Question

2. A 1-D mechanical bar (see Figure) consisting of two materials is clamped at x-0 while subjected to a displacement (u 10 microns) at x-2mm as shown in the figure. The materials are isotropic and perfectly bonded together. The Youngs modulus for the two materials is E1 -10 GPa and E-50GPa respectively. For the mechanical bar below assume a simple 1D problem and answer the following questions What are the stresses (o) strains (Ex) and displacements (u) in each of the bars. Note some of these maybe a function of (x) Plot the energy per unit volume in material 1 [i.e. energy per unit volume 1/2 (o,&x)] as function of the Young's modulus in material 2 for five cases, i.e. E2 1,5, 10, 25, and 50 GPa. Comment on the trend. a. b. a 1mm 1mm 10 microns 2 1 X

Answer 1

Pues otuton 1 mm 710 mucHen C 1o Xlo N E1 5 0 5o x1o3 E2= The Fan feuiesonneatn Question &Taktal hence toroeation aoraa (As Az) 62 S= FL 2 Total l0 mucH On AE O xlo mm L1 1 mm * W w T a Pasut Values Lhe aing get all enation ale - A2 EZ A,E 61 LI 62 L2 = E2 A 1 61 =lo x lo3 Joxio 5ox3 (4G 50 10 500xlo3 = lo x lo-3

3 3 5000 x1o xio 60 83. 333ma 5o0 83.333 m2 2 83-333 N mm2 62 Digplacamunt = Basn each Fa TL Ferr Bar 83-333x 1 loxio3 d1 0.008333 mm. Вая 2 Fay 62 L2 83.333 x E2 O. oo16666 mm- Staun Fao each Basr Tay Bas 83.333 - = T3 |оx103 E, O.Oo8333 Far Bas 83.333 2 E2 50x103 O. oo16666. E2 =

b Paot Farr matesual elnit Voluma Per Energy / 6 I 2 Energy Volume 2 2 E Fasy 티= logPa 10 xi03mPa E2 and SPa=103 mPa 1 E2 -(부· 부) 1 -3 lo xio3 0/XO L+1o loxio3 Energy Volume at E 2 E 1 X 2 x oxio3 Eneyy 4.1322 31XIO at E2- 1gfa Joule Vaume

2 Fooy and E loxlD3MPa 5 xio3 Pa ) OXIO-3 5 xlo3 Oio3 15 =1OxID 50xio3 500 mRabaN/mm2 Enegy Voluune 672 2 E 5 efa at E, = 2 500 2XIDXID 3 Enesgy Vou Joele ar E 5gPa O. 05555 mm3 and E s oxio mpa Fasy =1ox/o3Pa E2= /o X + - {Ox|ס? oxio3 20 !ס 3- xסן Dox1D3 61 got 50 mPa OR Imm2 Eney at (ogla 2 E Voume X 2 Xloxio Enesgy Jaule Oo125 Ez: lo gPa at Voume

and E loxio3 E225gfa ΙΟΥ1ο -3 25 X103 10x1o3 25 1o 10χ10 X 250 X(03 2500 /mm2 25 Enegy 25 9Pa at Er Volume 2 E 2 2 500 X 2 xlox lo3 3S Enegy O. 255 oule at 25e Pa E2 mm 3 Voume Fasy 50 e Pa and E1 o Pa E2 83-333 mPa COR) atrandy Cefe 7m2 Enegy 2E Volume (83.3s3)2 2x(0XID3 Eneigy Uolume O 34 7222 Joule mm 3 at E2509Pa Enegyyac Volume/Bar 1 blw Energy Volyme Bayi ghaph ED E acf.

-0:4- (50, 0.3472) -0:3- (25, 0.2551) 0:2- (10, 0.125) -01 (5, 0.0556) 40 (1,0.0041)20

Comment liends Trily eneigy of Bar D tre Slope ofEnergy Vs intial Vales asith Ine yreriasing cDn the Value E2) the Jnaeases Fasto hemce ncea E oaph On Ineieasing E) the enegy imera But dater On Sloadyhenae Goaph is duaaning ne tope emegy Vs E) Ahe der