Non-Linear Elastomeric Spring Design Using Mooney-Rivlin Constants Amir Khalilollahi Brian P. Felker Justin W. Wetzel Pennsylvania State University, The Behrend College Abstract A new product design for an engine mount requires very specific nonlinear load deflection curve and geometrical constraints. It was found that a specific geometrical design of an elastomeric material can be used to minimize the cost and complexities in manufacturing this product. Natural rubber was chosen for its transitional properties in shear/compression, low hysterisis, and setting losses in cyclical compression. ANSYS was extremely helpful in approaching the final design and evaluation of performance in all stages of development. Stress/strain tests such as uniaxial tension, biaxial tension, and shear were conducted to refine the nine Mooney -Revlin constants for the ANSYS material model. The final design was evaluated favorably by ANSYS and then was fabricated and tested to yield the desired characteristic curve. The test results were in close agreement with ANSYS predictions especially in the first 70 percent of the maximum deflection. Introduction Certain class of heavy engine mounts may require very customized non-linear performance that must satisfy limiting geometrical constraints. It is known that current technology utilizes metal springs and linkages to achieve a non-linear force versus deflection performance. However a well designed non-linear spring that mainly is made of rubber (elastomer) can have many advantages over the more complex spring linkage mounts. The elastomer mounts are more economical, durable, and easily replaceable. A typical non-linear elastomeric spring contains at least 50% (weight) of elastomeric material, and satisfies a specified force verses deflection curve depending on the desired application. The present mount product needs to posses a specified force verses deflection curve that can lie within a certain range at different stages (A,B, and C) as shown in Figure 1. Some important performance specs based on the desired plot (Figure 1) are The performance of the design is determined after the part has been allowed to set at the pre-load deflection for 5 minutes. At the Pre-Load deflection, defined by Section A of Figure 1, the returning force must be 350 lbs (+30% - 0%). In Section B of Figure 1, the force-deflection curve must have a spring rate (slope) of 0 to 50 lbs/in at any point in this region. In Section C of Figure 1, the force-deflection curve must have a spring rate greater than or equal to the spring rate in Section B at any point in that region. In addition the product must have the following size constraints: Depth: 1.125 inches +.002 Width: 3.375 inches +.002 Height: 7.500 inches +.002
Figure 1 - Specified Load-Deflection Curve for the Mount The use of elastomeric materials is well justified since they offer a favorable transitional trend between compression and shear deformation. This transition between compression and shear can be approximated by using Mooney-Rivlin hyper-elastic material constants (and elements). Mooney-Rivlin constants can approximate the stress verses strain curve for the elastomeric material [3]. To attain these constants, the following three stress-strain tests as visualized in Figure 2 are required: 1) Uniaxial Tension and Compression Test 2) Biaxial Tension and Compression Test 3) Shear Test. These stress-strain tests can then be used to calculate several Mooney-Rivlin material constants for the elastomer. Figure 2 - Uniaxial Tension, Biaxial Tension, And Simple Shear
Preliminary Model The initial use of ANSYS utilized only one of the nine Mooney-Rivlin constants. The one constant was approximated as half of the shear modulus for the elastomeric material. The use of only one constant was found to be inadequate since the results were accurate within 10% of the tested force and only for the deflections lower than.5 inches. It was decided to attempt to determine and input all nine Mooney-Rivlin constants for a realism and improved accuracy. Next some theoretical work was done using stress classical formulas for the rubber [1,2] in order to come up with a preliminary but sound design for the mount. After considering and analyzing some design possibilities using theory, a final geometrically acceptable symmetric V -shaped design was selected and checked for the desired force deflection behavior. The ANSYS model based on the half mount geometry (one leg) is shown in Figure 3. Figure 3 - FE Model of Preliminary Design To model the initial spring design in ANSYS using only one constant, the finite-element s geometry was constructed. HYPER56 4-nodes were used for the finite-element in the elastomer, while PLANE45 4-nodes were used for the steel. Hyperelastic elements require the Mooney-Rivlin constants instead of the Young s modulus and Poission s ratio to define the material properties. Because of the size constraints, two dimensional plane stress elements were created for the geometries designs. The high stress/deflection also required the strain stiffening effects and large deformation effects to be turned on. The steel elements were required, because of the unique load placement on the bonded regions of steel and elastomer. The loading of the spring consisted of constrained vertical shearing of the steel and elastomer bond. The steel and elastomer bond was created using the sharing of common nodes. The spring deflection was established by constraining one side of the spring in the two planes, and allowing the loaded side to only displace vertically. This assumption was reasonable since the applied deflection to the loaded central bar is without torsional effects, or any horizontal force component. The force verses deflection analysis required ANSYS to solve reaction forces at intermittent deflections during the loading. The FE model was deflected at 0.1-inch increments to its full deflection of about 2.5 inches. The reaction forces during this deflection were listed and plotted in the TIMEHIST under the POST PROCESSOR in the ANSYS tool bar.
Hyper-elastic Elastomer Constants After an initial model was constructed and evaluated using only one Mooney-Rivlin constant, the results were not satisfactory for the range of deflections as required. Then more experimental data based on many tests (as mentioned above) were incorporated into ANSYS to evaluate the nine constants for the recommended rubber elastomer. The left most strain column requires the uniaxial data, the middle requires the biaxial data, and the right most requires the shear data. The stress columns must contain corresponding input that compliments the strain columns entries. A CONST parameter is also displayed in ANSYS, which refers to the desired number of constants (2, 5, or 9). To calculate the constants, ANSYS requires that LAB = MOONEY and TBOPT = 1 on the tool bar command. The nine constants calculated by ANSYS were then inputted under the material properties as shown in Figure 4. Figure 4 - Elastomer s Mooney-Rivlin Constants Final Design and Experimental Evaluation The final design for the spring was achieved mostly through changing the geometrical parameters for the rubber, and the types of the rubber that were available and feasible to manufacture. Here we will present the optimal design as the results of evaluation of many ANSYS models. The final design was fabricated and tested in two phases. In the first phase the geometry was established and tests on a prototype of a member (a leg of the mount, Figure 3) were conducted to assure the accuracy of the final model performance. The force vs. deflection was plotted for the compression test conducted in the lab and compared with the results of the ANSYS model, as shown in Figure 5. The ANSYS model incorporates the nine Mooney-Revlin constants that the ANSYS model as noted above. The results were very encouraging since the ANSYS model was accurate to within 6.5 % of the force for the 71% range of deflections. The deviation after the 71% deflection can be contributed to the fact that the constants were based on the biaxial stress-strain test (one of the three tests) that was limited up to about 60% deflection. But as a whole this finding was a significant conclusive step towards gaining confidence in ANSYS as an analysis and more importantly a design tool.
Figure 5 - Curve for the Preliminary Model To finalize our model fabrication and its evaluation, a complete model, comprising of two V shaped rubber components, each V component having two members (legs), and in parallel was manufactured and tested to assure the desired spec on the performance of the mount. Figure 6 compares (1) the mount behavior in ANSYS, (2) tests results on the real prototype which was fabricated after some manufacturing difficulties, and (3) the recommended lower and upper limits as specified by the potential customer in this project.
Figure 6 - Load-Deflection Comparison for the Final Design It was needed to obtain also the unloading trend of the mount and to show that the hysteresis effect in unloading is acceptable. The loading force at a 1-inch deflection was 333.4 lbs., which is below the upper limit of the specifications. The linear region of the loading curve contained 38.5% of the section below the specifications maximum slope. At the full required deflection of 2.8 inches, the maximum force experience by the final prototype was 717 lbs. After the loading force completed the full deflection, the unloading force was evaluated. The linear region of the unloading curve contained 54.0% of the section below the specifications maximum slope. Overall the performance was very favorable in ANSYS model and the loading test but perhaps close to acceptable in the unloading part. It must be mention that at conditions conducive of slower loading/unloading times and selection of different rubber compositions this deviation may be remedied. Figure 7 illustrates the loading effects on the model at three sample times in ANSYS. Figure 7 - Stress and Deformation at Three Loading Stages
Conclusion A design for an engine mount was completed and manufactured that followed a specific nonlinear load deflection performance and geometrical constraints. Use of ANSYS models was found to be extremely significant in reaching and evaluating the final prototype. Natural rubber was chosen for its transitional properties in shear/compression, low hysterisis, and setting losses in cyclical compression. Stress/strain tests such as uniaxial tension, biaxial tension, and shear were conducted to refine the nine Mooney -Revlin constants for the ANSYS material model. The final design performance by the real prototype agreed well with ANSYS predictions especially within the 70 % of the deflection range. Overall the loading performance of stress/strain curve was well between the desired limits. Reference: 1) Lindley, P.B., Engineering Design with Natural Rubber, The Malaysian Rubber Producer s Research Association: Fifth Edn, 1992. 2) R.C. Weast, M. J. Astle, CRC Handbook of Chemistry and Physics, CRC Press, 1980, Boca Raton, FL. 3) ANSYS, Inc., Theoretical and User s Manual, Release 5.7, Southpointe, Canonsburg, PA