Closed-Form Solutions for Gradient Elastic Beams with Geometric Discontinuities by Laplace Transform

Abstract

The static bending solution of a gradient elastic beam with external discontinuities is presented by Laplace transform. Its utility lies in the ability to switch differential equations to algebraic forms that are more easily solved. A Laplace transformation is applied to the governing equation which is then solved for the static deflection of the microbeam. The exact static response of the gradient elastic beam with external discontinuities is obtained by applying known initial conditions when the others are derived from boundary conditions. The results are given in a series of figures and compared with their classical counterparts. The main contribution of this paper is to provide a closed-form solution for the static deflection of microbeams under geometric discontinuities.

1. Introduction

The microbeams are one of the major structures used widely in the fields of MEMS such as those in actuators [1], microswitches [2], microresonators [3], Atomic Force Microscopes [4], and sensors [5] in which thicknesses and lengths of microbeams are typically on the order of microns and submicrons. A large number of those applications utilized the dynamic mechanical properties of thin films materials for targeted performance specifications such as those vibration atomic force microscopes [6].

Microbeams used in MEMS have the dimensions in order of microns and submicrons. It is experimentally observed that not only does the conventional strain-based mechanics underestimate the stiffness of microscale components such as sensors but also it is unable to justify the size effects appearing in these structures [7, 8]. Due to the lacking of internal length scale parameters, classical strain-based mechanics theories fail to characterize those size effects phenomenon when the structural size is in micron and submicron scale. However, these size dependences can be successfully modeled by employing higher-order continuum theories, in which constitutive equations introduce additional length scale parameters in addition to conventional material parameters [9, 10]. There have been considerable studies on size effects in problems of vibration, bending, buckling, and torsion of microbeams based on nonlocal elasticity theory [11–13]. Yang et al. [14] have proposed the modified couple stress theory recently and only one length scale parameter is included. After this, the modified couple stress and the strain gradient elasticity theories have been widely applied to static and dynamic analysis of microbeams [9, 15–17].

In this study, the static behavior of gradient elastic beams subjected to geometric discontinuties is theoretically investigated in the Laplace domain. Its utility lies in the ability to switch differential equations to algebraic forms that are more easily solved. The material and microbeam models are assumed to obey the strain gradient theory, as developed in [18, 19]. A Laplace transformation is applied to the governing equation which is then solved for the static deflection of the microbeam. The solutions obtained are transformed to the time domain using the inverse Laplace transform. Static deflection curve is computed exactly and expressed in closed form.

2. Review of Laplace Transformation and Gradient Elasticity Theory

2.1. Laplace Transform Applied to Differential Equations

Laplace transformation is a very useful tool for computation of higher order differantial equations. This transformation helps to transform differantial equations into the form of algebraic equations which is easier to manipulate. Let Ω(𝑡) is a function on the interval [0,∞). The Laplace transform of Ω(𝑡) is the function in terms of 𝑠 as follows: ℒ_𝑡{Ω(𝑡)}=^∞₀𝑒^−𝑠𝑡Ω(𝑡)𝑑𝑡. (1) Since the Laplace transform yields a function of 𝑠, the notation ℒ[Ω(𝑡)]=𝑌(𝑠) is used to denote the Laplace transform of Ω(𝑡). Laplace transforms can be used to solve initial value problems, by transforming a differantial equation to an algebraic equation. Consider the following initial value problem: 𝜛_𝑛𝑑^𝑛Ω𝑑𝑡^𝑛+𝜛_𝑛−1𝑑^𝑛−1Ω𝑑𝑡^𝑛−1+𝜛_𝑛−2𝑑^𝑛−2Ω𝑑𝑡^𝑛−2+⋯+𝜛₀(𝑡),Ω(0)Ω=Ψ=Ω₀,Ω^′(0)=Ω₁,Ω^′′(0)=Ω₂,…,Ω^(𝑛−1)(0)=Ω_(𝑛−1), (2)(3) in which 𝜛_𝑛, 𝜛_𝑛−1, 𝜛_𝑛−2,…,𝜛₀ denote the constants and (3) represents the initial conditions. Taking the Laplace transforms of each term yields 𝜛_𝑛ℒ𝑑^𝑛Ω𝑑𝑡^𝑛+𝜛_𝑛−1ℒ𝑑^𝑛−1Ω𝑑𝑡^𝑛−1+𝜛_𝑛−2ℒ𝑑^𝑛−2Ω𝑑𝑡^𝑛−2+⋯+𝜛₀ℒ{Ω}=ℒ{Ψ(𝑡)}. (4) Using the property of the Laplace transformation of derivatives, one can obtain 𝜛_𝑛𝑠^𝑛𝑌(𝑠)−𝑠^𝑛−1Ω(0)−⋯−Ω^𝑛−1(0)+𝜛_𝑛−1𝑠^𝑛−1𝑌(𝑠)−𝑠^𝑛−2Ω(0)−⋯−Ω^𝑛−2(0)+⋯+𝜛₀𝑌(𝑠)=𝐺(𝑠). (5) The above algebraic equation can be solved for 𝑌(𝑠) and then take the inverse Laplace transform (ℒ⁻¹) to obtain the expression for Ω(𝑡).

2.2. Strain Gradient Elastic Euler Bernoulli Beam Model

In the present study the following gradient elasticity theory is employed which is suitable for investigating the static deflections of microbeams [18]. Consider 𝜎_𝑖𝑗=𝐶_{𝑖𝑗𝑘𝑙}𝜖_𝑘𝑙−𝛾²𝜖_{𝑘𝑙,𝑚𝑛}, (6) in which 𝜎_𝑖𝑗 and 𝜖_𝑖𝑗 represent the stress and strain tensors, 𝐶_{𝑖𝑗𝑘𝑙} represents the elastic moduli, and 𝛾 denotes the simply gradient elastic modulus, representing the internal or characteristic length of the material microstructure. It is assumed that the microbeam subjected to a concentrated load, as shown in Figure 1. The potential energy of the beam is defined as 1𝑈=2EI^𝐿₀𝑑²𝑣𝑑𝑧²²−𝛾²𝑑²𝑣𝜕𝑧²𝑑⁴𝑣𝑑𝑧⁴𝑑𝑧, (7) where EI is the bending stiffness of the microbeam. The variation of the external concentrated load is obtained as: 𝜑𝑊_𝐸=^𝐿₀𝑃₀𝛿(𝑧−𝑎)𝜑𝑣𝑑𝑧, (8) where 𝑃₀ is a concentrated force at an arbitrary point. In order to obtain the governing differantial equation of the system, the above equations substituted into the Hamilton principle resulting in [18] 𝑑EI⁴𝑣𝑑𝑧⁴−𝛾²𝜕⁶𝑣𝜕𝑧⁶−𝑃₀𝛿(𝑧−𝑎)=0. (9)

Figure 1 

Schematic diagram for a microbeam fixed at two ends with concentrated loading.

2.3. Weak-Form Formulation of the Microbeam Equation and Nonclassical Boundary Conditions

The starting point for deriving the weak form is to multiply the differential equation of gradient elastic beam with a test function and integrate it over the domain, ^𝐿₀𝜓(𝑧)𝑑EI⁴𝑣𝑑𝑧⁴−𝛾²𝑑⁶𝑣𝑑𝑧⁶−𝑃₀𝛿(𝑧−𝑎)𝑑𝑧=0, (10) where 𝜓(𝑧) is the test function and is integrating by parts as follows: 0=^𝐿₀𝑑EI²𝜓𝑑𝑧²𝑑²𝑣𝑑𝑧²−𝛾²𝑑³𝜓𝑑𝑧³𝑑³𝑣𝑑𝑧³−𝜓(𝑧)𝑃₀𝛿()+𝜓(𝑧)𝑑𝑧−𝑎𝑑𝑧EI³𝑣𝑑𝑧³−𝛾²𝑑⁵𝑣𝑑𝑧^5𝐿₀−𝑑𝜓(𝑧)𝑑𝑑𝑧EI²𝑣𝑑𝑧²−𝛾²𝑑⁴𝑣𝑑𝑧^4𝐿₀−𝛾²𝑑²𝜓(𝑧)𝑑𝑧²𝑑EI³𝑣𝑑𝑧^3𝐿₀. (11) The presented (11) is a weak derivation of governing equation and the boundary conditions for the microbeam based on gradient elasticity theory. It should be noted that when the scale parameter 𝛾 is taken to be zero in (11), the gradient elastic beam model decreases to the classical beam model. The boundary conditions for the microbeam fixed at two ends (see Figure 1) are 𝑣|_𝑧=0=0,𝜕𝑣𝜕𝑧_𝑧=0𝜕=0,³𝑣𝜕³𝑧_𝑧=0=0,𝑣|_𝑧=𝐿=0,𝜕𝑣𝜕𝑧_𝑧=𝐿𝜕=0,³𝑣𝜕³𝑧_𝑧=𝐿=0. (12)(13)

3. Application of Laplace Transform in Gradient Elasticity

In this section, a Laplace transformation is performed to the governing equation in gradient elasticity which is then solved for the static deflection of the microbeam.

3.1. A Microbeam Subjected to a Concentrated Load

Consider a microbeam subjected to a concentrated load. The concentrated load at 𝑧=𝑎 can be represented by 𝑃₀𝛿(𝑧−𝑎), where 𝛿(⋅) is the Dirac delta function or impulse function. Then the differential equation for the static deflection of the gradient elastic beam is 𝑑EI⁴𝑣𝑑𝑧⁴−𝛾²𝑑⁶𝑣𝑑𝑧⁶=𝑃₀𝛿(𝑧−𝑎). (14) Taking the Laplace transforms of both sides, 𝑃₀1−𝑒^−𝑎𝑠EI𝑠=−𝛾²𝑠⁶ℒ_𝑧[𝑣(𝑧)](𝑠)−𝑣⁽⁵⁾(0)−𝑣⁽⁴⁾(0)𝑠−𝑣⁽³⁾(0)𝑠²−𝑠³𝑣^′′(0)−𝑠⁴𝑣^′(0)−𝑣(0)𝑠⁵+𝑠⁴ℒ_𝑧[𝑣(𝑧)](𝑠)−𝑣⁽³⁾(0)−𝑠𝑣^′′(0)−𝑠²𝑣^′(0)−𝑣(0)𝑠³ (15) or ℒ_𝑧[𝑣(𝑧𝑠)𝑒)](=−^−𝑎𝑠EI𝑠𝑠⁴−𝛾²𝑠⁶EI𝛾²𝑣⁽⁵⁾(0)𝑠𝑒^𝑎𝑠+EI𝛾²𝑣⁽⁴⁾(0)𝑠²𝑒^𝑎𝑠+EI𝛾²𝑣⁽³⁾(0)𝑠³𝑒^𝑎𝑠−EI𝑣⁽³⁾(0)𝑠𝑒^𝑎𝑠+EI𝛾²𝑠⁴𝑒^𝑎𝑠𝑣^′′(0)−EI𝑠²𝑒^𝑎𝑠𝑣^′′(0)+EI𝛾²𝑠⁵𝑒^𝑎𝑠𝑣^′(0)−EI𝑠³𝑒^𝑎𝑠𝑣^′(0)+EI𝛾²𝑣(0)𝑠⁶𝑒^𝑎𝑠−EI𝑣(0)𝑠⁴𝑒^𝑎𝑠−𝑃₀𝑒^𝑎𝑠+𝑃₀. (16) Using the three known initial conditions in (12), ℒ_𝑧[𝑣(𝑧)](𝑠)=−𝑃₀𝑒^−𝑎𝑠−EI𝛾²𝐶₂𝑠²+EI𝐶₁𝑠²−𝛾²𝑠⁴−EI𝛾²𝐶₃𝑠+𝑃₀×EI𝑠⁵−𝛾²𝑠⁷⁻¹, (17) where 𝐶₁=𝑣^′′(0),𝐶₂=𝑣⁽⁴⁾(0),𝐶₃=𝑣⁽⁵⁾(0). (18) The problem is to find the solution of the differantial equation, as a function of 𝐶₁, 𝐶₂, and 𝐶₃ and then determine these constants by using the information at the point 𝑧=𝐿. The inverse Laplace transform (ℒ⁻¹) of (17) yields ℒ⁻¹{𝑣(𝑠)}=ℒ⁻¹−𝑃₀𝑒^−𝑎𝑠−EI𝛾²𝐶₂𝑠²+EI𝐶₁𝑠²−𝛾²𝑠⁴−EI𝛾²𝐶₃𝑠+𝑃₀×EI𝑠⁵−𝛾²𝑠⁷⁻¹,𝑣(𝑧)=1EI−𝑃₀𝜃()×−1𝑧−𝑎2𝛾⁴𝑒^{(𝑎−𝑧)/𝛾}−12𝛾⁴𝑒^{(𝑧−𝑎)/𝛾}+12𝛾²(𝑎−𝑧)²+1()24𝑎−𝑧⁴+𝛾⁴−16𝛾²𝐶₃EI𝑧³−3𝛾³𝑒^−𝑧/𝛾𝑒^2𝑧/𝛾−1+6𝛾²𝑧−12𝛾²𝐶₂×𝑧EI₂−𝛾²𝑒^−𝑧/𝛾𝑒^𝑧/𝛾−1²+12𝐶₁EI𝑧²−𝛾²𝑒^−𝑧/𝛾𝑒^𝑧/𝛾−1²+12𝛾²𝐶₁EI𝑒^−𝑧/𝛾𝑒^𝑧/𝛾−1²+1𝑃24⁰𝑧⁴+12𝛾²𝑡²−12𝛾⁴𝑒^−𝑧/𝛾𝑒^𝑧/𝛾−1², (19)(20) where 𝜃(⋅) is the Heaviside step function. Note that the Heaviside step function in (20) turns on at the right edge (𝑧=𝑎⁺) and the Dirac delta function turns on and off at the same point so we can write the above equation in two parts. For 0≤𝑧≤𝑎, 𝑣(𝑧)^{−𝑎}=1𝐶EI¹12EI𝑧²−𝛾²𝑒^−𝑧/𝛾𝑒^𝑧/𝛾−1²+12𝛾²EI𝑒^−𝑧/𝛾𝑒^𝑧/𝛾−1²−16𝛾²𝐶₃𝑧³−3𝛾³𝑒^−𝑧/𝛾𝑒^2𝑧/𝛾−1+6𝛾²𝑧−12𝛾²𝐶₂𝑧²−𝛾²𝑒^−𝑧/𝛾𝑒^𝑧/𝛾−1² (21) and for 𝑎≤𝑧≤𝐿, 𝑣(𝑧)^{+𝑎}=1𝐶EI¹12EI𝑧²−𝛾²𝑒^−𝑧/𝛾𝑒^𝑧/𝛾−1²+12𝛾²EI𝑒^−𝑧/𝛾𝑒^𝑧/𝛾−1²−16𝛾²𝐶₃𝑧³−3𝛾³𝑒^−𝑧/𝛾𝑒^2𝑧/𝛾−1+6𝛾²𝑧−12𝛾²𝐶₂×𝑧²−𝛾²𝑒^−𝑧/𝛾𝑒^𝑧/𝛾−1²+1𝑃EI⁰−12𝛾⁴𝑒^{(𝑎−𝑧)/𝛾}−12𝛾⁴𝑒^{(𝑧−𝑎)/𝛾}+12𝛾²(𝑎−𝑧)²+1()24𝑎−𝑧⁴+𝛾⁴. (22)𝐶₁, 𝐶₂, and 𝐶₃ are constants to be calculated with the aid of other boundary conditions and (22). For simplicity, the following quantities are defined: 𝜂₁=𝑒^𝐿/𝛾;𝜂₂=𝜂²₁𝜂−1;₃=𝜂₁+1;𝜂₄=𝜂₁𝛼−1;₁=12𝛾³−𝐿³+6𝛾²𝛼𝐿;₂=24𝛾⁴+𝐿⁴−8𝛾²𝐿²;𝛼₃=−144𝛾⁵+𝐿⁵+6𝛾𝐿⁴−12𝛾²𝐿³−96𝛾³𝐿²−216𝛾⁴𝛼𝐿;₄=144𝛾⁵+𝐿⁵−6𝛾𝐿⁴−12𝛾²𝐿³+96𝛾³𝐿²−216𝛾⁴𝛼𝐿;₅=12𝛾³+𝐿³−6𝛾²𝛽𝐿;₁=−12𝛾³+𝐿³−12𝛾²𝛽𝐿;₂=−24𝛾⁴+𝐿⁴−12𝛾²𝐿²−24𝛾³𝛽𝐿;₃=−24𝛾⁴+𝐿⁴−12𝛾²𝐿²+24𝛾³𝛽𝐿;₄=24𝛾⁴+𝐿⁴−12𝛾²𝐿²;𝛽₅=12𝛾³+𝐿³−12𝛾²𝐿. (23) By using (23), 𝐶₁, 𝐶₂, and 𝐶₃ parameters can be calculated as follows: 𝐶₁1=−12EI𝐿24𝛾³𝜂²₄+𝜂₂𝐿³−12𝛾²𝜂₂𝐿×𝑃₀𝑒^−𝑎/𝛾×3𝑎⁴𝜂₄𝐿𝑒^𝑎/𝛾×−2𝛾𝜂₁+2𝛾+𝜂₁𝐿+𝐿−4𝑎³𝑒^𝑎/𝛾×12𝛾³𝜂²₄+2𝜂₂𝐿³−3𝛾𝜂²₄𝐿²−6𝛾²𝜂₂𝐿+6𝑎²𝐿𝑒^𝑎/𝛾12𝛾³𝜂²₄+𝜂₂𝐿³−6𝛾²𝜂₂𝐿−12𝑎𝛾𝐿𝑒^𝑎/𝛾×12𝛾³𝜂₂+𝜂²₁+1𝐿³−6𝛾²𝜂²₃𝐿+𝐿−12𝛼₁𝛾²𝑒^2𝑎/𝛾+𝛼₃𝑒^𝑎/𝛾−12𝛼₅𝛾²𝜂²₁+12𝛼₂𝛾𝑒^{(𝑎+𝐿)/𝛾}−𝛼₄𝑒^{(𝑎+2𝐿)/𝛾}+72𝛾⁴2𝛾−𝐿×𝑒^{(2𝑎+𝐿)/𝛾}+72𝛾⁴𝜂₁,𝐶2𝛾+𝐿₂1=−2𝛾EI24𝛾³𝜂²₄+𝜂₂𝐿³−12𝛾²𝜂₂𝐿×𝑃₀𝑒^−𝑎/𝛾×𝑎⁴𝜂²₄−𝑒^𝑎/𝛾+2𝑎³𝜂²₄𝐿𝑒^𝑎/𝛾−12𝑎²𝛾²𝜂²₄𝑒^𝑎/𝛾+2𝛽₁𝛾𝑒^2𝑎/𝛾+𝛽₂𝑒^𝑎/𝛾−2𝑎𝑒^𝑎/𝛾×12𝛾³𝜂₂+𝜂²₁+1𝐿³−12𝛾²𝜂²₁+1𝐿+𝛽₃𝑒^{(𝑎+2𝐿)/𝛾}+2𝛽₄𝑒^{(𝑎+𝐿)/𝛾}+24𝛾⁴𝑒^{(2𝑎+𝐿)/𝛾}−2𝛽₅𝛾𝜂²₁+24𝛾⁴𝜂₁,𝐶₃=𝑃₀𝑒^−𝑎/𝛾2𝛾²EI24𝛾³𝜂₁−24𝛾³+𝜂₃𝐿³−12𝛾²𝐿×𝑒^𝑎/𝛾−𝑎𝜂₃𝑎³−2𝑎²𝐿+12𝑎𝛾²+2𝐿³−24𝛾²𝐿+24𝛾⁴𝑒^𝑎/𝛾+24𝑎𝛾³+𝜆₁+𝜆₂𝑒^𝐿/𝛾−24𝛾³𝜂₁𝑎𝑒^𝑎/𝛾−𝛾. (24) Consequently, substituting (24) into (21) and (22), one can get the higher-order deflection functions.

3.2. A Microbeam Subjected to Discontinuous Loading

In this section, it is assumed that the microbeam carries a discontinuous load, as shown in Figure 2. Loading functions are expressed in terms of multiple expressions in different regions (25). Consider 𝑞(𝑧)=𝑞₀,0<𝑧<𝑎,0,𝑎<𝑧<𝐿. (25) This multiple expressions can be simplified by writing a single expression using singularity functions. It is clear that the microbeam is composed of two microbeam segments in the intervals [0,𝑎] and [𝑎,𝐿] so the above equation can be written in terms of Heaviside’s unit function as 𝑞(𝑧)=𝑞₀{𝜗(𝑧)−𝜗(𝑧−𝑎)}. (26) Then, the governing differantial equation is 𝑑EI⁴𝑣𝑑𝑧⁴−𝛾²𝑑⁶𝑣𝑑𝑧⁶=𝑞₀{𝜗(𝑧)−𝜗(𝑧−𝑎)}. (27) Taking the Laplace transforms, 𝑞₀1−𝑒^−𝑎𝑠EI𝑠=−𝛾²𝑠⁶ℒ_𝑧[𝑣(𝑧)](𝑠)−𝑣⁽⁵⁾(0)−𝑣⁽⁴⁾(0)𝑠−𝑣⁽³⁾(0)𝑠²−𝑠³𝑣^′′(0)−𝑠⁴𝑣^′(0)−𝑣(0)𝑠⁵+𝑠⁴ℒ_𝑧[𝑣(𝑧)](𝑠)−𝑣⁽³⁾(0)−𝑠𝑣^′′(0)−𝑠²𝑣^′(0)−𝑣(0)𝑠³ (28) or ℒ_𝑧[𝑣(𝑧𝑠)𝑒)](=−^−𝑎𝑠EI𝑠𝑠⁴−𝛾²𝑠⁶EI𝛾²𝑠⁶𝑣(0)𝑒^𝑎𝑠+EI𝛾²𝑠⁵𝑒^𝑎𝑠𝑣^′(0)+EI𝛾²𝑠⁴𝑒^𝑎𝑠𝑣^′′(0)−EI𝑠⁴𝑣(0)𝑒^𝑎𝑠+EI𝛾²𝑠³𝑣⁽³⁾(0)𝑒^𝑎𝑠−EI𝑠³𝑒^𝑎𝑠𝑣^′(0)𝛾²𝑠²𝑣⁽⁴⁾(0)𝑒^𝑎𝑠−EI𝑠²𝑒^𝑎𝑠𝑣^′′(0)+EI𝛾²𝑠𝑣⁽⁵⁾(0)𝑒^𝑎𝑠−EI𝑠𝑣⁽³⁾(0)𝑒^𝑎𝑠−𝑞₀𝑒^𝑎𝑠+𝑞₀. (29) Using the initial conditions in (12), ℒ_𝑧[𝑣(𝑧𝑠)=𝑒)](_−𝑎𝑠𝑒^𝑎𝑠EI𝑠𝜉₁𝑠𝛾²𝑠²−1+𝛾²𝜉₂𝑠+𝜉₃−𝑞₀+𝑞₀EI𝑠⁵𝛾²𝑠²,−1 (30) where 𝜉₁=𝑣^′′(0);𝜉₂=𝑣⁽⁴⁾(0);𝜉₃=𝑣⁽⁵⁾(0). (31) Similarly, the inverse Laplace transform (ℒ⁻¹) of (30) yields ℒ⁻¹{𝑣(𝑠)}=ℒ⁻¹𝑒^−𝑎𝑠×𝑒^𝑎𝑠EI𝑠𝜉₁𝑠𝛾²𝑠²−1+𝛾²𝜉₂𝑠+𝜉₃−𝑞₀+𝑞₀×EI𝑠⁵𝛾²𝑠²−1⁻¹,𝑣(𝑧)=1𝑞24EI⁰()×−𝜗𝑧−𝑎−24𝛾⁴cosh𝑎−𝑧𝛾+12𝛾²(𝑎−𝑧)²+(𝑎−𝑧)⁴+24𝛾⁴+24𝛾⁴+𝑧⁴+12𝛾²𝑧²−24𝛾⁴𝑧cosh𝛾+12𝜉₁EI𝑧²+4𝛾²×EI−𝜉₃𝑧³−6𝛾³𝑧sinh𝛾+6𝛾²𝑧−3𝜉₂2𝛾²+𝑧²−2𝛾²𝑧cosh𝛾, (32) where 𝜗(⋅) is the Heaviside step function. Using the boundary conditions mentioned in the previous section, 𝜉₁=𝑚₁+𝑚₂+𝑚₃+𝑚₄𝑞₀12EI𝐿𝑚₅,𝜉₂=𝑚₆−𝑚₇+𝑚₈𝑞₀EI𝑚₉,𝜉₃=𝑚₁₀+𝑚₁₁𝑞₀𝑒^𝐿/2𝛾𝛾²EI𝑚₁₂, (33) where 𝑚₁=6𝛾𝑎³𝐿(𝑎−2𝐿)+4𝑎𝛾²2𝑎²−3𝑎𝐿+3𝐿²−24𝛾⁴𝐿,𝑚₂𝐿=6𝛾cosh𝛾×4𝑎𝛾²−2𝑎²+3𝑎𝐿+3𝐿²−𝑎𝐿𝑎³−2𝑎²𝐿+2𝐿³+2𝛾𝐿²𝐿²−6𝛾²𝑎sinh𝛾+24𝛾⁴𝐿,𝑚₃=𝐿𝑎²𝐿3𝑎²−8𝑎𝐿+6𝐿²−72𝛾⁴(2𝑎+𝐿)+12𝛾²(𝑎−𝐿)²()𝐿2𝑎+𝐿×sinh𝛾,𝑚₄=−12𝛾²𝑎𝐿cosh𝛾×𝐿𝐿²−6𝛾²𝐿sinh𝛾+12𝛾³𝐿cosh𝛾−1−72𝛾⁴𝑎𝐿sinh𝛾𝐿𝐿−2𝛾sinh𝛾,𝑚₅=𝐿𝐿²−12𝛾²𝐿sinh𝛾+24𝛾³𝐿cosh𝛾,𝑚−1₆=𝑎⁴−2𝑎³𝐿+12𝑎²𝛾²+24𝛾⁴𝑎cosh𝛾−24𝛾⁴,𝑚₇𝐿=cosh𝛾𝑎⁴−2𝑎³𝐿+24𝛾⁴𝑎cosh𝛾+2𝑎𝐿³−2𝛾𝐿𝐿²−12𝛾²𝑎sinh𝛾+12𝑎𝛾²(𝑎−2𝐿)−24𝛾⁴,𝑚₈𝐿=2𝛾sinh𝛾12𝛾³𝑎sinh𝛾−𝐿𝐿²−12𝛾²𝑎×cosh𝛾−12𝛾²(𝑎+𝐿)+𝐿³,𝑚₉=2𝛾𝐿𝐿²−12𝛾²𝐿sinh𝛾+48𝛾⁴𝐿cosh𝛾,𝑚−1₁₀𝐿=cosh𝑎2𝛾⁴−2𝑎³𝐿−24𝛾⁴𝑎cosh𝛾+2𝑎𝐿³+12𝑎𝛾²(𝑎−2𝐿)+24𝛾⁴,𝑚₁₁=24𝛾³𝑎𝛾sinh𝛾𝐿+𝑎sinh,𝑚2𝛾₁₂=−24𝛾³+𝑒^𝐿/𝛾24𝛾³+𝐿³−12𝛾²𝐿+𝐿³−12𝛾²𝐿. (34)

Figure 2 

Schematic diagram for a microbeam fixed at two ends with discontinuous loading.

4. Numerical Examples

In order to demonstrate the efficiency of the present formulation, some illustrative examples are solved and the results are compared with the classical elasticity results. In the numerical results, material scale parameter is taken as 𝛾=17.6 μm for homogeneous epoxy beam [8]. These examples demonstrate the usefulness of Laplace transform to the study of discontinuous loading.

4.1. Example 1

In this example, the microbeam have fixed boundary conditions at (𝑧=0,𝐿) and concentrated loading at (𝑧=𝑎), as depicted in Figure 1. For the comparison purposes, the following parameters are used in computing the numerical results: (𝑎=88,120,176,264), (𝐿=2𝑎), (𝛾=17.6 μm), and (𝑃₀/EI=1/1000 μN/(GPa ×  μm)). Figures 3, 4, and 5 show that the present deflection values agree very well with the existing data available in the literature. In order to investigate the material size effect, the dimensionless static deflection ratios are also plotted as a function of the dimensionless material parameter in Figure 6. This figure indicates that the strain gradient theory predicts the stiffer behavior for the microbeams compared to the classical theory.

(a)

(b)

Figure 3 

The effect of the material size parameter 𝛾 on the deflection for 𝑎=88,120 μm.

(a)

(b)

Figure 4 

The effect of the material size parameter 𝛾 on the deflection for 𝑎=176,264 μm.

(a)

(b)

Figure 5 

The effect of the material size parameter 𝛾 on the deflection for 𝐿/𝛾=500, 𝑎/𝛾=150,250,300, and 𝛾=17.6 μm.

Figure 6 

The effect of dimensionless length on the deflection ratios for 𝐿/𝑎=2 and 𝛾=17.6 μm.

4.2. Example 2

In this example, a microbeam subject to a discontinuous load is considered. For the comparison purposes, the following parameters are used in computing the numerical results: (𝑎=88 μm, 176 μm, 200 μm, 240 μm), (𝐿=2𝑎), (𝛾=17.6 μm), (𝑞₀/EI=1/1000 μN/(GPa ×  μm²)). Figures 7 and 8 display the effect of material parameter on the static behavior of the microbeam. It is clear that material parameter has a significant effect on the static deflection. Present results are in consistency with that for the gradient elastic model in [18].

(a)

(b)

Figure 7 

The effect of the material size parameter 𝛾 on the deflection for 𝑎=88,176 μm.

(a)

(b)

Figure 8 

The effect of the material size parameter 𝛾 on the deflection for 𝑎=200,240 μm.

It can be noted that the results predicted by the gradient elasticity theory are always smaller than those of the classical beam theory. Further, the difference between the two results is remarkable for the microbeams with 𝐿/𝛾≤10. It is well-known in the literature that gradient elasticity (the modified couple stress theory) evaluates the microbeams stiffer than evaluations of the classical theory.

5. Conclusion

Based on strain gradient elasticity theory with surface energy, the static bending solutions of the gradient elastic beams with external discontinuities are presented by Laplace transform. Explicit expressions are derived for static deflections for clamped-clamped boundary conditions. Derived expressions are verified for several numerical examples and the results show that the Laplace transform gives reliable results. Comparison of classical elasticity theory also performed to verify the present solutions. It can be concluded that the difference between the results predicted by the classical beam theories and those evaluated by the strain gradient theories are significant when 𝐿/𝛾 is low. The main contribution of this paper is to provide a closed-form solution for the static deflection of microbeams under geometric discontinuities.

Conflict of Interests

The author declare that there is no conflict of interests regarding the publication of this paper.