COMSOL Modeling and Tensile Loading of AluminumMaterial Test SamplesJoseph L. Palladino, Ph.D.Department of Engineering, Trinity College, Hartford, CT 06106, [email protected]— COMSOL Multiphysics was used tomodel “dog bone” aluminum material test samplesfor tensile loading. A 3D linear solid model wasstudied to quantify axial and transverse strains underaxial tensile loading conditions for both 6061 and7075 aluminum alloys. General purpose Constantanalloy strain gages were installed in both axial andtransverse directions at the midpoint of the sampletest section. Axial and transverse strain was measured for applied loads ranging from 0–2000 lb for6061 aluminum and 0–4000 lb for 7075 aluminum,applied as a tensile load with an Instron material testmachine. Model strains were compared to measuredstrains and were found to agree within 2%. Theelastic modulus was calculated for each test sampleby linear regression of the axial stress and strain, andthe Poisson ratio by linear regression of the axial andtransverse strains, which were within 2% and 3%,respectively, of the model parameter values.Keywords: COMSOL Multiphysics, solid mechanics, material testing, strain gage, elastic modulus,Poisson ratio, Saint-Venant’s principle.I. I NTRODUCTIONATERIAL test samples are commonly subjected to tensile testing, with an extensometer used to accurately measure changes in lengthof the material to determine strain. Lacking anextensometer, stain gages were used to measureaxial and transverse strain, with the location andmagnitudes of strain guided by a COMSOL finiteelement model. Test samples were loaded with anInstron tensile test machine (model 5500R), andmodel strains were validated by comparison withmeasured strains in both the axial and transversedirections for different aluminum alloys.MII. M ETHODS1) Equations: Modeling the material test samplerequires three equations: an equilibrium balance, aconstitutive relation relating stress and strain, anda kinematic relation relating displacement to strain.Newton’s second law serves as the equilibriumequation, which in tensor form is: · σ Fv ρ ü(1)where σ is stress, Fv is body force per volume, ρis density, and ü is acceleration. For static analysis,the right-hand side of this equation goes to zero.The constitutive equation relating the stress tensor σ to strain is the generalized Hooke’s lawσ C: (2)where C is the fourth-order elasticity tensor and: denotes the double dot tensor product. In COMSOL, this relation is expanded toσ σ0 C : ( 0 inel )(3)For this application, initial stress σ0 , initial strain 0 , and inelastic strain inel are all zero. Forisotropic material, the elasticity tensor reduces tothe 6 6 elasticity matrix: 2µ λλλλ2µ λλλλ2µ λ000000000000µ000000µ000000µ (4)where λ and µ are the Lamé constants, E is theelastic modulus, and ν is Poisson’s ratio, withmaterial properties listed in Table I.The final required equation is the kinematicrelation between displacements u and strains . Intensor formi1h u ( u)T(5) 2
where T denotes the tensor transpose. For rectangular Cartesian coordinates the strain tensor maybe written in indicial notation " ui uα uα1 uj ij 2 xi xj xi xj#(6)where α 1,2,3,. . . . For small deformations thehigher order terms are negligible and ij reducesto Cauchy’s infinitesimal strain tensor:"1 uj ui ij 2 xi xj#(7)bridge amplifier (Vishay P3 Strain Indicator andRecorder) . The bridge amplifier provides theexcitation voltage, completion resistances and, afterthe bridge is balanced and the gage factor is input,produces an output voltage directly in units ofmicro-strain (10 6 ).The instrumented aluminum test samples wereloaded in axial tension using an Instron 5500Rmaterial test machine, and loaded between 0–2000 lb, and 0–4000 lb for the 6061 and 7075alloys, respectively, below their yield stresses.III. R ESULTSTABLE IM ATERIAL PROPERTIES FOR 6061 AND 7075 ALUMINUMALLOYS USED IN THE COMSOL MODEL .ParameterElastic ModulusPoisson RatioSymbolEν6061707510,000 ksi0.3310,400 ksi0.332) COMSOL Multiphysics Model: A three dimensional Solid Mechanics model was built, withthe test sample profile geometry drawn using aCAD program (Ashlar-Vellum Graphite) as shownin Fig. 1. The profile was imported using theCOMSOL CAD Import Module, extruded into a1/8 in. thick 3D bar and modeled as homogeneous,linearly elastic 6061 or 7075 aluminum.An extra fine physics-controlled mesh was generated (Fig. 2) and a stationary analysis was performed, using default solver settings.3) Model Verification: Aluminum test samples milled from 6061 and 7075 alloys wereused. Strain gages used were general purposeCEA series polyimide encapsulated Constantan alloy (Vishay Micro-Measurements CEA-13-240UZ120) with 120 ohm resistance and a 2.2 gagefactor. The gages were installed using standardsurface preparation: degreasing, abrading, layout,conditioning, and neutralizing steps, following themethods in . They were bonded to the aluminum test samples with M-Bond 200 methyl-2cyanoacrylate adhesive.The strain gages were wired with 27 AWGpolyurethane insulated solid copper wire and gagelead wires were kept of uniform length to preventunwanted lead resistance differences. The gageswere wired as quarter bridges and connected to aFigure 3 shows first principal strain arising froman applied load of 600 lb. As expected, the stress inthe test region is uniform. Figure 4 shows a contourplot of first and second principal strains. It is clearfrom this plot that despite complex local stressesnear the grips (ends), there is a uniform centraltest region, 2 in. long, where strain gages may beplaced for accurate strain measurements.Figure 5 shows measured axial and transversestrains for a 6061 test sample loaded with 600 lb intension. Over the entire range of 0–2000 lb appliedload, measured strain was highly linear.A linear regression of the measured axial stressand strain gives the test sample’s elastic modulus,as shown in Fig. 6. Measured elastic modulus Ewas 9.785 106 ksi for 6061 aluminum, which iswithin 2% of the literature value of 10.0 106 ksi.Similar agreement between experiment and theorywas found for the 7075 alloy samples. A linearregression of the measured transverse strain andaxial strain gives the test sample’s Poisson ratio,as shown in Fig. 7. Measured Poisson ratio ν was0.3217 for 6061 aluminum, which is within 3%of the literature value of 0.33. Similar agreementbetween experiment and theory was found for the7075 alloy samples.COMSOL FEA models are useful for allowingstudents to “look inside” of structures. For example,introductory textbooks in mechanics of materialsuse a constant average normal stress across thewidth of an axially loaded bar. Intuition suggeststhat for an applied point load, there should besignificantly different local stresses depending onlocation. Saint-Venant’s principle predicts that thedifference between the effects of two different
Fig. 1. Test sample profile geometry drawn in Graphite, then imported into COMSOL with CAD Import Module. Units are ininches.Fig. 2. COMSOL extra fine mesh of material test sample, yielding 2,192 tetrahedral elements with 13,368 degrees of freedom.but statically equivalent loads becomes small atsufficiently large distance from the load . Figure 8 shows axial stress across a uniform bar ofaluminum with an applied axial point load. Theblue curve was calculated from the FEA model ata distance b/4 away from the applied load, whereb is the bar’s width, and shows substantial stressvariation. The green curve, measured at b/2, ismuch more uniform, while the red curve, measuredat a distance b away from the applied load, isnearly uniform. The average normal stress, equalto the applied force divided by the bar’s crosssectional area, is equal to the areas under thesecurves. For this particular example, the bar widthwas 1 m, thickness was 0.1 m, and applied forcewas 10,000 N, giving average normal stress of1 105 Pa.
Fig. 3. First principal strain for a 6061 aluminum test sample with a 600 lb tensile load. The test section width for thisparticular sample was 0.6 in. giving a test section stress of 8000 psi and first principal strain of 0.0008.Fig. 4. Modeled first and second principal strain contours for a 6061 sample with a 600 lb load, showing complex local strainsnear the test sample grip ends and a uniform central test region 2 in. long.
0.003Axial StrainTransverse StrainAxial Strain, Transverse 0010001500Axial Load [lb]2000Fig. 5. Measured axial and transverse strains of a 6061 test sample for the load range 0-2000 lb, showing a high degree oflinearity.30000Axial Stress 002Axial Strain0.00250.003Fig. 6. Elastic modulus, E, calculated from a linear regression of measured axial stress and strain for a 6061 test sample.Measured E was 9.785 106 ksi, which compares favorably to the literature value of 10.0 106 ksi.
0Transverse 010.00150.002Axial Strain0.00250.003Fig. 7. The Poisson ratio, ν, calculated from a linear regression of measured transverse strain and axial strain for a 6061 testsample. Measured ν was 0.3217, which compares favorably to the literature value of 0.33.Fig. 8. Axial stress calculated from a COMSOL model of a uniform bar of aluminum of width b subjected to an axial pointload. Results show that stress is not uniform near the applied load, but becomes so at a distance b from the load (red curve),demonstrating Saint-Venant’s principle. The negative sign in the plot denotes compression.
IV. D ISCUSSION AND C ONCLUSIONSR EFERENCESDespite its simplicity, this COMSOL model wasuseful in predicting and visualizing experimentalstresses and strains, particularly the stress concentrations between the grip regions at the testsample ends. Model results predicted a uniformstress region of 2 in., which guided placement ofthe strain gages. The experimentally determinedelastic modulus was found to be within 2% ofliterature values, and the Poisson ratio within 3%,showing that this method can accurately measureboth material properties for aluminum test samples.The model permits visualization of phenomenasuch as Saint-Venant’s principle, a topic that nowhas meaning for students. Close agreement betweentheory, model, and experiment validates the model,giving students confidence in this approach beforemoving on to more complex multiphysics models. Fung, Y.C., A First Course in Continuum Mechanics(2ed), Prentice-Hall, Englewood Cliffs, NJ, 1977. Davis, J.R. ed., Metals Handbook Desk Edition (2ed),ASM International, Materials Park, OH, 1998. Student Manual for Strain Gage Technology, Bulletin309E, Vishay Measurements Group, Raleigh, NC, 1992. Model P3 Strain Indicator and Recorder, instructionmanual, Vishay Micro-Measurements Group, Raleigh,NC, 2005. Saint-Venant, A.J.C.B., Memoire sur la Torsion desPrismes, Mem. Divers Savants, 14:233–560, 1855.
CAD program (Ashlar-Vellum Graphite) as shown in Fig. 1. The proﬁle was imported using the COMSOL CAD Import Module, extruded into a 1/8 in. thick 3D bar and modeled as homogeneous, linearly elastic 6061 or 7075 aluminum. An extra ﬁne physics-controlled mesh w