The influence of the mechanical damage on a viscothermoelastic nanobeam due to ramp-type heating under Green-Naghdi theory type-II

Abstract

The main object of this work is to analyze viscothermoelastic homogeneous and isotropic nanobeams under the effect of mechanical damage. A Green-Naghdi type-II model of thermoelasticity has been constructed under simply supported conditions. Moreover, the Laplace transforms have been applied for the governing differential equations. Tzou iteration method with approximation has been applied to calculate the Laplace inverse transforms. The numerical results are presented and validated for use in a viscothermoelastic rectangular nanobeam of silicon nitride when it has been thermally loaded by ramp-type heating and simple support. The numerical results have been demonstrated in different figures to stand on the effects of the mechanical damage variable, mechanical relaxation times parameters, and ramp-time heat parameter on all the studied functions. We conclude that the effect of the mechanical damage variable, ramp-time heating parameter, and mechanical relaxation times parameters are highly influential on all the studied functions and thermomechanical waves.

Keywords

Damage viscothermoelasticity nanobeam silicon nitride ramp-type heat Green-Naghdi

Introduction

Heat conduction has been described in a variety of mathematical and engineering models, such as dual-phase lag (DPL) assumed by Tzou and Transfer.^1,2 This model is common in the study of heat transfer problems and applications.^3–8 The linked thermoelasticity hypothesis is an additional theory. This theory analyses the classical Fourier law for heat conduction in two partial differential equations: the energy conservation equation and the motion equation.^9–12 By including the relaxation period on an isotropic substance, Lord and Shulman (L-S) modified the conventional Fourier law of heat conduction.¹³ As such, the law is modified in a way that includes both the heat flux and time derivative (We may refer to this as Cattaneo’s heat conduction). If we situate our claims in this theory, we may classify the heat equation as the hyperbolic type, where the paradox of infinite propagation speeds is omitted.^14–18

The most crucial aspect of micro/nanomechanics is the vibration of the nanobeams and resonators. By using dual-phase-lag (DPL), Alghamdi addressed the thermal damping vibration with voids of nanobeam resonators based on the theory of generalized thermoelasticity.⁹ Sharma and Grover investigated the transverse vibrations produced by a narrow beam resonator that has voids and is isotropic, homogeneous, and thermoelastic.¹⁹ The heat-damping vibration on a microplate and circular plate resonators was also studied by Sun and Saka.²⁰ They modified their formula by including a new and distinct component that depends on Poisson’s ratio and differs from the one presented in Lifshitz and Roukes.²¹ Numerous writers have researched and written on the vibration and heat-transfer process of nanoresonators.^22–30 Youssef and Al-Lehaibi, for instance, looked at the vibration of a gold nanobeam that had experienced a thermal shock.²³ Kidawa, on the other hand, was interested in the effects of internal and external damping on a beam’s transverse vibrations induced by a moving heat source, and he addresses this issue using the properties of the Green functions.²⁵ Boley talked about the vibrations that are dispersed over a rectangular nanobeam’s span when it experiences a thermal shock.²⁴ Additionally, Manolis and Beskos used a computational technique to examine the thermal characteristics of the elastic dynamic response of beam structure to thermal loading while dealing with vibrations that are thermally caused in nanobeam structures.²⁶ Al-Huniti et al. used a moving laser beam and high power to evaluate the strains and displacements that are thermally caused by a heated rod. Then, using Laplace’s transformation technique, they looked at the rod’s dynamic behavior.²²

Generalized thermoelasticity was approached differently by Green and Naghdi. They referred to it as thermoelasticity without dissipation of energy.³¹ By utilizing the broad definition of entropy balancing, they put out the central argument. They created three different response function types as a result. Type I employs a linear theoretical structure and is the same as conventional heat conduction (or conventional Fourier’s law). Thermoelastic disturbances can propagate with a limited speed in type-II and type-III systems. Type-II is the only type that prevents energy loss, even if type-III and type-II are utilized to flow heat in inflexible solids with limited wave speed.³²

A nonlinear large-amplitude free vibration response of nano-shells prepared from functionally graded porous materials has been investigated by considering the surface stress size effects and vibrational mode interactions.³³ To investigate the thermal post-buckling properties of porous composite nanoplates formed of a functionally graded material with a central cutout of various shapes, a surface elastic-based three-dimensional nonlinear formulation has been developed.³⁴ Thermomechanical vibration of bi-directional functionally graded non-uniform Timoshenko nanobeam has been studied through many applications.^35,36

In the earliest phases of actual materials, flaws like microcracks and voids are frequently present. Internal voids or gaps may widen and merge with deformation. Additionally, total separation happens when the arrangement of fresh micro-changes at the stress concentrators. Damage, the last step, results in a total loss of material integrity and the formation of microscopic fissures. According to macroscopic phenomena, a material’s damage can be categorized as brittle, ductile, creep, and fatigue damage.^30,37 In continuum mechanics, conclusions from the description of the macroscopic behavior of a damaged material are still pending. For us, damage may unite fracture mechanics with conventional continuum mechanics.

Damage variables can be introduced in a variety of methods. Assessing any injured body’s cross-section, we observe an element’s area $dA$ with unit normal vector “n.” We may denote the area of defects by $d A_{D}$ . The amount of damage may take the form^30,37:

D_{n} = \frac{d A_{D}}{dA}, 0 \leq D_{n} \leq 1

(1)

The case $D_{n}$ gives the undamaged state of the material, while the case $D_{n} = 1$ illustrates the completely, damaged state of the material and loss of stress on carrying capacity (this case is called fracture). The damage processes are valid when $D_{n} \approx 0.2 . . . 0.5$ This is an inevitable failure. If the value of the damage is constant through a ﬁnite area that is subjected to uniaxial tension, becomes $D_{n} = A_{D} / A$ .

We cannot describe the effects of the cross-section microcracks in the same way. Otherwise, our conclusions would be fallacious. On the other hand, if we use the isotropic damage, then, D is independent of n, thus, the eﬀective stresses become^30,37:

σ_{ij} = (1 - D) {\tilde{σ}}_{ij}

(2)

${\tilde{σ}}_{ij}$ denote the average stresses in the undamaged material. This definition proved to be useful as several applications and problems utilized it.^30,38–42

Therefore, the primary goal of this work is to analyze homogeneous, isotropic viscothermoelastic nanobeam of silicon nitride under the influence of mechanical damage. Under simply supported conditions, a thermoelasticity Green-Naghdi type-II model will be developed.

The impacts of the mechanical damage variable, mechanical relaxation times parameters, and ramp-time heat parameter on all the analyzed functions will be supported by the numerical results in various figures.

Basic equations

A material that adheres to the Kelvin-Voigt viscothermoelastic type in the well-known Cartesian coordinate, is homogeneous, isotropic, thermally conducting, and undeformed at a constant temperature $T_{0}$ The governing differential equations of motion and heat conduction are constructed based on the generalized thermoelasticity under the non-Fourier heat conduction law.

The displacement components are given by $U = (U_{x}, U_{y}, U_{z}) = (u, v, w)$ and the absolute temperature $T$ .

Create the governing equations as follows if forces and heat sources are not present⁴³:

The equations of motion take the form:

σ_{ij, j} = ρ {\overset{\cdot\cdot}{U}}_{i}

(3)

The stress-strain constitutive equations, taking into consideration damage mechanics, take the following form⁴⁴:

σ_{ij} = (1 - D) [λ δ_{ij} e_{kk} + 2 μ e_{ij} - β δ_{ij} (T - T_{0})]

(4)

The Green-Naghdi type-II of heat conduction equation under the effect of the mechanical damage takes the following form⁴⁴:

K^{*} T_{, ii} = ρ C_{υ} \overset{\cdot\cdot}{T} + T_{0} β δ_{ij} (1 - D) {\overset{\cdot\cdot}{e}}_{ij}

(5)

The deformation-displacement relations can take the following form:

e_{ij} = \frac{1}{2} (U_{i, j} + U_{j, i})

(6)

$i, j = x, y, z$ , $β = (3 λ + 2 μ) α_{T}$ , $α_{T}$ denotes the thermal linear expansion’s coefficient, $ρ$ is the density, $τ_{0}$ denotes the thermal relaxation time, $λ, μ$ denotes Lamè’s parameter under normal circumstances, $δ_{ij}$ is called the Kronecker delta function, $K^{*} = \frac{C_{υ} (λ + 2 μ)}{4}$ is called the characteristic of Green-Naghdi theory, $C_{υ}$ is the specified heat at constant strain, and $D$ is the mechanical damage variable.

Lamè’s parameters for viscothermoelastic materials have the following forms:

\begin{matrix} λ = λ_{0} (1 + λ_{1} \frac{\partial}{\partial t}), μ = μ_{0} (1 + μ_{1} \frac{\partial}{\partial t}), \\ β = (3 λ + 2 μ) α_{T} \end{matrix}

(7)

Problem formulation

We take that there are small deflections in a thin thermoelastic nanobeam of length $ℓ (0 \leq x \leq ℓ)$ , width $- b / 2 \leq y \leq b / 2$ , and thickness $- h / 2 \leq z \leq h / 2$ . We take that x, y, and z-axes are regarded as the longitude, width, and thickness of the beam, respectively as in Figure 1.

Figure 1.

Isotropic rectangular thermoelastic nanobeam.

In a state of equilibrium, the beam is not strained, not stressed, and free of mechanical damping, and the temperature is $T_{0}$ .⁶

The Euler-Bernoulli equation is considered in this study. While a result, we start with a plane’s cross-section that is initially perpendicular to the x-axis and stays level and perpendicular to the neutral surface as bending occurs.⁴⁵

Then, the displacement components can be formed as:

(u, v, w) = (- z \frac{\partial w (x, t)}{\partial x}, 0, w (x, t))

(8)

The equation of motion is given as:

\frac{\partial^{2} M (x, t)}{\partial x^{2}} + ρ A \frac{\partial^{2} w (x, t)}{\partial t^{2}} = 0

(9)

where $M (x, t)$ is the cross-flexural section’s moment and is given as follows:

M (x, t) = (1 - D) (λ + 2 μ) I \frac{\partial^{2} w (x, t)}{\partial x^{2}} + (1 - D) β M_{T} (x, t)

(10)

Moreover, $M_{T}$ denotes the moment of the thermal effect about the x-axis and is given by:

M_{T} (x, t) = b \int_{- h / h 2 2}^{h / h 2 2} T (x, z, t) zdz

(11)

Where $I = \frac{b h^{3}}{12}$ gives the moment of inertia of the cross-section about the x-axis.

Therefore, we may put the beam’s differential equation of lateral vibration that was induced thermally as⁴⁵:

\begin{matrix} (1 - D) (λ + 2 μ) I \frac{\partial^{4} w (x, t)}{\partial x^{4}} + (1 - D) β \frac{\partial^{2} M_{T} (x, t)}{\partial x^{2}} \\ + ρ A \frac{\partial^{2} w (x, t)}{\partial t^{2}} = 0 \end{matrix}

(12)

Where $w (x, t)$ is the lateral deflection, while $A = hb$ is the area of the beam’s cross-section.

The heat conduction equation based on non-Fourier law can be formed as⁴⁵:

\begin{matrix} \frac{\partial^{2} T (x, z, t)}{\partial x^{2}} + \frac{\partial^{2} T (x, z, t)}{\partial z^{2}} = \frac{ρ C_{υ}}{K^{*}} \overset{\cdot\cdot}{T} (x, z, t) \\ + \frac{β T_{0}}{K^{*}} (1 - D) \overset{\cdot\cdot}{e} (x, z, t) \end{matrix}

(13)

and

e (x, z, t) = \frac{\partial u (x, z, t)}{\partial x} + \frac{\partial v (x, z, t)}{\partial y} + \frac{\partial w (x, z, t)}{\partial z}

(14)

This volumetric strain is given as:

e (x, z, t) = - z \frac{\partial^{2} w (x, t)}{\partial x^{2}}

(15)

Based on the relation (7), we get:

\begin{matrix} λ + 2 μ = (λ_{0} + 2 μ_{0}) (1 + β_{1} \frac{\partial}{\partial t}), (3 λ + 2 μ) \\ = (3 λ_{0} + 2 μ_{0}) (1 + β_{2} \frac{\partial}{\partial t}) \end{matrix}

(16)

where $β_{1} = \frac{λ_{0} λ_{1} + 2 μ_{0} μ_{1}}{λ_{0} + 2 μ_{0}}$ and $β_{2} = \frac{3 λ_{0} λ_{1} + 2 μ_{0} μ_{1}}{3 λ_{0} + 2 μ_{0}}$ are the aggregation relaxation times parameters of the viscothermoelastic material.

Since heat does not transfer across the beam’s bottom and upper surfaces, we conclude that $\frac{\partial}{\partial z} T (x, \pm \frac{h}{2}, t) = 0$ .

If a thin beam is considered, we set variable temperature in terms of a $\sin (pz)$ function across the direction of thickness, thus, we have⁴⁶:

θ (x, z, t) = T (x, z, t) - T_{0} = ϑ (x, t) \sin (pz)

(17)

$θ (x, z, t)$ is the temperature increment and $p = \frac{π}{h}$ .

Thus, we obtain:

\begin{matrix} (1 - D) (λ_{0} + 2 μ_{0}) (1 + β_{1} \frac{\partial}{\partial t}) \frac{\partial^{4} w (x, t)}{\partial x^{4}} + \\ \frac{12 (3 λ_{0} + 2 μ_{0}) α_{T} (1 - D)}{h^{3}} (1 + β_{2} \frac{\partial}{\partial t}) \frac{\partial^{2} ϑ (x, t)}{\partial x^{2}} \\ \int_{- h / h 2 2}^{h / h 2 2} \sin (pz) zdz + \frac{12 ρ}{h^{2}} \overset{\cdot\cdot}{w} (x, t) = 0 \end{matrix}

(18)

Moreover, we obtain:

\begin{array}{l} (\frac{\partial^{2} ϑ (x, t)}{\partial x^{2}} - p^{2} ϑ (x, t)) \sin (p z) = ε \overset{\cdot\cdot}{ϑ} (x, t) \sin (p z) - \\ \frac{T_{0} (3 λ_{0} + 2 μ_{0}) α_{T} (1 - D)}{K^{*}} (1 + β_{2} \frac{\partial}{\partial t}) z \frac{\partial^{2} \overset{\cdot\cdot}{w} (x, t)}{\partial x^{2}} \end{array}

(19)

and

\begin{matrix} σ_{xx} (x, z, t) = (1 - D) \\ [e (x, z, t) - (3 λ_{0} + 2 μ_{0}) α_{T} (1 + β_{2} \frac{\partial}{\partial t}) ϑ (x, t) \sin (pz)] \end{matrix}

(20)

where $ε = \frac{ρ C_{υ}}{K^{*}}$ .

Upon integrating the equations, we get:

\begin{array}{l} (1 - D) (λ_{0} + 2 μ_{0}) (1 + β_{1} \frac{\partial}{\partial t}) \frac{\partial^{4} w (x, t)}{\partial x^{4}} + \\ \frac{24 (1 - D) (3 λ_{0} + 2 μ_{0}) α_{T}}{h π^{2}} (1 + β_{2} \frac{\partial}{\partial t}) \frac{\partial^{2} ϑ (x, t)}{\partial x^{2}} + \frac{12 ρ}{h^{2}} \overset{\cdot\cdot}{w} (x, t) = 0 \end{array}

(21)

We multiply both sides of the equation (19) by z and integrate with respect to z from $- \frac{h}{2} to \frac{h}{2}$ , then we get:

\begin{matrix} \frac{\partial^{2} ϑ (x, t)}{\partial x^{2}} - p^{2} ϑ (x, t) = ε \overset{\cdot\cdot}{ϑ} (x, t) \\ - \frac{T_{0} h π^{2} (1 - D) (3 λ_{0} + 2 μ_{0}) α_{T}}{24 K^{*}} (1 + β_{2} \frac{\partial}{\partial t}) \frac{\partial^{2} \overset{\cdot\cdot}{w} (x, t)}{\partial x^{2}} \end{matrix}

(22)

Then, the equation (18) becomes:

(x^{'}, w^{'}, h^{'}, ℓ^{'}) = \frac{ε c_{0}^{2}}{ℓ} (x, w, h, ℓ), (t^{'}, {β^{'}}_{1}, {β^{'}}_{2}) = \frac{ε c_{0}^{3}}{ℓ} (t, β_{1}, β_{2}), σ^{'} = \frac{σ}{λ + 2 μ}, ϑ^{'} = \frac{ϑ}{T_{0}}, c_{0}^{2} = \frac{λ + 2 μ}{ρ}

(23)

To keep matters simple, we use the following dimensionless variables¹⁹:

\begin{matrix} (x', w', h', ℓ') = \frac{ε c_{0}^{2}}{ℓ} (x, w, h, ℓ), (t', {β'}_{1}, {β'}_{2}) \\ = \frac{ε c_{0}^{3}}{ℓ} (t, β_{1}, β_{2}), σ = \frac{σ}{λ + 2 μ}, ϑ^{'} \\ = \frac{ϑ}{T_{0}}, c_{0}^{2} = \frac{λ + 2 μ}{ρ} \end{matrix}

(24)

Then:

\begin{matrix} (1 + β_{1} \frac{\partial}{\partial t}) \frac{\partial^{4} w (x, t)}{\partial x^{4}} + ε_{2} (1 + β_{2} \frac{\partial}{\partial t}) \frac{\partial^{2} ϑ (x, t)}{\partial x^{2}} \\ + ε_{1} \overset{\cdot\cdot}{w} (x, t) = 0 \end{matrix}

(25)

\begin{matrix} \frac{\partial^{2} ϑ (x, t)}{\partial x^{2}} - ε_{3} ϑ (x, t) = ε_{4} \overset{\cdot\cdot}{ϑ} (x, t) \\ - ε_{5} (1 + β_{2} \frac{\partial}{\partial t}) \frac{\partial^{2} \overset{\cdot\cdot}{w} (x, t)}{\partial x^{2}} \end{matrix}

(26)

and

\begin{matrix} σ_{xx} (x, z, t) = (1 - D) \\ [e (x, z, t) - γ (1 + β_{2} \frac{\partial}{\partial t}) ϑ (x, t) \sin (pz)] \end{matrix}

(27)

where

\begin{matrix} ε_{1} = \frac{12}{h^{2} (1 - D)}, ε_{2} = \frac{24 γ}{π^{2} h}, ε_{3} = p^{2}, \\ ε_{4} = ε c_{0}^{2}, ε_{5} = \frac{π^{2} c_{0}^{2} h α_{T} (1 - D) (3 λ_{0} + 2 μ_{0})}{24 K^{*}}, \\ γ = \frac{α_{T} T_{0} (3 λ_{0} + 2 μ_{0})}{λ_{0} + 2 μ_{0}} . \end{matrix}

(The prime has been omitted here just for convenience)

We apply Laplace transform for both sides of the equations (25)–(27) and (15), which is defined by:

\bar{f} (x; s) = \int_{0}^{\infty} e^{- s t} f (x; t) d t

(28)

Where the inversion of the Laplace transform takes the following form¹:

\begin{matrix} L^{- 1} (\bar{f} (x; s)) = f (t) \approx \frac{e^{κ t}}{t} \\ [\frac{1}{2} \bar{f} (x; κ) + Re \sum_{n = 1}^{N} {(- 1)}^{n} \bar{f} (x; κ + \frac{i n π}{t})] \end{matrix}

(29)

where “Re” is the real part and “i” is the imaginary unit.

To make convergence faster, numerous numerical experiments demonstrate that $κ$ satisfies $κ \approx 4.7$ .²

As a result, the following system of ordinary differential equations is obtained:

\begin{matrix} (1 + β_{1} s) \frac{\partial^{4} \bar{w} (x, s)}{\partial x^{4}} + ε_{2} (1 + β_{2} s) \frac{\partial^{2} \bar{ϑ} (x, s)}{\partial x^{2}} \\ + ε_{1} s^{2} \bar{w} (x, s) = 0 \end{matrix}

(30)

\begin{matrix} \frac{\partial^{2} \bar{ϑ} (x, s)}{\partial x^{2}} - ε_{3} \bar{ϑ} (x, s) = s^{2} ε_{4} \bar{ϑ} (x, s) \\ - ε_{5} (1 + β_{2} s) s^{2} \frac{\partial^{2} \bar{w} (x, s)}{\partial x^{2}} \end{matrix}

(31)

\begin{matrix} {\bar{σ}}_{xx} (x, z, s) = (1 - D) \\ [\bar{e} (x, z, s) - γ (1 + β_{2} s) \bar{ϑ} (x, s) \sin (pz)] \end{matrix}

(32)

and

\bar{e} (x, z, s) = - z \frac{\partial^{2} \bar{w} (x, s)}{\partial x^{2}}

(33)

Among applying the Laplace transforms these initial conditions have been used:

ϑ (x, 0) = w (x, 0) = \frac{\partial ϑ (x, 0)}{\partial t} = \frac{\partial w (x, 0)}{\partial t} = 0

(34)

Hence, we obtain:

\frac{\partial^{4} \bar{w} (x, s)}{\partial x^{4}} + α_{1} \bar{w} + α_{2} \frac{\partial^{2} ϑ (x, t)}{\partial x^{2}} = 0

(35)

\frac{\partial^{2} ϑ (x, t)}{\partial x^{2}} = α_{3} \bar{ϑ} (x, s) - α_{4} \frac{\partial^{2} \bar{w} (x, s)}{\partial x^{2}}

(36)

where $α_{1} = \frac{ε_{1} s^{2}}{1 + β_{1} s^{2}}, α_{2} = ε_{2} (\frac{1 + β_{2} s}{1 + β_{1} s}), α_{3} = ε_{3} + s^{2} ε_{4},$ $α_{4} = ε_{5} s^{2} (1 + β_{2} s)$ .

The system (35) and (36) gives us the following equation:

(\frac{\partial^{6}}{\partial x^{6}} - L \frac{\partial^{4}}{\partial x^{4}} + M \frac{\partial^{2}}{\partial x^{2}} - N) {\bar{ϑ} (x, s), \bar{w} (x, s)} = 0

(37)

where $L = α_{3} + α_{2} α_{4}, M = α_{1}, N = α_{1} α_{3}$ .

\bar{ϑ} (x, s) = - α_{4} \sum_{i = 1}^{3} A_{i} k_{i}^{2} \sinh (k_{i} (ℓ - x))

(38)

and

\bar{w} (x, s) = \sum_{i = 1}^{3} A_{i} (k_{i}^{2} - α_{3}) \sinh (k_{i} (ℓ - x))

(39)

We take $\pm k_{1}, \pm k_{2}, \pm k_{3}$ as the roots of the primary equation:

k^{6} - L k^{4} + M k^{2} - N = 0

(40)

The constants $A_{i} = A_{i} (s), i = 1, 2, 3$ are key here. To obtain them, we apply any set of boundary conditions. This allows us to regard the beam as simply supported and thermally loaded:

w (0, t) = \frac{\partial^{2} w (0, t)}{\partial x^{2}} = 0, θ (0, t) = ϑ_{0} g (t)

(41)

and

w (ℓ, t) = \frac{\partial^{2} w (ℓ, t)}{\partial x^{2}} = θ (ℓ, t) = 0

(42)

where $ϑ_{0}$ remains constant while $g (t)$ denoting the thermal loading function.

Equations (17) and (41) produce:

ϑ (0, t) = ϑ_{0} g (t)

(43)

Then, by using the Laplace transform, we obtain:

\bar{w} (0, s) = \frac{\partial^{2} \bar{w} (0, s)}{\partial x^{2}} = 0

(44)

\bar{ϑ} (0, s) = ϑ_{0} \bar{g} (t)

(45)

and

\bar{w} (ℓ, s) = \frac{\partial^{2} \bar{w} (ℓ, s)}{\partial x^{2}} = \bar{ϑ} (ℓ, s) = 0

(46)

Therefore, we get a system of linear equations:

\sum_{i = 1}^{3} A_{i} k_{i}^{2} \sinh (k_{i} ℓ) = - \frac{ϑ_{0} \bar{g} (t)}{α_{4}}

(47)

\sum_{i = 1}^{3} A_{i} (k_{i}^{2} - α_{3}) \sinh (k_{i} ℓ) = 0

(48)

and

\sum_{i = 1}^{3} A_{i} (k_{i}^{2} - α_{3}) k_{i}^{2} \sinh (k_{i} ℓ) = 0

(49)

Upon solving the above, we can obtain the solutions in the domain of the Laplace transform:

\bar{θ} (x, s) = \frac{ϑ_{0} \bar{g} (s) \sin (p z)}{α_{3}} [\begin{array}{l} \frac{(α_{3} - k_{2}^{2}) (α_{3} - k_{3}^{2}) k_{1}^{2}}{(k_{1}^{2} - k_{2}^{2}) (k_{1}^{2} - k_{3}^{2}) \sinh (k_{1} ℓ)} \sinh (k_{1} (ℓ - x)) + \\ \frac{(α_{3} - k_{1}^{2}) (α_{3} - k_{3}^{2}) k_{2}^{2}}{(k_{2}^{2} - k_{1}^{2}) (k_{2}^{2} - k_{3}^{2}) \sinh (k_{2} ℓ)} \sinh (k_{2} (ℓ - x)) + \\ \frac{(α_{3} - k_{1}^{2}) (α_{3} - k_{2}^{2}) k_{3}^{2}}{(k_{3}^{2} - k_{1}^{2}) (k_{3}^{2} - k_{2}^{2}) \sinh (k_{3} ℓ)} \sinh (k_{3} (ℓ - x)) \end{array}]

(50)

The lateral deflection is:

\bar{w} (x, s) = ϑ_{0} \bar{g} (s) α_{2} α_{3} [\begin{array}{l} \frac{1}{(k_{1}^{2} - k_{2}^{2}) (k_{1}^{2} - k_{3}^{2}) \sinh (k_{1} ℓ)} \sinh (k_{1} (ℓ - x)) + \\ \frac{1}{(k_{2}^{2} - k_{1}^{2}) (k_{2}^{2} - k_{3}^{2}) \sinh (k_{2} ℓ)} \sinh (k_{2} (ℓ - x)) + \\ \frac{1}{(k_{3}^{2} - k_{1}^{2}) (k_{3}^{2} - k_{2}^{2}) \sinh (k_{3} ℓ)} \sinh (k_{3} (ℓ - x)) \end{array}]

(51)

and the strain takes the form:

\bar{e} (x, s) = - z ϑ_{0} \bar{g} (s) α_{2} α_{3} [\begin{array}{l} \frac{1}{(k_{1}^{2} - k_{2}^{2}) (k_{1}^{2} - k_{3}^{2}) \sinh (k_{1} ℓ)} \sinh (k_{1} (ℓ - x)) + \\ \frac{1}{(k_{2}^{2} - k_{1}^{2}) (k_{2}^{2} - k_{3}^{2}) \sinh (k_{2} ℓ)} \sinh (k_{2} (ℓ - x)) + \\ \frac{1}{(k_{3}^{2} - k_{1}^{2}) (k_{3}^{2} - k_{2}^{2}) \sinh (k_{3} ℓ)} \sinh (k_{3} (ℓ - x)) \end{array}]

(52)

Using Refs.,^11,47,48 the strain-energy density function generated on the beam is:

ϖ (x, z, t) = \sum_{i, j = 1}^{3} \frac{1}{2} σ_{ij} e_{ij} = \frac{1}{2} σ_{xx} (x, z, t) e (x, z, t)

(53)

Thus, we have:

ϖ (x, z, t) = \frac{1}{2} [L^{- 1} ({\bar{σ}}_{x x} (x, z, s))] [L^{- 1} (\bar{e} (x, s))]

(54)

At this point, complete solutions are needed to proceed. We start by looking at the function of thermal loading $g (t)$ . We put the thermal loading ramp-type heat:

g (t) = {\begin{matrix} \frac{t}{t_{0}} & 0 < t < t_{0} \\ 1 & t \geq t_{0} \end{matrix}}

(55)

where $t_{0}$ remains constant and is called the ramp-type heat parameter.

Therefore:

\bar{g} (s) = \frac{1 - e^{- s t_{0}}}{s^{2} t_{0}}

(56)

Numerical results and discussion

Based on the previous discussion, we take a computational result as a numerical example. As such, silicon nitride has been used as the material as follows^43,45,49,50:

\begin{matrix} α_{T} = 2.71 {(10)}^{- 6} K^{- 1}, ρ = 3200 kg m^{- 3}, T_{0} 293 K, \\ C_{υ} = 630 J k g^{- 1} K^{- 1}, λ_{0} = 217 {(10)}^{9} N m^{- 2}, \\ λ_{0} = 108 {(10)}^{9} N m^{- 2}, τ_{0} = 4.32 {(10)}^{- 13} s, \\ λ_{1} = μ_{1} = 6.89 {(10)}^{- 13} s . \end{matrix}

(56)

The aspect ratios of the nanobeam are fixed as $\frac{ℓ}{h} = 10$ and $b = \frac{h}{2}$ . For the nanoscale beam, we will take the range of the beam length $ℓ$ in the nanometer scale and the original time t and the relaxation time $τ_{0}$ in the picosecond scale, respectively. The figures prepared to use non-dimensional variables for beam length $ℓ = 0.1, θ_{0} = 1.0, z = \frac{h}{3}, and t = 2.0$ . Figure 2(a) to (d) demonstrate the temperature increment, the lateral deflection, the stress, and the strain-energy density function for various values of the mechanical damage parameter $D = (0.0, 0.3)$ for the two cases of viscothermoelasticity when $(β_{1} \neq 0, β_{2} \neq 0)$ and non-viscothermoelasticity when $(β_{1} = β_{2} = 0)$ , respectively, for the situation $t_{0} < t$ .

Figure 2.

The studied functions when $t_{0} < t$ : (a) the temperature increment distribution, (b) the lateral deflection distribution,(c) the stress distribution, and (d) the stress-energy density distribution.

Figure 2(a) gives that neither the mechanical damage variable nor the viscothermoelastic parameter has any effect on the distribution of temperature increment.

Figure 2(b) shows that the mechanical damage variable and the viscothermoelastic parameter affect the lateral deflection distribution greatly. The four curves start and end at zero, noting that each curve has a peak. This aligns with the boundary conditions. The absolute values of the peak points of the lateral deflection take the following order:

\begin{matrix} | u_{Max} (\begin{matrix} D = 0.0 \\ β_{1} = β_{2} \neq 0 \end{matrix}) | > | u_{Max} (\begin{matrix} D = 0.0 \\ β_{1} = β_{2} = 0 \end{matrix}) | \\ > | u_{Max} (\begin{matrix} D = 0.3 \\ β_{1} = β_{2} \neq 0 \end{matrix}) | > | u_{Max} (\begin{matrix} D = 0.3 \\ β_{1} = β_{2} = 0 \end{matrix}) | \end{matrix}

(57)

If we increase the viscoelastic parameter, the absolute value of the lateral deflection for the damaged and undamaged situation increases. On the other hand, if we increase the mechanical damage parameter, the absolute value of the lateral deflection for the viscoelastic and non-viscoelastic cases decreases.

Figure 2(c) represents that the mechanical damage variable and the viscothermoelastic parameter have significant effects on the stress distribution. Each of the four curves starts with different values. They all end at zero though. By increasing the viscoelastic parameter, the absolute value of the stress for the damaged and undamaged situation increases. On the other hand, increasing the mechanical damage parameter decreases the absolute value of the stress for the viscoelastic and non-viscoelastic cases.

Figure 2(d) shows that the mechanical damage variable and the viscothermoelastic parameter have significant effects on the strain-energy density function. The four curves start and end at zero, each with their peak points, agreeing with the boundary conditions. The peak points of the stress-strain distribution assume the following order:

\begin{matrix} ϖ_{Max} (\begin{matrix} D = 0.0 \\ β_{1} = β_{2} \neq 0 \end{matrix}) > ϖ_{Max} (\begin{matrix} D = 0.0 \\ β_{1} = β_{2} = 0 \end{matrix}) \\ > ϖ_{Max} (\begin{matrix} D = 0.3 \\ β_{1} = β_{2} \neq 0 \end{matrix}) > ϖ_{Max} (\begin{matrix} D = 0.3 \\ β_{1} = β_{2} = 0 \end{matrix}) \end{matrix}

(58)

Similarly, increasing the viscoelastic parameter increases the absolute value of the lateral deflection for the damaged and undamaged situation. On the other hand, if we increase the mechanical damage parameter, the absolute value of the lateral deflection for the viscoelastic and non-viscoelastic cases decreases.

Figure 3(a) to (d) represent the temperature increment, the lateral deflection, the stress, and the stress-strain energy functions with different values of mechanical damage parameter $D = (0.0, 0.3)$ for the two cases of viscothermoelasticity $(β_{1} \neq 0, β_{2} \neq 0)$ and non-viscothermoelasticity $(β_{1} = β_{2} = 0)$ , respectively, when $t_{0} < t$ . We see that the behaviors of the functions under study are the same as in Figure 1. Based on the undamaged case, the viscothermoelastic and non-viscothermoelastic are very close. As for the damaged case, the viscothermoelastic and non-viscothermoelastic are different. This indicates that the heating’s ramping time parameter has a significant impact on the functions being studied.

Figure 3.

The studied functions when $t = t_{0}$ : (a) the temperature increment distribution, (b) the lateral deflection distribution,(c) the stress distribution, and (d) the stress-energy density distribution.

Figure 4(a) to (d) represent the temperature increment, the lateral deflection, the stress, and the strain-energy density function with different values of mechanical damage parameter $D = (0.0, 0.3)$ for the two cases of viscothermoelasticity $(β_{1} \neq 0, β_{2} \neq 0)$ and non-viscothermoelasticity $(β_{1} = β_{2} = 0)$ , respectively, when $t_{0} > t$ . Once again, we see that the behaviors are the same as in Figures 1 and 2. Based on the undamaged case, the viscothermoelastic and non-viscothermoelastic are very close. However, the viscothermoelastic and non-viscothermoelastic are different for the damaged case. This means that the ramping time parameter of heating affects the functions understudy greatly.

Figure 4.

The studied functions when $t < t_{0}$ : (a) The temperature increment distribution, (b) The lateral deflection distribution,(c) The stress distribution, and (d) The stress-energy density distribution.

Figure 5(a) to (d) show the temperature increment, the lateral deflection, the stress, and the stress-strain energy distributions for various values of ramping time parameter $t_{0} = (1.0, 2.0, 3.0)$ based on viscothermoelasticity $(β_{1} = β_{2} \neq 0)$ and damaged situation $D = 0.3$ . It is noted that the ramping time parameter has notable effects on the functions understudy. Figure 4(a) shows the temperature increment distribution. Any increase in the ramping time heat parameter decreases the temperature increment. The cases $t < t_{0}$ and $t = t_{0}$ are very close.

Figure 5.

The studied functions when $β_{1} = β_{2} - 0.02 and D = 0.3$ and various values of ramping heat parameter: (a) the temperature increment distribution, (b) the lateral deflection distribution, (c) the stress distribution, and (d) the stress-energy density distribution.

Figure 5(b) represents the lateral deflection where the three curves start and end at zero. Every curve has a peak point that assumes the following order:

| u_{Max} (t_{0} = 2.0) | > | u_{Max} (t_{0} = 1.0) | > | u_{Max} (t_{0} = 3.0) | >

(59)

Figure 5(c) demonstrates the distribution of stress. We see that the curves of the cases $t_{0} < t$ and $t = t_{0}$ start with the same value. As for the case $t_{0} > t$ , it has a different starting point. The absolute value of the starting point of the cases $t_{0} < t$ and $t = t_{0}$ is higher than the absolute value of the starting point of the case $t_{0} > t$ .

Figure 5(d) represents the strain-energy density function where the three curves start and end at zero. Each curve peaks in the following order:

ϖ_{Max} (t_{0} = 2.0) > ϖ_{Max} (t_{0} = 1.0) > ϖ_{Max} (t_{0} = 3.0)

(60)

We may conclude that the ramp-type heating parameter is impactful on all functions understudy, and it can be used as a controller or tuner to the energy generated on the beam.

For validation, the results of the current work agree with the results of the reference.⁵¹

Conclusion

Based on the previous discussion, we have examined a viscothermoelastic nanobeam when it is simply supported and thermally loaded using ramp-type heating under the influence of damage mechanics. We conclude that the damage parameters, the mechanical relaxation times parameters, and the ramp-time heat parameter affect the reaction and the vibration of the nanobeam greatly.

Therefore, we conclude the followings:

Increasing the viscoelastic parameter increases the absolute value of the lateral deflection (vibration), the absolute value of stress, and the strain-energy density function for the damaged and undamaged situation.

On the other hand, if the mechanical damage parameter increases, the absolute value of the lateral deflection, absolute value of stress, and strain-energy density function for the viscoelastic and non-viscoelastic situations decrease.

The ramp-type heating parameter is essential in the energy damping of the nanobeam and could be used to tune the energy which has been generated through the thermoelastic nanobeam and plays a vital role in its damping.

Footnotes

Acknowledgements

Not Applicable.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hamdy M Youssef

Data availability statement

Data sharing does not apply to this article as no datasets were generated or analyzed during the current study.

References

Tzou

. On the thermal shock wave induced by a moving heat source. J Heat Transf 1989; 111(2): 232–238.

Tzou

. Macro- to microscale heat transfer: the lagging behavior. Washington, DC: Taylor Francis, 1997.

Guo

Wang

, et al. Thermal wave interference as the origin of the overshooting phenomenon in dual-phase-lagging heat conduction. Int J Therm Sci 2011; 50(5): 825–830.

Al-Huniti

Al-Nimr

. Thermoelastic behavior of a composite slab under a rapid dual-phase-lag heating. J Therm Stresses 2004; 27(7): 607–623.

J-R

Kuo

C-P

Jiaung

W-S

. Study of heat transfer in multilayered structure within the framework of dual-phase-lag heat conduction model using lattice Boltzmann method. Int J Heat Mass Transf 2003; 46(1): 55–69.

Lee

Y-M

Tsai

T-W

. Ultra-fast pulse-laser heating on a two-layered semi-infinite material with interfacial contact conductance. Int Commun Heat Mass Transf 2007; 34(1): 45–51.

Liu

K-C

. Numerical analysis of dual-phase-lag heat transfer in a layered cylinder with nonlinear interface boundary conditions. Comput Phys Commun 2007; 177(3): 307–314.

Ramadan

. Semi-analytical solutions for the dual phase lag heat conduction in multilayered media. Int J Therm Sci 2009; 48(1): 14–25.

Alghamdi

. Dual-phase-lagging thermoelastic damping vibration in micro-nano scale beam resonators with voids. Int J Multidiscip Curr Res 2017; 5: 71–78.

10.

Alghamdi

Youssef

. Dual-phase-lagging thermoelastic damping in-extensional vibration of rotating nano-ring. Microsyst Technol 2017; 23(10): 4333–4343.

11.

Biot

. Thermoelasticity and irreversible thermodynamics. J Appl Phys 1956; 27(3): 240–253.

12.

Youssef

Alghamdi

. Thermoelastic damping in nanomechanical resonators based on two-temperature generalized thermoelasticity theory. J Therm Stresses 2015; 38(12): 1345–1359.

13.

Lord

Shulman

. A generalized dynamical theory of thermoelasticity. J Mech Phys Solids 1967; 15(5): 299–309.

14.

Dhaliwal

Sherief

. Generalized thermoelasticity for anisotropic media. Q Appl Math 1980; 38(1): 1–8.

15.

Hoang

. Thermoelastic damping depending on vibration modes of nanobeam resonator. Commun Phys 2016; 25(4): 317.

16.

Naik

Hanay

Hiebert

, et al. Towards single-molecule nanomechanical mass spectrometry. Nat Nanotechnol 2009; 4(7): 445–450.

17.

O’Connell

Hofheinz

Ansmann

, et al. Quantum ground state and single-phonon control of a mechanical resonator. Nature 2010; 464(7289): 697–703.

18.

van Beek

JTM

Puers

. A review of MEMS oscillators for frequency reference and timing applications. J Micromech Microeng 2012; 22(1): 013001.

19.

Sharma

Grover

. Thermoelastic vibrations in micro-/nano-scale beam resonators with voids. J Sound Vib 2011; 330(12): 2964–2977.

20.

Sun

Saka

. Thermoelastic damping in micro-scale circular plate resonators. J Sound Vib 2010; 329(3): 328–337.

21.

Lifshitz

Roukes

. Thermoelastic damping in micro- and nanomechanical systems. Phys Rev B 2000; 61(8): 5600–5609.

22.

Al-Huniti

Al-Nimr

Naji

. Dynamic response of a rod due to a moving heat source under the hyperbolic heat conduction model. J Sound Vib 2001; 242(4): 629–640.

23.

Al-Lehaibi

EAN

Youssef

. Vibration of gold nano-beam with variable Young’s modulus due to thermal shock. World J Nano Sci Eng 2015; 05(04): 194–203.

24.

Boley

. Approximate analyses of thermally induced vibrations of beams and plates. J Appl Mech 1972; 39(1): 212–216.

25.

Kidawa-Kukla

. Application of the Green functions to the problem of the thermally induced vibration of a beam. J Sound Vib 2003; 262(4): 865–876.

26.

Manolis

Beskos

. Thermally induced vibrations of beam structures. Comput Methods Appl Mech Eng 1980; 21(3): 337–355.

27.

Hadi

Nejad

Rastgoo

, et al. Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory. Steel Compos Struct 2018; 26(6): 663–672.

28.

Sahmani

Safaei

. Influence of homogenization models on size-dependent nonlinear bending and post-buckling of bi-directional functionally graded micro/nano-beams. Appl Math Model 2020; 82: 336–358.

29.

Dangi

Saini

Lal

, et al. Size dependent FEM model for Bi-directional functionally graded nano-beams. Mater Today Proc 2020; 24: 1302–1311.

30.

Zhang

Wang

, et al. Mechanical behavior of coupled elastoplastic damage of clastic sandstone of different burial depths. Energies 2020; 13(7): 1640.

31.

Green

Naghdi

. Thermoelasticity without energy dissipation. J Elast 1993; 31(3): 189–208.

32.

Green

Naghdi

. A re-examination of the basic postulates of thermomechanics. Proc R Soc Lond A Math Phys Sci 1991; 432(1885): 171–194.

33.

Hongwei

Sahmani

Safaei

. On size-dependent large-amplitude free oscillations of FGPM nanoshells incorporating vibrational mode interactions. Arch Civil Mech Eng 2020; 20: 1–23.

34.

Fan

Cai

Sahmani

, et al. Isogeometric thermal post-buckling analysis of porous FGM quasi-3D nanoplates having cutouts with different shapes based upon surface stress elasticity. Compos Struct 2021; 262: 113604.

35.

Chinika

Lal

Sukavanam

. Effect of surface stresses on the dynamic behavior of bi-directional functionally graded nonlocal strain gradient nanobeams via generalized differential quadrature rule. Eur J Mech-ASolids 2021; 90: 104376.

36.

Lal

Dangi

. Dynamic analysis of bi-directional functionally graded Timoshenko nanobeam on the basis of Eringen's nonlocal theory incorporating the surface effect. Appl Math Comput 2021; 395: 125857.

37.

Gross

Seelig

. Fracture mechanics: with an introduction to micromechanics. Berlin: Springer, 2017.

38.

Öchsner

. Continuum damage mechanics. In: Mooney MJ (ed.) Continuum damage and fracture mechanics. Berlin: Springer, 2016, pp.65–84.

39.

Voyiadjis

. Handbook of damage mechanics: nano to macro scale for materials and structures. Berlin: Springer, 2015.

40.

Yao

Keer

, et al. A continuum damage mechanics-based unified creep and plasticity model for solder materials. Acta Mater 2015; 83: 160–168.

41.

Voyiadjis

Kattan

. Introducing damage mechanics templates for the systematic and consistent formulation of holistic material damage models. Acta Mech 2017; 228(3): 951–990.

42.

Khatir

Tehami

Khatir

, et al. Multiple damage detection and localization in beam-like and complex structures using co-ordinate modal assurance criterion combined with firefly and genetic algorithms. J Vibroengineering 2016; 18(8): 5063–5073.

43.

Grover

. Transverse vibrations in micro-scale viscothermoelastic beam resonators. Arch Appl Mech 2013; 83(2): 303–314.

44.

Saanouni

Mariage

J-F

Cherouat

, et al. Numerical prediction of discontinuous central bursting in axisymmetric forward extrusion by continuum damage mechanics. Comput Struct 2004; 82(27): 2309–2332.

45.

Grover

. Viscothermoelastic vibrations in micro-scale beam resonators with linearly varying thickness. Can J Phys 2012; 90(5): 487–496.

46.

Youssef

Al-Ghamdi

. Vibration of gold nano beam in context of two-temperature generalized thermoelasticity without energy dissipation. In: ICTEA: international conference on thermal engineering, 2017.

47.

Biot

. Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena. J Appl Phys 1954; 25(11): 1385–1391.

48.

Biot

. Variational principles in irreversible thermodynamics with application to viscoelasticity. Phys Rev 1955; 97(6): 1463–1469.

49.

Grover

. Damping in thin circular viscothermoelastic plate resonators. Can J Phys 2015; 93(12): 1597–1605.

50.

El-Bary

Youssef

Nasr

MAE

. Hyperbolic two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to the non-Gaussian laser beam. J Umm Al-Qura Univ Eng Archit 2022; 13(1–2): 62–69.

51.

Youssef

El-Bary

. Generalized fractional viscothermoelastic nanobeam under the classical Caputo and the new Caputo–Fabrizio definitions of fractional derivatives. Waves Random Complex Media 2023; 33: 545–566.