On the numerical solution of a class of problems in a linear first strain-gradient theory of elasticity

1970 ◽  
Vol 2 (2) ◽  
pp. 159-174 ◽  
Author(s):  
J. T. Oden ◽  
D. M. Rigsby ◽  
D. Cornett
Keyword(s):  
Numerical Solution ◽  
Strain Gradient ◽  
Theory Of Elasticity ◽  
Strain Gradient Theory ◽  
Gradient Theory

scholarly journals 1D HERMITE ELEMENTS FOR C1-CONTINUOUS SOLUTIONS IN SECOND GRADIENT ELASTICITY

2016 ◽  
Vol 7 ◽  
pp. 33-37 ◽  
Author(s):  
Christian Liebold ◽  
Wolfgang H. Müller
Keyword(s):  
Finite Element ◽  
Size Effect ◽  
Numerical Solution ◽  
Stiffness Matrix ◽  
Gradient Elasticity ◽  
Theory Of Elasticity ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Additional Material ◽  
Hermite Finite Elements

We present a modified strain gradient theory of elasticity for linear isotropic materials in order to account for the so-called size effect. Additional material length scale parameters are introduced and the problem of static beam bending is analyzed. A numerical solution is derived by means of a finite element approach. A global C1-continuous displacement field is applied in finite element solutions because the higher-order strain energy density additionally depends on second gradients of displacements. So-called Hermite finite elements are used that allow for merging gradients between elements. The element stiffness matrix as well as the global stiffness matrix of the problem is developed. Convergence, C1-continuity and the size effect in the numerical solution is shown. Experiments on bending stiffnesses of different sized micro beams made of the polymer SU-8 are performed by using an atomic force microscope and the results are compared to the numerical solution.

scholarly journals Thermo-electro-mechanical behaviour of Nano-sized structures

2020 ◽  
Vol 310 ◽  
pp. 00060
Author(s):  
Miroslav Repka ◽  
Ladislav Sator
Keyword(s):  
Size Effect ◽  
Mechanical Behaviour ◽  
Strain Gradient ◽  
Mechanical Response ◽  
Scale Parameter ◽  
Theory Of Elasticity ◽  
Strain Gradient Theory ◽  
Length Scale Parameter ◽  
Gradient Theory ◽  
Length Scale

Thermo-electro-mechanical behaviour of the nano-sized structures is analysed by the finite element method (FEM). The mechanical response of the nano-sized structures cannot be modelled with classical continuum theories due to the size effect phenomenon. The strain gradient theory with one length scale parameter has been applied to study size effect phenomenon. The coupled theory of thermo-electricity has been used together with strain gradient theory of elasticity. The governing equations have been derived and incorporated into the commercial software Comsol via weak form module. The influence of the length scale parameter on mechanical response of the structures is investigated by some numerical examples.

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