Numerical Simulation of Thermomechanical Problems in Applied Mechanics: Application to Solidification Problem

Transcription

1 Leoardo Joural of Scieces ISSN Issue 9, July-December 2006 p Numerical Simulatio of Thermomechaical Problems i Applied Mechaics: Applicatio to Solidificatio Problem Vicet Obiajulu OGWUAGWU Mechaical Egieerig Departmet, Federal Uiversity of Techology, Mia, Nigeria ovogwuagwu@yahoo.com Abstract Thermomechaical couplig problems are commo pheomea i the field of Solid Mechaics. I these problems, the mechaical respose of such solids depeds o the thermal behaviour of the medium ad vice versa. The preset study presets the two-dimesioal umerical simulatio of stress distributio i a coolig igot durig cotiuous castig of steel. Results obtaied from the simulatio show good represetatio of stress distributio i igot coolig ad compares well with experimetal results from past studies. Keywords Thermomechaical couplig; Igot coolig; Temperature stresses; Phase trasformatio Itroductio May problems exist i the physical scieces i which there exist couplig betwee mechaical ad thermal pheomea. Icrease i the temperature of a body ca lead to chages i deformatio behaviour of the body which i tur may lead to variatio i stresses (commoly referred to as temperature stresses). This is a commo pheomeo i the solidificatio of metals i moulds as well as coolig of welds etc. I geeral, the mechaical respose of such bodies depeds o their thermal behaviour. 25

2 Numerical Simulatio of Thermomechaical Problems i Applied Mechaics: Applicatio to Solidificatio Problem Vicet Obiajulu OGWUAGWU I this paper, a thermomechaical problem ivolvig the solidificatio ad coolig of steel igot is preseted. Numerical aalysis of material processig is usually very complicated due to couplig amog the temperature field, stress-strai history ad the material phase trasformatios ([1]-[3]). Mathematical Modellig Figure 1 shows a schematic diagram of the couplig betwee temperature, stress-strai ad phase trasformatio. Temperature Phase Trasformatio Stress - Strai History Figure 1. Couplig betwee temperature, stress-strai ad phase trasformatio Couplig betwee temperature ad phase trasformatio Materials phase trasformatios are basically govered by the temperature profile, as idicated i path 1 i figure 1. For slow chage i temperature, the equilibrium phase diagram ca be used to describe the phase trasformatio. I this case, oly the temperature determies the material costitutio. For fast coolig rates, however, the cotiuous coolig trasformatio is most appropriate. The temperature field ca be affected by the phase trasformatio through the release or absorptio of the latet heat of trasformatio as show by path 2 i figure 1. The latet heat effect is basically a heat icrease give by the relatio: T k p T x EαT t x 1 2ν t t i v ρc = + q + S i (1) 26

3 Leoardo Joural of Scieces ISSN Issue 9, July-December 2006 p where k is the thermal coductivity, α is the thermal expasio coefficiet, q is the iteral heat geeratio rate, v is the elastic volume strai, S is the deviatory stress compoets ad p is the plastic strai compoet. The third term o the right had side of equatio (1) represets the elastic expasio work while the last term is the eergy dissipatio due to plastic deformatio. Couplig betwee temperature ad stress-strai histories The chage of temperature field causes thermal strai, represeted by path 3 i figure 1. Also, the stress-strai histories ca affect the temperature due to the work of volume chage ad plastic deformatio. Geerally, these terms usually have little effects o the temperature filed for small deformatios. Couplig betwee trasformatio ad stress-strai histories The total strai ca be divided ito several compoets give by: e th p tr tp = (2) The phase trasformatio icludes a trasformatio strai tr i equatio (2) above. This is usually caused by the differece of the specific volume of the phases ivolved i the phase chage. There is also trasformatio plasticity tp caused by ielastic deformatio eve at stresses below yield stregth. The chage i volume fractios durig phase trasformatio leads to chage i mechaical properties as idicated by path 4 i figure 1. The elastic strai ad stress are related by the followig costitutive equatio: e 1 = [(1 +ν) σ δνσ mm] (3) E To obtai the icremetal strai, we subtract all the other strai compoets from the total strai icremet as: = (4) e th tr p tp E ad ν are both temperature ad material compositio depedet. For coexistig phases, the followig relatios hold: E(T,v )= k vkt k(t), k=1 k ν(t,v )= v t (T) (5) where the subscript is the umber of the phase ad v k is the volume fractio of the phase. k=1 k k 27

4 Numerical Simulatio of Thermomechaical Problems i Applied Mechaics: Applicatio to Solidificatio Problem Vicet Obiajulu OGWUAGWU The total thermal strai is give as: th =δ( v kαk)(t T ref ) (6) I the rate form, equatio (6) ca be writte i the form: th α T! =δ( (T T ref ) +α) (7) t t where α is the average values of the thermal expasio coefficiet give by: k k (8) k=1 α= v α Geerally, a mea value of α from a referece temperature to ay temperature withi the rage of iterest ca be used to calculate the thermal strai, which is defied as: α= 1 T αθdθ ( ) T T (9) Tref ref The thermal strai icremet is calculated as: ( ) ( - ) =α α (10) th + 1 T+ 1 Tref + 1 TTref which accout for chage i thermal strai i a time icremet. For phase trasformatio ivolvig a umber of phases, the trasformatio strais for a isotropic material is give by: th tr =δ vkk k=1 (11) The trasformatio plasticity ca be determied from the total trasformatio plastic strai formulatio as: tr 5σ v = (12) 6Y v E where Y E is the yield stregth of the weaker of the two phases ivolved ad v/v is the volume strai. The rate formulatio is give as: d tr d 1 = K(1 v k) v k( σ δσ mm) (13) dt dt 3 The vo Mises yield criterio is used with liear isotropic hardeig rule for the plastic strai. Now, the stress model for two-dimesioal plae stress problems is derived from the equatio of equilibrium give as: 28

5 Leoardo Joural of Scieces ISSN Issue 9, July-December 2006 p σ x + xy + x = x τ τ f 0, x σ xy y + + y = f 0 where the relatio betwee strai ad displacemet are give by: x u v u v =, y =, xy = + x y x (14) (15) which i matrix form becomes: = B u (16) where B is the matrix defied as: 0 x B = 0 (17) x Utilizig the priciple of virtual work for equilibrium coditios gives: δ σ dω δu f dω δu b dγ = 0 (18) T T T Ω Ω Γ The for small icremets σ, b, ad f the equilibrium coditio still holds ad equatio (22) is still satisfied. The fiite elemet discretio for displacemet ad strai are: u = ΣN i d i -N T d, δu = N T δd, = ΣB i d i -B T d; δ = Bδd which o substitutio ito equatio (18) yields: ( ) (19) T T T T d B d N fd N bd 0 Ω Ω Γ δ σ Ω Ω Γ = Or sice equatio (19) is satisfied for arbitrary values of δd, we obtai: (20) T T T B σdω N fdω N bdγ = 0 Ω Ω Γ Also, for isotropic materials, equatio (19) is give i matrix form as: σ = D 1 + D 2 ΩT which o substitutio ito equatio (20) yields: K d = R = R th + R m (21) where: K = Ω B T D 1 BdΩ R th = Ω B T D 2 TdΩ R m = Ω N T fbdω + Γ N T fbdγ 29

6 Numerical Simulatio of Thermomechaical Problems i Applied Mechaics: Applicatio to Solidificatio Problem Vicet Obiajulu OGWUAGWU Results ad Discussios From the stress histories of figures 3 ad 4 respectively, the effect of phase trasformatio zoe ca be traced. The outer surfaces express tesile stresses immediately after solidificatio ad the reversal to compressio. The symmetrical axes of the cast are subjected to tesile stress state after complete solidificatio as expected. The temperature cotours ad pricipal stresses alog the axis of symmetry ad at the outer surfaces are show i Figures 2, 3, 4, ad 5 respectively. Figure 2. Temperature histories alog the axis of symmetry The cast completely solidified after about 900 secods. The liquids core of the cast gave rise to small tesile stresses while the outer solid shell is i compressio. The high trasiet tesile stress at the surface combied with some other solidificatio factors may cause some defects. Uloadig to a elastic state caused a correspodig deep gradiet i the stress field. These uloadig coditios correspodig to the stress reversal are evidet i figures 3 ad 4. 30

7 Leoardo Joural of Scieces ISSN Issue 9, July-December 2006 p Figure 3. Temperature histories at the outer surface Figure 4. Stress histories alog the axis of symmetry 31

8 Numerical Simulatio of Thermomechaical Problems i Applied Mechaics: Applicatio to Solidificatio Problem Vicet Obiajulu OGWUAGWU Figure 5. Stress histories at the outer surface Based o the iformatio obtaied from the developed pricipal stresses, it ca be predicted if a give process coditios would lead to some defects such as cracks. For istace, if the maximum pricipal stress at the cetre of the castig exceeds some criteria for crack formatio, the at that temperature, there may be the possibility of iitiatio ad propagatio of logitudial cracks. This is as a result of the radom orietatio of the grais at the cetre. Refereces [1] Ioue T., Wag Z., Couplig betwee Stresses, Temperature, ad Metallic Structure Durig Processes Ivolvig Phase Trasformatios, Materials Sciece Techology, 1, p , [2] Sjostrom S., Iteractios ad Costitutive Models for Calculatig Quech Stresses i Steel, Material Sciece ad Techology, 1, p , [3] Wag Z. Ioue T., Viscoplastic Costitutive Relatio Icorporatig Phase Trasformatio: Applicatio to Weldig, Material Sciece ad Techology, 1, p , [4] Vaz Jr. M. Owe D. R. J., Thermo-mechaical Couplig ad Large Scale Elasto-plastic Problems, Applicatio of Numerical Methods i Egieerig,