3. Modeling A

3.1. Characteristics of modeling

AXIS, \(\mathit{Q8}\) mesh

Breakdown:	10 elements depending on the length
	1 element in the thickness

Boundary conditions:

in \(A\), \(B\)	DDL_IMPO = (GROUP_NO = “A”, DY= 0.)
	DDL_IMPO = (GROUP_NO = “B”, DY= 0.)

Pressure + shape effect: field \(U\)

PRES_REP:	(GROUP_MA = cont_pr, PRES = 2.E8)
FORCE_CONTOUR:	(GROUP_MA = effond, FY= 1.95E9)

Thermal expansion: field \({U}_{1}\)

char_no:
CREA_CHAMP	(AFFE = (TOUT = “OUI”, NOM_CMP = “TEMP”, VALE = 100.))

char_th:
AFFE_MATERIAU	(AFFE_VARC = F (TOUT = “OUI”, CHAM_GD = “”, = CHAR_NO, VALE_REF = 0., NOM_VARC = “TEMP”,)

Pre-deformations: field \({U}_{2}\)

PRE_EPSI:	(TOUT = “OUI”, EPXX = 1.2E-3, EPYY = 1.2E-3,
	EPZZ = 1.2E-3, EPXY = 0.)

Node names:

\(A\mathrm{=}\mathit{N1}\)

\(B\mathrm{=}\mathit{N2}\)

\(C\mathrm{=}\mathit{N3}\)

\(D\mathrm{=}\mathit{N4}\)

3.2. Characteristics of the mesh

Number of knots: 53

Number of meshes and types: 10 QUAD8, 22 SEG3

3.3. Tested sizes and results

Results for fields \({U}_{1}\), \({U}_{2}\), \(U\)

Field	Location	Variables	Reference
Thermal field \({U}_{1}\)	\(A\)		\(\mathit{Ur}(\mathit{DX})\)	5.7 x 10-5
	\(B\)	\(\mathit{Ur}(\mathit{DX})\)	6 x 10-5
	\(C\)	\(\mathit{Ur}(\mathit{DX})\)	6 x 10-5
		\(\mathit{DY}\)	1.2 x 10-3
	\(D\)	\(\mathit{Ur}(\mathit{DX})\)	5.7 x 10-5
		\(U(\mathit{DY})\)	1.2 x 10-3
	\(A\), mesh \(\mathit{M1}\)	\({\varepsilon }_{\mathit{rr}}\)	1.2 x 10-3
		\({\varepsilon }_{\theta \theta }\)	1.2 x 10-3
		\({\varepsilon }_{\mathit{zz}}\)	1.2 x 10-3
	\(B\), mesh \(\mathit{M1}\)	\({\varepsilon }_{\mathit{rr}}\)	1.2 x 10-3
		\({\varepsilon }_{\theta \theta }\)	1.2 x 10-3
		\({\varepsilon }_{\mathit{zz}}\)	1.2 x 10-3
	\(C\), mesh \(\mathit{M10}\)	\({\varepsilon }_{\mathit{rr}}\)	1.2 x 10-3
		\({\varepsilon }_{\theta \theta }\)	1.2 x 10-3
		\({\varepsilon }_{\mathit{zz}}\)	1.2 x 10-3
	\(D\), mesh \(\mathit{M10}\)	\({\varepsilon }_{\mathit{rr}}\)	1.2 x 10-3
		\({\varepsilon }_{\theta \theta }\)	1.2 x 10-3
		\({\varepsilon }_{\mathit{zz}}\)	1.2 x 10-3
Pressure field \(U\)	\(C\)		\({U}_{\theta }(\mathit{DY})\)	3.714 x 10-3
	\(D\)	\({U}_{\theta }(\mathit{DY})\)	3.714 x 10-3
	\(C\), mesh \(\mathit{M10}\)	\({\varepsilon }_{\theta \theta }\)	3.714 x 10-3
	\(D\), mesh \(\mathit{M10}\)	\({\varepsilon }_{\theta \theta }\)	3.714 x 10-3
Field \({U}_{2}\)	\(C\)	\({U}_{\theta \theta }\)	4.914 x 10-3
	\(D\)	\({U}_{\theta \theta }\)	4.914 x 10-3
	\(C\), mesh	\({\varepsilon }_{\theta \theta }\)	4.914 x 10-3
	\(D\), mesh	\({\varepsilon }_{\theta \theta }\)	4.914 x 10-3

3.4. notes

• The aim of the test is not to obtain high precision in terms of the results, but simply to verify the relationship: \({U}_{2}\mathrm{=}U+{U}_{1}\); therefore, the calculation was only carried out with a rough mesh.

• It can be seen that the relationship sought is well verified at the free end of the cylinder.

• It is also verified that the deformation field resulting from thermal expansion is uniformly equal to \(1.2\mathrm{\times }{10}^{\mathrm{-}3}\).