Prestress Effect on the Thermomechanical Response of Viscoelastic Plate Under Harmonic Loading

A statement of the coupled thermomechanical problem on forced resonant vibrations and dissipative heating of hinged viscoelastic elastomeric plate is given with account of prestresses present in the plate. It is assumed the prestress is generated as a result of the manufacturing process or preliminary plate service. The problem statement is based on the standard Kirchhoff-Love hypotheses and concept of complex moduli that are used to describe the viscoelastic material response to harmonic loading. Under these circumstances, the prestress manifests itself as a membrane forces applied in the plane of the rectangular plate. Therefore, the problem of in-plane stress state and problem of forced transverse vibration of the plate can be solved separately. Both steady-state and transient thermal response is investigated. Influence of the prestress is studied in details. Dissipative heating temperature histories are calculated for the variety of the prestress and loading parameters. Temperature criterion is adopted to determine the critical state. The data obtained are used for the plate fatigue life prediction as well as for the investigation of prestress effect on the plate response. The reliability of the values of frequencies on the several lowest resonances was checked. For the most energy-intensive first mode of transverse vibrations, the influence of the preliminary tensile stress state, as well as the amplitude of the transverse distributed load on the amplitude–frequency characteristics and temperature evolution was studied.


Introduction
Thermomechanical coupling problems are common phenomena in the field of Solid Mechanics. Mechanical engineering equipment [1], aerospace industry structures and energy generation machinery as well as transport engine/structure components can be subjected to severe mechanical and thermal cyclic loads. Assessment of component life depends on the behavior of materials used. Elastomers are widely used as materials for the structural components mentioned. Under intensive cyclic loading, the effect of dissipative heating can be particularly profound for the polymers. The dissipative heating is caused by an appearance of a mechanical hysteresis resulting from the viscoelastic nature of the most of industrial polymers. Part of the dissipated mechanical energy leads to generation of thermal energy [2][3][4][5], while the rest of the energy transferred to the structure is both partly stored in the material [6] and dissipated as a result of plastic deformation [7,8], viscoplastic deformation, and variations of yield limit caused by isotropic and kinematic hardening [9].
It is generally adopted that dissipative heating is usually attributed to the internal friction of the material [10], which causes the hysteretic response. Energy dissipated as a heat leads to the increase of the temperature of the loaded structure. Small temperature advance over one separate cycle can result in high heating level for prolonged operational period [11,12].
The effect of thermomechanical coupling described above should be taken account for when the adequate models for fatigue life predictions are developed. This implies a necessity for the deep investigation of dissipative heating, including its physical nature, the consequences, and the relations with degradation mechanisms in structural elements, as well as the estimation of safe temperature ranges.
Additional factor that can significantly influence the fatigue life prediction of the polymeric structures is the residual stress which can arise as a result of manufacturing processes. For the subsequent cyclic loading problem, these stresses should be considered as prestresses. The effect of the prestresses, as well, has to be taken into account to improve the reliability of the fatigue life prediction technique.
To address the issue of fatigue life prediction with account of thermomechanical coupling, the theories based on the relations between the amplitudes of main field variables are usually used to describe the thermomechanical response of viscoelastic solids under cyclic loading. Application of complex amplitudes and complex moduli concept in the case of harmonic loading yields the complete system of amplitude equations for linearized coupled thermoviscoelasticity in terms of complex amplitudes. This system is complemented with the complex analogues of the equation of motion along with the stationary or non-stationary equation of heat transfer. In the former case, the temperature-frequency characteristics of vibration can be obtained and the critical loading parameters can be specified. To describe the over critical heating and, therefore, to build the fatigue S-N curves the latter equation should be used instead. The detailed reviews of the theories and approaches developed along with the numerous applications can be found in [13][14][15][16][17][18].
In this paper, a fatigue model for prediction of a number of cycles to thermal failure is developed, taking into consideration stationary dissipative heating and structure prestress that can significantly affect the fatigue life characteristics of the viscous rectangular plate subjected to transverse harmonic loading. Special attention is paid to the influence of membrane prestress on the amplitude-frequency and temperature-frequency characteristics as well as on the non-stationary histories of dissipative heating for cyclically excited plate.

Theoretical Concepts
Let us consider a hingedly supported plate of thickness h with planform dimensions a and b. It is assumed that the plate is composed of the linear viscoelastic material which material properties are frequency, amplitude and temperature independent. The rectangular Cartesian coordinates (x, y, z) with the xy-plane coinciding with the geometric middle plane of the plate and the z-coordinate taken positive upward is introduced. The plate contour is formed by the edges x = 0, a; y = 0, b. It is also assumed that prestresses are generated in the plate as a result of manufacturing process.
and frequency ω which is close to one of the resonances. The plate edges are assumed to be thermally insulated while the convective heat transfer to the ambient medium of temperature T C is supposed to take place at the face surfaces of the plate.
To simulate the mechanical response of the plate under consideration, the classical Kirchhoff-Love hypotheses are made use of [19,20]. The integral operators of linear viscoelasticity are used to describe the viscoelastic properties of the plate material [13][14][15]. In the particular case of harmonic deformation, these operators are reduced to multiplication of complex quantities where the operator and complex quantities are marked with asterisk and wave respectively. It is also assumed that temperature is constant through the plate thickness.

Main Equations
In general case, the equations of transverse vibration of the plate are obtained in the following form [19,20] ( ) , where u, v and w are components of the displacement field along Ox, Oy and Oz axis respectively; ij N and ij M , , , i j x y = are thickness-integrated forces and moments, known as the stress resultants [19,20]; k is the foundation modulus (for the cases where it has to be taken account for), ρ is the material density.
In system (4), 0 J and 2 J are the mass moments of inertia . Under these assumptions, the in-plane problem for the plate becomes statically determinate. Thus, the first and second equations from (4) are satisfied identically. Classical plate theory formalism for isotropic elastic material yields the 3 rd equation from (4) in the form ( ) where ( ) is the plate bending stiffness coefficient, E and ν are the material Young's modulus and Poisson ratio respectively; As a result, in the case of temperature independent material properties and assumption of a temperature constancy through the plate thickness, the mechanical problem is reduced to two uncoupled problems: (i) plane stress problem, (ii) pure bending problem. The former one defines the prestress while the latter problem describes the varying harmonically in time pure bending state which is superimposed over the prestressed state of the plate.
Boundary conditions for the linear bending of a rectangular plate with edges parallel to the x and y coordinates for the particular case of hingedly supported have the form 2 2 Under harmonic loading, investigation of steady-state vibration regime is usually performed with the use of complex moduli and complex amplitudes concept [13][14][15][16]. These yields the equation (7) as follows ( ) are complex bending stiffness coefficient, deflection of the plate and load; 1 i = − . It is worth mentioning here that expressions for the boundary conditions should have the same form (9) with substitution of complex variable w ɶ instead of real one w .
To obtain the time variation laws for the beam deflection and load, one should apply the formulas , , , cos , sin .
Here ( ) ′ ⋅ and ( )′′ ⋅ are the real and imaginary parts of the complex amplitude. The complex equation (10) is equivalent to system of two real equations with respect to the complex amplitudes in the form with the boundary conditions (9) rewritten in terms of the complex amplitudes as well

The Dissipative Function
Evolution of the temperature field is described by the heat conduction equation averaged over the plate thickness and vibration period complemented with thermal initial and boundary conditions 0 at 0; 0 at 0, ; 0 at 0, .
In the equations (14) and (15), θ is the dissipative heating temperature, θ = T − T C ; c, λ and α S are the specific heat, heat conductivity coefficient and heat transfer coefficient at the face surfaces of the plate respectively; T C is the ambient temperature; D W is the dissipative function averaged over the period of vibrations and over the plate thickness ; In the expressions (17), the quantities ij M ɶ , 1 ε ɶ ij , and 1 γ ɶ xy , , , i j x y = are complex-value analogues of moments from equations (4) and bending strains (curvatures and twisting); ν ɶ is the complex Poisson ratio, i . Let us emphasize here that version of the cyclic viscoelasticity with real Poisson ratio and complex bulk modulus is chosen for the materials under consideration in this problem. Thus, relations ′ ν = ν , 0 ′′ ν = are assumed to be valid further on.
As a result, a statement of the coupled thermomechanical problem on forced vibrations and dissipative heating of hinged viscoelastic plate with account of prestresses applied is given by the equations (12) and (14) with the expression (16) for the dissipative function complemented with correspondent mechanical boundary conditions (13) along with thermal initial and boundary conditions (15).

Solution Technique
Let us study the case when the prestress state can be a result of presence of the normal preliminary traction only: It is worth mentioning here that for cyclically stable materials, mechanical processes stabilize much faster than the thermal ones [13]. It enables us to consider mechanical part of the problem as the steady-state process in contrast to the thermal part which is studied in the frame of transient analysis at the aforementioned assumptions with respect to the material constants. Thus, solution of the steady state vibration problem for the hingedly supported plate can be written in the form [13,20] where mn w′ and mn w′′ are unknown complex amplitudes of the eigenmode mn corresponding to the natural frequency ω mn . This frequency is determined by the well-known formula taking account for the prestresses [13,20] where 0 D is the plate bending stiffness coefficient calculated for elastic response of the plate material. In the case of viscoelastic material, it should be calculated as If the transverse loading varies according to sine or cosine law over the time then this relation can be modified in the obvious way.
In this case, the plate response will be formed by the resonating mode mn, and the sums in the expression (18) can be discarded. Substitution of the resultant into (17) with accounting for 0 0 xy N = yields the system of two linear algebraic equations with respect to mn w′ and mn w′′ of the following form (21) is determined by the expression 1 2 ,

Solution of the system of equations
Substitution of (24) into (16) with accounting for the first line from (17) The transient heat conductivity problem formed by the equation (14) complemented by the expression for the dissipative function (25) along with the thermal initial and boundary conditions (15) can be solved numerically.

Stationary Problem
The stationary problem was first solved.      Figure 3 illustrates the effect of shifting the resonant frequency (to the region of higher frequency values) with an increase in preliminary tensile membrane tractions in the plate, which corresponds to the known theoretical and experimentally confirmed result. A similar effect is obtained for the case of temperature frequency response ( Figure 4). The shift of the resonance region to the right is accompanied by an increase in temperature. As it can be seen from figure 5, the behavior of the amplitude-frequency response corresponds to the typical behavior of the characteristics in the case of a linear viscoelastic material and is accompanied by an increase in the amplitude of deflection with an increase in the amplitude of the transverse loading. The temperature-frequency response corresponding to this case is presented in figure 6. It illustrates a sharp increase in temperature at resonance frequency, which, a critical value of the loading parameter is reached, leads to dangerous levels of heating. The maximum temperature is equal to 90.1°C.
For the elastomer chosen as the plate material, the melting point is 120°С. Therefore, in this work, the value of 100°С was chosen as the critical value of the heating temperature θ cr , at which a noticeable softening of the material is observed.

Non-stationary Problem
Next, the non-stationary problem for the square plate was considered. The heat conduction equation (14) complemented with thermal initial and boundary conditions (15) was solved.
The ambient temperature is chosen to be equal to T C = 20°C. The dissipative heating temperature equals θ = T − T C .
In figure 7, the temperature-time dependence at the resonant frequency f = 468.     Figure 8 illustrates the distribution of the temperature in the first eigenmode (m = 1, n = 1). The maximum temperature is reached at x = a/2 and equals θ max = 79.5°C.
The first eigenmode is the most energy-intensive vibration mode.
In figure 9, the Wöhler diagram at frequency f = 468.1 Hz, which is resonant for the membrane force = xx N 45 kN/m, was constructed. In figure 9, the green, blue, violet, and red lines corresponds to = xx N 20, 30, 40, and 45 kN/m respectively. This is called a falling branch. In figure 10, the Wöhler diagram at frequency f = 337.9 Hz, which is resonant for the membrane force = xx N 0, was constructed. In figure 10, the red, violet, blue, and green lines corresponds to = xx N 0, 5, 10, and 20 kN/m respectively. This is called a raising branch.
To provide the efficiency of the plate, it is necessary to choose such modes on vibration in which the level of heating does not exceed a critical value θ cr . Critical levels of prestresses and loads are determined by the points of intersection of the temperature-frequency response with a line of θ cr = 100°С.

Conclusion
Under the assumption of the Kirchhoff-Love hypotheses validity, the formulation and solution of the stationary and non-stationary problems of forced resonant vibrations and vibroheating of a viscoelastic polyethylene plate with preliminary normal membrane force in case of a rectangular plate, hinged along the contour are presented. For the most energy-intensive first mode of bending vibrations of a pivotally supported plate, the influence of the indicated preliminary state, as well as the level of the transverse distributed load, material properties, heat transfer conditions on the frequency dependences of the maximum deflection amplitude and dissipative heating temperature was studied.
The critical values of prestresses are determined at which the temperature of the vibroheating reaches a value at which the plate material begins to soften. Temperature-time dependences for a given elastomeric plate are shown. The material fatigue limits are found for which the Wöhler diagrams are constructed.