Dynamic stability analysis of Mindlin viscoelastic plates subjected to constant and harmonic in-plane compressions based on free vibration analysis of elastic plates

Abstract

In this paper, a dynamic stability analysis of moderately thick viscoelastic plates is performed employing the Boltzmann integral law with constant bulk modulus. The remarkable and new point of the proposed method is that the frequency of a Mindlin viscoelastic plate subjected to simultaneous constant and harmonic in-plane compressive loads is explicitly predictable based on free vibration analysis of an elastic plate. Moreover, the damped part of frequency is easily calculated. Also, the critical excitation for which the system becomes unstable is determined. This method is completely new and significantly reduces the computational cost. The obtained results are compared with other existing results to show the efficiency and accuracy of the proposed method. This method is used to investigate the effects of viscoelastic properties and in-plane compressions on the steady-state responses of Mindlin viscoelastic plates under time-dependent compressive loads.

Appendix: Dynamic stability analysis of a simply supported viscoelastic column

The equation of a viscoelastic column subjected to a time-dependent compressive load $p\left( t \right)$, as illustrated in Fig. 10, is defined as [2]:

$$\frac{{\partial^{2} M}}{{\partial x^{2} }} - p\left( t \right)\frac{{\partial^{2} w}}{{\partial x^{2} }} = m\frac{{\partial^{2} w}}{{\partial t^{2} }}$$

(80)

in which $x$ is the longitudinal axis, $w$ is the transverse displacement, and $m$ denotes the mass per unit length.

The bending moment $M$ can be expressed as:

$$M = \mathop \int \limits_{A}^{{}} \sigma \left( {x,y,t} \right)y{\text{d}}A$$

(81)

where $y$ is the transverse axis.

Fig. 10

A viscoelastic column subjected to time-dependent axial compressive load

The constitutive equation of a linear viscoelastic material based on the Boltzmann integral can be given as [24]:

$$\sigma \left( {x,y,t} \right) = E\left( t \right)\varepsilon \left( 0 \right) + \mathop \int \limits_{0}^{t} E\left( {t - \tau } \right)\dot{\varepsilon }\left( \tau \right){\text{d}}\tau$$

(82)

where $E\left( t \right)$ is the relaxation function. It is noted that in case of steady harmonic vibrations employing Eq. (82) is not necessary, and the problem reduces to the study of the storage and loss modulus [30]. But, since the goal of this paper is to study the dynamic stability of columns subjected to axial compression, using Eq. (83) is necessary.

For a Bernoulli beam, the relation between the strain $\varepsilon$ and the deflection $w$ can be written as [2]:

$$\varepsilon \left( {x,y,t} \right) = - y\frac{{\partial^{2} w\left( {x,t} \right)}}{{\partial x^{2} }}.$$

(83)

Considering simply supported boundary conditions and by supposing that all points of the column move in phase, the deflection may be represented by [1, 4]:

$$w\left( {x,t} \right) = F\left( t \right){\text{ sin}}\frac{\pi x}{l}.$$

(84)

The relaxation function can be defined as follows:

$$E\left( t \right) = E_{0} \eta \left( t \right), \eta \left( t \right) = c_{1} + c_{2} {\text{e}}^{ - \lambda t}$$

(85)

where $E_{0}$ is the elasticity modulus at time zero.

Substituting Eqs. (82–85) into Eq. (80), Eq. (86) is obtained:

$$E\left( t \right)\frac{{I\pi^{4} }}{{l^{4} }}F\left( 0 \right) + E_{0} \frac{{I\pi^{4} }}{{l^{4} }}\mathop \int \limits_{0}^{t} \eta \left( {t - \tau } \right)\dot{F}\left( \tau \right){\text{d}}\tau - \frac{{\pi^{2} }}{{l^{2} }}p\left( t \right)F\left( t \right) + m\ddot{F}\left( t \right) = 0$$

(86)

in which $I$ is the column moment of inertia.

The time-dependent compressive load, $p\left( t \right)$, may be given as:

$$p\left( t \right) = \left( {\alpha_{1} + \beta_{1} \cos \varphi t} \right)P_{{\text{e}}} , P_{{\text{e}}} = \frac{{\pi^{2} E_{0} I }}{{l^{2} }}, 0 \le \alpha_{1} < 1, \beta_{1} \ge 0, 0 < \varphi < \pi /2$$

(87)

in which $\alpha_{1}$ and $\beta_{1}$ are constant, and $\varphi$ is the excitation frequency. Replacing Eq. (87) in Eq. (85), Eq. (88) is obtained:

$$\eta \left( t \right)F\left( 0 \right) + \mathop \int \limits_{0}^{t} \eta \left( {t - \tau } \right)\dot{F}\left( \tau \right){\text{d}}\tau - \alpha_{1} F\left( t \right) - \beta_{1} \cos \varphi tF\left( t \right) + \frac{{\ddot{F}\left( t \right)}}{{\Omega^{2} }} = 0$$

(88)

where the fundamental natural frequency at time zero is defined as follows:

$${\Omega }^{2} = \frac{{E_{0} I\pi^{4} }}{{ml^{4} }}.$$

(89)

For a viscoelastic column, the time function can be defined as:

$$F\left( t \right) = {\text{e}}^{{s_{0} t}}.$$

(90)

So, one can write Eq. (91) as follows:

$$\dot{F}\left( t \right) = s_{0} F\left( t \right), \ddot{F}\left( t \right) = s_{0}^{2} F\left( t \right).$$

(91)

Replacing Eq. (91) in Eq. (88), Eq. (92) is obtained:

$$\eta \left( t \right) + s_{0} \mathop \int \limits_{0}^{t} \eta \left( {t - \tau } \right)F\left( \tau \right){\text{d}}\tau - \alpha_{1} F\left( t \right) - \beta_{1} \cos \varphi tF\left( t \right) + \left( {\frac{{s_{0} }}{\Omega }} \right)^{2} F\left( t \right) = 0.$$

(92)

If Laplace transform is taken from Eq. (92), Eq. (93) is obtained:

$$\eta^{*} + s_{0} \mathop \int \limits_{0}^{\infty } \mathop \int \limits_{0}^{t} \eta \left( {t - \tau } \right)F\left( \tau \right){\text{d}}\tau {\text{e}}^{{ - {\text{st}}}} {\text{d}}t - \alpha_{1} F^{*} - \beta_{1} \left( {\cos \varphi tF\left( t \right)} \right)^{*} + \frac{{\ddot{F}^{*} }}{{\Omega^{2} }} = 0$$

(93)

in which $\eta^{*}$, $F^{*}$, $\ddot{F}^{*},$ and $\left( {\cos \varphi tF\left( t \right)} \right)^{*}$ are the Laplace transformations of $\eta \left( t \right)$, $F\left( t \right)$, $\ddot{F}\left( t \right)$, and $\cos \varphi tF\left( t \right)$, respectively.

According to Laplace transform properties, the second term of Eq. (93) can be calculated as:

$$s_{0} \mathop \int \limits_{0}^{\infty } \mathop \int \limits_{0}^{t} \eta \left( {t - \tau } \right)F\left( \tau \right){\text{d}}\tau {\text{e}}^{{ - {\text{st}}}} {\text{d}}t = s_{0} \mathop \int \limits_{0}^{\infty } \mathop \int \limits_{0}^{t} \eta \left( \tau \right)F\left( {t - \tau } \right){\text{d}}\tau {\text{e}}^{{ - {\text{st}}}} {\text{d}}t = s_{0} \mathop \int \limits_{0}^{\infty } \mathop \int \limits_{0}^{t} \eta \left( \tau \right){\text{e}}^{{s_{0} \left( {t - \tau } \right)}} {\text{d}}\tau {\text{e}}^{{ - {\text{st}}}} {\text{d}}t = s_{0} \mathop \int \limits_{0}^{\infty } \left( {\mathop \int \limits_{0}^{t} \eta \left( \tau \right){\text{e}}^{{ - s_{0} \tau }} {\text{d}}\tau } \right){\text{e}}^{{ - {\text{st}} + s_{0} t}} {\text{d}}t.$$

(94)

On the other hand, the convolution integral of Eq. (94) can be calculated as:

$$s_{0} \mathop \int \limits_{0}^{\infty } \left( {\mathop \int \limits_{0}^{t} \eta \left( \tau \right){\text{e}}^{{ - s_{0} \tau }} {\text{d}}\tau } \right){\text{e}}^{{ - {\text{st}} + s_{0} t}} {\text{d}}t = s_{0} \mathop \int \limits_{0}^{\infty } \eta \left( \tau \right){\text{e}}^{{ - s_{0} \tau }} {\text{d}}\tau \mathop \int \limits_{0}^{\infty } e^{{s_{0} t}} {\text{e}}^{{ - {\text{st}}}} {\text{d}}t = s_{0} \eta_{{s_{0} }}^{*} F^{*} ,$$

$$\eta_{{s_{0} }}^{*} = \left( {\frac{{c_{1} }}{{s_{0} }} + \frac{{c_{2} }}{{s_{0} + \lambda }}} \right), F^{*} = \mathop \int \limits_{0}^{\infty } {\text{e}}^{{s_{0} t}} {\text{e}}^{{ - {\text{st}}}} {\text{d}}t = \frac{1}{{s - s_{0} }}.$$

(95)

Using Eqs. (90) and (95), Eq. (93) can be simplified as:

$$\eta^{*} + s_{0} \eta_{{s_{0} }}^{*} F^{*} - \alpha_{1} F^{*} + \left( {\frac{{s_{0} }}{\Omega }} \right)^{2} F^{*} - \beta_{1} \left( {\cos \varphi tF\left( t \right)} \right)^{*} = 0.$$

(96)

Alternatively:

$$\eta^{*} + \left( {s_{0} \eta_{{s_{0} }}^{*} - \alpha_{1} + \left( {\frac{{s_{0} }}{\Omega }} \right)^{2} } \right)\frac{1}{{s - s_{0} }} - \frac{{\beta_{1} }}{2}\left( {\frac{1}{{s - s_{0} + i\varphi }} + \frac{1}{{s - s_{0} - i\varphi }}} \right) = 0.$$

(97)

If Eq. (92) is solved in the time domain too, Eq. (98) is derived:

$$\left( {s_{0} \left( {\frac{{c_{1} }}{{s_{0} }} + \frac{{c_{2} }}{{s_{0} + \lambda }}} \right) - \alpha_{1} + \left( {\frac{{s_{0} }}{{\Omega }}} \right)^{2} } \right)\frac{1}{{i\varphi + \alpha_{0} }} - \beta_{1} \frac{{i\varphi + \alpha_{0} }}{{2i\varphi \alpha_{0} + \alpha_{0}^{2} }} = 0.$$

(98)

Comparing Eq. (97) and Eq. (98), the Laplace parameter in Eq. (97) can be replaced by $i\left( {\omega_{0} + \varphi } \right){ },$ that is, $s = i\left( {\omega_{0} + \varphi } \right)$.

Replacing $s = i\left( {\omega_{0} + \varphi } \right)$ and ignoring $\eta^{*} = \frac{{c_{1} }}{{i\left( {\omega_{0} + \varphi } \right)}} + \frac{{c_{2} }}{{i\left( {\omega_{0} + \varphi } \right) + \lambda }}$, Eq. (97) can be rewritten as follows:

$$s_{0} \eta_{{s_{0} }}^{*} - \alpha_{1} + \left( {\frac{{s_{0} }}{{\Omega }}} \right)^{2} - \beta_{1} \frac{{\left( {i\varphi + \alpha_{0} } \right)^{2} }}{{2i\varphi \alpha_{0} + \alpha_{0}^{2} }} = 0, s_{0} \eta_{{s_{0} }}^{*} = c_{1} + \frac{{c_{2} s_{0} }}{{s_{0} + \lambda }} , s_{0} = i\omega_{0} - \alpha_{0}$$

(99)

Replacing $s_{0} = i\omega_{0} - \alpha_{0}$ in Eq. (99) and separating the real and imaginary parts, Eqs. (100) and (101) are obtained:

$$c_{1} + c_{2} \frac{{\omega_{0}^{2} + \alpha_{0}^{2} - \alpha_{0} \lambda }}{{\omega_{0}^{2} + \left( { - \alpha_{0} + \lambda } \right)^{2} }} - \alpha_{1} - \left( {\frac{{\omega_{0} }}{{\Omega }}} \right)^{2} + \left( {\frac{{\alpha_{0} }}{{\Omega }}} \right)^{2} - \beta_{1} \frac{{3\varphi^{2} \alpha_{0}^{2} + \alpha_{0}^{4} }}{{4\varphi^{2} \alpha_{0}^{2} + \alpha_{0}^{4} }} = 0,$$

(100)

$$c_{2} \frac{{\lambda \omega_{0} }}{{\omega_{0}^{2} + \left( { - \alpha_{0} + \lambda } \right)^{2} }} - 2\frac{{\omega_{0} }}{{\Omega }}\frac{{\alpha_{0} }}{{\Omega }} - \beta_{1} \frac{{2\varphi^{3} \alpha_{0} }}{{4\varphi^{2} \alpha_{0}^{2} + \alpha_{0}^{4} }} = 0.$$

(101)

Since $\alpha_{0} \ll {\Omega },{ }\lambda \ll {\Omega }$, by ignoring $\left( {\alpha_{0/} {\Omega }} \right)^{2}$, $\left( {\lambda /{\Omega }} \right)^{2}$, and $\left( {\lambda /{\Omega }} \right)\left( {\alpha_{0/} {\Omega }} \right)$, Eqs. (102–103) are derived:

$$c_{1} + c_{2} - \alpha_{1} - \left( {\frac{{\omega_{0} }}{{\Omega }}} \right)^{2} - \frac{3}{4}\beta_{1} \approx 0 or \frac{{\omega_{0} }}{{\Omega }} \approx \sqrt {1 - \alpha_{1} - \frac{3}{4}\beta_{1} }.$$

(102)

Inserting $\omega_{0}$ in Eq. (103), $\alpha_{0}$ is obtained:

$$c_{2} \frac{\lambda }{{\omega_{0} }} - 2\frac{{\omega_{0} }}{{\Omega }}\frac{{\alpha_{0} }}{{\Omega }} - \beta_{1} \frac{\varphi }{{2\alpha_{0} }} \approx 0{ }.$$

(103)

Alternatively:

$$\frac{{\alpha_{0} }}{{\Omega }} \approx \frac{{\frac{{c_{2} \lambda }}{{\Omega }} + \sqrt {\left( {\frac{{c_{2} \lambda }}{{\Omega }}} \right)^{2} - 4\beta_{1} \frac{\varphi }{{\Omega }}\left( {1 - \alpha_{1} - \frac{3}{4}\beta_{1} } \right)^{3/2} } }}{{4 \times \left( {1 - \alpha_{1} - \frac{3}{4}\beta_{1} } \right)}}.$$

(104)

If the free vibration is studied ($\alpha_{1} = 0,\beta_{1} = 0$), then $\omega_{0} \approx {\Omega }$ and $\alpha_{0} \approx \frac{{c_{2} \lambda }}{2}$, which is consistent with the results of Ref. [2].

Substituting Eqs. (102) and (103) into Eq. (84), Eq. (105) is derived

$$w\left( {x,t} \right) = {\text{e}}^{{\left( {i\omega_{0} - \alpha_{0} } \right)t}} {\text{sin}}\frac{\pi x}{l} = {\text{e}}^{{ - \alpha_{0} t}} \left( {\cos \omega_{0} t + i\sin \omega_{0} t} \right){\text{sin}}\frac{\pi x}{l}.$$

(105)

Considering Eq. (104), the viscoelastic column is always stable if the conditions below are satisfied:

$$1 - \alpha_{1} - \frac{3}{4}\beta_{1} > 0, \left( {\frac{{c_{2} \lambda }}{{\Omega }}} \right)^{2} \ge 4\beta_{1} \frac{\varphi }{{\Omega }}\left( {1 - \alpha_{1} - \frac{3}{4}\beta_{1} } \right)^{\frac{3}{2}} ,$$

$$0 \le \alpha_{1} < 1, \beta_{1} \ge 0, 0 < \varphi < \pi /2.$$

(106)

Finally, one can solve Eq. (107) for calculating $\beta_{{{\text{cr}}}}$ as follows:

$$\left( {\frac{{c_{2} \lambda }}{{\Omega }}} \right)^{2} = 4\beta_{{{\text{cr}}}} \frac{\varphi }{{\Omega }}\left( {1 - \alpha_{1} - \frac{3}{4}\beta_{{{\text{cr}}}} } \right)^{3/2}.$$

(107)