Code_Aster. HSNV120 - Tension hyper elastic of a bar under thermal loading

Transcription

1 Titre : HSNV - Traction hperélastique d'un barreau sou[...] Date : 4/8/ Page : / HSNV - Tension hper elastic of a bar under thermal loading Abstract: This quasi-static thermomechanical test consists in heating a parallelepipedic bar uniforml, to subject it to an important tension for finall letting it return in a discharged state. One validates thus the kinematics of the large deformations hper elastics (command STAT_NON_LINE, ke word COMP_ELAS) for a nonlinear elastic behavior model (ELAS_VMIS_LINE and ELAS_VMIS_TRAC) with thermal loading. The bar is modelled b an element voluminal (HEXA, modelization A) or quadrangular (QUAD8, assumption of the plane stresses, modelization B). The results got b Aster do not differ from the theoretical solution.

2 Titre : HSNV - Traction hperélastique d'un barreau sou[...] Date : 4/8/ Page : / Problem of reference. Geometr (m m ) 4 z 3 (m m ) x. Material properties the material obes a constitutive law isotropic nonlinear hper elastic with isotropic linear hardening. S σ E E T E. 5 MPa E T. 3 MPa σ 3 MPa ν,3 α 4 K E

3 Titre : HSNV - Traction hperélastique d'un barreau sou[...] Date : 4/8/ Page : 3/.3 Boundar conditions and loadings the bar blocked in the direction Ox on the face [,] is subjected to a uniform temperature T and a tractive effort F distributed on the face [3,4]. The sequences of loading are the following ones: 4 T unif F 3 T ( C) F (MPa) 98 3 t(s) 3 t(s) Reference temperature: T réf C.

4 Titre : HSNV - Traction hperélastique d'un barreau sou[...] Date : 4/8/ Page : 4/ Reference solution. Method of calculating used for the reference solution One seeks the field of displacement U in the form: ( ) U x,, z ux v vz The gradient of the transformation, the strain and its mechanical share are then: with: F + u + v + v E u( u + ) ( + ) T ( F F ) v v v( v + ) m E E α T a b b a b u( u + ) α T v( v + ) α T Note: ( ) a b a b ( on suppose que a > b) E m eq

5 Titre : HSNV - Traction hperélastique d'un barreau sou[...] Date : 4/8/ Page : 5/ The behavior model is written: with: Sxx K( a + b) + G( a b) 3 S Szz K( a + b) G( a b) 3 3K E ν module de compressibilité To determine G b taking account of linear hardening, one introduces: the shear modulus: µ the hardening modulus: R E + ν E ET ' E E T, The pseudonm local variable p is worth then: p ( ) ( ) µ E m σ eq + 3µ µ a b σ + 3µ Finall, G is written: B taking account of the boundar conditions: G σ + p a b S S xx F + u (charge morte) (bord libre) The sstem to be solved is written: µ ( a b) σ F K( a + b) + σ µ + u ( a b) K( a b) µ σ + σ µ

6 Titre : HSNV - Traction hperélastique d'un barreau sou[...] Date : 4/8/ Page : 6/ He is also written: 3K( a + b) µ ( a b) F + u F 3 µ 3 µ + σ + u A F fixed, it is thus about a nonlinear sstem in u and v, since a is quadratic in u and b quadratic in v. Nevertheless, one can choose to fix u (thus a ) and to solve a linear sstem in F and b (from which one deduces p and v ): a u( u + ) α T u F K b K a 3 µ R + u F b a µ µ σ ' µ ( a b) σ p + 3µ v ( b a T ) + + It then remains to express the stress of Cauch: That is to sa here: µ σ Det ( F ) F S FT σ σ xx + u ( + v) σ zz S xx As for the force exerted on the face [3,4], because of assumption of died loads, she is written simpl: F F F [ ] x o où o : surface initiale de la face 3,4 z F S S

7 Titre : HSNV - Traction hperélastique d'un barreau sou[...] Date : 4/8/ Page : 7/. Results of reference One will adopt like results of reference displacements, the stress of Cauch and the force exerted on the face [3,4 ] (in 3D onl): At time t s ( T C, tension F ) makes some, one seeks F such as lengthening: u, K MPa MPa R ' MPa a F b F b F 98 MPa b. 46 p 8.9 v σxx MPa σx x 98 N σ σxz F σzz σ z Fz At time t3 s ( T, F ) the bar returned in its initial state: F U σ p.3 Uncertaint on the solution the solution is analtical. With the rounding errors near, one can consider it exact..4 Bibliographical references One will be able to refer to: ) E. LORENTZ: A nonlinear behavior model hper elastic - internal Note EDF DER HI-74/95//

8 Titre : HSNV - Traction hperélastique d'un barreau sou[...] Date : 4/8/ Page : 8/ 3 Modelization A 3. Characteristic of the voluminal modelization Modelization: z 5 7 mesh HEXA nets QUAD8 8 (mm) x Boundar conditions: N : N : N6 : U x U U z U x U z U U x N9 N3 N4 N5 N7 : U x Charge: Tension on the face [ ] 3. Characteristics of the mesh Man nodes: Number of meshes: HEXA QUAD8

9 Titre : HSNV - Traction hperélastique d'un barreau sou[...] Date : 4/8/ Page : 9/ 3.3 Quantities tested and results Identification Reference Aster % difference t Displacement DX ( N8 )... t Displacement DY ( N8 ) t Displacement DZ ( N8 ) t Stresses SIGXX ( PG ) t Stresses SIGYY ( PG ) / t Forced SIGZZ ( PG ) - / t Forced SIGXY ( PG ) - / t Forced SIGXZ ( PG ) - / t Forced SIGYZ ( PG ) - / t Variable p VARI ( PG ) t3 Displacement DX ( N8 ) -3 / t3 Displacement DY ( N8 ) -3 / t3 Displacement DZ ( N8 ) -4 / t3 Forced SIGXX ( PG ) - / t3 Forced SIGYY ( PG ) - / t3 Forced SIGZZ ( PG ) - / t3 Forced SIGXY ( PG ) - / t3 Forced SIGXZ ( PG ) - / t3 Forced SIGYZ ( PG ) - / t3 Variable p VARI ( PG ) / t nodal Force DX ( N8 ) t nodal Force DY ( N8 ) -5 / t nodal Force DZ ( N8 ) -6 / 3.4 Remarks Computation of the nodal force: The applied force on the face [3,4 ] F x, is distributed between the various nodes according to following weighting: nodes tops: / F x nodes mediums: 4 / F x / 4/ / 4/ 4/ / 4/ /

10 Titre : HSNV - Traction hperélastique d'un barreau sou[...] Date : 4/8/ Page : / 4 Modelization B 4. Characteristic of the modelization Modelization D plane stresses: mesh QUAD8 nets SEG x Boundar conditions: N : N : N5 : Loading: U x U x U x Tension on the face [3 47] (mesh SEG3) U 4. Characteristic of the mesh Man nodes: 8 Number of meshes: QUAD8 SEG3 4.3 Quantities tested and results Identification Reference Aster % difference t Displacement DX ( N4 ). t Displacement DY ( N4 ) t Stresses SIGXX ( PG ) t Stresses SIGYY ( PG ) - / t Forced SIGXY ( PG ) Variable - t / p VARI ( PG ) t3 Displacement DX ( N4 ) -4 / t3 Displacement DY ( N4 ) -3 / t3 Forced SIGXX ( PG ) - / t3 Forced SIGYY ( PG ) - / t3 Forced SIGXY ( PG ) Variable - t3 / p VARI ( PG ) /

11 Titre : HSNV - Traction hperélastique d'un barreau sou[...] Date : 4/8/ Page : / 5 Summar of the results the numerical and analtical results coincide remarkabl. One can however be astonished b the execution time manifestl longer for the modelization in plane stresses ( 3,8s ) that for 3D ( 47, s ). The difference is explained b a discretization in time much finer for the plane stresses, related to problems of convergence (the algorithm of resolution of the nonlinear scalar equation in p is still rudimentar).