TY - JOUR

T1 - On the analytical and meshless numerical approaches to mixture stress gradient functionally graded nano-bar in tension

AU - Faghidian, S. Ali

AU - Żur, Krzysztof Kamil

AU - Pan, Ernian

AU - Kim, Jinseok

N1 - Publisher Copyright:
© 2021 Elsevier Ltd

PY - 2022/1/1

Y1 - 2022/1/1

N2 - The mixture stress gradient theory of elasticity is conceived via consistent unification of the classical elasticity theory and the stress gradient theory within a stationary variational framework. The boundary-value problem associated with a functionally graded nano-bar is rigorously formulated. The constitutive law of the axial force field is determined and equipped with proper non-standard boundary conditions. Evidences of well-posedness of the mixture stress gradient problems, defined on finite structural domains, are demonstrated by analytical analysis of the axial displacement field of structural schemes of practical interest in nano-mechanics. An effective meshless numerical approach is, moreover, introduced based on the proposed stationary variational principle while employing autonomous series solution of the kinematic and kinetic field variables. Suitable mathematical forms of the coordinate functions are set forth in terms of the modified Chebyshev polynomials, satisfying the required classical and non-standard boundary conditions. An excellent agreement between the numerical results of the axial displacement field of the functionally graded nano-bar and the analytical solution counterpart is confirmed on the entire span of the nano-sized bar, in terms of the mixture parameter and the stress gradient characteristic parameter. The effectiveness of the established meshless numerical approach, demonstrating a fast convergence rate and an admissible convergence region, is hence ensured. The established mixture stress gradient theory can effectively characterize the peculiar size-dependent response of functionally graded structural elements of advanced ultra-small systems.

AB - The mixture stress gradient theory of elasticity is conceived via consistent unification of the classical elasticity theory and the stress gradient theory within a stationary variational framework. The boundary-value problem associated with a functionally graded nano-bar is rigorously formulated. The constitutive law of the axial force field is determined and equipped with proper non-standard boundary conditions. Evidences of well-posedness of the mixture stress gradient problems, defined on finite structural domains, are demonstrated by analytical analysis of the axial displacement field of structural schemes of practical interest in nano-mechanics. An effective meshless numerical approach is, moreover, introduced based on the proposed stationary variational principle while employing autonomous series solution of the kinematic and kinetic field variables. Suitable mathematical forms of the coordinate functions are set forth in terms of the modified Chebyshev polynomials, satisfying the required classical and non-standard boundary conditions. An excellent agreement between the numerical results of the axial displacement field of the functionally graded nano-bar and the analytical solution counterpart is confirmed on the entire span of the nano-sized bar, in terms of the mixture parameter and the stress gradient characteristic parameter. The effectiveness of the established meshless numerical approach, demonstrating a fast convergence rate and an admissible convergence region, is hence ensured. The established mixture stress gradient theory can effectively characterize the peculiar size-dependent response of functionally graded structural elements of advanced ultra-small systems.

KW - Chebyshev polynomials

KW - Meshless method

KW - Nano-bar

KW - Numerical approach

KW - Stationary variational principle

KW - Stress gradient theory

UR - http://www.scopus.com/inward/record.url?scp=85119264924&partnerID=8YFLogxK

U2 - 10.1016/j.enganabound.2021.11.010

DO - 10.1016/j.enganabound.2021.11.010

M3 - Article

AN - SCOPUS:85119264924

VL - 134

SP - 571

EP - 580

JO - Engineering Analysis with Boundary Elements

JF - Engineering Analysis with Boundary Elements

SN - 0955-7997

ER -