1 - 2 v^{S}} α^{S} Δ T

(10)

The force and moment fields of the sandwich stiffened plate with auxetic core layer can be defined as:

\begin{array}{l} (N_{i}, M_{i}) = (N_{i}^{GRA}, M_{i}^{GRA}) + (N_{i}^{A U}, M_{i}^{A U}) + (N_{i}^{FGM}, M_{i}^{FGM}) + (N_{i}^{S}, M_{i}^{S}), i = x, y, x y, \\ (Q_{i}) = (Q_{i}^{GRA}) + (Q_{i}^{A U}) + (Q_{i}^{FGM}), i = x, y \end{array}

(11)

Substituting equations (8) to (10) into equation (11), the forces and moments of the sandwich stiffened plate can be expressed:

\begin{array}{l} N_{x} = A_{11} ε_{x}^{0} + A_{12} ε_{y}^{0} + A_{13} ϕ_{x, x} + A_{14} ϕ_{y, y} + Φ_{1}, \\ N_{y} = A_{21} ε_{x}^{0} + A_{22} ε_{y}^{0} + A_{23} ϕ_{x, x} + A_{24} ϕ_{y, y} + Φ_{2}, \\ N_{x y} = A_{31} γ_{x y}^{0} + A_{32} (ϕ_{x, y} + ϕ_{y, x}), \\ M_{x} = A_{13} ε_{x}^{0} + A_{23} ε_{y}^{0} + A_{43} ϕ_{x, x} + A_{44} ϕ_{y, y} + Φ_{3}, \\ M_{y} = A_{23} ε_{x}^{0} + A_{24} ε_{y}^{0} + A_{44} ϕ_{x, x} + A_{45} ϕ_{y, y} + Φ_{4}, \end{array}

\begin{array}{l} M_{x y} = A_{32} γ_{x y}^{0} + A_{52} (ϕ_{x, y} + ϕ_{y, x}), \\ Q_{x} = A_{61} (\frac{\partial w}{\partial x} + ϕ_{x}), \\ Q_{y} = A_{71} (\frac{\partial w}{\partial y} + ϕ_{y}) \end{array}

(12)

with expressions

A_{11}, \dots A_{71}, Φ_{1}, \dots Φ_{4}

are mentioned in Appendix 1.

The nonlinear motion equation of the sandwich stiffened plate resting on an elastic medium will satisfy identically by using a stress function as [14]

N_{x} = \frac{\partial^{2} f}{\partial y^{2}}, N_{y} = \frac{\partial^{2} f}{\partial x^{2}}, N_{xy} = - \frac{\partial^{2} f}{\partial x \partial y}

(13)

Replacing this stress function in equation (12), the normal and the shear strain on the midplane of the plate components are

\begin{array}{l} ε_{x}^{0} = \frac{A_{12}}{Δ} f_{, x x} - \frac{A_{22}}{Δ} f_{, y y} + \frac{(A_{22} A_{13} - A_{12} A_{23})}{Δ} ϕ_{x, x} + \frac{(A_{22} A_{14} - A_{12} A_{24})}{Δ} ϕ_{y, y} + \frac{A_{22}}{Δ} Φ_{1}^{} - \frac{A_{12}}{Δ} Φ_{2}^{}, \\ ε_{y}^{0} = \frac{A_{21}}{Δ} f_{, y y} - \frac{A_{11}}{Δ} f_{, x x} + \frac{(A_{11} A_{23} - A_{21} A_{13})}{Δ} ϕ_{x, x} + \frac{(A_{11} A_{24} - A_{21} A_{14})}{Δ} ϕ_{y, y} - \frac{A_{21}}{Δ} Φ_{1}^{} + \frac{A_{11}}{Δ} Φ_{2}^{}, \\ γ_{x y}^{0} = - \frac{1}{A_{31}} f_{, x y} - \frac{A_{32}}{A_{31}} (ϕ_{x, y} + ϕ_{y, x}) \end{array}

(14)

these are important quantities used in the geometrical compatibility equation for an imperfect plate. In addition, the new quantities that appear in these above equations will be found in the Appendix 1.

Solution procedure

An imperfect sandwich stiffened plate considered in this study is assumed to be simply supported and subjected to uniformly distributed pressure of axial compression of intensities $P_{x}$ and $P_{y}$ respectively at its cross section. Two cases of boundary conditions will be considered [14]

FM: The sandwich plate edges are simply supported and freely movable

\begin{array}{l} w = ϕ_{y} = M_{x x} = N_{x y} = 0, N_{x} = N_{x 0}, at : x = {0, a}, \\ w = ϕ_{x} = M_{y y} = N_{x y} = 0, N_{y} = N_{y 0}, at : y = {0, b} \end{array}

(15)

IM: The sandwich plate edges are simply supported and immovable

\begin{array}{l} w = u = ϕ_{y} = M_{x x} = 0, N_{x} = N_{x 0}, at : x = {0, a}, \\ w = v = ϕ_{x} = M_{y y} = 0, N_{y} = N_{y 0}, at : y = {0, b} \end{array}

(16)

The approximate solutions satisfying the mentioned conditions are chose:

\begin{array}{l} w (x, y, t) = W (t) \sin α x \sin β y, \\ w^{*} (x, y, t) = μ h \sin α x \sin β y, \\ ϕ_{x} (x, y, t) = Φ_{x} (t) \cos α x \sin β y, \\ ϕ_{y} (x, y, t) = Φ_{y} (t) \sin α x c os β y \end{array}

(17)

where

α = \frac{m π}{a}, β = \frac{n π}{b}

m, n = 1, 2, \dots

are the natural numbers of half waves in the corresponding direction

x, y

;

W (t)

is the time dependent total amplitude and

Φ_{x}, Φ_{y}

are the amplitudes which are functions dependent on time, and

μ

is a quantity that is characteristic of the structural imperfection.

With this approximate solutions, the stress function obtained has a form

f (x, y, t) = A_{1} (t) \cos 2 α x + A_{2} (t) \cos 2 β y + A_{3} (t) \sin α x \sin β y + \frac{1}{2} N_{x 0} y^{2} + \frac{1}{2} N_{y 0} x^{2}

(18)

in which

A_{1} = - W (W + 2 μ h) \frac{β^{2}}{α^{2}} \frac{Δ}{32 A_{11}}, A_{2} = - W (W + 2 μ h) \frac{α^{2}}{β^{2}} \frac{Δ}{32 A_{22}}, A_{3} = D_{1} Φ_{x} + D_{2} Φ_{y},

and

D_{1}, D_{2}

are mentioned in Appendix 1.

After a series of necessary transformations, the basic equations written for the sandwich stiffened plate are rewritten as follows

\begin{array}{l} L_{11} (w) + L_{12} (ϕ_{x}) + L_{13} (ϕ_{y}) + P_{1} (w, f) + P_{1}^{*} (w^{*}, f) + q = I_{0} \frac{\partial^{2} w}{\partial t^{2}} \\ L_{21} (w) + L_{22} (ϕ_{x}) + L_{23} (ϕ_{y}) + P_{2} (f) + L_{21} (w^{*}) = [I_{2} - \frac{I_{1}^{2}}{I_{0}}] \frac{\partial^{2} ϕ_{x}}{\partial t^{2}} \\ L_{31} (w) + L_{32} (ϕ_{x}) + L_{33} (ϕ_{y}) + P_{3} (f) + L_{31} (w^{*}) = (I_{2} - \frac{I_{1}^{2}}{I_{0}}) \frac{\partial^{2} ϕ_{y}}{\partial t^{2}} \end{array}

(19)

expressions

L_{ij}, P_{i}, I_{0, 1, 2}, (i, j) = (1, 2, 3)

are mentioned in Appendix 2.

Substituting equations (17) and (18) into equation (19) and applying the Bubnov–Galerkin method, obtained

\begin{array}{l} l_{11} W+ l_{12} Φ_{x} + l_{13} Φ_{y} + l_{14} (W+ μ h) Φ_{x} + l_{15} (W+ μ h) Φ_{y} + l_{16} (W+ μ h) \\ + l_{17} W (W+ μ h) (W + 2 μ h) + F (W+ μ h) + l_{18} q = I_{0} W_{, t t} \\ l_{21} Φ_{x} + l_{22} Φ_{y} + l_{23} (W+ μ h) + l_{24} W (W + 2 μ h) = [I_{2} - \frac{I_{1}^{2}}{I_{0}}] Φ_{x, t t} \\ l_{31} Φ_{x} + l_{32} Φ_{y} + l_{33} (W+ μ h) + l_{34} W (W + 2 μ h) = (I_{2} - \frac{I_{1}^{2}}{I_{0}}) Φ_{y, t t} \end{array}

(20)

with expressions

l_{ij}, (i = (1, 2, 3), j = (1, 2, 3, 4))

and

F

are mentioned in Appendix 2.

Pre-loaded axial compression

Assume that the sandwich stiffened plate is loaded under uniform compressive loads $P_{x}$ and $P_{y}$ (Pascal) on the edges $x = (0, a)$ and $y = (0, b)$ (corresponding to FM case) with

N_{x 0} = - P_{x} h, N_{y 0} = - P_{y} h .

(21)

Under the effect of these loads, the report will analyze the impact of blast and mechanical load on the dynamic behavior of the sandwich stiffened plate in next section.

Thermal environment

The effect of temperature on the structure will be considered in case 2 (IM), corresponding to the condition: the plate with all edges which are simply supported and immovable. Under the influence of thermal load, the condition expressing the immovability on the edges, $u = 0$ (on $x = 0, a$ ) and $v = 0$ (on $y = 0, b$ ), is satisfied in an average sense as

\int_{0}^{b} \int_{0}^{a} \frac{\partial u}{\partial x} d x d y = 0, \int_{0}^{a} \int_{0}^{b} \frac{\partial v}{\partial y} d x d y = 0

(22)

In accordance with this average sense, the respective force components are calculated as follows

\begin{array}{l} N_{x 0} = c_{11} Φ_{x} + c_{12} Φ_{y} + c_{14} W^{2} + c_{15} W μ h + Φ_{1}^{}, \\ N_{y 0} = c_{21} Φ_{x} + c_{22} Φ_{y} + c_{24} W^{2} + c_{25} W μ h + Φ_{2}^{} \end{array}

(23)

Substituting equation (23) into equation (20), leads to the basic equations used to investigate the nonlinear dynamic response of sandwich stiffened plates in the case all IM edges

\begin{array}{l} l_{11} W+ l_{12} Φ_{x} + l_{13} Φ_{y} + [l_{14} - (c_{11} α^{2} + c_{21} β^{2})] (W+ μ h) Φ_{x} + \\ + [l_{15} - (c_{12} α^{2} + c_{22} β^{2})] (W+ μ h) Φ_{y} + l_{16} (W+ μ h) + l_{17} W (W+ μ h) (W + 2 μ h) + \\ - [(c_{14} α^{2} + c_{24} β^{2}) W^{2} (W+ μ h) + (c_{15} α^{2} + c_{25} β^{2}) W μ h (W+ μ h) + (α^{2} Φ_{1}^{} + β^{2} Φ_{2}^{}) (W+ μ h)] \\ + l_{18} q = I_{0} W_{, t t}; \\ l_{21} Φ_{x} + l_{22} Φ_{y} + l_{23} (W+ μ h) + l_{24} W (W + 2 μ h) = [I_{2} - \frac{I_{1}^{2}}{I_{0}}] Φ_{x, t t}; \\ l_{31} Φ_{x} + l_{32} Φ_{y} + l_{33} (W+ μ h) + l_{34} W (W + 2 μ h) = (I_{2} - \frac{I_{1}^{2}}{I_{0}}) Φ_{y, t t} \end{array}

(24)

Nonlinear dynamic and vibration analysis

To simplify the process of calculations, a hypothesis is used in this report, in particular, to assume rotations $Φ_{x}, Φ_{y}$ exist, but the inertial forces caused by the rotation angles $Φ_{x}, Φ_{y}$ are small so can be ignored. The system of equation (19) can be written as follows

\begin{array}{l} I_{0} W_{, t t} - a_{1} W - a_{2} (W+ μ h) - a_{3} {(W+ μ h)}^{2} + \\ - a_{4} W (W + 2 μ h) - a_{5} W (W+ μ h) (W + 2 μ h) = a_{6} Q \sin (Ω t) \end{array}

(25)

In the case of ignoring the effect of imperfections in the original plate, i.e. $μ h = 0$ , the equation (25) can be written as follows

I_{0} W_{, t t} - (a_{1} + a_{2}) W - (a_{3} + a_{4}) W^{2} - a_{5} W^{3} = a_{6} Q \sin (Ω t)

(26)

The fundamental frequencies of sandwich stiffened plates can be determined approximately by

ω_{m n} = \sqrt{\frac{- (a_{1} + a_{2})}{I_{0}}}

(27)

The nonlinear vibration of sandwich stiffened plate will be surveyed through considering uniformly distributed transverse load $q = Q \sin Ω t$ , then, the equation (27) is written

W_{, t t} + ω_{m n}^{2} (W + M W^{2} + N W^{3}) - R \sin (Ω t) = 0

(28)

where

M = \frac{a_{3} + a_{4}}{a_{1} + a_{2}}, N = \frac{a_{5}}{a_{1} + a_{2}}, R = \frac{a_{6}}{I_{0}} Q

For finding amplitude-frequency characteristics based on the method of harmonic balance of nonlinear vibration, the time dependent total amplitude are selected $W (t) = A \sin Ω t,$ and substitute it into equation (28) and applying procedure like Gakerkin method $(\int_{0}^{π / 2 Ω} X \sin Ω t d t = 0)$ , the frequency-amplitude relation of nonlinear forced vibration is obtained

α^{2} - (1 - \frac{8}{3 π} M A + \frac{3}{4} N A^{2}) + \frac{R}{A ω_{m n}^{2}} = 0, α = \frac{Ω}{ω_{m n}}

(29)

If $R = 0$ , i.e. no external force acting on the plate, the frequency-amplitude relation of free nonlinear vibration is obtained

ω_{N L}^{2} = ω_{m n}^{2} (1 - \frac{8}{3 π} M A + \frac{3}{4} N A^{2})

(30)

Results and discussion

Validation of results

The comparative results will be shown in this section to assess the reliability of the research methodology in the report. The first comparison is shown in Table 1 for the results of calculating the natural frequency between the results obtained in this report based on the FSDT and the study [33] based on the Reissner–Mindlin plate theory, study [34] based on a higher-order deformation theory, [35] base on the FSDT by using the element-free kp-Ritz method, and research [36] base on the FSDT. The value of the $h / a$ ratio and the power law index are selected to calculate the fundamental frequency and the lowest third frequency parameters: $h / a = {0.05, 0.1, 0.2}$ and $N = {0, 0 .5, 1, 4, 10, \infty}$ respectively. For convenience in comparison, the plates are made of a mixture of aluminum (Al) and alumina (Al₂O₃). It is easy to see that the results shown in Table 1 show the similarity with Hashemi et al.’s comparison results in [33].

Table 1.

Comparison of the natural frequency parameter $\overset{⌢}{β} = ω h \sqrt{ρ_{c} / E_{c}}$ for Al/ Al₂O₃ plates.

$h / a$	$(m, n)$	Method	N
$h / a$	$(m, n)$	Method	0	0.5	1	4	10	$\infty$
0.05	(1,1)	Present	0.0148	0.0126	0.0114	0.0099	0.0095
		Ref. [28]	0.0148	0.0125	0.0113	0.0098	0.0094
		Ref. [30]	0.0146	0.0124	0.0112	0.0097	0.0093
		Ref. [31]	0.0148	0.0128	0.0115	0.0101	0.0096
0.1	(1,1)	Present	0.0581	0.0495	0.0448	0.0390	0.0371	0.0296
	(1,1)	Ref. [28]	0.0577	0.0490	0.0442	0.0382	0.0366	0.0293
		Ref. [29]	0.0577	0.0492	0.0443	0.0381	0.0364	0.0293
		Ref. [30]	0.0568	0.0482	0.0435	0.0376	0.0359
		Ref. [31]	0.0577	0.0492	0.0445	0.0383	0.0363	0.0294
	(1,2)	Present	0.1398	0.1194	0.1082	0.0935	0.0885	0.0711
		Ref. [28]	0.1376	0.1173	0.1059	0.0911	0.0867	0.0701
		Ref. [30]	0.1381	0.1180	0.1063	0.0904	0.0859	0.0701
		Ref. [31]	0.1354	0.1154	0.1042		0.0850
	(2,2)	Present	0.2158	0.1851	0.1682	0.1448	0.1362	0.1098
		Ref. [28]	0.2112	0.1805	0.1631	0.1397	0.1324	0.1076
		Ref. [29]	0.2121	0.1819	0.1640	0.1383	0.1306	0.1077
		Ref. [30]	0.2063	0.1764	0.1594		0.1289
0.2	(1,1)	Present	0.2158	0.1851	0.1682	0.1448	0.1362	0.1098
	(1,1)	Ref. [28]	0.2112	0.1805	0.1631	0.1397	0.1324	0.1076
		Ref. [29]	0.2121	0.1819	0.1640	0.1383	0.1306
		Ref. [30]	0.2055	0.1757	0.1587	0.1356	0.1284
		Ref. [31]	0.2112	0.1806	0.1650	0.1371	0.1304	0.1075
	(1,2)	Present	0.4776	0.4132	0.3765	0.3198	0.2967	0.2431
		Ref. [28]	0.4618	0.3978	0.3604	0.3049	0.2856	0.2352
		Ref. [29]	0.4658	0.4040	0.3644	0.3000	0.2790	0.2365
	(2,2)	Present	0.6929	0.6037	0.5521	0.4653	0.4270	0.3527
		Ref. [28]	0.6676	0.5779	0.5245	0.4405	0.4097	0.3399
		Ref. [29]	0.6753	0.5891	0.5444	0.4362	0.3981	0.3429

Comparison results of the first and second natural frequency parameters of isotropic plate as provided in Table 2 are compared with the solutions of Hashemi et al. in [33] and Hashemi and Arsanjani in [37]. The results are not unexpected, it can be seen that a good agreement is obtained in this comparison.

Table 2.

Comparison study of frequency parameter $\overset{⌢}{β} = ω a^{2} \sqrt{ρ_{c} / E_{c}} / h$ for isotropic plates.

Material		First mode			Second mode
		$h / a$			$h / a$
		0.05	0.1	0.2	0.05	0.1	0.2
Fully ceramic	Present	5.9317	5.8117	5.3957	14.677	13.98	11.941
	Ref. [28]	5.9212	5.7748	5.2972	14.616	13.794	11.615
	Ref. [32]	5.9214	5.7751	5.2974	14.616	13.795	11.616
Fully metallic	Present	3.0192	2.9581	2.7463	7.4704	7.1158	6.0778
	Ref. [28]	3.0329	2.9575	2.7115	7.4858	7.0622	5.9416
	Ref. [32]	3.0139	2.9394	2.6963	7.4396	7.0212	5.9123

By homogenizing the material layers of the plate structure are made of FGM material, the following subsequent comparisons also give results that can show the reliability of the method study in the paper.

The results of the first two non dimensional fundamental frequencies for thick simply supported Si₃N₄/SUS304 FGM plate are compared with the results given by Singh and Harsha [38] and by Huang and Shen [39], where E(Si₃N₄) = 322.27 MPa, E(SUS304) = 207.78 MPa, and $α$ (Si₃N₄) = 0.000007475(1/C), $α$ (SUS304) = 0.00001532 (1/C) represents material properties of FGM at room temperature. Poisson's ratio is considered as 0.28. The results are in excellent agreement with an average error less than 1% as exhibited in Table 3 for different temperature differences. The slight difference in results at higher temperature differences occurs due to different temperature profile employed in the present formulation.

Table 3.

Comparison of non-dimensional linear frequency for Si₃N₄/SUS304 FGM square plate without effect of elastic foundation.

		Mode 1			Mode 2
$Δ T$		Fully ceramic	N=1	Fully metallic	Fully ceramic	N=1	Fully metallic
0	Ref. [39]	12.495	7.555	5.405	29.131	17.649	12.602
	Ref. [38]	12.507	7.561	5.410	29.256	17.716	12.655
	Present	12.506	7.564	5.410	29.250	17.718	12.653
100	Ref. [39]	12.382	7.514	5.335	29.243	17.694	12.587
	Ref. [38]	12.449	7.505	5.352	29.197	17.659	12.596
	Present	12.251	7.330	5.181	28.985	17.476	12.416
300	Ref. [39]	12.213	7.305	5.104	28.976	17.486	12.342
	Ref. [38]	12.332	7.392	5.232	29.087	17.545	12.475
	Present	11.724	6.838	4.690	28.446	16.980	11.929
		Mode 3			Mode 4
$Δ T$		Fully ceramic	N=1	Fully metallic	Fully ceramic	N=1	Fully metallic
0	Ref. [39]	43.845	26.606	18.967	52.822	32.081	22.850
0	Present	44.207	26.814	19.123	53.411	32.407	23.104
100	Ref. [39]	44.072	26.717	19.008	53.105	32.242	22.908
100	Present	43.930	26.561	18.876	53.126	32.148	22.851
300	Ref. [39]	43.797	26.506	18.763	52.821	31.970	22.658
300	Present	43.371	26.048	18.374	52.553	31.622	22.337

Figure 2 shows the next comparison of nonlinear dynamic responses of the FGM plate without elastic foundations subjected to blast load between the results obtained in this report and the study [40] by Duc et al. As can be seen, a good agreement is obtained in this comparison.

Nonlinear dynamic and vibration response of sandwich stiffened plate

In this section, the nonlinear dynamic and vibration response of sandwich stiffened plate with an auxetic core layer in thermal environment supported by elastic medium and subjected to blast and mechanical loads are analyzed and discussion. For convenience in surveying and assessing the effect of initial parameters and boundary conditions on the structure, except for the case of surveying the effect of temperature on the sandwich plate, the remaining cases are all considered in the field with an environment temperature variation $Δ T = 0$ . In addition, if no further comment is given, the chosen GPL distribution will be UD (uniformly distributed) and characteristic for blast loading with parameters $P_{s} = 290 kPa, T_{s} = 0.01, B_{s} = 2.$

The well- known Friedlander wave equation (30) defines the rise and fall of the static over pressure $P_{s}$ with time as shown [16]

q (t) = P_{s} (t) = 1.8 P_{s} (1 - \frac{t}{T_{s}}) \exp^{- \frac{B_{s} t}{T_{s}}}

(31)

where the “1.8” factor accounts for the effects of a hemispherical blast

The parameters are selected for sandwich plate configuration as follows $a / b = 1, a / h_{P} = 25, h_{A U} = 2 h_{GRA}, h_{FGM} = h_{GRA},$ $h_{s} = h_{FGM}, b_{s} = h_{s}, d_{s} = 2 h_{s}$ . For each different material layer, the corresponding parameters are selected: for FGM layer: $N = 1$ ; for GPL layer: $V_{GPL}^{*} = 0.11,$ and for auxetic layer: $h_{a u} / l_{a u} = 1.8,$ $θ = 30 °,$ $t_{a u} / l_{a u} = 0.0138571.$

Figure 3 describes the impact of stiffeners on the amplitude – time curves for nonlinear dynamic response of sandwich plate with negative Poisson’s ratio in core layer. As can be seen, over time, the amplitude of stiffened plate decreases compared to the unstiffened plate. This result demonstrates that, similar to blast resistance of stiffened sandwich panels with aluminum cenosphere syntactic foam as indicated in [1], the stiffened sandwich panels with auxetic core layer also offers superior advantages compared to cases without stiffened reinforcement.

Figures 4 and 5 show the effects of volume fraction index $N (0.5, 1, 2)$ and size of imperfections $μ (0, 0.01, 0.02)$ on the amplitude – time curves for nonlinear dynamic response of sandwich plate with negative Poisson’s ratio in core layer (mode $(m, n) = (1, 1))$ . It can be seen that the amplitude of the nonlinear dynamic response of plate increases when increasing the power law index $N$ and the imperfection coefficient $μ$ . This proves that these two indexes of the FGM material layer greatly affects the blast resistance of the plate.

Figure 6 shows the effects of the elastic medium on the nonlinear dynamic response of the sandwich plates with negative Poisson’s ratio under blast loading. The effect of elastic substrate is similar to that of other structures, it can be seen that the elastic medium make the vibration amplitude of sandwich plate reduce when increasing the coefficients $K_{W}, K_{p}$ . That shows the positive influent of elastic medium. In addition, Winkler’s elastic medium $K_{W}$ is weaker than Pasternak’s medium $K_{p}$ .

Impact of blast and mechanical loads (pre-loaded axial compressive force) on the amplitude – time curves for nonlinear dynamic response of the sandwich plates with negative Poisson’s ratio are shown in Figures 7 and 8. It is easy to see that, if the change in amplitude under the influence of the mechanical load is not much difference, then when under the influence of blast load, this amplitude changes significantly. However, a general rule is that the nonlinear dynamic response amplitude increases with increasing pre-loaded axial compression and blast loading value.

The influent of parameter characterizing the duration of the blast pulse on nonlinear response of sandwich plate for three cases Ts = (0.005, 0.01, 0.02) is surveyed in Figure 9. Easy to see, the amplitude of vibration increases with increase in the value of the blast pulse Ts and vice versa.

The effect of GPL volume fraction $V_{GPL}^{*}$ on the amplitude – time curves of the sandwich plates is shown in Figure 10 with UD types of GPL distribution. As can be observed that, the value of amplitude decreases when the volume fraction increases.

The next evidence of the impact of GPL layer on the nonlinear dynamic response of the sandwich plates with negative Poisson’s ratio under blast load is illustrated in Figure 11, with the parameter is chosen to survey is three types of GPL reinforcements (UD, FG-V and FG-X). From Figure 11 can be seen that the various types of GPL distributions make the negligible changed amplitude. Besides, the amplitude – time curves of FG-X and UD types of GPL distribution is almost no difference.

Subsequent surveys are conducted to show the effect of proportions $a / b$ , $a / h_{p}$ , $h_{p} / h_{A U}$ (the parameters of the sandwich plate) on the nonlinear dynamic behavior of the structure (Figures 12 to 14). As can see, the amplitude of the amplitude – time curves is very sensitive to the change in these ratios. In other words, the value of amplitude increases when these ratios increase. The results of these influences survey are significant in optimizing the parameters between the material layers as well as the parameters of the plate.

Based on equation (29), the relationships between the frequency ratio and the amplitude of the vibration can be obtained. The Influent of the elastic medium on the frequency-amplitude relation of the sandwich plate with negative Poisson’s ratio under blast load is considered in Figure 15. Similar to the results shown in [14] by Duc and Cong, namely, with the same amplitude, the sandwich plate with negative Poisson’s ratio on elastic medium has smaller frequency than the structure without elastic medium, and the Pasternak medium influences faster and more powerful than Winkler medium on the frequency-amplitude relations of the sandwich plate with negative Poisson’s ratio.

Figure 16 shows the impact of pre-loaded on frequency-amplitude curve of the sandwich plate with auxetic core layer, obviously with the same amplitude, when the pre-loaded increases, the frequency of vibration becomes larger.

The influence of temperature on the behavior of the sandwich stiffened plate with negative Poisson’s ratio is further analyzed in this section. Firstly, Figure 17 shows the effects of uniform temperatures rise with $Δ T = (0, 100, 200^{o} C)$ on the nonlinear displacement-time response of sandwich stiffened plate under blast load. The dynamic response amplitude raises as $Δ T$ increases.

The effect of temperature and elastic foundations on natural frequencies $ω (s^{- 1})$ of sandwich stiffened plate is showed in Table 4. From this table, it can be seen that the value of the natural frequency increases when the values $K_{1}$ and $K_{2}$ increase and decrease when the value $Δ T$ increases. Similar to the survey on the effect of coefficient $K_{W}, K_{p}$ above, the elastic medium have positive effect whilst temperature has negative effect on the natural frequency value. Furthermore, the Pasternak elastic foundation influences on the natural frequency larger than the Winkler foundation.

Table 4.

Effect of temperature and coefficient $K_{W}, K_{p}$ on natural frequencies $ω (s^{- 1})$ of sandwich stiffened plate.

$Δ T$	50		100		200
Mode $K_{W}, K_{P}$	1	2	1	2	1	2
$K_{W} = 0 GPa / m, K_{P} = 0 GPa . m$	0.0228	0.0602	0.0210	0.0589	0.0160	0.0559
$K_{W} = 0.3 GPa / m, K_{P} = 0 GPa . m$	0.0294	0.0630	0.0281	0.0618	0.0251	0.0591
$K_{W} = 0.3 GPa / m, K_{P} = 0.02 GPa . m$	0.0357	0.0706	0.0348	0.0698	0.0328	0.0678

$a / b = 1, a / h_{P} = 25, h_{A U} = 0.5 h_{P}, h_{FGM} = h_{GRA}, U D, N = 1, λ_{o m} = ω \times h_{P} \times \sqrt{ρ_{m} / E_{m}}$ .

The Influent of temperature and $a / h_{P}$ ratio on natural frequencies $ω (s^{- 1})$ of sandwich plate with and without stiffened is listed in Table 5. As can be seen, increasing in both temperature $Δ T$ and $a / h_{P}$ ratio will contribute to the decrease of natural frequency values $ω (1 / s)$ . In addition, in the case of plate reinforced with stiffened reinforcement, this frequency has a higher value than the case without stiffened.

Table 5.

Influent of temperature and $a / h_{P}$ ratio on natural frequencies $ω (s^{- 1})$ of sandwich plate with and without stiffened

$Δ T$		50		100		200
Mode		1	2	1	2	1	2
$a / h_{P} = 15$	Stiffened	0.0673	0.1654	0.0659	0.1651	0.0629	0.1645
$a / h_{P} = 15$	Unstiffened	0.0594	0.1439	0.0584	0.1438	0.0559	0.1433
$a / h_{P} = 20$	Stiffened	0.0531	0.1344	0.0508	0.1331	0.0453	0.1301
$a / h_{P} = 20$	Unstiffened	0.0447	0.1107	0.0429	0.1096	0.0386	0.1071
$a / h_{P} = 25$	Stiffened	0.0443	0.1147	0.0411	0.1124	0.0326	0.1071
$a / h_{P} = 25$	Unstiffened	0.0357	0.0894	0.0332	0.0876	0.0268	0.0833

$a / b = 1, h_{A U} = 0.5 h_{P}, h_{FGM} = h_{GRA}, U D, N = 1, K_{W} = 10 MPa / m, K_{P} = 1 MPa . m, h_{s} = 0.015, b_{s} = h_{s}, d_{s} = 2 h_{s}; λ_{o m} = ω \times a^{2} \times \sqrt{ρ_{m} / E_{m}} / h_{s} / 10^{3}$ .

Conclusion

The impact of blast and mechanical loads on the stiffened sandwich plate with an auxetic core layer in thermal environment has been investigated in this report. Few conclusions are drawn as follows:

By using the FSDT, the analytic method used in this report gives results similar to the finite element method. This is reflected in the comparison section with almost identical results with the other 3 reports indicated in the references. Meanwhile, the obtained results in this study are expressed explicitly in terms of the input parameters of the material and the structure, so when we change these parameters we can actively control the behaviour of the structures.

The combination of material layers and additional reinforcement stiffened in the plate with auxetic core layer also offers superior advantages compared to case without stiffened reinforcement or homogeneous plates, even when the plate is under blast load.

Elastic medium have a positive impact on the amplitude - time curve and vibration, specifically making amplitude dynamic response decrease and the natural frequency increase.

The amplitude of the dynamic response is very sensitive with the change in the ratio of parameters of sandwich structure.

Supplemental Material

sj-pdf-1-jsm-10.1177_10996362211021912 - Supplemental material for Impact of blast and mechanical loads on the shear deformable stiffened sandwich plate with an auxetic core layer in thermal environment

Supplemental material, sj-pdf-1-jsm-10.1177_10996362211021912 for Impact of blast and mechanical loads on the shear deformable stiffened sandwich plate with an auxetic core layer in thermal environment by Vu Thi Thuy Anh, Vu Dinh Quang, Nguyen Dinh Duc and Pham Ngoc Thinh in Journal of Sandwich Structures & Materials

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

ORCID iD

Pham Ngoc Thinh

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