A Spring-Layer Model for a Bi-Layered Plate-Strip with Initial Stress Through Imperfect Contact Interface

In this paper, we present our report on the forced vibration of a bilayered plate-strip with initial stress resting on a rigid foundation induced by a time-harmonic force. The investigation is carried out according to the piecewise homogeneous body model with utilizing the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB). The materials of the body are chosen to be linearly elastic, homogeneous, and isotropic. The interface between the layers is assumed to be imperfect, and is simulated by the springlayer model. A similar degree of imperfection on the interface is realized in the normal and tangential directions. The mathematical model for the problem under consideration is designed, and the system of the equations of motion is approximately solved by employing the finite element method (FEM). The numerical results explaining the influence of the parameter that characterizes the degree of corresponding imperfectness on the dynamic response of the plate-strip are presented. In particular, we demonstrate that the distributions of the normal stress become flat, as the normal-spring parameter increases.


Introduction
Multilayered structures have received substantial attention because they are regularly encountered on a daily basis. The mechanical properties of the interface adhesive bonds play a key role in investigating the dynamic response of the body under consideration. Therefore, the imperfect contact conditions, which cause discontinuity in the displacement components, are of increasing importance. For instance, with regard to the assumption that stresses are continuous, but displacements are discontinuous across the interface, the widely used spring-type imperfectness of the interface is modeled, and it is observed that the jumps in displacement components are linearly proportional to the respective interface stress components. Different imperfect contact conditions are also encountered in the designation and production of the materials.
Many known factors determine the dynamical behavior of multilayered systems, including its geometry, the initial static stress, and the frequency of the dynamic force exerted. The influence of the initial stresses cannot be investigated with regard to classical linear theory of elasticity since it is nonlinear. According to the preceding mechanical consideration, when the amplitudes of the deformations by a dynamic force of a system with initial deformation are gradually smaller than those of the static initial force, appropriate investigations can be conducted within the scope of the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB). This theory has been developed within the scope of elastodynamics, upon which more detailed information is available in the monographs by Guz [8,9] and Akbarov [5].
Due to technological demands and the need for economic use of materials, some interesting phenomena have been observed within the scope of TLTEWISB and its other version. Kepceler [11] investigated the torsional wave propagation in the bi-material compounded cylinder with an imperfect interface in the absence of initial stresses. Hu et al. [13] considered the influence of identically applied initial pressures on the radial surfaces of a hollow cylinder composed of materials with a first power hypo-elastic constitutive model. Akbarov and Negin [4] studied the generalized Rayleigh wave dispersion in a system consisting of two axially pre-stressed covering layers and half-spaces with an imperfect interface. Kurt et al. [12] examined the extensional and flexural Lamb waves in the sandwiched plate consisting of a metal elastic middle-core layer imperfectly bonded to piezoelectric face layers. By analysis of this research, it is clear that the problems under consideration have complex structures. The forced vibration of a pre-stressed plate-strip with finite length resting on a rigid foundation under a timeharmonic force has been comprehensively studied employing the finite element method (FEM) by Akbarov et al. [2] (for plate-strip subjected to a perpendicular force) and [1] (for bi-layered plate-strip subjected to a perpendicular force), Eröz [7] for plate-strip subjected to an inclined force), Akbarov et al. [3] (for bi-layered plate-strip with shear-spring interface), and Daşdemir and Eröz [7] (for bi-layered plate-strip subjected to an inclined force).
The forced vibration of a pre-stressed bi-layered plate-strip subject to a time-harmonic external force resting on a rigid foundation, where shear-and normal-spring types of imperfect contact at the interface exist, has yet to be analyzed. The mathematical modeling used to present fundamental insights for characterizing the influence of the imperfection of the contact conditions between the layers is missing. To address this issue, mathematical modeling is designed within the scope of the piecewise homogeneous body model with the utilization of TLTEWISB, to investigate the imperfect contact conditions and the frequency response of the plate-strip with finite length. The numerical calculations are done by using the FEM. The problems studied by Akbarov et al. [1,2,3,6,7] are special cases of the current problem. Therefore, the findings presented in this paper can also be regarded as a development of the aforementioned papers, because imperfect contact conditions between the layers of the plate-strip for both the shear-and normal-spring types are satisfied.

Problem Statement
The problem is investigated in the plane-strain state within the scope of TLTEWISB by utilizing the piecewise homogeneous body model. The mathematical modeling is constructed according to the following theory.
Consider a bi-layered plate-strip with length 2a and thickness h ( ) 1 2 h h = + , where 1 h and 2 h denote the thickness of the upper and lower layers, respectively. We assume that the materials of the layers are chosen to be linear, elastic, homogeneous, and isotropic. As shown in Fig. 1, the plate-strip is resting on a rigid foundation and is subjected to the action of a time-harmonic force. The Cartesian coordinates given by i x are associated with the initial state and coincide with the Lagrange coordinates i x′ in the natural state. The length of the body along the direction of the 3 Ox axis is infinite. Since the force being applied to the body under consideration extends to infinity along this axis, the plane-stress-deformation state appears on the 1 2 Ox x plane. To represent the quantities related to the upper and lower layers, the upper indices "(1)" and "(2)" are respectively used, and the values for the initial state are denoted by the additional superscript 0. According to Fig. 1, the plate-strip lies in the domain 1 2 Each layer of plate-strip interacts with one another and with the rigid foundation after each of them is exposed by a homogeneous normal tension or compression force separately (only along the direction of the 1 Ox axis). Therefore, the uniaxial homogenous initial stress state occurs in the plate-strip. This initial stress is represented according to the linear theory of the elasticity, and written as where ( ) m q is a constant for each layer.
Guz [1,2] and Akbarov [5] express the general forms of the governing field equations under consideration as follows: where ; ; ; is given, where ( ) δ ⋅ is a Dirac delta function.
Because of the above initial stress that is applied to the edge of the plate-strip, the following boundary condition is satisfied.
The complete interaction at the interface plane between the considered body and rigid foundation are given by ( ) In addition to the preceding assumptions, it is believed that an imperfect contact interaction exists between the elastic layers. For the analysis presented herein, the imperfectness is described with reference to a spring-layer model. This modeling is based on the assumption that stresses are continuous but the displacements are discontinuous across the interface. This means that there are jumps in the displacement components, namely these jumps are proportional to the tension components of the corresponding interface in terms of the spring-type interface parameters. Hence, it is believed that the displacement u + and force f + at one surface of an interface are proportional to the displacement u − and force f − at the other surface of the same interface, respectively. This approach can be written as where A , B , F , and G are square matrices, and [ ] . indicates the jump in the corresponding quantities at the respective interface. In this paper, we consider the following case. Depending on the selection of the matrices A , B , F , and G different incomplete contact situations can occur. Neglecting the matrix G and assuming that the matrices A and B are zero, we obtain the case investigated by Jones and Whittier [10], in matrix form: where F is a constant diagonal matrix whose entries are identified in terms of the thickness and elastic constants of the investigated body. As a result, the imperfect contact conditions can be given mathematically as where 1 F and 2 F are the shear-and normal-spring imperfect parameters, respectively and they are dimensionless varying over the ranges of The spring imperfectness creates three limiting cases. When 1 , the displacements are continuous at the interface. Therefore, the spring imperfectness approaches a welded interface. If 1 F → ∞ and 2 F → ∞ , an unrestricted interaction occurs at the interface without mechanical contact. In the case wherein 1 F → ∞ and 2 0 F = , the displacements This completes the presentation of the governing field equations and the corresponding boundarycontact conditions for the plate-strip shown in Fig. 1.

Solution Procedure
Since the problem under consideration has a complex structure and boundary-contact conditions, an analytical solution of the problem cannot be achieved. Therefore, the solution to this problem is approximated by using the FEM. Recall that the external force applied to the plate-strip is assumed to be time-harmonic with frequency ω as ( ) where the superimposed bar represents the amplitude of the corresponding quantity. The dimensionless coordinate system is also introduced as To obtain FEM modeling of our last problem, the functional is proposed, where To carry out the FEM modeling of the problem under consideration, the virtual work principle and the standard Rayleigh-Ritz method are adopted, as stated by Zienkiewicz and Taylor [15]. With this approach, the domain B is divided into a finite element of sub-domains whose structures are nine-node smooth rectangular elements. The number of these finite elements is determined by the desired numerical convergence requirement. After certain mathematical adjustments, a system of algebraic equations is obtained. In Eq. (16), K is the stiffness matrix, M is the mass matrix, u  is the unknown nodal displacement vector, and F is the force vector. The explicit forms of the matrices and vectors are not provided herein; the explicit forms are derived from Eq. (13) using the proposed procedure. Therefore, the FEM modeling of the problem under consideration is completed.

Results and Discussions
The key objective of this study is to investigate the influence of the spring-type imperfect parameters (in both the normal and tangential directions) on the dynamic behavior of the plate-strip under consideration. Before giving the analysis of the numerical results, some explanations are required. The following notations are introduced:  ν ν = = and F 0 = is examined. Uflyand [14] considered the case for the plate with infinite length, and the problem was resolved. It can be demonstrated that the geometry of the plate-strip under consideration begins to resemble the one investigated by Uflyand as / 2 0 h a → . In this situation, the numerical results given by the current FEM algorithm must converge to the corresponding ones from the one, which is the asterisked graph in Fig. 2, given by Uflyand [14]. This speculation is proven by the graphs in Fig. 2, and hence the validity and trustworthiness of the algorithm and programs.  These discussions are made without reference to the initial stress in each layer. Therefore, in Fig. 6, the influence of the initial stress parameter η on the dependence between 22 and Ω is analyzed. Correspondingly, numerical results are given for various values of the imperfect parameter F . A comparison of the graphs in Fig. 6 reveals that the initial tension of the layers causes a decrease; however, the initial compression causes an increase in the absolute values of the normal stress 22 The influence of the initial stresses on the frequency response of the stress 22 in the case wherein the spring-type imperfection exists between the layers is more considerable compared with when the layers are in complete contact. The imperfect parameter F changes the character of the influence of the initial stress of the layers on the frequency response of the plate-strip. It can be demonstrated that an increase in the values of initial tension parameter leads to the decrease in its resonance by contrast with the influence of the compression parameter. In addition, the number of the parametric resonance of 22 arisen in certain values of the initial stress parameter η decreases as the imperfect parameter F increases.

Conclusions
In this study, the forced vibration induced by a time-harmonic external force of a pre-stressed platestrip with two imperfectly bonded elastic layers, which rest on a rigid foundation, has been investigated. This has been done based on the fundamental principles of the TLTEWISB in the case where both shearand normal-spring-type imperfections exist between the layers. The mathematical problem is created for consideration and numerically solved using FEM. Numerical results have demonstrated the influence of the change in imperfectness on the frequency response of the normal stress, acting on the interface planes between the plate-strip and rigid foundation. Numerical investigations have shown the following: • the imperfect interaction between the layers affects the character of the influence of the initial stresses on the frequency response of plate-strip; • and the numbers of the local maxima and minima of the stress 22 0 / h p σ versus the dimensionless frequency Ω decrease with increasing the ratio / 2 h a. The numerical results listed above have been presented under two different cases (for example a pair of Al + St), but note that they also have a general validity in a qualitative sense.