Modeling the dynamic bending of rigid-plastic fiber-reinforced laminated curvilinear doubly connected thin plates with free outer contour

Abstract

A theoretical model of the dynamic bending of rigid-plastic hybrid composite, arbitrary curvilinear doubly connected thin plates is developed. Inner contour of the plate is simply supported or clamped and outer one is free. The plates are on a viscous basis and under the action of uniformly distributed loads of explosive type. The plates are laminated and fibrous, with layers arranged symmetrically with respect to the middle surface. In each layer the reinforcing fibers, made of different materials, are located in directions parallel or normal to inner contour of plate. The structural model of the reinforced layer considering the plane stress state in fibers is used. The equations of the dynamic deformation of plate and simple analytical formula for the limit load are obtained. Numerical examples are given for a fiber-reinforced four-layered curvilinear plate with a supported hole in the form of ellipse and super-ellipse at the same total amount of reinforcement.

Keywords

Bending of fiber-metal laminate curvilinear plate rigid-plastic model structural model of composite plane stress state doubly connected plates viscous foundation explosive load limit load final deflection

1. Introduction

Structures of composite materials are widely used in modern engineering. The problem of studying the dynamic plastic deformation of fiber-reinforced plates and shells is topical at present, but, despite its importance, theoretical research in this direction is insufficient [1 –8]. The study of the dynamic behavior of various thin-walled structural elements under the influence of high-intensity loads is of great importance for assessing their damage, as well as in the creation of such elements by the methods of pulse stamping. The rigid-plastic body model is widely used for solving of dynamic problems under explosive-type loads due to its simplicity and convenience of calculated schemes [1, 9 –13]. In Yankovskii [1], the current state of the art in calculating the inelastic dynamic deformation of shell constructions with a layered fibrous structure is discussed, and the use of the model of rigid-plastic bodies in such calculations is substantiated.

Within the rigid-plastic model the body begins to deform when the load exceeds a certain limit value. In this model, the solutions are obtained without performing complete elastoplastic stepwise analysis under proportionally increasing loading, i.e. without considering loading history. The rigid-plastic theory solutions are quite simple and give quick and qualitatively correct information about the deformation of the structure. In the framework of this model, the first study to assess the residual damage of rectangular plates under the influence of explosive loads was performed by Gvozdev [14], in which the plate is deformed in form of the set of rigid areas separated by rectilinear yield lines. This approach is also widely known under the name “yield line theory” and has been successfully used for limit analysis of various polygonal homogeneous and reinforced concrete plates [15 –18].

The merits and demerits of the structural model of a reinforced layer with a one-dimensional stress state in fibers [19] are discussed in Yankovskii [1]. In Romanova and. Yankovskii [20] the models of unidirectionally and orthogonally fiber-reinforced rigid-plastic media of a regular periodic structure allowing one to determine the yield loci for these composites are constructed considering the plane stress state in fibers. In Wang et al. [21], within the framework of the rigid-plastic body model the limit load of circular sandwich panels with foam aluminum filler under the action of quasi-static local load is estimated. The dynamic behavior of rigid-plastic homogeneous doubly connected plates supported along inner contour is considered in Kostrik et al. [22] for circular hole and in Nemirovsky and Romanova [23] for curved hole. In Nemirovsky and Fedorova [24], on the basis of a structural model [19], complex structures of reinforcement along curvilinear coordinates for elastic plates loaded in the plane are considered.

The reinforcement of structural elements is used to create designs with optimal technological and operational properties. The reinforced doubly connected thin plates can be elements of various hatch covers, stoppers, dampers, and screens separating areas with different pressures, and also can be used in other loaded elements of designs in aerospace, automobile, and structural engineering. The annular plates are used as internal watertight bulkheads in submersibles in order to isolate a damaged compartment in the case of an accident. Axisymmetric plates clamped at the inner edge can be considered as simplified models of flanges used in pipe connections [25]. Reinforced panels of varying configurations, supported on several inner columns, can deform as double-connected plates [26, 27]. Curvilinear plates with a supported circular hole, reinforced in radial and circumferential directions, are analyzed in Romanova [28].

Currently, the plates in which the reinforcing fibers have a curvilinear shape are of great interest [24, 29 –31]. Composite materials, where the reinforcement is purposely made variable, may lead to more efficient designs [32]. By selecting the type of reinforcement structure, combinations of binder, and reinforcement materials, it is possible to create a structure with predetermined strength characteristics. Different kinds of reinforcement in the region of plastic deformations have their own peculiarities; therefore, they have to be analyzed separately. In the present work, the arbitrary curvilinear doubly connected hybrid thin laminates, supported on inner contour and reinforced with fibers in directions parallel (equidistant) and normal to the supported contour, under action of short-term intense dynamic loads of explosive type, are considered. The plates are on a viscous basis. A technique to examine the dynamic bending of such plates is developed. It is based on a rigid-plastic analysis in a geometrically linear formulation with the use of the structural model of a reinforced layer taking into account the plane stress state in fibers [20]. It is assumed that the mechanical characteristics of the materials included in the layer package are commensurate (differ no more than by one order of magnitude). Across the thickness, all layers have a regular and quasi-homogeneous structure, and on borders between layers, the condition of an ideal mechanical contact is satisfied. In this case, for the layered thin plates the traditional Kirchhoff hypotheses are acceptable. The transverse shears and through-the-thickness deformations are discarded.

The proposed model of deformation is essentially a development and modification of the yield line theory in the case of fiber-reinforced plates of arbitrary curvilinear contour under the influence of explosive loads. In this case, the appearing yield lines are curvilinear and their position changes in time.

2. Formulation of problem, basic geometrical relations, and assumptions

Let us consider a doubly connected, thin Kirchhoff plate simply supported or fully clamped on its inner contour $L_{1}$ of arbitrary smooth convex form. The outer contour $L_{2}$ of the plate is free and has an arbitrary form (Figure 1). The plate is subjected to a uniformly distributed explosive surface load of intensity $P (t)$ that reaches its maximum $P_{max} = P (0)$ at the initial instant of time $t = 0$ and then decreases rapidly. Deflections of the plate are considered small. The effect of membrane forces and vertical shears in the plasticity criterion and geometrical changes are neglected.

The inner contour $L_{1}$ of the plate is specified in the parametric form

x = x_{1} (φ), y = y_{1} (φ), 0 \leq φ < 2 π .

The radius of curvature of the $L_{1}$ is

R (φ) = \frac{B^{3} (φ)}{(\partial x_{1} / \partial φ) (\partial^{2} y_{1} / \partial φ^{2}) - (\partial^{2} x_{1} / \partial φ^{2}) (\partial y_{1} / \partial φ)}

where $B (φ) = \sqrt{{(\partial x_{1} / \partial φ)}^{2} + {(\partial y_{1} / \partial φ)}^{2}}$ . To be specific, we assume that the plate is a symmetrical with respect to $x$ -axis.

Let’s introduce the curvilinear orthogonal coordinate system $(ν_{1}, ν_{2})$ related to the Cartesian coordinate system by the relations

x = x_{1} (ν_{2}) + ν_{1} y'_{1} (ν_{2}) / B (ν_{2}), y = y_{1} (ν_{2}) - ν_{1} x'_{1} (ν_{2}) / B (ν_{2})

(1)

where the prime denotes the derivative with respect to coordinate $ν_{2}$ . The curves $ν_{1} = const \geq 0$ are at the distance $ν_{1}$ from the contour $L_{1}$ toward the outer contour and have the radius of the curvature $r_{1} = R (ν_{2}) + ν_{1}$ . The straight lines $ν_{2} = const$ are perpendiculars to the inner contour $L_{1}$ and their radius of the curvature is $r_{2} = \infty$ (see Figure 1). In coordinates $(ν_{1}, ν_{2})$ the elements of length and area are $B (ν_{2}) [1 + ν_{1} / R (ν_{2})] d ν_{2}$ and $B (ν_{2}) [1 + ν_{1} / R (ν_{2})] d ν_{1} d ν_{2}$ . Then the equation of the supported contour $L_{1}$ has the form

ν_{1} = 0, 0 \leq ν_{2} < 2 π .

The angle $ψ$ between the $ν_{1}$ -direction and the $x$ -axis and its differential are

ψ = - arctg [x'_{1} (ν_{2}) / y'_{1} (ν_{2})], d ψ = [B (ν_{2}) / R (ν_{2})] d ν_{2},

and therefore the following equalities are valid

\int_{0}^{π} \frac{B (ν_{2})}{R (ν_{2})} d ν_{2} = \int_{0}^{π} d ψ = π .

(2)

Let the outer contour $L_{2}$ of the plate is specified in the explicit form

y = F_{2} (x) .

We designate the distance from the contour $L_{1}$ to the outer contour $L_{2}$ along the normal to the contour $L_{1}$ as $D_{2} (ν_{2})$ . Then from relations (1) we have

x = x_{1} (ν_{2}) + D_{2} (ν_{2}) y'_{1} (ν_{2}) / B (ν_{2}), y = y_{1} (ν_{2}) - D_{2} (ν_{2}) x'_{1} (ν_{2}) / B (ν_{2}) .

From the last expressions taking into account the explicit form of the outer contour $L_{2}$ , we obtain the equation for calculation of function $D_{2} (ν_{2})$

y_{1} (ν_{2}) - D_{2} (ν_{2}) x'_{1} (ν_{2}) / B (ν_{2}) = F_{2} (x_{1} (ν_{2}) + D_{2} (ν_{2}) y'_{1} (ν_{2}) / B (ν_{2})) .

Then the contour $L_{2}$ will be considered to be specified in the coordinates $(ν_{1}, ν_{2})$ as

ν_{1} = D_{2} (ν_{2}), 0 \leq ν_{2} < 2 π .

The plate is made of a hybrid laminated material with its layers distributed symmetrically with respect to the middle surface of the plate. It is assumed that each layer contains a large number of reinforced elementary layers and isotropic binding interlayers, which are described by the model of a perfectly rigid-plastic material. In each layer, the reinforcing fibers can be located parallel (on the lines $ν_{1} = const$ ) or normal (on the lines $ν_{2} = const$ ) to the contour $L_{1}$ of the plate (see Figure 1). In different layers, the materials of fibers and binder are different. The structures of plate reinforcement created using the same materials can be different depending on the arrangement of reinforced layers. It is assumed that the component materials have the same tensile and compressive yield points.

For the plate considered, the surface density of the material is

ρ (ν_{1}) = 2 \sum_{k = 1}^{n} ρ_{k} (h_{k} - h_{k - 1}), ρ_{k} = (1 - ω_{1 k} - ω_{2 k}) ρ_{0 k} + ω_{1 k} ρ_{1 k} + ω_{2 k} ρ_{2 k}

(3)

where n is half of the total number of layers; $ρ_{0 k}$ is density of the binder material in a kth layer; $ρ_{1 k}$ is the density of material of the fibers of a kth layer in the direction normal to the contour $L_{1}$ (along the lines $ν_{2} = const$ ); $ρ_{2 k}$ is the density of material of the fibers of a kth layer in the direction parallel to the contour $L_{1}$ (along the lines $ν_{1} = const$ ); $ω_{jk} (v_{1})$ is the reinforcement density (the relative volume content of fibers in a kth layer) of the fibers located in the kth layer along the lines $ν_{3 - j} = const$ ( $j = 1, 2$ ), which depends on v₁ in the general case; $h_{k}$ are coordinates of layers boundaries; $h_{0} = 0$ ; $h_{n} = H$ ; $2 H$ is thickness of the plate.

For the type of reinforcement considered, the limit linear bending moments $M_{01}$ and $M_{02}$ , normal and tangential to the plate contour $L_{1}$ , are (according to the structural model of a reinforced layer taking into account the plane stress state in fibers [20])

M_{0 j} (ν_{1}) = \sum_{k = 1}^{n} σ_{0 jk} (ν_{1}) (h_{k}^{2} - h_{k - 1}^{2}), (j = 1, 2)

(4)

σ_{0 jk} (ν_{1}) = [1 - ω_{jk} (ν_{1})] σ_{0 k} + ω_{jk} (ν_{1}) σ_{jk}

where $σ_{0 k}$ is the yield point of binder material in a kth layer; $σ_{jk}$ is the yield point of material of fibers located along the directions $ν_{j}$ in a kth layer.

In the framework of rigid-plastic theory, it is necessary to construct kinematically acceptable velocity fields to obtain a solution. In the dynamics of the composite plate under consideration, depending on the value $P_{max}$ two mechanisms of deformation are possible. Under loads not exceeding the limit load $P_{0}$ (“low” loads, $0 < P_{max} \leq P_{0}$ ), the plate is at rest. For loads slightly higher than the limit load (“moderate” loads, $P_{0} < P_{max} \leq P_{1}$ ), in the case of simple support of the contour, the plane of the plate rotates around the supported contour $L_{1}$ through the same angle of rotation $α_{1}$ and deformed into a ruled surface. As in Kostrik et al. and Nemirovsky and Romanova [22, 23], we assume that the angle $α_{1}$ does not depend on the parameter $ν_{2}$ . For the reinforced plates, in contrast to homogeneous ones [23], in the case of a clamped contour $L_{1}$ , the plane of the plate may rotate not around the contour $L_{1}$ , but around the formed plastic hinged closed curve $L_{d}$ (see Figure 1), having the equation

ν_{1} = d (0 \leq ν_{2} < 2 π, 0 \leq d < D_{2}) .

On the contour $L_{d}$ the normal bending moment is equal to $M_{01} (d)$ . The area between the contours $L_{1}$ and $L_{d}$ is not deformed. The distance $d$ is unknown and is determined from the condition of minimum of limit load of the plate. When $d = 0$ the hinge $L_{d}$ coincides with the contour $L_{1}$ . The scheme of deformation at the “moderate” loads we name scheme 1.

At sufficiently great values of $P_{max}$ (“high” loads, $P_{max} > P_{1}$ ), scheme 2 is possible, in which, as in the exact rigid-plastic solution for homogeneous annular plates [22] and in the approximate solution for curved homogeneous and composite plates with an hole [23, 32], the deformation of the plate occurs with the appearance of a non-stationary plastic hinged closed curve $l$ moving translationally (see Figure 2). Curve $l$ divides the plate into two ruled surfaces $Z_{1}$ and $Z_{2}$ , which move independently. The difference between schemes 1 and 2 is that in scheme 1, one area is deformed, and in scheme 2, two areas $Z_{1}$ and $Z_{2}$ are deformed independently.

3. The equations of motion and analysis of behavior of plate in scheme 1

The equations of motion of the plate will be deduced from the principle of virtual power with the use of d’Alembert’s principle:

K = A - N,

(5)

K = \int \int_{S} ρ \frac{dV}{dt} V^{*} ds, A = \int \int_{S} [P (t) - β V] V^{*} ds,

(6)

N = \sum_{m} \int_{l_{m}} M_{m} {[θ^{*}]}_{l_{m}} d l_{m} + \int \int_{S} (M_{01} | κ_{1}^{*} | + M_{02} | κ_{2}^{*} |) ds .

(7)

Here, $K$ , $A$ , and $N$ are the powers of inertial, external, and internal forces, respectively; $S$ and $ds$ are the area of the plate and its area element; $u$ is deflection; $V = du / dt$ ; $V$ is deflection velocity; $β$ is the coefficient of viscous resistance of the base; $m$ is the number of lines of discontinuity of the angular velocity; $l_{m}$ and $d l_{m}$ are lines of discontinuity of the angular velocity and its line element; ${[θ]}_{l_{m}}$ is discontinuity of the angular velocity on the line $l_{m}$ ; $M_{m}$ is normal bending moment on the line $l_{m}$ ; $κ_{1}$ and $κ_{2}$ are the principal curvatures of the surface of $V$ . The superscript “∗” designates kinematically admissible quantities. In the case of blast waves on ship structures, the coefficient $β$ is assumed equal to $ρ_{s} c$ , where $ρ_{s}$ is the density of the surrounding medium, с is the speed of sound in it [33]. If there is no resistance foundation, equation (5) coincides with the equation of motion in Jones [34], where the axial forces being assumed to equal zero means that geometrical changes are ignored. Note that in Jones [34] this equation was suggested for use with plates of an arbitrary contour and arbitrary edge conditions. But to date it has been used in the literature for circular and rectangular plates only.

In scheme 1, deflection velocities of the plate (see Figures 1, 3) are

V (ν_{1}, ν_{2}, t) = {\begin{matrix} 0, 0 \leq ν_{1} \leq \bar{d} \\ χ_{1} (t) (ν_{1} - \bar{d}), ν_{1} > \bar{d} \end{matrix} (0 \leq ν_{2} < 2 π)

(8)

where $χ_{1} = d α_{1} / dt$ ; $\bar{d} = d (1 - η)$ ; $η = 0$ for the clamped contour $L_{1}$ and $η = 1$ for its simply supporting. The principal curvatures of the surface of $V (ν_{1}, ν_{2}, t)$ in the area $ν_{1} > \bar{d}$ ( $0 \leq ν_{2} < 2 π$ ) are

κ_{1} = - \partial^{2} V / \partial ν_{1}^{2} = 0, κ_{2} = (- \partial V / \partial ν_{1}) / r_{1} = - χ_{1} (t) / [R (ν_{2}) + ν_{1}] .

(9)

Expressions (6) take the form:

K = \frac{d χ_{1}}{dt} χ_{1}^{*} \int \int_{S} ρ (ν_{1}) {(ν_{1} - \bar{d})}^{2} ds = 2 \frac{d χ_{1}}{dt} χ_{1}^{*} \int_{0}^{π} \int_{\bar{d}}^{D_{2} (ν_{2})} B (ν_{2}) (1 + \frac{ν_{1}}{R}) ρ (ν_{1}) {(ν_{1} - \bar{d})}^{2} d ν_{1} d ν_{2},

(10)

\begin{matrix} A = χ_{1}^{*} \int \int_{S} [P (t) - β χ_{1} (ν_{1} - \bar{d})] (ν_{1} - \bar{d}) ds \end{matrix}

(11)

\begin{matrix} = 2 χ_{1}^{*} [P (t) \int_{0}^{π} \int_{\bar{d}}^{D_{2} (ν_{2})} B (1 + \frac{ν_{1}}{R}) (ν_{1} - \bar{d}) d ν_{1} d ν_{2} - β χ_{1} \int_{0}^{π} \int_{\bar{d}}^{D_{2} (ν_{2})} B (1 + \frac{ν_{1}}{R}) {(ν_{1} - \bar{d})}^{2} d ν_{1} d ν_{2}] \\ = \frac{χ_{1}^{*}}{6} [2 P (t) \int_{0}^{π} \frac{B}{R} {(D_{2} - \bar{d})}^{2} (3 R + 2 D_{2} + \bar{d}) d ν_{2} - β χ_{1} \int_{0}^{π} \frac{B}{R} {(D_{2} - \bar{d})}^{3} (4 R + 3 D_{2} + \bar{d}) d ν_{2}] . \end{matrix}

The normal bending moment on the free contour $L_{2}$ is equal to zero, on contour $L_{d}$ it is

M_{m} = - (1 - η) M_{01} (\bar{d}) .

The discontinuity of the angular velocity on contour $L_{d}$ is $[θ]_{L_{d}} = χ_{1}$ . The internal power (7), accounting formulas (2) and (9), is

N = 2 χ_{1}^{*} {(1 - η) M_{01} (\bar{d}) [\int_{0}^{π} B (ν_{2}) d ν_{2} + π \bar{d}] + \int_{0}^{π} \frac{B}{R} \int_{\bar{d}}^{D_{2} (ν_{2})} M_{02} (ν_{1}) d ν_{1} d ν_{2}} .

(12)

Inserting equations (10)–(12) into equation (5) and taking into account that $χ_{1}^{*}$ is an independent function, we obtain the equation of motion of plate in scheme 1

\frac{d χ_{1}}{dt} \int_{0}^{π} \int_{\bar{d}}^{D_{2} (ν_{2})} B (ν_{2}) [1 + \frac{ν_{1}}{R (ν_{2})}] ρ (ν_{1}) {(ν_{1} - \bar{d})}^{2} d ν_{1} d ν_{2}

(13)

\begin{matrix} + \frac{β χ_{1}}{12} \int_{0}^{π} \frac{B}{R} {(D_{2} - \bar{d})}^{3} (4 R + 3 D_{2} + \bar{d}) d ν_{2} = \frac{P (t)}{6} \int_{0}^{π} \frac{B}{R} {(D_{2} - \bar{d})}^{2} (3 R + 2 D_{2} + \bar{d}) d ν_{2} \\ - (1 - η) M_{01} (\bar{d}) (\int_{0}^{π} B d ν_{2} + π \bar{d}) - \int_{0}^{π} \frac{B}{R} \int_{\bar{d}}^{D_{2} (ν_{2})} M_{02} (ν_{1}) d ν_{1} d ν_{2} . \end{matrix}

The initial conditions are

χ_{1} (0) = α_{1} (0) = 0 .

(14)

The limit load $P_{0}$ is determined from equation (13) assuming that $d χ_{1} / dt = 0$ at the beginning of deformation $t = 0$ :

P_{0} = 6 min_{0 \leq \bar{d} < min_{0 \leq ν_{2} < π} D_{2} (ν_{2})} \frac{(1 - η) M_{01} (\bar{d}) (\int_{0}^{π} B d ν_{2} + π \bar{d}) + \int_{0}^{π} \frac{B}{R} \int_{\bar{d}}^{D_{2} (ν_{2})} M_{02} (ν_{1}) d ν_{1} d ν_{2}}{\int_{0}^{π} \frac{B}{R} {(D_{2} - \bar{d})}^{2} (3 R + 2 D_{2} + \bar{d}) d ν_{2}}

(15)

= 6 \frac{(1 - η) M_{01} ({\bar{d}}_{0}) [\int_{0}^{π} B (ν_{2}) d ν_{2} + π {\bar{d}}_{0}] + \int_{0}^{π} \frac{B (ν_{2})}{R (ν_{2})} \int_{{\bar{d}}_{0}}^{D_{2} (ν_{2})} M_{02} (ν_{1}) d ν_{1} d ν_{2}}{\int_{0}^{π} \frac{B}{R} {(D_{2} - {\bar{d}}_{0})}^{2} (3 R + 2 D_{2} + {\bar{d}}_{0}) d ν_{2}}

where ${\bar{d}}_{0} = d_{0} (1 - η)$ . The distance $d_{0}$ determines position of the contour $L_{d}$ at the limit state.

We write equation (13) as

\frac{d χ_{1} (t)}{dt} + E_{1} χ_{1} (t) = E_{2} [P (t) - P_{0}],

(16)

E_{1} = [β \int_{0}^{π} \frac{B}{R} {(D_{2} - {\bar{d}}_{0})}^{3} (4 R + 3 D_{2} + {\bar{d}}_{0}) d ν_{2}] / [12 \int_{0}^{π} \int_{{\bar{d}}_{0}}^{D_{2} (ν_{2})} B (1 + \frac{ν_{1}}{R}) ρ {(ν_{1} - {\bar{d}}_{0})}^{2} d ν_{1} d ν_{2}],

E_{2} = [\int_{0}^{π} \frac{B}{R} {(D_{2} - \bar{d})}^{2} (3 R + 2 D_{2} + \bar{d}) d ν_{2}] / [6 \int_{0}^{π} \int_{{\bar{d}}_{0}}^{D_{2} (ν_{2})} B (1 + \frac{ν_{1}}{R}) ρ {(ν_{1} - {\bar{d}}_{0})}^{2} d ν_{1} d ν_{2}] .

Integrating equation (16), we have

χ_{1} (t) = E_{2} e^{- E_{1} t} \int_{0}^{t} e^{E_{1} τ} [P (τ) - P_{0}] d τ,

(17)

α_{1} (t) = E_{2} \int_{0}^{t} e^{- E_{1} λ} \int_{0}^{λ} e^{E_{1} τ} [P (τ) - P_{0}] d τ d λ .

The plate moves up to a stop at the moment $t_{f}$ , when

χ_{1} (t_{f}) = 0 .

(18)

From equations (17) and (18) it follows that $t_{f}$ determined from the equation

\int_{0}^{t_{f}} e^{E_{1} τ} P (τ) d τ = \frac{P_{0}}{E_{1}} (e^{E_{1} t_{f}} - 1) .

The maximum final deflection is on the outer contour of the plate and it is

u_{f} = α_{1} (t_{f}) [max_{0 \leq ν_{2} \leq π} D_{2} (ν_{2}) - {\bar{d}}_{0}] .

All deflections of plate are calculated from equations (8).

For the simply supported contour $L_{1}$ ( $η = 1$ ) the parameters of reinforcement in the $ν_{1}$ -direction are not included in the expression (15), but they are in the coefficients $E_{1}$ , $E_{2}$ of equation (16). Therefore, in this case, reinforcement in the direction normal to the inner contour does not influence the limit load, but it affects the dynamic behavior of the plate and its final deflection.

A simple analytical formula (15) for $P_{0}$ allows us to quickly assess the limit load of the reinforced plates considered. For example, from formulas (15) and (4) in the case $η = 1$ and $D_{2} = const$ we have

P_{0} = \frac{6 \sum_{k = 1}^{n} [σ_{0 k 2} + (σ_{2 k} - σ_{0 k}) Ω_{k}] (h_{k}^{2} - h_{k - 1}^{2})}{D_{2}^{2} [\frac{3}{π} \int_{0}^{π} B (ν_{2}) d ν_{2} + 2 D_{2}]}, Ω_{k} = \int_{0}^{D_{2}} ω_{2 k} (ν_{1}) d ν_{1} .

From the last formula it can be seen that if the different reinforced multilayered plates considered, simply supported on the inner contour, have the same for each layer integral reinforcement characteristics $Ω_{k}$ ( $k = 1, . . ., n$ ) with possibly different functions $ω_{2 k} (ν_{1})$ , they have the same limit loads $P_{0}$ (in case of identical other geometrical and physical parameters). For example, the limit load will be the same for the following piecewise-linear reinforcement densities $ω_{2 k} (ν_{1})$ with $Ω_{k} = C_{k} D_{2}$ ( $C_{k} = const for k th layer$ ):

ω_{2 k} (ν_{1}) = C_{k}, 0 \leq ν_{1} \leq D_{2},

ω_{2 k} (ν_{1}) = 2 C_{k} ν_{1} / D_{2}^{2}, 0 \leq ν_{1} \leq D_{2},

ω_{2 k} (ν_{1}) = 2 C_{k} (1 - ν_{1} / D_{2}) / D_{2}, 0 \leq ν_{1} \leq D_{2},

ω_{2 k} (ν_{1}) = C_{k} / 2, 0 \leq ν_{1} \leq D_{2} / 2 and ω_{2 k} (ν_{1}) = 3 C_{k} / 2, D_{2} / 2 \leq ν_{1} \leq D_{2},

ω_{2 k} (ν_{1}) = 3 C_{k} / 2, 0 \leq ν_{1} \leq D_{2} / 2 and ω_{2 k} (ν_{1}) = C_{k} / 2, D_{2} / 2 \leq ν_{1} \leq D_{2} .

In this example the inequality $0 \leq C_{k} \leq max (2 / 3; D_{2} / 2)$ must be true because of the inequality $0 \leq ω_{2 k} (ν_{1}) \leq 1$ .

4. The equations of motion and analysis of behavior of plate in scheme 2

In scheme 2 the plate deforms so that a non-stationary plastic hinged closed curve $l$ appears and moves translationally; as a result, the plate is divided into two ruled surfaces $Z_{1}$ and $Z_{2}$ (Figure 2). The region $Z_{1}$ rotates around the supported contour $L_{1}$ . The region $Z_{2}$ moves translationally and rotates around the curve $l$ . Due to the translational motion of the curve $l$ , all its points move at the same velocity, which is denoted by $V_{c} (t)$ . As in Nemirovsky and Romanova [23] and Romanova [28], from the continuity of velocities on curve $l$ we obtain that the normal to the line $l$ is normal to the contour $L_{1}$ and that the distance $D$ between $L_{1}$ and $l$ time-dependent, but does not depend on the coordinate $ν_{2}$ . The equation of the contour $l$ is

ν_{1} = D (t) (0 \leq ν_{2} < 2 π) .

Due to the continuity condition of the velocities on the curve $l$ , it follows that the region $Z_{1}$ rotates around the supported contour $L_{1}$ through the same angular velocity, which is denoted by $χ_{1} (t)$ . Let us assume that the region $Z_{2}$ rotates around the movable hinge curve $l$ through an angular velocity $χ_{2} (t)$ that also does not depend on the coordinate $ν_{2}$ . The discontinuity of the angular velocity on curve $l$ is equal to $| χ_{1} - χ_{2} |$ .

In scheme 2, deflection velocities of the plate (see Figures 2, 4) are

\begin{matrix} (ν_{1}, ν_{2}) \in Z_{1} : V (ν_{1}, ν_{2}, t) = χ_{1} (t) ν_{1} \\ (ν_{1}, ν_{2}) \in Z_{2} : V (ν_{1}, ν_{2}, t) = V_{c} (t) + χ_{2} (t) (ν_{1} - D) \end{matrix}

(19)

where the areas $Z_{1}$ and $Z_{2}$ are defined as

Z_{1} : 0 \leq ν_{1} \leq D, 0 \leq ν_{2} < 2 π; Z_{2} : D \leq ν_{1} \leq D_{2} (ν_{2}), 0 \leq ν_{2} < 2 π .

The principal curvatures of the surface of deflections velocities in the area $Z_{1}$ determined by the expression (9), but in the area $Z_{2}$ , they are

κ_{1} = 0, κ_{2} = - χ_{2} (t) / [(R ν_{2}) + ν_{1}] .

The equations of motion of the plate for deformation scheme 2 will be deduced from (5). Expressions (6) take the form

K = \frac{d χ_{1}}{dt} χ_{1}^{*} \int \int_{Z_{1}} ρ ν_{1}^{2} ds + \int \int_{Z_{2}} [\frac{d χ_{2}}{dt} (ν_{1} - D) + \frac{d V_{c}}{dt}] [χ_{2}^{*} (ν_{1} - D) + V_{c}^{*}] ρ ds

(20)

\begin{matrix} = χ_{1}^{*} \frac{d χ_{1}}{dt} \int \int_{Z_{1}} ρ ν_{1}^{2} ds + χ_{2}^{*} [\frac{d χ_{2}}{dt} \int \int_{Z_{2}} (ν_{1} - D)^{2} ρ ds + \frac{d V_{c}}{dt} \int \int_{Z_{2}} (ν_{1} - D) ρ ds] \\ + V_{c}^{*} [\frac{d χ_{2}}{dt} \int \int_{Z_{2}} (ν_{1} - D) ρ ds + \frac{d V_{c}}{dt} \int \int_{Z_{2}} ρ d s], \\ A = \int \int_{Z_{1}} [P (t) - β V] V^{*} ds + \int \int_{Z_{2}} [P (t) - β V] V^{*} ds = χ_{1}^{*} [P (t) \int \int_{Z_{1}} ν_{1} d s - β χ_{1} \int \int_{Z_{1}} ν_{1}^{2} ds] \end{matrix}

(21)

\begin{matrix} + χ_{2}^{*} {P (t) \int \int_{Z_{2}} (ν_{1} - D) ds - β [V_{c} \int \int_{Z_{2}} (ν_{1} - D) ds + χ_{2} \int \int_{Z_{2}} (ν_{1} - D)^{2} ds]} \\ + V_{c}^{*} {P (t) \int \int_{Z_{2}} d s - β [V_{c} \int \int_{Z_{2}} d s + χ_{2} \int \int_{Z_{2}} (ν_{1} - D) ds]} . \end{matrix}

The power of internal forces $N$ (7) is equal to sum:

N = \sum_{i = 1}^{4} N_{i}

where $N_{1}$ , $N_{2}$ , $N_{3}$ and $N_{4}$ are the powers of internal forces on the lines $L_{1}$ and $l$ and in the areas $Z_{1}$ and $Z_{2}$ , respectively. With the account (2), the powers are

N_{1} = (1 - η) M_{01} (0) \int_{L_{1}} {[θ^{*}]}_{L_{1}} d L_{1} = 2 χ_{1}^{*} (1 - η) M_{01} (0) \int_{0}^{π} B d ν_{2},

N_{2} = M_{01} (D) \int_{l} {[θ^{*}]}_{l} dl = 2 | χ_{1}^{*} - χ_{2}^{*} | M_{01} (D) \int_{0}^{π} B (1 + \frac{D}{R}) d ν_{2}

= 2 | χ_{1}^{*} - χ_{2}^{*} | M_{01} (D) [\int_{0}^{π} B (ν_{2}) d ν_{2} + π D],

N_{3} = \int \int_{Z_{1}} M_{02} (ν_{1}) | κ_{2}^{*} | ds = 2 χ_{1}^{*} \int_{0}^{π} \int_{0}^{D} \frac{B}{R} M_{02} (ν_{1}) d ν_{1} d ν_{2},

N_{4} = \int \int_{Z_{2}} M_{02} (ν_{1}) | κ_{2}^{*} | ds = 2 χ_{2}^{*} \int_{0}^{π} \int_{D}^{D_{2} (ν_{2})} \frac{B (ν_{2})}{R (ν_{2})} M_{02} (ν_{1}) d ν_{1} d ν_{2} .

The full power $N$ of internal forces of the plate in scheme 2 is

N = χ_{1}^{*} m_{1} (D) + χ_{2}^{*} m_{2} (D),

(22)

m_{1} (D) = 2 {(1 - η) M_{01} (0) \int_{0}^{π} B (ν_{2}) d ν_{2} \pm M_{01} (D) [\int_{0}^{π} B (ν_{2}) d ν_{2} + π D]

+ \int_{0}^{π} \int_{0}^{D} \frac{B (ν_{2})}{R (ν_{2})} M_{02} (ν_{1}) d ν_{1} d ν_{2}},

m_{2} (D) = 2 {\int_{0}^{π} \int_{D}^{D_{2} (ν_{2})} \frac{B (ν_{2})}{R (ν_{2})} M_{02} (ν_{1}) d ν_{1} d ν_{2} - [\pm M_{01} (D) (\int_{0}^{π} Bd ν_{2} + π D)]}

where the upper sign corresponds to $χ_{1}^{*} > χ_{2}^{*}$ (scheme 2a), and the lower sign to $χ_{1}^{*} < χ_{2}^{*}$ (scheme 2b) (see Figure 4). If $χ_{1}^{*} = χ_{2}^{*}$ scheme 2 is modified into scheme 1.

Substituting (20)–(22) into (6), considering that the functions $χ_{1}^{*}$ , $χ_{2}^{*}$ and $V_{c}^{*}$ are independent, we obtain the equations of motion of the plate

(d χ_{1} / dt) Σ_{11} (D) + χ_{1} β Σ_{12} (D) = P (t) Σ_{13} (D) - m_{1} (D),

(23)

(d χ_{2} / dt) Σ_{21} (D) + χ_{2} β Σ_{22} (D) = [P (t) - V_{c} β] Σ_{23} (D) - m_{2} (D) Σ_{24} (D),

(24)

(d V_{c} / dt) Σ_{21} (D) + χ_{2} β Σ_{32} (D) = [P (t) - V_{c} β] Σ_{33} (D) + m_{2} (D) Σ_{34} (D)

(25)

where

\begin{matrix} Σ_{11} (D) = \int \int_{Z_{1}} ρ (ν_{1}) ν_{1}^{2} ds, Σ_{12} (D) = \int \int_{Z_{1}} ν_{1}^{2} ds, Σ_{13} (D) = \int \int_{Z_{1}} ν_{1} d s, \\ Σ_{21} (D) = \int \int_{Z_{2}} ρ (ν_{1}) ds \int \int_{Z_{2}} ρ (ν_{1}) ν_{1}^{2} ds - [\int \int_{Z_{2}} ρ (ν_{1}) ν_{1} {ds]}^{2}, \\ Σ_{22} (D) = \int \int_{Z_{2}} {(ν_{1} - D)}^{2} ds \int \int_{Z_{2}} ρ (ν_{1}) ds - \int \int_{Z_{2}} (ν_{1} - D) ds \int \int_{Z_{2}} ρ (ν_{1}) (ν_{1} - D) ds, \\ Σ_{23} (D) = \int \int_{Z_{2}} ν_{1} ds \int \int_{Z_{2}} ρ (ν_{1}) ds - \int \int_{Z_{2}} ds \int \int_{Z_{2}} ρ (ν_{1}) ν_{1} ds, Σ_{24} (D) = \int \int_{Z_{2}} ρ (ν_{1}) ds, \\ Σ_{32} (D) = \int \int_{Z_{2}} ρ (ν_{1}) {(ν_{1} - D)}^{2} ds \int \int_{Z_{2}} (ν_{1} - D) ds - \int \int_{Z_{2}} {(ν_{1} - D)}^{2} ds \int \int_{Z_{2}} ρ (ν_{1}) (ν_{1} - D) ds, \\ Σ_{33} (D) = \int \int_{Z_{2}} ds \int \int_{Z_{2}} ρ (ν_{1}) {(ν_{1} - D)}^{2} ds - \int \int_{Z_{2}} (ν_{1} - D) ds \int \int_{Z_{2}} ρ (ν_{1}) (ν_{1} - D) ds, \\ Σ_{34} (D) = \int \int_{Z_{2}} ρ (ν_{1}) (ν_{1} - D) ds . \end{matrix}

The continuity condition for the velocities on the boundary of the areas $Z_{1}$ and $Z_{2}$ yields

χ_{1} D = V_{c} .

(26)

The system of equations (23)–(26) describes the behavior of the plate according to scheme 2. The initial conditions have the form (14) and

χ_{2} (0) = α_{2} (0) = V_{c} (0) = u_{c} (0) = 0, D_{0} = D (0)

(27)

where $u_{c} (t)$ is deflection on $l$ . The initial value $D_{0}$ is determined depending on $P_{max}$ , as will be shown further.

To further analyze the system (23)–(26) it is necessary to evaluate the sign of the expression $Σ_{21} (D)$ . From the Cauchy–Bunyakovsky–Schwarz inequality for a real Hilbert space of functions $f$ and $g$ , defined on $Z_{2}$ and integrable together with their squares and scalar product $\int \int_{Z_{2}} fgds$ , it follows that if the domain $Z_{2}$ is not degenerate, therefore

(\int \int_{Z_{2}} fg ds)^{2} \leq (\int \int_{Z_{2}} f^{2} ds) \int \int_{Z_{2}} g^{2} ds .

If $f = ν_{1} \sqrt{ρ} > 0$ and $g = \sqrt{ρ} > 0$ , then $Σ_{21} (D) > 0$ .

The condition $χ_{1} = χ_{2}$ yields the minimum value $P_{1}$ of $P_{max}$ at which the plate is deformed according to scheme 2. The value of the function $D$ corresponding to the load $P_{1}$ is denoted by $D_{1}$ . If $t \geq 0$ and $χ_{1} = χ_{2}$ , then (14) and (27) yield

(P Σ_{13} - m_{1} - β χ_{1} Σ_{12}) Σ_{21} = [P Σ_{23} - m_{2} Σ_{24} - β (χ_{1} Σ_{22} + V_{c} Σ_{23})] Σ_{11} .

Taking into account the initial conditions (14) and (27), the above equality takes the form

P_{max} = \frac{m_{1} (D_{0}) Σ_{21} (D_{0}) - m_{2} (D_{0}) Σ_{11} (D_{0}) Σ_{24} (D_{0})}{Σ_{13} (D_{0}) Σ_{21} (D_{0}) - Σ_{11} (D_{0}) Σ_{23} (D_{0})} .

(28)

Differentiating (26) with respect to time and eliminating the functions $d χ_{1} / dt$ and $d V_{c} / dt$ from obtained expression with the help (23) and (25), we get

- χ_{1} (t) \frac{dD (t)}{dt} = D \frac{P Σ_{13} - m_{1} - χ_{1} β Σ_{12}}{Σ_{11}} - \frac{P Σ_{33} + m_{2} Σ_{34} - β (χ_{2} Σ_{32} + V_{c} Σ_{33})}{Σ_{21}} .

(29)

With (14) and (27), equation (29) yields

P_{max} = \frac{m_{1} (D_{0}) D_{0} Σ_{21} (D_{0}) + m_{2} (D_{0}) Σ_{11} (D_{0}) Σ_{34} (D_{0})}{D_{0} Σ_{13} (D_{0}) Σ_{21} (D_{0}) - Σ_{11} (D_{0}) Σ_{33} (D_{0})} .

(30)

Setting $P_{max} = P_{1}$ and $D = D_{0} = D_{1}$ in (28) and (30), we obtain

P_{1} = \frac{m_{1} (D_{1}) D_{1} Σ_{21} (D_{1}) + m_{2} (D_{1}) Σ_{11} (D_{1}) Σ_{34} (D_{1})}{D_{1} Σ_{13} (D_{1}) Σ_{21} (D_{1}) - Σ_{11} (D_{1}) Σ_{33} (D_{1})}

(31)

where $D_{1}$ satisfies the equation

\frac{m_{1} (D_{1}) Σ_{21} (D_{1}) - m_{2} (D_{1}) Σ_{11} (D_{1}) Σ_{24} (D_{1})}{Σ_{13} (D_{1}) Σ_{21} (D_{1}) - Σ_{11} (D_{1}) Σ_{23} (D_{1})}

(32)

= \frac{m_{1} (D_{1}) D_{1} Σ_{21} (D_{1}) + m_{2} (D_{1}) Σ_{11} (D_{1}) Σ_{34} (D_{1})}{D_{1} Σ_{13} (D_{1}) Σ_{21} (D_{1}) - Σ_{11} (D_{1}) Σ_{33} (D_{1})} .

Let us analyze the dynamic behavior of the plate under “high” loads: $P_{max} > P_{1}$ . The movement will start according to scheme 2 with a line $l$ , the initial location of which is determined by the value $D_{0}$ from equation (30). Whether the plate follows scheme 2a or 2b can be ascertained by using the minimum load $P_{1}$ from equation (31) with the upper and lower signs in expression (22) for $m_{1}$ and $m_{2}$ . If the reinforcement is such that the function $ρ (ν_{1})$ is constant or varies only near the contour $L_{1}$ and is constant in the other part of plate, then $ρ (ν_{1})$ is constant in the area $Z_{2}$ , hence, $Σ_{23} = 0$ and $Σ_{22} = Σ_{21} > 0$ . From (24) it follows that since $Σ_{24} > 0$ and $d χ_{2} (0) / dt$ cannot be negative, then in the expression for $m_{2}$ the upper sign should be selected, therefore, the plate deforms according to scheme 2a. In addition, the coefficient $β$ is not included in the equation (31); therefore, the viscous resistance of the base does not affect the choice of the variant of scheme 2.

The motion of the plate during the first phase ( $0 \leq t \leq t_{1}$ ) is described by equations (23)–(26) with the initial conditions (14) and (27). The motion of the hinge $l$ is described by equation (29).

For both schemes 2a and 2b, either there is an instant $t_{1}$ at which $χ_{1} (t_{1}) = χ_{2} (t_{1})$ and the motion follows pattern 1 during the second phase, or the motion stops at an instant $t_{f}$ , determined from equations (24)–(26) under condition $χ_{1} (t_{f}) = χ_{2} (t_{f}) = 0$ (similarly as the obtaining of the equation (30))

\begin{matrix} P (t_{f}) {D (t_{f}) Σ_{13} (D (t_{f})) Σ_{21} (D (t_{f})) - Σ_{33} (D (t_{f})) Σ_{11} (D (t_{f}))} \\ = m_{1} (D (t_{f})) D (t_{f}) Σ_{21} (D (t_{f})) + m_{2} (D (t_{f})) Σ_{11} (D (t_{f})) Σ_{34} (D (t_{f})) . \end{matrix}

In the first phase, the change of the variants of scheme 2 cannot occur, because it is only possible under condition $χ_{1} = χ_{2}$ , but after this equality the plate will continue to move according to scheme 1 (due to the fact that the load does not increase).

If the plate moves in the second phase ( $t_{1} < t \leq t_{f}$ ), its deformation is described by equation (16) with the initial conditions $χ_{1} (t_{1})$ and $α_{1} (t_{1})$ defined at the end of the first phase. The plate stops at $t_{f}$ defined by (18). The deflections of plate are calculated from equations (19) and (8) for both phases.

5. Numerical examples

5.1. Curvilinear plate with an elliptic hole

To demonstrate the capabilities of the proposed technique, let us consider a curvilinear plate with an elliptic supported hole (Figure 5), as an example. The inner contour $L_{1}$ of this plate is described by the parametric equation

x = R_{1} \cos φ, y = γ_{1} R_{1} \sin φ (R_{1} > 0, γ_{1} \leq 1, 0 \leq φ < 2 π) .

Relations (1) become

x = R_{1} [1 + v_{1} γ_{1} / B (v_{2})] \cos v_{2}, y = R_{1} [γ_{1} + v_{1} / B (v_{2})] \sin v_{2} .

In this case

R (ν_{2}) = B^{3} (ν_{2}) / (γ_{1} R_{1}^{2}), B (v_{2}) = R_{1} \sqrt{\sin^{2} v_{2} + γ_{1}^{2} \cos^{2} v_{2}} .

Let the contour $L_{2}$ is described by the following equation in coordinates $(v_{1}, v_{2})$

D_{2} (v_{2}) = [c_{1} + c_{2} \sin (4 v_{2} - π / 2)] R_{1} (c_{1} > 0, c_{2} \geq 0, 0 \leq v_{2} < 2 π) .

Let us consider a four-layered plate (Figure 6) with the following parameters:

γ_{1} = 0.6, c_{2} = 0.1, n = 2, h_{1} = 0, 85 H, h_{2} = H, ρ_{0 k} = ρ_{0}, ρ_{jk} / ρ_{0} = 3,

σ_{jk} = σ_{j}, σ_{0 k} = σ_{0} (j, k = 1, 2), σ_{1} = σ_{2} = 10 σ_{0} .

Such a relation between reinforcement parameters is possible, for example, if the binder material is aluminum and that of fibers is steel.

The condition of the same total amount of reinforcement is

h_{1} \int_{0}^{π} \int_{0}^{D_{2} (ν_{2})} [ω_{11} (ν_{1}) + ω_{21} (ν_{1})] B (ν_{2}) [1 + ν_{1} / R (ν_{2})] d ν_{1} d ν_{2}

(33)

+ (h_{2} - h_{1}) \int_{0}^{π} \int_{0}^{D_{2} (ν_{2})} [ω_{12} (ν_{1}) + ω_{22} (ν_{1})] B (ν_{2}) [1 + ν_{1} / R (ν_{2})] d ν_{1} d ν_{2} = const .

Assuming that the functions $ω_{jk}$ do not depend on the coordinate $v_{1}$ , the condition (33) was taken as

[h_{1} (ω_{11} + ω_{21}) + (h_{2} - h_{1}) (ω_{12} + ω_{22})] / H = const = 0.12 .

(34)

Note that to fulfill the condition of constancy of the reinforcement density $ω_{1 k}$ in each layer the thickness of the reinforcing fibers must be variable.

In Figures 7 –10 the results of calculations at condition (34) are given, where curves 1–7 correspond to the following reinforcement densities:

\begin{matrix} (1) ω_{11} = 0.12 / 0.85 \approx 0.141, ω_{21} = ω_{12} = ω_{22} = 0, \\ (2) ω_{21} = 0.12 / 0.85, ω_{11} = ω_{12} = ω_{22} = 0, \\ (3) ω_{12} = 0.8, ω_{11} = ω_{21} = ω_{22} = 0, (4) ω_{22} = 0.8, ω_{11} = ω_{12} = ω_{21} = 0 \end{matrix}

(35)

(5) ω_{12} = ω_{22} = 0.4, ω_{11} = ω_{21} = 0, (6) ω_{11} = ω_{21} = ω_{12} = ω_{22} = 0.06,

(7) ω_{11} = ω_{21} = ω_{12} = ω_{22} = 0 .

The case 7 is the absence of reinforcement, which corresponds to homogeneous plate. In cases 5 and 6, for limiting bending moments (4) the equality $M_{01} = M_{02}$ is satisfied, that corresponds to quasi-isotropic reinforcement.

In Figures 7 and 8 for the simply supported inner contour and for the clamped one, respectively, the relations between the dimensionless limit load $p_{0}$ and the dimensionless size $c_{1}$ of external contour, calculated by (15), are illustrated (where $p_{i} = P_{i} R_{1}^{2} / (σ_{0} H^{2})$ , $i = 0; 1$ ). In Figure 7 lines 1, 3, and 7 are the same, because the reinforcement in the direction normal to the inner contour does not affect the limit loads of the simply supported plates. In Figure 8 the maximum limit load at $c_{1} < 0.8$ is in case 3 (reinforcement of the outer layers in the directions normal to the inner contour) and at $c_{1} > 0.8$ is in case 4 (reinforcement of the outer layers in the directions parallel to the inner contour). It is also obtained that the limit load decreases with increasing of the parameter $c_{2}$ .

The dimensionless final deflections $w = {uR}_{1}^{2} ρ_{0} / (σ_{0} H T^{2})$ in the section $y = 0$ of a considered clamped plate with $γ_{1} = 0.6$ , $c_{1} = 3.1$ are shown on Figures 9 and 10 at $\bar{β} = β T / ρ_{0} = 0.2$ and at $\bar{β} = 0.5$ , respectively. The plate is under the action of load with the rectangular form of pulse

P (t) = 10 σ_{0} H^{2} / R_{1}^{2} at 0 \leq t \leq T, P (t) = 0 at t > T .

(36)

The loads $p_{0}$ and $p_{1}$ , and the distances $d_{i} = D_{i} / R_{1}$ ( $i = 0, 1$ ) for the cases of reinforcement (35) are given in Table 1. For quasi-isotropic reinforcement (cases 5 and 6) and for homogeneous material (case 7) the distance $d_{1}$ is the same. The load (36) will be “high” in cases 1, 2, 4–7, but “moderate” in case 3. In Figure 9 lines 8 and 9 correspond to the deflections of non-reinforced plate (case 7) at $t = T$ and at $t = t_{1}$ when $χ_{1} (t_{1}) = χ_{2} (t_{1})$ .

Table 1.

Dimensionless loads p₀, p₁ and distances d₀, d₁ for cases of reinforcement (35).

Case	$P_{0}$	$P_{1}$	$d_{1}$	$d_{0}$
1	0.34	6.64	1.51	1.39
2	0.49	3.98	1.97	1.61
3	0.4	10.3	1.43	–
4	0.73	4.63	2.21	1.99
5	0.57	7.15	1.68	1.59
6	0.44	5.5	1.68	1.5
7	0.28	3.57	1.68	1.29

Numerically it is found that when increasing $P_{max}$ of the initial value $D_{0}$ decreases. From the data of Figures 9 and 10 it can be seen that at the increase of coefficient of viscous resistance of the basis, the final deflections of the plate decrease. It is also seen that a change in reinforcement densities considerably affects both the size and shape of the final deflections of plates. The governing system of equations was solved numerically by the Runge–Kutta method and at “high” loads the deformation occurred according to scheme 2a.

If $γ_{1} = 1$ , $c_{1} > 0$ , $c_{2} = 0$ the considered plate becomes an annular laminated plate reinforced with fibers in radial and circumferential directions, with the radius of the outer contour $R_{2} = γ R_{1}$ ( $γ = 1 + c_{1} > 1$ ) and the radial coordinate $r = ν_{1} + R_{1}$ . For this, in the case of a simply supported inner contour, the limit load calculated on the basis of an exact solution obtained at the rectangular plasticity criterion [35], is

P_{0} = \frac{6}{R_{1}^{3} {(γ - 1)}^{2} (1 + 2 γ)} \int_{R_{1}}^{γ R_{1}} M_{02} (r) dr .

(37)

When $η = 1$ , the limit load (15) coincides with the exact solution (37), where $M_{02} (r)$ is defined in (4) with account $r = ν_{1} + R_{1}$ .

The analysis of Figures 7 –10 shows that the quasi-isotropic reinforcement (cases 5 and 6) is not the best with respect to criterions of the minimum limit load and the maximum final deflection.

5.2. Circular plate with a supper-elliptic hole

As another example, let us calculate limit loads of a circular laminated plate with a hole $L_{1}$ in the form of a super-ellipse [36] (Figure 11), which is reinforced with fibers of constant cross-section. The contour $L_{1}$ is described by the equations (for eighth part of the plate):

x = R_{1} \cos^{1 / 3} φ, y = R_{1} \sin^{1 / 3} φ (R_{1} > 0, 0 \leq φ < π / 4) .

Relations (1) become

\begin{matrix} x = R_{1} \cos^{1 / 3} v_{2} [1 + v_{1} \frac{\cos^{4 / 3} v_{2}}{R_{1} U (v_{2})}], y = R_{1} \sin^{1 / 3} v_{2} [1 + v_{1} \frac{\sin^{4 / 3} v_{2}}{R_{1} U (v_{2})}], \\ U (v_{2}) = \sqrt{\sin^{10 / 3} v_{2} + \cos^{10 / 3} v_{2}} . \end{matrix}

In this example

B (v_{2}) = R_{1} U (v_{2}) / (3 \sin^{2 / 3} v_{2} \cos^{2 / 3} v_{2}), R (ν_{2}) = R_{1} U^{3} (v_{2}) / (5 \sin^{4 / 3} v_{2} \cos^{4 / 3} v_{2}) .

The contour $L_{2}$ is circle of radius $R_{2}$ ( $R_{2} > \sqrt{2} R_{1}$ ) and it is described by the following equation in coordinates $(v_{1}, v_{2})$ :

v_{1} = D_{2} (v_{2}) = \frac{- R_{1}}{U (v_{2})} + \sqrt{\frac{R_{1}^{2}}{U^{2} (v_{2})} + R_{2}^{2} - R_{1}^{2} (\sin^{2 / 3} v_{2} + \cos^{2 / 3} v_{2})} (0 < v_{2} < π / 4) .

The parameters $n$ , $h_{k}$ , $ρ_{0 k}$ , $ρ_{jk}$ , $σ_{jk}$ , $σ_{0 k}$ $(j, k = 1, 2)$ are the same as in the first example.

For fibers of constant cross-section, located in the direction of $v_{1} = const$ , we have

ω_{2 k} (v_{1}) = ω_{2 k} (0) = const .

(38)

For fibers located in the direction $v_{2} = const$ from the relation

ω_{1 k} (0) \int_{0}^{π / 4} B (v_{2}) d v_{2} = ω_{1 k} (v_{1}) \int_{0}^{π / 4} B (v_{2}) [1 + \frac{v_{1}}{R (v_{2})}] d v_{2}

and from formula (2) we obtain

ω_{1 k} (v_{1}) = \frac{ω_{1 k} (0)}{1 + v_{1} / Q}, Q = \frac{1}{π / 4} \int_{0}^{π / 4} B (v_{2}) d v_{2} = const .

(39)

The condition of the same total amount of reinforcement (33), considering relations (38) and (39), in calculations was taken in the form

{[h_{1} ω_{11} (0) + (h_{2} - h_{1}) ω_{12} (0)] s_{q} + h_{1} ω_{21} + (h_{2} - h_{1}) ω_{22}} / H = 0.1,

(40)

\begin{matrix} s_{q} = \int_{0}^{π / 4} \int_{0}^{D_{2} (v_{2})} \frac{B (1 + v_{1} / R)}{(1 + v_{1} / Q)} d v_{1} d v_{2} / \int_{0}^{π / 4} \int_{0}^{D_{2} (v_{2})} B (1 + v_{1} / R) d v_{1} d v_{2} \\ = 2 Q \int_{0}^{π / 4} \frac{B}{R} [(R - Q) \ln (1 + D_{2} / Q) + D_{2}] d v_{2} / \int_{0}^{π / 4} \frac{B}{R} D_{2} (2 R + D_{2}) d v_{2} . \end{matrix}

In Figures 12 (for simply supported inner contour) and 13 (for clamped one) the results of calculations of limit load $p_{0}$ (15) as function of on dimensionless size $a_{2} = R_{2} / R_{1}$ of external contour at condition (40) are given, where lines 1–8 correspond to the following reinforcement densities:

(1) ω_{21} = 0.1 / 0.85 \approx 0.118, ω_{11} (0) = ω_{12} (0) = ω_{22} = 0,

(2) ω_{22} = 0.1 / 0.15 \approx 0.667, ω_{11} (0) = ω_{12} (0) = ω_{21} = 0,

(3) ω_{11} (0) = 0.1 / (0.85 s_{q}), ω_{12} (0) = ω_{21} = ω_{22} = 0,

(4) ω_{12} (0) = 0.1 / (0.15 s_{q}), ω_{11} (0) = ω_{21} = ω_{22} = 0,

(5) ω_{21} = ω_{22} = 0.1, ω_{11} (0) = ω_{12} (0) = 0,

(6) ω_{11} (0) = ω_{12} (0) = 0.1 / s_{q}, ω_{21} = ω_{22} = 0,

(7) ω_{12} (0) = ω_{22} = 0.1 / [0.15 (1 + s_{q})], ω_{11} (0) = ω_{21} = 0,

(8) ω_{11} (0) = ω_{12} (0) = ω_{21} = ω_{22} = 0 (non - reinforced plate) .

In Figure 12 lines 3, 4, 6, and 8 are the same, because the reinforcement in the direction normal to the inner contour does not affect the limit loads of the simply supported plates. In Figure 13 lines 1 and 5 are close. In Figure 13 for case 4 the limit load is not calculated for all values $a_{2}$ because with the constant selected for the example in the condition (40) the density of reinforcement $ω_{12} (0)$ exceeds the value equal to one at $a_{2} \geq 2.36$ , but $ω_{12} (0) \leq 1$ within the meaning of the problem.

Figures 7, 8, 12, and 13 show that the reinforcement parameters significantly affect the limit load even at same total amount of reinforcement.

It is numerically obtained that in the considered examples $d_{0} \equiv 0$ and the plastic hinge $L_{d}$ coincide with the contour $L_{1}$ .

The calculation results showed that at the same total amount of reinforcement a laminated plate will be the strongest (from the criterions of maximum limit load and minimum final deflection) if fibers are arranged in the layers closer to its faces. The conclusion agrees with the idea embodied in sandwich panels [37, 38]. Also it indicates that the proposed model gives qualitatively correct results.

6. Conclusion

In this research on the basis of the model of a perfectly rigid-plastic material in a geometrically linear formulation a theoretical solution is developed for analysis of dynamic bending of hybrid composite, arbitrary curvilinear doubly connected thin plates. The plates are under the action of high-intensity short-term dynamic loads of explosive type, distributed uniformly on the surface. The structural model of a reinforced layer and plane stress state in fibers are taken into account. The plates are fully clamped or simply supported on the inner contour and free on the outer contour. The plates are on a viscous basis. They are laminated and fibrous, with layers arranged symmetrically with respect to the middle surface. The fibers are made of various materials and are located parallel or normally to the inner contour of the plate. Depending on the loading amplitude, two deformation schemes of the plates are possible. On the basis of a principle of virtual power, in combination with the d’Alembert principle, the equations of dynamic deformation are obtained for each of the patterns, and conditions of their realization are analyzed. To simplify the calculation of double integrals over curvilinear areas, an orthogonal curvilinear system of coordinates connected with the inner contour of plates is introduced. A simple analytical formula for estimating the limit loads of plates is found. Numerical examples are given for a reinforced laminated curvilinear plate with an elliptical hole and for a circular plate with a hole in the form of a super-ellipse. Cases of constant reinforcement densities and reinforcement of fibers of constant cross-section, when the reinforcement densities are variable, are considered. It is shown that a change in reinforcement parameters considerably affects both the load-carrying ability and the final deflections of plates. It is found that, at the same total amount of reinforcement, a laminated plate will be the strongest from the criterions of maximum limit load and minimum final deflection if the fibers are arranged in the layers close to its faces. It is obtained that quasi-isotropic reinforcement is not the best one.

The approximate solutions suggested can be useful in modeling protective reinforced planar metal-composite laminated doubly connected elements of structures, and in estimating their damage at the actions of explosive loads as well as in the creation of such elements by the methods of pulse stamping. In addition, this research may serve as a benchmark for future assessments in these fields of engineering.

Footnotes

Funding

The author disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research was carried out within the framework of the Program of Fundamental Scientific Research of the state academies of sciences in 2017–2020 (project No. 23.4.1 – Mechanics of deformation and destruction of materials, media at mechanical loads, under the influence of physical fields and chemically active media).

ORCID iD

Tatiana Pavlovna Romanova

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