Equations of elasticity: Cartesian grid Equations In this model u = (U, V, W) is vector of displacement. The equations of X-force balance are:

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1 Equations of elasticit: Cartesian grid Equations In this model u = (U, V, W) is vector of displacement. The equations of X-force balance are: x x 0 f x x The displacements and the stresses are connected b Hooke s law: G LD HT, x G LD HT, G LD HT, U V U W V W x G, x G, G, x x D, U V W,, x here D - dilatation (relative changing of volume at deformation), G,L - first () and second () Lame's constant: E - Young s module, - oisson s coefficient, - thermal-expansion coefficient, E E G,L H E 1 () (1) 1 Use () can be written (1) as U U U ( G L) G G fx x x (3) V W L HT G G 0 x x x The Equations for u and u have a similar tpe. The terms in figured bracket {} is standard div(grad) term hoenics with anisotrop diffusion. For two-dimensional (x,) problems, the U-equation simplifies to: x x x f 0, U G L L L H t, x U V x G G x Stresses and strains in the -direction are linked b the equation: b (3a)

2 where b is a parameter which expresses the extent to which the material is free to expand in that direction, which ma var between ero (for free expansion) and infinit (for absolute constraint). With (3a) can be written as 1 L H t L G b For what is known as the lane-strain model, b, 0, L( ) H, t while for what is known as the lane-stress model, 1 b 0, 0, L H t L G For one-dimensional (x) problems b, b and 1 1 d L H, t L G b 1 dd 1 1 d L H, d t L G b 1 dd L L, d L G b L G b FVE We shall use standard approximation for fluxes: U U G L AE G L A x x he ne E U U G Aen G A U U G A G A S f V 0 se eh el int,e x,e e Z le es (4) where

3 S L A L A int,e E,E,E E,, H T A H T A E E E V AnGn AEnG En V A G A G x en x es s s Es Es x eh x el V VEn Vn V VEs Vs,,... x en xe x es xe N G n,... N G G W AhGh AEhGEh W AlGl AElG El, N (5) 3 Fig. 1 Internal u x -Cell Boundar Condition Tangential boundar (en, Fig. 1) : u u x G 0 G x At a FVE-level possible to unite two members: U G Aen ne and V AnGn AEnG En x en Summar we shall get on boundar

4 4 U V A G A G n n En En G Aen x ne en U V G Aen 0,neAen x ne I.e. on boundar b means of patch it is necessar to assign tangential stresses itself! (6) Normal boundar (e, Fig. ) : Fig. Boundar u x -Cell Boundar condition : u x G L L L HT x Uniting members FVE, shall get U G L AE LE,E,E AE HETE AE x E 0,eAE (7) I.e. on boundar b means of patch it is necessar to assign normal stresses itself!

5 5 Equations of elasticit: olar grid Initial equations - a Hooke s law ( U = u, V = u r, W = u ): V rr Grr Le HT, e rr, r 1 U V G Le HT, e, r r W G Le HT, e, 1 V U U r G, r r r U W r G, r 1 W U G, r e err e e and balance of force equation 1 1r r rrr fr 0, r r r r 1 1 r rr f 0, r r r r 1 1 rr f 0, r r r (8) (9) With (4.1) and (4.) can be V-equations written as 1 V 1 1 V V r(g L) G G r r r r r 1 1 U U W rl(e e ) HT G G r r r r r r 1 1 U V G L rr Le e HT fr 0, r r r OR

6 6 1 V 1 1 V V r(g L) G G r r r r r 1 U U W L(e e ) HT G G r r r r r 1 1 U V G Lrr G f r 0, r r r (10) This is polar analog main Cartesian equation (3). Similar equations possible to write for U and W component: 1 U 1 1 U U rg G L G r r r r r 1 V 1 1 V U 1 W L(err e ) HT G L rg G r r r r r r r 11 V U U G f 0, r r r r (11) 1 W 1 1 W W rg G G L r r r r r 1 V 1 U L(err e ) HT rg G f 0, r r r (1) Boundar Condition on FGE-level Displacement U (u ) tangential boundar (N/S and H/L). Similar doing we have, either as in Cartesian case - two members unite on boundaries: U V A G A G G A U r n n En En en en ne en rn U 1 V U G A A r r r ne en 0,ne en Displacement U (u ) normal boundar (E/W). Uniting members FVE, shall get 1 U V G L A L A H T A A r r E E E rr,e,e E E E E 0,e E Displacement V (u r ) tangential boundar ( E/W and H/L). Similar doing

7 7 1 V U A G A G G A U r r r e e Ne Ne tan tan tan e U 1 V U G A A r r r tan tan 0,tan tan Displacement V (u r ) normal boundar (N/S). Here appears the problem with association of the members. This is connected with difference b form of the members in V-equation 1 V r(g L) and L(e e ) HT r r r r On FGE-level V V G L AN G L A and N AN AL H T L H T A N,N,N N N,, On N-boundar can to write V An A G L An Ln,n,n HnT n n V A A G L L HT A L H T n n n,n,n n n n A A A L n 0,n n n,n,n n n H T Displacement W (u ) tangential and normal boundar. Similarl Cartesian coordinate sstem

8 Add VIA Colocate Displacements Mehod 8 We shall consider FVE for D(x,) cartesian grid when displacements are determined in the centre scalar cell. W wn N en EN E ws S es x The equations of X-force balance are: x fx 0, x The displacements and the stresses are connected b Hooke s law: (G L ) L H T, m x m m (G L ) L H T, m m x m x b, U V G, x U V,, x 1 L HT, x L G b G b G b Lm L, Hm H G b L G b L Integrating equation (13) on scalar cell, possible get FVE U U G Lm Ae G Lm Aw x x n e U U G An G As Sint, fx,v 0 s w (13) (14) (15) Where internal source consists of gradient and tangential terms:

9 9 Ven Ves Vwn Vws Sint, Lm,e Hm,eTe Ae Lm,w Hm,wTw Aw (16) Ven Vwn Ves Vws Gn An Gs As x x For calculation all propert-coefficients on faces of cells is used harmonic interpolation, for instance N N, Gn G GN For calculation of the temperature on faces is used linear interpolation TE T x Te T, xe For calculation of displacements in corners (the centre edges in -direction), for instance Ven shall consider "red"-rectangle -E-E-N. We Shall consider that in this rectangle V V V V V x x x x x x and V VE V V VN V,, x xe n VN VEN V V xe x x n Boundar Condition The Boundar conditions are assigned similarl mate the staggered displacements!