# Finite-Element Analysis of Stress Concentration in ASTM D 638 Tension Specimens

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2 2 JOURNAL OF TESTING AND EVALUATION Finite-Element Modeling The two-dimensional plane stress six-node and twelve degreeof-freedom triangular linear elastic element was selected to model the flat-plate tension specimens. The commercially available AN- SYS finite-element analysis software and its mesh generator were used. Figure 2 shows the finite-element meshes used to model the four types of ASTM D 638 tension specimens. Due to the symmetry of the specimen geometry, only one quarter of the specimen is modeled. Point O is the center of the specimen. Lines OA and OE are the symmetry planes of the specimen. Point O is fixed in both the X- and Y-directions. Nodes on Line OA have zero displacement in the X-direction and are free to move in the Y-direction. Nodes on Line OE have zero displacement in the Y-direction and are free to move in the X-direction. Nodes on Line ED are prescribed a uniform 0.1-mm displacement in the X-direction and zero displacement in the Y-direction. This is intended to simulate the holding of both ends in the wide section of the tension specimen using perfectly rigid grips. The extension of the specimen under this condition is 0.2 mm. In actual tensile tests, the specimen is clamped and pulled using a set of mechanical or hydraulic grips. The actual loading condition near the gripped area is therefore the combination of the normal compressive stress and in-plane tensile and shear stresses. Dimensions of finite-element meshes and the number of elements and nodes are summarized in the table in Fig. 2 for each type of ASTM D 638 specimen. OA is one half of the width of the gage FIG. 1 Pictures of ASTM D Type IV tension specimens: (a) Nylon-11 bonded Nd-Fe-B (59.7 vol%) material tested with at 100 C, and (b) pure Nylon-11 tested at 100 C. FIG. 3 Geometric model of the single and double radius transitional area. FIG. 2 Finite-element mesh for four types of specimen in ASTM D

3 GARRELL ET AL. ON ASTM 638 TEST SPECIMENS 3 FIG. 4 Distribution of von Mises stress in ASTM D 638 tensile test specimens. (Value by the node represents the ratio of nodal von Mises stress relative to the maximum von Mises stress at node MX in Pa.) section, DE is one half of the width of the wide section, OE is one half of the length of the specimen, and AB is one half of the length of the gage section. Between Points B and C, the specimen changes from the narrow to the wide section. As shown in Figs. 2a, 2b, and 2c, Type I, II, and III specimens consist of a circular arc of radius r 1 between Points B and C. These are known as single-arc specimens. As shown in Fig. 2d, two radii, r 1 and r 2, are used to blend two arcs in the transitional region between Points B and C for the Type IV specimen. This is designated as the double-arc specimen. The geometrical relationships of the single- and double-arc shape between Points B and C are shown in Fig. 3; w 1 and w 2 are widths of the narrow and wide section, respectively. As shown in Fig. 3a, the single-radius specimen has a continuous slope at Point B and a discontinuous slope at Point C. The double-arc specimen, as shown in Fig. 3b, has a continuous slope at all three transitional points: B, F, and C. Arcs BF and FC have the same angle, indicated by. The center of Arc BF, denoted as C 1 in Fig. 3a, has the same X-coordinate as Point B. Similarly, the center of Arc CF, denoted as C 2, has the same X-coordinate as Point C. These geometrical constraints together ensure a continuous slope at Points B, F, and C for the double-arc specimen. A hydrostatic stress-dependent yield criterion is usually used to analyze stress concentrations in plastics [12,13]. The yield criterion for magnetic particle reinforced plastics has not yet been studied. The maximum von Mises stress was used to calculate the stress concentration factor and to identify the location that possibly initializes the fracture. In addition to the von Mises stress, the maximum principal stress, which is more suitable for the analysis of brittle materials, was also applied to calculate the stress concentration factor. The finite-element analysis provided results of stress components and the von Mises and principal stress for each node. The nodal stress was used to calculate the stress concentration factor, K f. max K f (1) nominal where max is either the maximum nodal von Mises stress or maximum first principal stress, and nominal is either the average von Mises stress or the average first principal stress on nodes in the gage section. Finite-element analysis results show that the variation of von Mises stress among nodes in the gage section is very small, less than 1.5%. An elastic modulus of 130 MPa and Poisson s ratio of 0.3 were used in the finite-element analysis.

4 4 JOURNAL OF TESTING AND EVALUATION Finite-Element Analysis of Four Types of Tension Specimens Figure 4 shows the distribution of von Mises stress for ASTM D 638 Type I, II, III, and IV standard tension specimens. For each specimen, an overview and another enlarged view of the von Mises TABLE 1 The stress concentration for different single- and doubleradius tension specimen geometry. Designation r 1 (mm) r 2 (mm) w 1 /r 1 K f ASTM Type I D 638 Type II Type III Type IV Single S-I radius S-II tension S-III specimen S-IV S-V S-VI S-VII Double D-I radius D-II* tension D-III specimen D-IV D-V D-VI D-VII * The same geometry as Type IV. stress distribution is presented. The enlarged view is centered about Point B to identify the increase in von Mises stress. In Fig. 4, MX and MN mark the nodes with maximum and minimum von Mises stress, respectively. As shown in the overview of the specimen, Node MN is close to Point C. In the enlarged view, the location of Node MX relative to Node B can be seen. Node MX is next to Node B, as in Type I and II, or a node away from Node B, as in Type III and IV. It is noted that the mesh shown in the enlarged view represents the exaggeratedly deformed finite-element mesh from the ANSYS output. Node B, originally located at the intersection of a straight line and an arc, can be seen on a slightly curved region after deformation. The ratio of nodal von Mises stress relative to the maximum von Mises stress at Node MX is also presented in the enlarged view on nodes around Points B and MX in Fig. 4. For Type I, II, and III specimens, the overview of von Mises stress distribution does not show the stress concentration. Upon closely examining the distribution of von Mises stress around Point MX in the enlarged view, all four types of ASTM D 638 tension specimens demonstrate different levels of stress concentration at Node MX. The von Mises stress at Node MX ( max ) and at nodes in the gage section ( nominal ) were extracted from the finite-element output and substituted into Eq 1 to calculate the stress concentration factor, K f. Results of K f are listed in Table 1. Type IV and II specimens have the highest and lowest K f, and 1.014, respectively. It was found that K f has an almost linear relationship versus the parameter w 1 /r 1, as shown in Fig. 5, for the four types of ASTM D 638 specimen. FIG. 5 The stress concentration factor versus ratio of gage section width and radius for five types of ASTM D 638 specimens and the single- and double-radius specimens.

5 GARRELL ET AL. ON ASTM 638 TEST SPECIMENS 5 Tension specimens with low w 1 /r 1 ratio usually have large arc radius and are longer than specimens with high w 1 /r 1. For example, as shown in Fig. 2 and Table 1, Type II and IV specimens have the same width in the gage section and different radius r 1. The Type II specimen has a larger arc radius, which makes the specimen longer and has lower w 1 /r 1 and K f. Longer and bigger tension specimens are unattractive because they require more material and a bigger die for injection molding and cost more to produce. Instead of using the von Mises stress, the maximum principal stress and nominal principal stress were extracted from the finiteelement output and substituted into Eq 1 to calculate K f. For all four types of specimens, the difference in K f based on von Mises and maximum principal stresses is very small, less than Finite-Element Analysis of Single- and Double-Arc Tension Specimens In this section, the investigation of stress distribution in alternative tension specimen geometries that maintain the same overall size as the Type IV specimen is discussed. The objective is to identify the geometric parameters that would reduce the magnitude of K f. The width (w 1 ) and length of the gage section are 6 and 13 mm, respectively. The width of the wide section, w 2, is 19 mm. Two groups of sample geometry were studied. One group consists of single-arc specimens, similar to those in Type I, II, and III. Another group has the double-arc geometry, similar to the Type IV specimen. Single-Arc Specimens As shown in Table 1, seven single radius curves are used to blend between Points B and C. The minimum radius of Arc BC is 6.5 mm ( (w 2 w 1 )/2). The specimens with radii of 6.5, 9, 13, 19, 28, 39, or 52 mm at Arc BC were designated as S-I to S-VII, respectively. The shape and finite-element mesh of three selected specimens, S-I, S-IV, and S-VII, are shown in Fig. 6. For specimens with arc radius equal to or less than 39 mm, the overall length of the original Type IV specimen can be maintained. The K f of the 39-mm radius single-arc specimen (S-VI), as shown in Table 1, is This is significantly lower than K f of of the Type IV specimen. Plots of K f versus w 1 /r 1 for these seven specimens are shown in Fig. 5. The larger radius of arc BC lowers both w 1 /r 1 and K f. Figure 5 also shows that the relationship between K f and w 1 /r 1 is almost linear and matches the trend of Type I to IV specimens. Double-Arc Specimens Seven double-arc specimens, denoted as D-I to D-VII, were studied as well. As shown in Fig. 3, the values of r 1 and r 2 in the doublearc geometry can be reversed and still maintain the continuity of both position and slope at Points B, F, and C. This special geometrical characteristic of the double-arc shape was used to define conjugate pairs of specimens. As shown in Table 1, D-I (r 1 9 mm, r 2 30 mm) and D-VII (r 1 30 mm, r 2 9 mm) form a conjugate pair of double-arc specimens. D-II (r 1 14 mm, r 2 25 mm) and D-VI (r 1 25 mm, r 2 14 mm) as well as D-III (r 1 17 mm, r 2 22 mm) and D-V (r 1 22 mm, r 2 17 mm) are another two sets of conjugate double-arc specimens. It is noted that the D-II specimen has the same geometry as the ASTM D 638 Type IV specimen. The intent is to investigate the change in K ƒ of a conjugate doublearc specimen. The D-III specimen has the same r 1 and r 2, both equal to 19.5 mm. The shape and finite-element mesh of three selected specimens, D-I, D-IV, and D-VII, are shown in Fig. 6. Finite-element analysis results of K f versus w 1 /r 1 for seven double-arc specimens are listed in Table 1 and plotted in Fig. 5. It is important to note that the value of r 2 does not affect K f. By just switching the two arcs of the original Type IV specimen, the K f was reduced from to By further increasing r 1 to 30 mm without changing the overall size of the Type IV specimen, K f was reduced to Modeling of the Stress Concentration Factor As shown in Fig. 5, a straight line can be used to determine K f for various w 1 /r 1. Linear regression analysis was performed on all 17 data points, including the four types of specimen specified by ASTM D 638 and the seven single-arc and six double-arc specimens. The following model is obtained: K f (w 1 /r 1 ) (2) Equation 2 can be used to predict the K f based on w 1 /r 1 and may be useful for new tension specimen design. FIG. 6 Finite-element meshes for selected single- and double-arc specimens (3 mm half-width in the gage section for all specimens).

6 6 JOURNAL OF TESTING AND EVALUATION The linear trend indicates the possibility that an analytical solution for the stress distribution in this geometrical configuration may exist. This requires further investigation. Concluding Remarks The stress distribution of four types of tension specimens recommended in ASTM D 638 was investigated through the use of the finite-element technique. It was found that the location of maximum von Mises stress for Type IV geometry coincided with the location of experimentally observed failures in brittle specimens made of Nylon-11 bonded Nd-Fe-B particles. This study identified a solution to reduce the magnitude of stress concentration by increasing the radius of the arc in the transitional area. A linear correlation of stress concentration factor K f versus w 1 /r 1 was identified and a linear model was established to predict the K f based on w 1 /r 1. Results obtained in this research can help establish standards and aid in the design of specimen geometry for tension testing of brittle particle-reinforced composites. Acknowledgments A portion of this research was sponsored by the User program of the High Temperature Material Laboratory of Oak Ridge National Laboratory and by National Science Foundation Grant (Dr. K. P. Rajurkar, Program Director). References [1] ASTM D : Standard Test Method for Tensile Strength of Plastics, Annual Book of ASTM Standards, ASTM International, West Conshohocken, PA, [2] Garrell, M. G., Ma, B. M., Lara-Curzio, E., Shih, A. J., and Scattergood, R. O., Mechanical Properties of Nylon Bonded Nd-Fe-B Permanent Magnets, Journal of Magnetism and Magnetic Materials, accepted for publication. [3] Garrell, M. G., Ma, B. M., Lara-Curzio, E., Shih, A. J., and Scattergood, R. O., Mechanical Properties of Polyphenylene-Sulfide (PPS) Bonded Nd-Fe-B Permanent Magnets, Materials Science and Engineering: A, submitted. [4] Garrell, M. G., Mechanical Properties of Injection Molded Permanent Magnets, M.S. thesis, North Carolina State University, Raleigh, NC, April [5] Pilkey, W. D., Peterson s Stress Concentration Factors, 2nd ed., John Wiley & Sons, New York, [6] Wilson, L. H. and White, D. J. Stress-Concentration Factors for Shoulder Fillets and Grooves in Plates, Journal of Strain Analysis, Vol. 8, 1973, pp [7] Frocht, M. M. Factor of Stress Concentration Photoelastically Determined, Transactions of ASME, 1935, Vol. 57, pp. A-67 A-68. [8] Worthem, D. W., Flat Tension Specimen Design for Advanced Composites, NASA Technical Report CR , NASA Lewis Research Center, Cleveland, OH, [9] Shi, J., Brady, L., and Ireson, A., Design of Room Temperature Tension Specimen Made of Ceramic Matrix Composites, Key Engineering Materials, Vol , Trans Tech Publications, 1999, pp [10] Wang, Z, Chiang, C. K., and Chuang, T.-J., Optimum Design of a Ceramic Tensile Creep Specimen Using a Finite Element Method, Journal of Research of the National Institute of Standards and Technology, Vol. 102, 1997, pp [11] Oplinger, D. W., Parker, B. S., Gandhi, K. R., Lamothe, R., and Foley, G., On the Streamline Specimen for Tension Testing of Composite Materials, Recent Advances in Composites in the United States and Japan, ASTM STP 864, J. R. Vinson and M. Taya, Eds., ASTM International, West Conshohocken, PA, 1985, pp [12] McCrum, N. G., Buckley, C. P., and Bucknall, C. B., Principles of Polymer Engineering, 2nd ed., Oxford University Press, Oxford, [13] Courtney, T. H., Mechanical Behavior of Materials, 2nd ed., McGraw-Hill, New York, 2000.

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