A unified reproducing kernel gradient smoothing Galerkin meshfree approach to strain gradient elasticity

Abstract

A unified reproducing kernel gradient smoothing formulation is presented for efficient Galerkin meshfree analysis of strain gradient elasticity problems with particular reference to high order basis functions, such as cubic and quartic basis functions. The integration constraint is elaborated for the Galerkin meshfree formulation of strain gradient elasticity, where the possible inconsistency issue associated with the separate first and second order gradient smoothing formalism is completely resolved. In this approach, the first order smoothed gradients of meshfree shape functions are expressed as a reproduced kernel form and the second order smoothed gradients are deduced from their first order counterparts through direct differentiation. Thereafter, the unknown coefficients in the first and second order smoothed gradients are simultaneously solved from the integration constraint. The resulting smoothed gradients automatically meet the integration consistency by construction. Subsequently, it is proven that the reproducing kernel gradient smoothing quadrature rules for the conventional elasticity problems are well suitable for the strain gradient elasticity problems. Numerical results clearly demonstrate the superior performance of the proposed approach regarding convergence, accuracy, as well as efficiency, in comparison with the Gauss integration-based meshfree formulation for strain gradient elasticity problems.

Appendix

For the reproducing kernel gradient smoothing integration using quartic meshfree approximation, the properties of different type 1D, 2D and 3D sample points as shown in Fig. 2 are described in Tables

Table 6 Properties for different type 1D sample points

6,

Table 7 Properties for different type 2D sample points

7,

Table 8 Properties for different type 3D sample points

8.

The three sets of independent equations arising from Eqs. (68) and (69) for 1D, 2D and 3D cases are attained as follows:

• (1) 1D case

  $$ \left\{ {\begin{array}{*{20}l} {\sum\limits_{{S = {\mathcal{S}}_{2} }} {\omega_{S}^{b} \langle \hat{\delta }_{2} \rangle } = 1} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1 - 3} }} {\omega_{S}^{i} \langle 1} \rangle = 1} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1,3} }} {\omega_{S}^{i} \langle \xi_{1} \xi_{2} } \rangle = \frac{1}{6}} \hfill \\ {\sum\limits_{{S = {\mathcal{S}}_{1,3} }} {\omega_{S}^{i} \langle \xi_{1}^{2} \xi_{2}^{2} } \rangle = \frac{1}{30}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1,3} }} {\omega_{S}^{i} \langle \xi_{1}^{3} \xi_{2}^{3} } \rangle = \frac{1}{140}} \hfill & {} \hfill \\ \end{array} } \right. $$

  (82)

• (2) 2D case

  $$ \left\{ {\begin{array}{*{20}l} {\sum\limits_{{S = {\mathcal{S}}_{2,3,5} }} {\omega_{S}^{b} \langle \hat{\delta }_{3} \rangle = 1} } \hfill & {\sum\limits_{{S = {\mathcal{S}}_{3,5} }}^{{}} {\omega_{S}^{b} \langle \xi_{1} \xi_{2} \hat{\delta }_{3} \rangle = \frac{1}{6}} } \hfill & {\sum\limits_{{S = {\mathcal{S}}_{3,5} }} {\omega_{S}^{b} \langle \xi_{1}^{2} \xi_{2}^{2} \hat{\delta }_{3} } \rangle = \frac{1}{30}} \hfill \\ {\sum\limits_{{S = {\mathcal{S}}_{3,5} }} {\omega_{S}^{b} \langle \xi_{1}^{3} \xi_{2}^{3} \hat{\delta }_{3} \rangle } = \frac{1}{140}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1 - 6} }} {\omega_{S}^{i} \langle 1} \rangle = 1} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1,3 - 6} }} {\omega_{S}^{i} \langle \xi_{1} \xi_{2} } \rangle = \frac{1}{12}} \hfill \\ {\sum\limits_{{S = {\mathcal{S}}_{1,4,6} }} {\omega_{S}^{i} \langle \xi_{1} \xi_{2} \xi_{3} } \rangle = \frac{1}{60}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1,3 - 6} }} {\omega_{S}^{i} \langle \xi_{1}^{2} \xi_{2}^{2} } \rangle = \frac{1}{90}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1,4,6} }} {\omega_{S}^{i} \langle \xi_{1}^{2} \xi_{2}^{2} \xi_{3}^{{}} } \rangle = \frac{1}{630}} \hfill \\ {\sum\limits_{{S = {\mathcal{S}}_{1,4,6} }} {\omega_{S}^{i} \langle \xi_{1}^{2} \xi_{2}^{2} \xi_{3}^{2} } \rangle = \frac{1}{2520}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1,3 - 6} }} {\omega_{S}^{i} \langle \xi_{1}^{3} \xi_{2}^{3} } \rangle = \frac{1}{560}} \hfill & {} \hfill \\ \end{array} } \right. $$

  (83)

• (3) 3D case

  $$ \left\{ {\begin{array}{*{20}l} {\sum\limits_{{S = {\mathcal{S}}_{2,3,5,7,8,10} }} {\omega_{S}^{b} \langle \hat{\delta }_{4} \rangle } = 1} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{3,5,7,8,10} }} {\omega_{S}^{b} \langle \xi_{1} \xi_{2} \hat{\delta }_{4} \rangle } = \frac{1}{12}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{3,8,10} }} {\omega_{S}^{b} \langle \xi_{1}^{{}} \xi_{2}^{{}} \xi_{3}^{{}} \hat{\delta }_{4} \rangle } = \frac{1}{60}} \hfill \\ {\sum\limits_{{S = {\mathcal{S}}_{3,5,7,8,10} }} {\omega_{S}^{b} \langle \xi_{1}^{2} \xi_{2}^{2} \hat{\delta }_{4} \rangle } = \frac{1}{90}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{3,8,10} }} {\omega_{S}^{b} \langle \xi_{1}^{2} \xi_{2}^{2} \xi_{3}^{{}} \hat{\delta }_{4} \rangle } = \frac{1}{630}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{3,5,7,8,10} }} {\omega_{S}^{b} \langle \xi_{1}^{3} \xi_{2}^{3} \hat{\delta }_{4} \rangle } = \frac{1}{560}} \hfill \\ {\sum\limits_{{S = {\mathcal{S}}_{3,8,10} }} {\omega_{S}^{b} \langle \xi_{1}^{2} \xi_{2}^{2} \xi_{3}^{2} \hat{\delta }_{4} \rangle } = \frac{1}{2520}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{3,8,10} }} {\omega_{S}^{b} \langle \xi_{1}^{{}} \xi_{2}^{2} \xi_{3}^{4} \hat{\delta }_{4} \rangle } = \frac{1}{3780}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1 - 11} }} {\omega_{S}^{i} \langle 1\rangle } = 1} \hfill \\ {\sum\limits_{{S = {\mathcal{S}}_{1,3 - 11} }} {\omega_{S}^{i} \langle \xi_{1} \xi_{2} \rangle } = \frac{1}{20}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1,3,4,6,8 - 11} }} {\omega_{S}^{i} \langle \xi_{1} \xi_{2} \xi_{3} \rangle } = \frac{1}{120}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1,3 - 11} }} {\omega_{S}^{i} \langle \xi_{1}^{2} \xi_{2}^{2} \rangle } = \frac{1}{210}} \hfill \\ {\sum\limits_{{S = {\mathcal{S}}_{1,4,6,9,11} }} {\omega_{S}^{i} \langle \xi_{1} \xi_{2} \xi_{3} \xi_{4} \rangle } = \frac{1}{840}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1,3,4,6,8 - 11} }} {\omega_{S}^{i} \langle \xi_{1}^{2} \xi_{2}^{2} \xi_{3}^{{}} \rangle } = \frac{1}{1680}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1,3,4,6,8 - 11} }} {\omega_{S}^{i} \langle \xi_{1}^{2} \xi_{2}^{2} \xi_{3}^{2} \rangle } = \frac{1}{7560}} \hfill \\ {\sum\limits_{{S = {\mathcal{S}}_{1,3 - 11} }} {\omega_{S}^{i} \langle \xi_{1}^{3} \xi_{2}^{3} \rangle } = \frac{1}{1680}} \hfill & {\sum\limits_{{S = {\mathcal{S}}_{1,4,6,9,11} }} {\omega_{S}^{i} \langle \xi_{1}^{2} \xi_{2}^{2} \xi_{3}^{{}} \xi_{4}^{{}} \rangle } = \frac{1}{15120}} \hfill & {} \hfill \\ \end{array} } \right. $$

  (84)