3 Applications for One-Dimensional Problems
3.2 Static Problems
3.2.1 Direct Approach to the Nonlocal Integral Model
A direct handling of the nonlocal integral constitutive equation is to deduce the strains by vanishing the dynamical terms of Eq. (33) and then integrating by parts twice.
The following equation concerning the strain is ergo obtained [80]:
L 0
A(|x−s|,e0a)d2w
ds2ds= ˜q(x)+C1x+C2 (40) whereq(x)˜ := E I1 0xq(y)d y
d xandC1,C2 ∈ .
A cantilever and a simply supported beam subjected to two types of loading (i.e., a concentrated load applied to the beam’s middle and a uniformly distributed load) are respectively studied. The essential BCs of a cantilever and a simply supported beam are respectively:w(L)=dw(L)/d x=0 andw(0)=w(L)=0.
The following values are selected for the concentrated load, the uniformly dis- tributed load, the length, the bending stiffness, the radius as well as the thick- ness of the SWCNT/nanobeam:P0=1×10−9N,q0=1×10−9N/1×10−9m,L= 10×10−9m,E I =4.41×10−26Nm2,r =0.34×10−9m andt=0.34×10−9m.
We note that A∈L2(0,L) (i.e., L 0
L
0 |A(|x−s|,e0a)|2ds<∞), since the attenuation function is continuous and the integral operator is compact. This entails the solution’s existence and uniqueness to the investigated problems. To solve numer- ically the integral Eq. (40), we use Gauss quadrature rule converging toward the analytical expression of the integral, since the points’ number tends to infinity [81].
Owing to the existence of the boundary layer, the number of Gauss points selected without a significant computational cost is 3000. A linear system of equationsKe=g is deduced. The coefficient matrix, the vector of unknowns and the vector of external loads are denoted byK,eandg, respectively.
The ratio of the beam’s length to the beam’s height is sufficiently large in all the above examples as well. Ergo, we assume that the thin beams’ theory is appropriate, and the fact that the EBBT does not take into consideration the shear deformation does not produce a significance error.
The next results of figures exhibit the normalized strainεx x := maxex x|ecl
x x|,ex x :=
ex x(x,z)= −zmaxd2w
d x2, with respect to the length of the beam for each explored case.
The classic strain is denoted byeclx x. The following figures display the strainεx x of the classic-local model, the TPNI stress model and the model of the modified kernel fore0a=0.01L. To display the strain in all the figures, a representative set of 3000 Gauss points is indicatively selected.
Equation (40) can be alternatively written as:
MN L(x)=E Iq˜(x)+ ˜C1x+ ˜C2 (41) whereC˜1,C˜2∈ .
The strains are deduced by the following equations for the case of a cantilever beam subjected to a uniformly distributed loadq0(BCs:MN L(0)=QN L(0)=0) [54]:
d2w
d x2ξ(x,e0a)+ L
0
A(|x−s|,e0a)d2w
ds2ds= q0x2
2E I (42)
k1
d2w d x2 +k2
L 0
A(|x−s|,e0a)d2w
ds2ds =q0x2
2E I (43)
What is more, the next equations give the strains for the case of a cantilever beam subjected to a concentrated load at an interior pointx0(BCs:MN L(0)=QN L(0)= 0) [54]:
d2w
d x2ξ(x,e0a)+ L
0
A(|x−s|,e0a)d2w
ds2ds = P0
E I(x−x0)H(x−x0) (44) k1
d2w d x2 +k2
L 0
A(|x−s|,e0a)d2w
ds2ds = P0
E I(x−x0)H(x−x0) (45) whereH(x−x0)is the step/Heaviside function at pointx0. Furthermore, the strains are obtained by the following equations for the case of a simply supported beam subjected to a uniformly distributed loadq0(BCs:MN L(0)=MN L(L)=0) [54]:
d2w
d x2ξ(x,e0a)+ L
0
A(|x−s|,e0a)d2w
ds2ds =q0x E I
x 2 −1
(46)
k1
d2w d x2 +k2
L 0
A(|x−s|,e0a)d2w
ds2ds =q0x E I
x 2 −1
(47) Moreover, the next equations yield the strains for the case of a simply supported beam subjected to a concentrated load at an interior point x0 (BCs: MN L(0)= MN L(L)=0) [54]:
d2w
d x2ξ(x,e0a)+ L
0
A(|x−s|,e0a)d2w ds2ds
= P0
E I(x−x0)H(x−x0)− P0 E I
1−x0
L
x (48)
k1d2w d x2 +k2
L 0
A(|x−s|,e0a)d2w ds2ds
= P0
E I(x−x0)H(x−x0)− P0
E I
1−x0
L
x (49)
The discrete mathematical expressions of Eq. (42)–(49) take the form [54]:
f(ti)ξ(ti,e0a)+ G
j=1
cjA(|ti−tj|,e0a)f(tj)=g(ti), i =1, . . . ,G (50)
f(ti)k1+k2
G j=1
cjA(|ti−tj|,e0a)f(tj)=g(ti). i =1, . . . ,G (51)
A cantilever beam is subjected to a uniformly distributed load in Fig.2. A dif- ference between the nonlocal integral models and the classic model is indicated.
Apart from an area near to the fixed point, the strains of the nonlocal integral models exceed that of the classic model (Fig.2b). Whenk1-parameter reduces, the integral models also tend to each other without including the area close to the fixed point (Fig.2b). Moreover, the boundary layer concerning the TPNI stress model intensi- fies close to the fixed point of the beam (Fig.2b). The boundary layer, in particular, decreases, yet its influence zone increases ask1-parameter augments. The classic model is approached by the TPNI stress model whenk1-parameter increases as well (Fig.2). Another conclusion is that the TPNI stress model converges toward the model of the modified kernel when k1-parameter decreases, and the points of the body are not too close to the fixed point (Fig.2). On the other hand, the boundary layer of the model of the modified kernel is indiscernible near to the fixed point.
Additionally, the corresponding strain exceeds that of the classic model when the distance between an arbitrary point and the fixed point increases. What is more, the strain of the modified kernel model reduces and is exceeded by that of the classic model when the free end is approached.
A concentrated load is applied to the middle of a cantilever beam in Fig.3. The stains of the nonlocal integral models are identical to that of the classic model except
Fig. 2 aThe normalized strainεx x of a cantilever beam, with a uniformly distributed load, for e0a/L=0.01.bDetails of an area close to the fixed point. (AmodH =Amod)
Fig. 3 aThe normalized strainεx xof a cantilever beam, with a concentrated load applied to the beam’s middle, fore0a/L=0.01. Details of an area of the load pointband an area close to the fixed pointc
for the areas close to the load and the fixed points of the beam. Whenk1-parameter reduces, the strain of the TPNI stress model decreases (Fig. 3a). As regards the TPNI stress model, a boundary layer, similar to the case of a cantilever beam with a uniformly distributed load, is developed near to the fixed point (Fig. 3b, c). To be more specific, the boundary layer decreases, but at the same time its influence zone increases as k1-parameter augments. Taking into account the increase ofk1- parameter, the TPNI stress model tends to the classic model (Fig.3). Nevertheless, the strain of the model of the modified kernel takes smaller values than those of the other models at the load points (Fig.3b). Furthermore, the strain of the TPNI stress model tends to that of the model of the modified kernel at the load point whenk1- parameter decreases. A significant conclusion is that the model of the modified kernel does not appear to have a discernible boundary layer at the fixed point (Fig.3c).
Another conclusion of importance is that the boundary layer depends only on the investigated governing equation, but not the exerted loading (Figs.2b and3c).
Two different types of loading are applied to a simply supported beam in Fig.4a, c. Similarities are presented between the models which are explored. However, the boundary layer regarding the TPNI stress model is less sharp and the strain takes
Fig. 4 The normalized strainεx xof a simply supported beam fore0a/L=0.01.aA concentrated load applied to the beam’s middle.bDetails of an area close to the fixed point.cA uniformly distributed load.dDetails of an area close to the fixed point
smaller values than that of a cantilever beam in an area close to the fixed points (Fig.4b, d). The influence zone of the boundary layer of the TPNI stress model enlarges ask1-parameter increases, yet the boundary layer compared to that of the modified kernel and being close to the fixed points decreases (Fig.4b, d). The strain of the model of the modified kernel, in particular, takes larger values than that of the TPNI stress model withk1=0.5 in areas close to the fixed points. This indicates the difficulty of the model of the modified kernel to handle the boundary layer of a simply supported beam.
To check the stability of the linear system of equations, the condition number of the coefficient matrixKshould be calculated. The condition number of a matrixK n×nis defined asκ(K)= ||K|| · ||K−1||.
The definitions of the norm are given by||K||∞=max1≤i≤n
n
j=1|Ki j|,||K||1= max1≤j≤n
n
i=1|Ki j|and||K||2=n 1≤i,j≤n
Ki j
21/2 .
IfKis a symmetric matrix, then||K||2=ρ(K), whereρ(K)=max{λi :K}and λi are the eigenvalues of the matrixK. In case thatKis a symmetric and a normal matrix, it then holds thatκ(K)= |λmax|/|λmin|.
Table 1 Cantilever beam: Condition number
κ(K)= ||K|| · ||K−1||
of the matrix of coeffi- cients,K
e0a/L Gauss points
Amod k1=0.01 k1=0.1 k1=0.2 k1=0.5
0.01 500 4875.431 100.9415 10.00192 4.984897 1.990991
1000 2335.448 101.0362 10.00464 4.985585 1.991034 2000 2240.871 101.1099 10.00701 4.986253 1.991106 3000 1663.168 101.1350 10.00786 4.986504 1.991137
0.1 [54] 500 93.13096 97.12981 9.651288 4.82809 1.951197
1000 92.71524 97.19989 9.653703 4.82881 1.951289
2000 92.54316 97.25277 9.655588 4.829385 1.951368
3000 92.51211 97.27045 9.65623 4.829583 1.951395
The linear system corresponding to the TPNI stress model is stable. The condition number of the coefficient matrixKtends to 1 whenk1-parameter increases. Also, the condition number is slightly affected by the nonlocal parameter (Table1).
What is more, the condition number of the coefficient matrixKcorresponding to the model of the modified kernel is acceptable for the case ofe0a/L=0.01. When the nonlocal parameter takes such a small value, the model of the modified kernel approaches the first kind Fredholm integral equation. On the other hand, the condition number gives excellent results whene0a/L =0.1, and the linear system deriving is stable (Table1).