Determination of the Young's modulus of an aluminium specimen

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1 Maria Teresa Restivo, Faculdade de Engenharia da Universidade do Porto, Portugal, Carlos Sousa, CATIM - Centro de Apoio à Industria Metaloecânica, Porto, Portugal, csousa@cati.pt Noveber, 2003

2 Two distinct ethods are eployed for deterining the Young's odulus of an aluiniu specien. The first is based on a procedure followed by testing laboratories, using standard ethodology and broad range test equipent. The second ethod is based on traditional ethodology used in R&D and teaching laboratories, using dedicated equipent. A bea-type specien of constant rectangular cross section was prepared and instruented with two electrical strain gauges, one on the top and the other on the botto surface, syetrically bonded respectively on the top and botto surfaces and oriented along the longitudinal ais of the bea. 1. Tensile test achine: plotting of the force vs. displaceent curve The specien diensions were carefully easured with specific equipent (following recoendations of EN ). The specien was then inserted in the testing achine grips for perforing the tensile test under autoated controlled conditions at roo teperature within standard abient teperature range. The free length between the grips was sufficient for the insertion of the echanical gauge. The aiu tensile force of the test was preset in the testing equipent. The test was perfored according to a standard procedure with a onotonic increase of the applied force up to the predefined aiu value, followed by a onotonic decrease to zero. The force evolution was at constant rate and within a recoended range. The echanical gauge, positioned at idlength of the specien, easures the elongation value for the current applied force which is deterined by the load cell, integrated into the tensile achine, figure 1. Load Cell -y recorder Mechanical gauge Tensile Machine y h b Fig. 1 The tensile test syste Pág.2

3 Hooke s law is used in Eq. 1 to copute the Young's odulus, taking into account the bea cross section diensions b and h, the applied tensile load increents F and the corresponding elongations L of the etensoeter gauge length L e : E = F L e /(b h L ) (1) Fro Eq. 1, the sensivity coefficient of E associated to each variable is given as: E/(F) = L e /(b h L) -2 (2) E/L e = F /(b h L) N. -3 (3) E/b = -F L e /(b² h L) N. -3 (4) E/h = -F L e /(b h² L) N. -3 (5) E/(L) = -F L e / (b h L²) N. -3 (6) Table I Tensile Test Machine - Uncertainty Coponents and Test Results i Physical quantity / uncertainty coponent Standard uncertainty 1 Specien width uncertainty (b) b = 25,04 Maiu adissible error of calliper (DIN 862) =±0,02 u b= 0,02 / F3 u b =1, Specien thickness uncertainty (h) h = 5,112 (ean value) Maiu adissible error of outside icroeter (DIN 863-1) =±0,004. u h= 0,004 / F3 u h =2, Load transducer of test equipent F=8500 N (Maiu tensile force = 9500 N) Equipent calibrated and certified according to NP EN , accuracy class 1, aiu adissible error = ±1,0% u F=(9500 1/100)/F3 u F = 54,8 N 4 Paper resolution (for the displaceent) Estiated resolution 0,2. u L(resol)=(0,02 0,005)/ F12 u L(resol)= 2, Displaceent transducer (fiing operation) L e = 50 This error could be originated by incorrect adjustent in fiing the transducer to the specien, with a aiu value of ±0,25. u Le=0,25/F3 u Le=1, Displaceent transducer (calibration characteristic) Calibration uncertainty = ±1,4 J, co k=2,4. Type B, noral distribution (t-student). u L(U)=0,0014/2,4 u L(U)= 5, Displaceent easureent repeatability L(ean)=0,05015 (3 repeated readings) u L = 2, /63 Type A, eperiental standard deviation s L=2, u L = 1, Pág.3

4 Coponent Table II Final Calculations for Tensile Test Machine Quantity Value Type u(i) c i u² i (y) GL 1 Width uncertainty b 2, BR 1, , ,32 10 N -3 N² Thickness uncertainty h 5, BR 2, , ,94 10 N -3 N² Load cell F 8500 N BR 54,8 N 7, ² 1, N² y recorder resolution L ---- BR 2, , ,45 10 N -3 N² Mechanical gauge position L e 50,0 BR 1, , ,65 10 N -3 N² Mechanical gauge uncertainty L ---- BN 5, , , N -3 N² -4 7 Elongation repeatability L 5, A 1, , , N -3 N² -4 E 66, N -2 u² (y) 8, N² -4 u (y) 9, N -2 v eff 90 k 2,03 U ±1, The final result, presented with up to two significant digits, is then: N -2 E = (66,2±1,9) GPa 2. Bending test: resistance strain gauge easureents The specien was then used in a siple cantilever bea ounting test syste. The electricalstrain gauges were integrated in a Wheatstone bridge circuit. The load is applied on the free end of the bea, at a distance L fro the geoetric centre of the strain gauges grid, figure 2. F L b h y gauges Fig. 2 Cantilever bea test setup Pág.4

5 For any aterials, loaded within a oderate stress range in a uniaial tension test, the relation between the stress along the -direction ( ) and the corresponding longitudinal strain ( ) can be established by Hooke s law: = /E (7) The bending oent produced by the load F applied at a distance L fro the geoetric centre of the strain gauge grid causes a longitudinal strain : = 6 F L /(E b h 2 ) (8) which leads to: E = 6 g L /( b h²) (9) where is the loading ass and g the acceleration of gravity (fro now on will be written as ). Fro Eq. 9, the sensivity of E associated to each variable is given as: E/F = 6 L /( b h² ) -2 (10) E/L = 6 F /( b h² ) N. -3 (11) E/ = -6 F / (² b h² ) N. -2 (12) E/b = -6 L F /( b² h² ) N. -3 (13) E/h = -12 F /( b h 3 ) N. -3 (14) Table III Bending Test - Uncertainty Coponents and Test Results i Physical quantity / uncertainty coponent Standard uncertainty 1 Force F Uncertainty associated to the loading ass Maiu adissible error = ±4010-6, by calibration. = 0,99468 kg; g = 9,8028.s -2. Type B, rectangular distribution 2 Length L L = 0,1608 Maiu adissible error = ± 0,2 (±0,210-3 ). Type B, rectangular distribution u F=( ,994689,8028)/ 3 u F = 2, N u L =0, / 3 u L = 1, Width b b = 0,02504 u b = (0, )/ 3 Maiu adissible error of calliper (DIN 862) =±0,02. u b = 1, Thickness h h=0, u h = (0, )/ 3 Maiu adissible error of Outside Microeter (DIN 863-1) = ±0,004. u h = 2, Strain easureents repeatability (ean) = 218,2 J/ ( ) Type A, eperiental standard deviation s = 1, u = 5, Pág.5

6 6 Volteter Measured value = 0,2182 V range = 1V Maiu adissible error (anufacturer): ± 30 pp reading ± 5 pp range. 7 Measureent Bridge Range= 40 J/ Maiu adissible error (anufacturer): ±0,02% range (range 10 4 J/). 8 Strain Resistance R Maiu adissible error (anufacturer) = ± 0,3% of noinal value (120 ). u V = ( , )/ 3 u V = 9, V u P = (0,02/ )/ 3 u P = 1, u R = (1200,3/100)/ 3 u R = 2, Gauge Factor GF Maiu adissible error (anufacturer) = ± 0,5% of noinal value (GF = 2,1). U GF = (2,10,5/100)/ 3 U GF= 6, Considering that: GF = (dr/r)/( dl/l) = (dr/r)/ (18), fro Eq. 18, the sensivity coefficient of associated to each variable is given as: /GF = - /GF (diensionless quantity) (19) )/R = - /R -1 (20) Table IV Bending Test - Uncertainty Coponents and Test Results associated to the strain Coponent Quantity Type u(i) c i u² i (y) Gauge Factor BR 6, , , Resistance R BR 2, , , Volteter V BR 9, V 1, V -1 8, Measureent Bridge BR 1, , u² (y) 1, u (y) 1, Pág.6

7 Table V Final Calculations for Bending Test Coponent Quantity Value Type u(i) c i u² i (y) GL 1 Force F 9,751 N BR 2, N 6, , N Distance to applied load L 0,1608 BR 1, , N. -3 2, N Specien width b 0,0250 BR 1, , N , N Specien thickness 5 Strain (equipent) h 0, BR 2, , N. -3 3, N ,2 BR 1, , N. -2 1, N Strain (repeatability) A 5, ,02 10 N. -2 2, N. -3 N u² (y) 1, E 65, N -2 N u (y) 4, N -2 v eff k 2 Presenting the final result with two significant digits we have: U 0, N -2 E = (65,88±0,84) GPa 0,8 0,7 0,6 0,5 GPa 0,4 0,3 0,2 0,1 0 width thickness load cell -y recorder gauge position gauge uncert. elongation repet. MPa 0,80 0,60 0,40 0,20 0,00 force distance load width thickness strain equip. strain repeat. Coparison of coponent uncertainties (Tensile test ethod) Coparison of coponent uncertainties (Resistance strain gauge ethod) Pág.7

2. A crack which is oblique (Swedish sned ) with respect to the xy coordinate system is to be analysed. TMHL

2. A crack which is oblique (Swedish sned ) with respect to the xy coordinate system is to be analysed. TMHL (Del I, teori; 1 p.) 1. In fracture echanics, the concept of energy release rate is iportant. Fro the fundaental energy balance of a case with possible crack growth, one usually derives the equation where