Shape-Free Finite Element Method: Another Way between Mesh and Mesh-Free Methods

Table 1

Eighteen fundamental analytical solutions of the Airy stress function and resulting stress and displacement solutions for plane stress problems*.

i	𝜙𝑖	𝜎𝑥𝑖	𝜎𝑦𝑖	𝜏𝑥𝑦𝑖	𝑈𝑖	𝑉𝑖
1	1	0	0	0	1/E	0
2	x	0	0	0	0	1/E
3	y	0	0	0	y/E	−x/E
4	x ^2	0	2	0	−2 𝜇x/E	2y/E
5	xy	0	0	−1	− (1 + 𝜇)y/E	−(1 + 𝜇)x/E
6	y ^2	2	0	0	2x/E	−2 𝜇y/E
7	x ^3	0	6x	0	−3(𝜇x ^2 + y ^2)/E	6xy/E
8	x ^2y	0	2y	−2x	−2 𝜇xy/E	[y ^2 − (2 + 𝜇)x ^2]/E
9	xy ^2	2x	0	−2y	[𝑥^2−(2+𝜇)𝑦^2]/𝐸	−2 𝜇xy/E
10	y ^3	6y	0	0	6xy/E	−3(𝜇y ^2 + x ^2)/E
11	x ^3y	0	6xy	−3x ^2	−(3 𝜇x ^2y + y ^3)/E	[3𝑥𝑦^2−(2+𝜇)𝑥^3]/𝐸
12	xy ^3	6xy	0	−3y ^2	[3x ^2y − (2 + 𝜇)y ^3]/E	−(3 𝜇x ^2y + y ^3)/E
13	𝑥^4−𝑦^4	−12y ^2	12x ^2	0	−4(3xy ^2+ 𝜇x ^3)/E	4(3x ^2y + 𝜇y ^3)/E
14	6𝑥^2𝑦^2−𝑥^4−𝑦^4	12(𝑥^2−𝑦^2)	−12(𝑥^2−𝑦^2)	−24xy	4(1 + 𝜇)(𝑥^3−3𝑥𝑦^2)/E	4(1 + 𝜇)(𝑦^3−3𝑥^2𝑦)/E
15	𝑥^3𝑦^2−𝑥𝑦^4	2x(𝑥^2−6𝑦^2)	6xy ^2	−2y(3𝑥^2−2𝑦^2)	[𝑥^4−6(2 + 𝜇)x ^2y ^2 + (3 + 2 𝜇)y ^4]/2E	[2(1 + 2 𝜇)𝑥𝑦^3−2 𝜇x ^3y]/E
16	5𝑥^3𝑦^2−𝑥^5	10x ^3	−10x(2𝑥^2−3𝑦^2)	−30x ^2y	2.5[(1 + 2 𝜇)𝑥^4−6 𝜇x ^2y ^2 − y ^4]/E	10xy[𝑦^2−(2+𝜇)𝑥^2]/E
17	𝑥^2𝑦^3−𝑥^4𝑦	6x ^2y	−2y(6𝑥^2−𝑦^2)	2x(2𝑥^2−3𝑦^2)	[2(1 + 2 𝜇)x ^3𝑦−2 𝜇xy ^3]/E	[𝑦^4−6(2 + 𝜇)x ^2y ^2 + (3+2 𝜇)x ^4]/2E
18	5𝑥^2𝑦^3−𝑦^5	10y(3𝑥^2−2𝑦^2)	10y ^3	−30xy ^2	10xy[x ^2 − (2 + 𝜇)y ^2]/E	2.5[(1 + 2 𝜇)𝑦^4−6 𝜇 x ^2y ^2 − x ^4]/E

∗Note: 𝐸→𝐸/(1−𝜇²) and 𝜇→𝜇/(1−𝜇) for plane strain problem. E and 𝜇 are Young’s modulus and Poisson’s ratio, respectively.