DYNAMICS OF MULTILAYERED ORTHOTROPIC VISCOELASTIC PLATES OF MAXWELL SOLIDS

This paper is concerned with a simplified dynamical analysis of orthotroplc viscoelastic plates that are made up of an arbitrary number of layers each of which is a Maxwell type solid. This study includes the case where some or all the layers are themselves constituted by thinly laminated materials with couple stresses. The recurrence equations for the shear stresses are obtained for an arbitrary number of layers and then applied to plates with two or three layers. The viscoelastic damping effect is determined by the process of llnearlzatlon and then illustrated by a plate composed of one, two or three layers. It is found that the damping increases with anlsotropy and wave number. These results are shown by graphical representations.

1980 AMS SUBJECT CLASSIFICATION CODE: 73F I. INTRODUCTION.Based upon the general theory of ultilayered continua in finite anisotroplc elasticity due to Blot [I-4], Pal Roy [5,6] has studied the problems of elastic wave propagation in a thinly layered laminated medium with stress couples under initial stresses.
In these studies the composite structures are assumed to be made up of multilayered elastic materials and hence some modifications are required for applications to real solids.It is almost impossible to study statics or dynamics of real solids without suitable approximations and/or assumptions.However, the Maxwell solid is generally believed to be one of the best practical examples of real solids.
It is assumed that the Maxwell solid consists of a series of spring with a viscous element known as 'dash pot' (see Biswas [7]), and the main factor which makes this model different from the elastic one is the relaxation time.
From a practical point of view, the study of statics or dynamics of Maxwell solids seems to be important in its own merit and has physical applications.
In this paper, we study a simplified dynamical analysis of orthotroplc viscoelastic plates that are made up of an arbitrary ntnber of layers each of which is a Maxwell type solid.This problem Includes the case where some or all the layers are themselves made of thlnly laminated materials with couple stresses.
The recurrence equations for the shear stresses are derived for an arbitrary number of layers and then applied to plates with two or three layers.
The effect of the viscoelastic d. amplng is examined with examples.The damping is found to increase with anlsotropy and wave number.Results are shown by graphical representations.

BASIC EQUATIONS AND ASSUMPTIONS
We consider a plane strain deformation of a plate of Maxwell solids of thickness h.The x-axls is chosen midway between the boundarles of the two faces and the y-axls is normal to them.
The stress-straln relations for the Maxwell solid with the effects of relaxation time, x are given by [7]: We assume that the dlsplacment fields are harmonic function of time t and slnusoldally distributed along the x-dlrectlon so that u U(y) sin mx e (2.7) where is the frequency, V is assumed to be a constant and equal to its average value across the thickness h.
The solution for the shearing stress A(y) is determined from equation (2.12) with the boundary conditions and A 2 for A(y) at the top, y h/2 and the bottom, y -h/2 of he plate.This solution still contains the unknown constant deflection V which is determined by integrating the equilibrium equation (2.6)  represents the total normal load applied to the same unit area.When this normal load is known, the deflection V is obtained from (2.14), and hence te shear stress distribution A(y).Thus is known from (2.9).Once we know o the value of U(y) is determined by combining (2.5)The above results can easily be extended to a multllayered plate constituted by a superposition of thin adherent homogeneous layers each of which is a Maxwell type solid.
Within each layer N is supposed to be a constant.
Considering first a single layer of thickness h with A and A 2 as the shear stresses at the top and bottom of the layer, it turns out from differential equation (2.12) that where A C cosh Bray + C 2 sinh Bray-mV The displacement amplitude U is given by equation (2.17).Its values U and U 2 at the top and bottom of the layer are found to be UI __I (Ala + A2b) + 2Vc We consider now a plate composed of n orthotropic homogeneous layers.The Jth layer (J 1,2...n) of thickness h is characterized by coefficients j, Tj and a mass density p j with the corresponding parameters aj bj cj M Nj.The displacement amplitude and shear stress at the top and bottom of the Jth layers are respectively (Uj, Aj) and (Uj+I, Aj+l).Now, at the interface between the layers J and (J+l) the displacement amplitude Uj+ as it is found for the lower face of the Jth layer and upper face of the (J+l)th layer, must be the same.We obtain from equations (3.Thus when the values of the shear stresses A and An+ are given at the outer boundaries, the recurrence equations (3.9) will lead to the evaulatulon of the (n-l) shear stresses A(j--l,3...n) at the interfaces.
The normal load Sj acting on the Jth layer is obtained from equations (2.14) and (3.1) and has the form c.
m2 V hj Sj -Mjcj(Aj + Aj+ I) + m2V hjMj(l -y.)-0j (3.11) The total load S applied to the multi-layered plate is obtained by summing the loads S. applied to each layer.We obtain n 2 S > .' S -ZjM c (Aj + + m2V m-m t j--I J The equation (3.12) will provide the only unknown V of the system.When this unknown V is known, the shear stresses Aj at the interfaces are also completely kn own. 4. VISCOELASTIC MATERIALS.
.3) that for Maxwell type solid there is only one coefficient instead of two coefficients for the elastic materials.
Viscoelastic properties of the layers may be taken into account by substituting the following general form of operator for the coefficient M (see Blot [8]): f IM(r) dr + M" + M'p (4.4) 0 p+r d where p--.For harmonic oscillations p i and the operator also becomes a complex quantity. 5. THINLY LAYERED LAMINATED MATERIALS.
We now consider a multi-layered plate in which the layers are themselves composed of thinly laminated materials of Maxwell solids.For a laminated medium composed of a repeated sequence of n thin layers each of which occupies a fraction j(J--l,2...n) of the total thickness h' of the laminated medium and characterized by the coefficient Mj j/(l --i )T The equivalent coefficient M i.e. the coefficient of the equivalent anisotropic mdium) is n j--1 J (5.1) x Oxy +y Oyy p --+t 2 b-Vx 4 (5.5) Consequently, equation (3.12) must be replaced by n 2 n S j=l" (Aj + Aj+ I) cjMj + m2Vk a Pt V + m4V j=ll bjhj C5. 6) where bj(j 1,2...n) is the couple stress coefficient of the Jth laminated medium.
The result (5.6) can immediately be extended to viscoelastic laminated media.

EVALUATION OF DAMPING.
An important task in the problem of design analysis is the determination of the effect of viscoelastic layers on vibration absorption at resonance.We consider a simply supported homogeneous anlsotroplc plate.The span p equal to half the wave length is m From equations (2.14) and (3.1) we have the expression for S which, after putting A 0 and A 2 0, becomes tanh 8'.The variation of F for different values of B (anlsotropy) and T are shown in figure I.It is seen that the damping increases with anisotropy and wave number.The result (6.3) is also applicable to the laminated plate of the equivalent continuum as described in section 5.In the case the inclusion of couple stress will contribute an 4 additional term m Vh Ab where Ab is the imaginary part of equation ( 5 The variations of P with are shown in Figure 2 which confirms the symmetric variation with respect to the relaxation time .As a particular example, we consider a plate composed of two layers.The material (obviously Maxwell type) of the first layer of thickness h' is characterized by the coefficient M and that the second layer of thickness h" is characterized by the coefficient M 2 and suppose h h' + h".
The applied shear stresses at the outerfaces are put equal to zero (i.e.
AI=A3ffiO.The shear stress at the interface is found by equating the interfacial displacement amplitude U 2 considered as belonging to the first and second layer.
Applying equations (3.3) we obtain ( a I) 2V c 2 (6.7) u 2- where Nland N 2 are the values of N for M M and M M 2 respectively.The total load q (ffi ql + q2 is given by 2(H cl+ H2c2) (c1+ c2) The total load q= ql + q2 in this case is given by 2 The second and third terms represent the normal load corresponding to superposed layers with perfect Interracial slip while the first term represents the effect of adherence between layers.
1.0  The foregoing results of a simply supported plate with a loading distributed as a half sine wave are obviously valid for an arbitrary loading provided it can be expanded in a series along the span.
In this case, the results are applied to the various Fourier components with a suitable value of m corresponding to each wave length.The procedures used for the simply supported case can easily be extended to the case where the plate is built-in (i.e. when both end points of the plate are rigidly attached so that the displacement of its faces are both zero.In this case, in order to satisfy the boundary conditions we must, instead of sinusoidal solutions, consider exponential solutions.Such solutions are empirically derived from the trigonometric solutions (2.7) and (2.8) if we replace m by ik.
TO MULTILAYERED PLATES AND VISCOELASTICITY.

3 )
material M is replaced by M + M where M is the purely imaginary term.The Imaginary part of the load S is represented by S and is obtained from equation (6.2) by linearlzing it with respect to M. We obtain determines the damping (longitudinal) on vibration attenuation.
the deflection V when the normal load applied to the structure is given.We next consider a plate composed of three layers.The first and third layers are identical in nature and are characterized by the coefficient M while the middle one is characterized by the coefficient M2.Because of the symmetry, the interfacial

Fig. 3 A
Fig.3A three-layered symmetric plates composed of a core of thickness h 2 sandwiched between two identical layers of thickness h I with respect to y