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

Authors: Mustafa Özgür Yayli
Journal: Mathematical Problems in Engineering
 (2013)

---

## 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.


---

## Body

## 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. LetΩ(t) is a function on the interval [0,∞). The Laplace transform of Ω(t) is the function in terms of s as follows:
(1)ℒt{Ω(t)}=∫0∞‍e-stΩ(t)dt.
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:
(2)ϖndnΩdtn+ϖn-1dn-1Ωdtn-1+ϖn-2dn-2Ωdtn-2+⋯+ϖ0Ω=Ψ(t),(3)Ω(0)=Ω0,Ω′(0)=Ω1,Ω′′(0)=Ω2,…,Ω(n-1)(0)=Ω(n-1),
in which ϖn, ϖn-1, ϖn-2,…,ϖ0 denote the constants and (3) represents the initial conditions. Taking the Laplace transforms of each term yields
(4)ϖnℒ{dnΩdtn}+ϖn-1ℒ{dn-1Ωdtn-1}+ϖn-2ℒ{dn-2Ωdtn-2}+⋯+ϖ0ℒ{Ω}=ℒ{Ψ(t)}.
Using the property of the Laplace transformation of derivatives, one can obtain
(5)ϖn[snY(s)-sn-1Ω(0)-⋯-Ωn-1(0)]+ϖn-1[sn-1Y(s)-sn-2Ω(0)-⋯-Ωn-2(0)]+⋯+ϖ0Y(s)=G(s).
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
(6)σij=Cijkl(ϵkl-γ2ϵkl,mn),
in which σij and ϵij represent the stress and strain tensors, Cijkl 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
(7)U=12EI∫0L‍{(d2vdz2)2-γ2(d2v∂z2d4vdz4)}dz,
where EI is the bending stiffness of the microbeam. The variation of the external concentrated load is obtained as:
(8)φWE=∫0L‍P0δ(z-a)φvdz,
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]
(9)EI{(d4vdz4)-γ2(∂6v∂z6)}-P0δ(z-a)=0.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,(10)∫0L‍ψ(z)(EI{(d4vdz4)-γ2(d6vdz6)}-P0δ(z-a))dz=0,
where ψ(z) is the test function and is integrating by parts as follows:
(11)0=∫0L‍(EI[d2ψdz2d2vdz2-γ2d3ψdz3d3vdz3]00000-ψ(z)P0δ(z-a)[d2ψdz2d2vdz2-γ2d3ψdz3d3vdz3])dz+{ψ(z)EI(d3vdz3-γ2d5vdz5)}0L-{dψ(z)dzEI(d2vdz2-γ2d4vdz4)}0L-{γ2d2ψ(z)dz2EId3vdz3}0L.
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
(12)v|z=0=0,∂v∂z|z=0=0,∂3v∂3z|z=0=0,(13)v|z=L=0,∂v∂z|z=L=0,∂3v∂3z|z=L=0.


## 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. LetΩ(t) is a function on the interval [0,∞). The Laplace transform of Ω(t) is the function in terms of s as follows:
(1)ℒt{Ω(t)}=∫0∞‍e-stΩ(t)dt.
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:
(2)ϖndnΩdtn+ϖn-1dn-1Ωdtn-1+ϖn-2dn-2Ωdtn-2+⋯+ϖ0Ω=Ψ(t),(3)Ω(0)=Ω0,Ω′(0)=Ω1,Ω′′(0)=Ω2,…,Ω(n-1)(0)=Ω(n-1),
in which ϖn, ϖn-1, ϖn-2,…,ϖ0 denote the constants and (3) represents the initial conditions. Taking the Laplace transforms of each term yields
(4)ϖnℒ{dnΩdtn}+ϖn-1ℒ{dn-1Ωdtn-1}+ϖn-2ℒ{dn-2Ωdtn-2}+⋯+ϖ0ℒ{Ω}=ℒ{Ψ(t)}.
Using the property of the Laplace transformation of derivatives, one can obtain
(5)ϖn[snY(s)-sn-1Ω(0)-⋯-Ωn-1(0)]+ϖn-1[sn-1Y(s)-sn-2Ω(0)-⋯-Ωn-2(0)]+⋯+ϖ0Y(s)=G(s).
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
(6)σij=Cijkl(ϵkl-γ2ϵkl,mn),
in which σij and ϵij represent the stress and strain tensors, Cijkl 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
(7)U=12EI∫0L‍{(d2vdz2)2-γ2(d2v∂z2d4vdz4)}dz,
where EI is the bending stiffness of the microbeam. The variation of the external concentrated load is obtained as:
(8)φWE=∫0L‍P0δ(z-a)φvdz,
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]
(9)EI{(d4vdz4)-γ2(∂6v∂z6)}-P0δ(z-a)=0.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,(10)∫0L‍ψ(z)(EI{(d4vdz4)-γ2(d6vdz6)}-P0δ(z-a))dz=0,
where ψ(z) is the test function and is integrating by parts as follows:
(11)0=∫0L‍(EI[d2ψdz2d2vdz2-γ2d3ψdz3d3vdz3]00000-ψ(z)P0δ(z-a)[d2ψdz2d2vdz2-γ2d3ψdz3d3vdz3])dz+{ψ(z)EI(d3vdz3-γ2d5vdz5)}0L-{dψ(z)dzEI(d2vdz2-γ2d4vdz4)}0L-{γ2d2ψ(z)dz2EId3vdz3}0L.
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
(12)v|z=0=0,∂v∂z|z=0=0,∂3v∂3z|z=0=0,(13)v|z=L=0,∂v∂z|z=L=0,∂3v∂3z|z=L=0.


## 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 atz=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
(14)EI{(d4vdz4)-γ2(d6vdz6)}=P0δ(z-a).
Taking the Laplace transforms of both sides,
(15)P0(1-e-as)EIs=-γ2(s6(ℒz[v(z)](s))-v(5)(0)-v(4)(0)s0000.0000-v(3)(0)s2-s3v′′(0)-s4v′(0)-v(0)s5)+s4(ℒz[v(z)](s))-v(3)(0)-sv′′(0)-s2v′(0)-v(0)s3
or
(16)ℒz[v(z)](s)=-e-asEIs(s4-γ2s6)EIγ2v(5)(0)seas+EIγ2v(4)(0)s2eas+EIγ2v(3)(0)s3eas-EIv(3)(0)seas+EIγ2s4easv′′(0)-EIs2easv′′(0)+EIγ2s5easv′(0)-EIs3easv′(0)+EIγ2v(0)s6eas-EIv(0)s4eas-P0eas+P0.
Using the three known initial conditions in (12),
(17)ℒz[v(z)](s)=(-P0e-as-EIγ2C2s200+EIC1(s2-γ2s4)-EIγ2C3s+P0)×(EI(s5-γ2s7))-1,
where
(18)C1=v′′(0),C2=v(4)(0),C3=v(5)(0).
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
(19)ℒ-1{v(s)}=ℒ-1{(EI(s5-γ2s7))-1(-P0e-as-EIγ2C2s2+EIC1(s2-γ2s4)00000000000-EIγ2C3s+P0)00000000×(EI(s5-γ2s7))-1},(20)v(z)=1EI((ez/γ-1)2-P0θ(z-a)000×(-12γ4e(a-z)/γ-12γ4e(z-a)/γ0000000+12γ2(a-z)2+124(a-z)4+γ4)000-16γ2C3EI(z3-3γ3e-z/γ(e2z/γ-1)+6γ2z)000-12γ2C2EI000×(z2-γ2e-z/γ(ez/γ-1)2)000+12C1EI(z2-γ2e-z/γ(ez/γ-1)2)000+12γ2C1EIe-z/γ(ez/γ-1)2000+124P0((ez/γ-1)2z4+12γ2t200000000000-12γ4e-z/γ(ez/γ-1)2)),
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,
(21)v(z){-a}=1EIC1(12EI(z2-γ2e-z/γ(ez/γ-1)2)00000000000+12γ2EIe-z/γ(ez/γ-1)2)-16γ2C3(z3-3γ3e-z/γ(e2z/γ-1)+6γ2z)-12γ2C2(z2-γ2e-z/γ(ez/γ-1)2)
and for a≤z≤L,
(22)v(z){+a}=1EIC1(12EI(z2-γ2e-z/γ(ez/γ-1)2)0000000000+12γ2EIe-z/γ(ez/γ-1)2)-16γ2C3(z3-3γ3e-z/γ(e2z/γ-1)0000000000000+6γ2z(e2z/γ-1))-12γ2C2×(z2-γ2e-z/γ(ez/γ-1)2)+1EIP0(-12γ4e(a-z)/γ-12γ4e(z-a)/γ200000000000+12γ2(a-z)2+124(a-z)4+γ4).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:
(23)η1=eL/γ;η2=η12-1;η3=η1+1;η4=η1-1;α1=12γ3-L3+6γ2L;α2=24γ4+L4-8γ2L2;α3=-144γ5+L5+6γL4-12γ2L3-96γ3L2-216γ4L;α4=144γ5+L5-6γL4-12γ2L3+96γ3L2-216γ4L;α5=12γ3+L3-6γ2L;β1=-12γ3+L3-12γ2L;β2=-24γ4+L4-12γ2L2-24γ3L;β3=-24γ4+L4-12γ2L2+24γ3L;β4=24γ4+L4-12γ2L2;β5=12γ3+L3-12γ2L.
By using (23), C1, C2, and C3 parameters can be calculated as follows:
(24)C1=-112EIL(24γ3η42+η2L3-12γ2η2L)×(×e(2a+L)/γ+72γ4η1(2γ+L)))P0e-a/γ000×(×e(2a+L)/γ+72γ4η1(2γ+L))3a4(η4)Lea/γ000000×(-2γη1+2γ+η1L+L)-4a3ea/γ000000×(12γ3η42+2η2L3-3γη42L2-6γ2η2L)000000+6a2Lea/γ(12γ3η42+η2L3-6γ2η2L)000000-12aγLea/γ000000×(12γ3η2+(η12+1)L3-6γ2η32L)000000+L(-12α1γ2e2a/γ+α3ea/γ00000000000-12α5γ2η12+12α2γe(a+L)/γ00000000000-α4e(a+2L)/γ+72γ4(2γ-L)00000000000×e(2a+L)/γ+72γ4η1(2γ+L)))),C2=-12γEI(24γ3η42+η2L3-12γ2η2L)×(000000-2β5γη12+24γ4η1)P0e-a/γ000×(a4η42(-ea/γ)+2a3η42Lea/γ000000-12a2γ2η42ea/γ+2β1γe2a/γ000000+β2ea/γ-2aea/γ000000×(12γ3η2+(η12+1)L30000000000-12γ2(η12+1)L)+β3e(a+2L)/γ000000+2β4e(a+L)/γ+24γ4e(2a+L)/γ000000-2β5γη12+24γ4η1)),C3=P0e-a/γ2γ2EI(24γ3η1-24γ3+η3(L3-12γ2L))×ea/γ(λ2eL/γ-aη3(a3-2a2L+12aγ2+2L3000000000000-24γ2L)+24γ4ea/γ000000+24aγ3+λ1+λ2eL/γ)-24γ3η1(aea/γ-γ).
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 Figure2. Loading functions are expressed in terms of multiple expressions in different regions (25). Consider
(25)q(z)={q0,0<z<a,0,a<z<L.
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
(26)q(z)=q0{ϑ(z)-ϑ(z-a)}.
Then, the governing differantial equation is
(27)EI{(d4vdz4)-γ2(d6vdz6)}=q0{ϑ(z)-ϑ(z-a)}.
Taking the Laplace transforms,
(28)q0(1-e-as)EIs=-γ2(s6(ℒz[v(z)](s))-v(5)(0)-v(4)(0)s000000000-v(3)(0)s2-s3v′′(0)-s4v′(0)-v(0)s5)+s4(ℒz[v(z)](s))-v(3)(0)-sv′′(0)-s2v′(0)-v(0)s3
or
(29)ℒz[v(z)](s)=-e-asEIs(s4-γ2s6)EIγ2s6v(0)eas+EIγ2s5easv′(0)+EIγ2s4easv′′(0)-EIs4v(0)eas+EIγ2s3v(3)(0)eas-EIs3easv′(0)γ2s2v(4)(0)eas-EIs2easv′′(0)+EIγ2sv(5)(0)eas-EIsv(3)(0)eas-q0eas+q0.
Using the initial conditions in (12),
(30)ℒz[v(z)](s)=e-as(eas(EIs(ξ1s(γ2s2-1)+γ2(ξ2s+ξ3))-q0)+q0)EIs5(γ2s2-1),
where
(31)ξ1=v′′(0);ξ2=v(4)(0);ξ3=v(5)(0).
Similarly, the inverse Laplace transform (ℒ-1) of (30) yields
(32)ℒ-1{v(s)}=ℒ-1{(EIs5(γ2s2-1))-1e-as0000000000×(eas(EIs(ξ1s(γ2s2-1)+γ2(ξ2s+ξ3))00000000000000000-q0(γ2s2-1))+q0(ξ1s(γ2s2-1)+γ2(ξ2s+ξ3)))0000000×(EIs5(γ2s2-1))-1},v(z)=124EI((2γ2+z2-2γ2cosh(zγ))q0((zγ)-ϑ(z-a)000000000×(-24γ4cosh(a-zγ)0000000000000+12γ2(a-z)2+(a-z)4+24γ4(a-zγ))000000000+24γ4+z4+12γ2z2000000000-24γ4cosh(zγ))+12ξ1EIz2+4γ2EI00000×(-ξ3(z3-6γ3sinh(zγ)+6γ2z)000000000-3ξ2(2γ2+z2-2γ2cosh(zγ)))),
where ϑ(·) is the Heaviside step function. Using the boundary conditions mentioned in the previous section,
(33)ξ1=(m1+m2+m3+m4)q012EILm5,ξ2=(m6-m7+m8)q0EIm9,ξ3=(m10+m11)q0eL/2γγ2EIm12,
where
(34)m1=6γ(a3L(a-2L)+4aγ2(2a2-3aL+3L2)-24γ4L),m2=6γcosh(Lγ)×((aγ)4aγ2(-2a2+3aL+3L2)-aL(a3-2a2L+2L3)000+2γL2(L2-6γ2)sinh(aγ)+24γ4L),m3=L(a2L(3a2-8aL+6L2)000-72γ4(2a+L)+12γ2(a-L)2(2a+L))×sinh(Lγ),m4=-12γ2Lcosh(aγ)×(L(L2-6γ2)sinh(Lγ)+12γ3(cosh(Lγ)-1))-72γ4Lsinh(aγ)(L-2γsinh(Lγ)),m5=L(L2-12γ2)sinh(Lγ)+24γ3(cosh(Lγ)-1),m6=a4-2a3L+12a2γ2+24γ4cosh(aγ)-24γ4,m7=cosh(Lγ)(a4-2a3L+24γ4cosh(aγ)0000000.000+2aL3-2γL(L2-12γ2)sinh(aγ)000000.0000+12aγ2(a-2L)-24γ4(aγ)),m8=2γsinh(Lγ)(12γ3sinh(aγ)-L(L2-12γ2)0000000000000×cosh(aγ)-12γ2(a+L)+L3),m9=2γL(L2-12γ2)sinh(Lγ)+48γ4(cosh(Lγ)-1),m10=cosh(L2γ)(a4-2a3L-24γ4cosh(aγ)00000000000+2aL3+12aγ2(a-2L)+24γ4(aγ)),m11=24γ3(γsinh(aγ)+a)sinh(L2γ),m12=-24γ3+eL/γ(24γ3+L3-12γ2L)+L3-12γ2L.Figure 2
Schematic diagram for a microbeam fixed at two ends with discontinuous loading.


## 3.1. A Microbeam Subjected to a Concentrated Load

Consider a microbeam subjected to a concentrated load. The concentrated load atz=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
(14)EI{(d4vdz4)-γ2(d6vdz6)}=P0δ(z-a).
Taking the Laplace transforms of both sides,
(15)P0(1-e-as)EIs=-γ2(s6(ℒz[v(z)](s))-v(5)(0)-v(4)(0)s0000.0000-v(3)(0)s2-s3v′′(0)-s4v′(0)-v(0)s5)+s4(ℒz[v(z)](s))-v(3)(0)-sv′′(0)-s2v′(0)-v(0)s3
or
(16)ℒz[v(z)](s)=-e-asEIs(s4-γ2s6)EIγ2v(5)(0)seas+EIγ2v(4)(0)s2eas+EIγ2v(3)(0)s3eas-EIv(3)(0)seas+EIγ2s4easv′′(0)-EIs2easv′′(0)+EIγ2s5easv′(0)-EIs3easv′(0)+EIγ2v(0)s6eas-EIv(0)s4eas-P0eas+P0.
Using the three known initial conditions in (12),
(17)ℒz[v(z)](s)=(-P0e-as-EIγ2C2s200+EIC1(s2-γ2s4)-EIγ2C3s+P0)×(EI(s5-γ2s7))-1,
where
(18)C1=v′′(0),C2=v(4)(0),C3=v(5)(0).
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
(19)ℒ-1{v(s)}=ℒ-1{(EI(s5-γ2s7))-1(-P0e-as-EIγ2C2s2+EIC1(s2-γ2s4)00000000000-EIγ2C3s+P0)00000000×(EI(s5-γ2s7))-1},(20)v(z)=1EI((ez/γ-1)2-P0θ(z-a)000×(-12γ4e(a-z)/γ-12γ4e(z-a)/γ0000000+12γ2(a-z)2+124(a-z)4+γ4)000-16γ2C3EI(z3-3γ3e-z/γ(e2z/γ-1)+6γ2z)000-12γ2C2EI000×(z2-γ2e-z/γ(ez/γ-1)2)000+12C1EI(z2-γ2e-z/γ(ez/γ-1)2)000+12γ2C1EIe-z/γ(ez/γ-1)2000+124P0((ez/γ-1)2z4+12γ2t200000000000-12γ4e-z/γ(ez/γ-1)2)),
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,
(21)v(z){-a}=1EIC1(12EI(z2-γ2e-z/γ(ez/γ-1)2)00000000000+12γ2EIe-z/γ(ez/γ-1)2)-16γ2C3(z3-3γ3e-z/γ(e2z/γ-1)+6γ2z)-12γ2C2(z2-γ2e-z/γ(ez/γ-1)2)
and for a≤z≤L,
(22)v(z){+a}=1EIC1(12EI(z2-γ2e-z/γ(ez/γ-1)2)0000000000+12γ2EIe-z/γ(ez/γ-1)2)-16γ2C3(z3-3γ3e-z/γ(e2z/γ-1)0000000000000+6γ2z(e2z/γ-1))-12γ2C2×(z2-γ2e-z/γ(ez/γ-1)2)+1EIP0(-12γ4e(a-z)/γ-12γ4e(z-a)/γ200000000000+12γ2(a-z)2+124(a-z)4+γ4).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:
(23)η1=eL/γ;η2=η12-1;η3=η1+1;η4=η1-1;α1=12γ3-L3+6γ2L;α2=24γ4+L4-8γ2L2;α3=-144γ5+L5+6γL4-12γ2L3-96γ3L2-216γ4L;α4=144γ5+L5-6γL4-12γ2L3+96γ3L2-216γ4L;α5=12γ3+L3-6γ2L;β1=-12γ3+L3-12γ2L;β2=-24γ4+L4-12γ2L2-24γ3L;β3=-24γ4+L4-12γ2L2+24γ3L;β4=24γ4+L4-12γ2L2;β5=12γ3+L3-12γ2L.
By using (23), C1, C2, and C3 parameters can be calculated as follows:
(24)C1=-112EIL(24γ3η42+η2L3-12γ2η2L)×(×e(2a+L)/γ+72γ4η1(2γ+L)))P0e-a/γ000×(×e(2a+L)/γ+72γ4η1(2γ+L))3a4(η4)Lea/γ000000×(-2γη1+2γ+η1L+L)-4a3ea/γ000000×(12γ3η42+2η2L3-3γη42L2-6γ2η2L)000000+6a2Lea/γ(12γ3η42+η2L3-6γ2η2L)000000-12aγLea/γ000000×(12γ3η2+(η12+1)L3-6γ2η32L)000000+L(-12α1γ2e2a/γ+α3ea/γ00000000000-12α5γ2η12+12α2γe(a+L)/γ00000000000-α4e(a+2L)/γ+72γ4(2γ-L)00000000000×e(2a+L)/γ+72γ4η1(2γ+L)))),C2=-12γEI(24γ3η42+η2L3-12γ2η2L)×(000000-2β5γη12+24γ4η1)P0e-a/γ000×(a4η42(-ea/γ)+2a3η42Lea/γ000000-12a2γ2η42ea/γ+2β1γe2a/γ000000+β2ea/γ-2aea/γ000000×(12γ3η2+(η12+1)L30000000000-12γ2(η12+1)L)+β3e(a+2L)/γ000000+2β4e(a+L)/γ+24γ4e(2a+L)/γ000000-2β5γη12+24γ4η1)),C3=P0e-a/γ2γ2EI(24γ3η1-24γ3+η3(L3-12γ2L))×ea/γ(λ2eL/γ-aη3(a3-2a2L+12aγ2+2L3000000000000-24γ2L)+24γ4ea/γ000000+24aγ3+λ1+λ2eL/γ)-24γ3η1(aea/γ-γ).
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 Figure2. Loading functions are expressed in terms of multiple expressions in different regions (25). Consider
(25)q(z)={q0,0<z<a,0,a<z<L.
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
(26)q(z)=q0{ϑ(z)-ϑ(z-a)}.
Then, the governing differantial equation is
(27)EI{(d4vdz4)-γ2(d6vdz6)}=q0{ϑ(z)-ϑ(z-a)}.
Taking the Laplace transforms,
(28)q0(1-e-as)EIs=-γ2(s6(ℒz[v(z)](s))-v(5)(0)-v(4)(0)s000000000-v(3)(0)s2-s3v′′(0)-s4v′(0)-v(0)s5)+s4(ℒz[v(z)](s))-v(3)(0)-sv′′(0)-s2v′(0)-v(0)s3
or
(29)ℒz[v(z)](s)=-e-asEIs(s4-γ2s6)EIγ2s6v(0)eas+EIγ2s5easv′(0)+EIγ2s4easv′′(0)-EIs4v(0)eas+EIγ2s3v(3)(0)eas-EIs3easv′(0)γ2s2v(4)(0)eas-EIs2easv′′(0)+EIγ2sv(5)(0)eas-EIsv(3)(0)eas-q0eas+q0.
Using the initial conditions in (12),
(30)ℒz[v(z)](s)=e-as(eas(EIs(ξ1s(γ2s2-1)+γ2(ξ2s+ξ3))-q0)+q0)EIs5(γ2s2-1),
where
(31)ξ1=v′′(0);ξ2=v(4)(0);ξ3=v(5)(0).
Similarly, the inverse Laplace transform (ℒ-1) of (30) yields
(32)ℒ-1{v(s)}=ℒ-1{(EIs5(γ2s2-1))-1e-as0000000000×(eas(EIs(ξ1s(γ2s2-1)+γ2(ξ2s+ξ3))00000000000000000-q0(γ2s2-1))+q0(ξ1s(γ2s2-1)+γ2(ξ2s+ξ3)))0000000×(EIs5(γ2s2-1))-1},v(z)=124EI((2γ2+z2-2γ2cosh(zγ))q0((zγ)-ϑ(z-a)000000000×(-24γ4cosh(a-zγ)0000000000000+12γ2(a-z)2+(a-z)4+24γ4(a-zγ))000000000+24γ4+z4+12γ2z2000000000-24γ4cosh(zγ))+12ξ1EIz2+4γ2EI00000×(-ξ3(z3-6γ3sinh(zγ)+6γ2z)000000000-3ξ2(2γ2+z2-2γ2cosh(zγ)))),
where ϑ(·) is the Heaviside step function. Using the boundary conditions mentioned in the previous section,
(33)ξ1=(m1+m2+m3+m4)q012EILm5,ξ2=(m6-m7+m8)q0EIm9,ξ3=(m10+m11)q0eL/2γγ2EIm12,
where
(34)m1=6γ(a3L(a-2L)+4aγ2(2a2-3aL+3L2)-24γ4L),m2=6γcosh(Lγ)×((aγ)4aγ2(-2a2+3aL+3L2)-aL(a3-2a2L+2L3)000+2γL2(L2-6γ2)sinh(aγ)+24γ4L),m3=L(a2L(3a2-8aL+6L2)000-72γ4(2a+L)+12γ2(a-L)2(2a+L))×sinh(Lγ),m4=-12γ2Lcosh(aγ)×(L(L2-6γ2)sinh(Lγ)+12γ3(cosh(Lγ)-1))-72γ4Lsinh(aγ)(L-2γsinh(Lγ)),m5=L(L2-12γ2)sinh(Lγ)+24γ3(cosh(Lγ)-1),m6=a4-2a3L+12a2γ2+24γ4cosh(aγ)-24γ4,m7=cosh(Lγ)(a4-2a3L+24γ4cosh(aγ)0000000.000+2aL3-2γL(L2-12γ2)sinh(aγ)000000.0000+12aγ2(a-2L)-24γ4(aγ)),m8=2γsinh(Lγ)(12γ3sinh(aγ)-L(L2-12γ2)0000000000000×cosh(aγ)-12γ2(a+L)+L3),m9=2γL(L2-12γ2)sinh(Lγ)+48γ4(cosh(Lγ)-1),m10=cosh(L2γ)(a4-2a3L-24γ4cosh(aγ)00000000000+2aL3+12aγ2(a-2L)+24γ4(aγ)),m11=24γ3(γsinh(aγ)+a)sinh(L2γ),m12=-24γ3+eL/γ(24γ3+L3-12γ2L)+L3-12γ2L.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 asγ=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.The effect of the material size parameterγ on the deflection for a=88,120 μm.
(a)
(b)The effect of the material size parameterγ on the deflection for a=176,264 μm.
(a)
(b)The effect of the material size parameterγ on the deflection for L/γ=500, a/γ=150,250,300, and γ=17.6 μm.
(a)
(b)Figure 6
The effect of dimensionless length on the deflection ratios forL/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].The effect of the material size parameterγ on the deflection for a=88,176 μm.
(a)
(b)The effect of the material size parameterγ on the deflection for a=200,240 μm.
(a)
(b)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 withL/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.


## 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.The effect of the material size parameterγ on the deflection for a=88,120 μm.
(a)
(b)The effect of the material size parameterγ on the deflection for a=176,264 μm.
(a)
(b)The effect of the material size parameterγ on the deflection for L/γ=500, a/γ=150,250,300, and γ=17.6 μm.
(a)
(b)Figure 6
The effect of dimensionless length on the deflection ratios forL/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].The effect of the material size parameterγ on the deflection for a=88,176 μm.
(a)
(b)The effect of the material size parameterγ on the deflection for a=200,240 μm.
(a)
(b)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 withL/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 whenL/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.



---

*Source: 129872-2013-11-21.xml*