Abstract
In the present work, we developed a new FEM framework to simulate the mechanical responses of the Euler–Bernoulli beam with a two-phase local/nonlocal mixed model. The shape function is a fifth-order polynomial and constitutive boundary conditions (CBCs) are treated as external loads. The main advantages of the present model are the efficient of convergence, simplicity of expressions, and the flexibility on handling various boundary conditions as well as the external loads. Several numerical tests, including static bending, free vibration, and elastic bulking, are carried out to validate the FEM framework. The results showed the complete agreement with the exact solution from Laplace transformation and indicated a simple and reliable scheme to deal with complicated nanosystems.
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Abbreviations
- \(\delta v, \delta v^\prime , \delta v^{\prime \prime }\) :
-
Test function and its gradient
- \({\mathbf {B}}^i\) :
-
Geometric matrix
- \({\mathbf {f}}^\mathrm {int},{\mathbf {f}}^\mathrm {ext}\) :
-
Internal and external nodal force array
- \({\mathbf {K}}^\mathrm {int}\) :
-
Jacobian matrix of the internal nodal force
- \({\mathbf {K}}_\varDelta\) :
-
Incremental stiffness matrix
- \({\mathbf {M}}\) :
-
Mass matrix
- \({\mathbf {N}}\) :
-
Shape function matrix
- \({\mathbf {u}}\) :
-
Displacement array
- \(\omega\) :
-
Frequency of vibration
- \({\overline{Q}},{\overline{M}},{\overline{T}}\) :
-
External shear force, moment, and high-order force
- \(\rho\) :
-
Density
- \(\sigma _{ij},\varepsilon _{ij}\) :
-
Components of stress and strain tensor
- \({\widehat{Q}},{\widehat{M}}\) :
-
React shear force and moment
- \(\xi ,\kappa\) :
-
Nonlocal parameters
- A :
-
The area of the cross section
- E :
-
Young’s module
- I :
-
The second axial moment of the cross-sectional area
- n :
-
External normal vector
- P :
-
Axial force
- Q, M :
-
Internal shear force and moment over cross section
- \(q,q^\prime ,q^{\prime \prime }\) :
-
Distributed pressure and its gradient
- T :
-
High-order force conjugated to the \(w^{\prime \prime }\)
- \(z,z^\prime ,z^{\prime \prime },z^{(3)},z^{(4)},z^{(5)}\) :
-
Deflection and its 1st, 2nd, 3rd, 4th, and 5th gradient with x
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Acknowledgements
The authors are grateful for the support of the present work by the National Natural Science Foundation of China (12172169) and the Priority Academic Program Development of Jiangsu Higher Education Institutions. Bian is also grateful for the scholarship provided by the China Scholarship Council for a 1-year study at the University of Stuttgart (201806830018).
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Appendices
A List of the terms in the weak form of the present model
Some terms in the the weak form of the present model from the theorem of divergence (integration by parts for one-dimension case) are listed here:
B Shape and geometric matrices of the present beam element
The explicit expression of the shape function and the ith geometric matrix of the fifth-order polynomial are listed (over the internal \([0,l_\mathrm {e}]\)):
C Table of coefficients of CBCs at both sides of the beam
D Table of locations and weights with Gaussian quadrature
Some low-order quadrature rules are tabulated below (over interval \([-1,1]\)) (Table 2).
where 7-points quadrature is only necessary for calculating \(\int _V{\mathbf {N}}^\mathrm {T}{\mathbf {N}}\mathrm {d}V\)
E Lagrange multiplier method for linear constrain in FEM
The solution of the differential equation with constraints is equivalent to finding the stationary points of the stationary points of :
The value of \(\lambda _I\) is the react-force or moment at boundaries.
F Derivation of the exact solution to free vibration analysis of the present nonlocal Euler–Bernoulli beam with Laplace transform
Ignoring the axial force P and performing Laplace transformation on Eq. (5) , one gets
Applying the Laplace transformation to both sides of Eq. (7) and taking into account the convolution theorem, one can derive
where
According to the derivative property on Laplace transformation, one can obtain
in which
Combining Eqs. (57), (58), and (60), and taking into account
where
The parameters \(a_0\), p and q are listed as follows:
One obtains Eq. (62)
For the nonlocal beam with clamped–clamped boundary conditions, we can obtain from Eq. (62) and Eq. (65)
where \(L_i(x),\ i=0,1,\ldots ,5\) are defined as
Combining the Eqs. (59) and (66) as well as clamped boundary at the both sides of the beam, we can obtain
where \(A_{ij}^\mathrm {CC}\) are listed as follows:
None-zero solution of homogeneous linear Eq. (68) requires the coefficient matrix of \(C_0\), \(C_1\), and \(C_2\) singular, e.g.,
The eigenvalues \(\omega\) can be thus obtained numerically.
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Bian, PL., Qing, H. Structural analysis of nonlocal nanobeam via FEM using equivalent nonlocal differential model. Engineering with Computers 39, 2565–2581 (2023). https://doi.org/10.1007/s00366-021-01575-5
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DOI: https://doi.org/10.1007/s00366-021-01575-5