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Abstract: The objective of this paper is to present a method for determining elastic modulus, yield strength and the true stress-true strain diagram of an aluminum alloy in a virtually non-destructive manner. Standard test methods for predicting mechanical properties require the removal of large material samples from the in-service component, which is impractical. To circumvent this situation, a new dumb-bell shaped miniature specimen has been designed and fabricated from the material of which the properties are to be determined. Also the test fixtures were developed to perform tension test on this proposed miniature specimen in the Zwick testing machine. An inverse finite element algorithm is proposed to find the material properties of the alloy using the load-elongation diagram obtained as miniature test output. The predicted results corroborate well with the experimental results. The proposed miniature specimen has an additional advantage in finite element modelling with respect to computational time and memory space. 43838
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Keywords: Inverse finite element; Miniature test; Tensile properties; Load-elongation
1. Introduction Inverse problems could be described as problems where the answer is known, but not the question. In other words, the results or consequences are known, but not the cause. This unknown cause can be found out using the inverse finite element procedure. The inverse simulation approach is of practical value for a number of reasons and has attracted particular attention in the solution of nonlinear problems where interest is focused upon the control action needed to achieve a particular form of output response [1]. Manahan et al [2] have demonstrated the use of large strain finite element analysis to measure material uniaxial tensile stress-strain behaviour from the small punch test. The approach consists of matching the observed small punch test load-displacement curve to a curve from a database of curves via a series of finite element analysis of the small punch test. Similarly, an inverse methodology to determine the elastic-plastic stress-strain curve of material in the form of constitutive law having unknown parameters was attempted from the small punch load-displacement curves [3]. Researchers also performed the finite element simulation of ball indentation test to estimate the flow properties based on the measurement of load (P) vs. penetration (h) curve [4, 5]. Although the over all trend of predicted true stress and strain is consistent with the input constitutive law, the constitutive behaviour at low strains is not always well estimated by the ball indentation method by analyzing P-h data. For the present study a dumb-bell shaped miniature specimen is designed and used for predicting the properties of any unknown material using the experimental setup described elsewhere [6]. The developed inverse finite element technique is used to determine the elastic modulus (E), yield strength (σy) and true stress-true strain diagram of the aluminum alloy. The computed values are compared against the uniaxial tensile test results. This may further be used to predict tensile and fracture properties of the material. 2. Experimental Work The aluminum alloy (AR66) used in the present work has the chemical composition as given in Table 1. Table 1. The chemical compositions (%by wt) of aluminum alloy Material Zn Al Cu ZrAR66 alloy 6.30 89.70 1.55 0.14 Figure 1 shows the configuration of the developed miniature specimen for the present study. The advantage of this miniature specimen is that it requires minimum number of machining operations. The specimens were mechanically polished and a final finish was given using 600 grit emery paper. The thickness of the specimen was maintained at 0.50 mm with an accuracy of 1%. Then the specimens were marked with gage length.
Fig. 1. Configuration of the miniature dumb-bell specimen. A special test fixture was used to hold the miniature specimen in the test machine as the small size of specimen does not permit the movable cross head to come very close to the top cross head. The complete assembly consisting of fixture along with the specimen holders, test specimen and loading pins is shown in Fig. 2. The complete assembly is installed in the Zwick testing machine having a load cell of 5kN capacity and testware named testXpert® interfaced with it. Then the fixture attached to the specimen holder is detached smoothly. An extensometer is attached with the testing machine for the measurement of elongation of the miniature specimen during the test. The sensor arm of the extensometer is placed just near the miniature specimen. During the miniature test the load-elongation diagram was recorded. The typical load-elongation diagram as obtained from the test is shown in Fig. 3. 3. Inverse finite element simulation The finite element modeling calculations were conducted using the finite element code, ABAQUS. The 2-dimensional analytical model of dumb-bell shaped miniature specimen used for the simulation is shown in Fig. 4. The test specimen is modeled with eight noded quadratic quadrilateral plane stress elements. The loading pins were treated as 2-D rigid bodies with a low friction coefficient of 0.01. The boundary condition used in the present case is as follows: The loading pin in the top fixture is fixed, and the one in the bottom fixture is constrained against translation in the X direction. However, it was allowed to experience displacement in the negative Y direction as in the experimental situation. A concentrated force was applied to the reference point of the bottom pin in the negative Y direction. Specimen holderFixtureCover plateSpecimen Specimen holderFixtureCover plateSpecimen Fig. 2. Assembly of fixture attaching specimen holders and the specimen 0501001502002503003500 0.05 0.1 0.15 0.2 0.25 0.3 0.35Elongation in mmLoad in NMiniature testInverse FEM Fig. 3. Comparison of load-elongation curve obtained from the miniature test and the inverse finite element for the specimen from the aluminum alloy Load PinSpecimenLoad PinSpecimen Fig. 4. Finite element model showing the mesh. 040801201602002402803200 0.05 0.1 0.15 0.2 0.25Elongation in mmLoad in NExperimentfem P1δ1040801201602002402803200 0.05 0.1 0.15 0.2 0.25Elongation in mmLoad in NExperimentfem P1δ1 Fig. 5. Comparison of load-elongation curve for the determination of elastic modulus In inverse finite element method the miniature test based experimental load-elongation curve is given as an input in a linear piecewise manner to the finite element software. The experimental load–elongation curve is used to compare with the curve produced from inverse finite element procedure. The algorithm used for the determination of properties of materials using miniature specimen test along with the inverse finite element procedure can be explained in step wise manner as follows. a) The first iteration of inverse finite element analysis was carried out with an initial Young’s modulus value Eas. In the present study Eas is taken as 70 GPa. The inputs to the inverse finite element analysis are Eas as well as load P1 and elongation δ1(for comparison purpose only), where the latter corresponds to the load point P1 on the miniature load–elongation (P-δ) curve where the linear portion ends or the non-linear portion starts. The inverse finite element analysis is carried out with the above parameters as inputs and after the elastic analysis is completed, the elongation of the specimen (δp) is noted. Then the value of δp is compared with δ1 (elongation corresponding to load P1 in the miniature test load-elongation curve) as follows: |δp-δ1|≤ α (1) where α has a very small value (tolerance) set by the user. In the present study this tolerance is set to 0.001 mm. If the above condition is not satisfied then the value of Eas is changed as follows, () ( )1111 pas as mmEδδβδ +⎡⎤ − ⎛⎞=+ ⎢ ⎥ ⎜⎟⎝⎠ ⎣⎦m E (2) where β is the convergence control parameter (0 <β ≤ 1), (Eas)m+1 is the value of E at (m+1)th iteration, (Eas)m is the value of E at mth iteration. Then the analysis is carried out till the above condition is satisfied. At this iteration, the Eas value is the Young’s modulus (E) of the material. Figure 5 shows the comparison between the load-elongation curve obtained from the FE analysis and the miniature test. A good agreement is noticed between the load-elongation curves obtained from both the methods up to the load point P1. The maximum von-Mises stress present in the specimen at load P1 is the yield strength of the material (σt1). The corresponding true plastic strain εt1 is zero. Hence we have determined the first point (σt1, εt1) on the true stress-true plastic strain curve. b) Additional points in the true stress-true plastic strain curve are obtained by considering the miniature test load-elongation curve. It was observed that the elongation at the maximum load in the miniature test and strain at the maximum load in the standard uniaxial test bears a fixed ratio and this ratio is termed as equivalent gage length of miniature specimen. δmax = leq εmax