Department of Materials and Metallurgical Engineering Bangladesh University of Engineering Technology, Dhaka MME 222 Materials Testing Sessional.50 Credits Laboratory 4 Bending Test of Materials. Objective Bend or flexure testing is common in springs and brittle materials whose failure behaviours are linear such as concretes, stones, woods, plastics, glasses and ceramics. Other types of brittle materials such as powder metallurgy processed metals and materials are normally tested under a transverse flexure. Bend test is therefore suitable for evaluating strength of brittle materials where conduction of tensile test and interpretation of test result of material is difficult and sometimes impossible. After completion of this experiment, students should be able to. conduct 3-point bend test and investigate responses of materials when subjected to bending.2 determine flexural bend strength and elastic modulus of brittle material, and.3 analyse statistical nature of the test data and determine the Weibull modulus of the brittle materials 2. Materials and Equipment 2. Bend samples 2.2 Universal testing machine 2.3 Vernia calliper 3. Experimental Procedure 3. Bend test of materials 3.. Measure the width and thickness of the specimen. Mark on the locations where the load will be applied under three-point bending. 3..2 Place the sample carefully on to the stage of 3-point bending fixture of a universal testing machine. Make sure that the loading point is placed on to the marked location. 3..3 Carry out the bend test until failure takes place. 3..4 Construct the load-extension or load-deflection curve to calculate the flexural bend strength and elastic modulus of the specimen. 3..5 Repeat steps 3.. to 3..4 to conduct bend tests of other specimens. 3.7 Complete the Data Sheet, Table 3.. 3.2 Statistical analysis of data 3.2. Conduct 3-point bend test of at least 5 glass specimens following steps 3.. to 3..3. 3.2.2 Note the breaking loads and then determine flexural bend strength (MOR) of these specimens.
3.2.3 Arrange them in descending order, rank them and determine their failure probability, P f. 3.2.5 Construct ln ( ln ( P f )) vs. ln (MOR) curve and determine Weibull constants m and 0. 4. Results 4. Display the test data in the Table. 4.2 Determine bend strength, elastic modulus and flexural strength (MOR) of the material. 4.3 Determine Weibul modulus of the material and comment on the degree of homogeneity of the material. 5. Discussion 5. Answer the following questions: (a) Give three examples of engineering applications that involve bending properties of materials. (b) What would you expect if the bending experiment has been carried out at elevated temperature? (c) Why does the presence of cracks in ductile materials present not much problem when compare with brittle material? (d) Explain why flexural strength of brittle material is always higher than tensile strength of the respective material. (e) (f) Determine the flexural strength of material testing using 4-point bend test. What conclusion can you draw from the Weibull modulus value of a test sample? (g) Compare and explain the Weibull modulus of cast and wrought aluminium alloys. 2
Table 3.: Data Sheet for Bend Test Sample identification Symbol Unit Value 2 3 Average Material Width of sample w mm Thickness of sample t mm Span length of sample L mm Fracture load P f kn Flexural Bend Strength (MOR) fb MPa Elastic Modulus E b GPa **Provide all calculations in separate pages. Signature with Date of the Instructor/Course Tutor 3
6.0 Theoretical Background 6. Introduction Preparing specimens from brittle materials, such as ceramics and glasses, for direct tension tests is difficult because of the problems involved in shaping and machining them to proper dimensions. Furthermore, such specimens are sensitive to surface defects and notches, and clamping brittle test specimens for testing is difficult. Also, improper alignment of the test specimen may result in a non-uniform stress distribution along the cross section and ultimate premature failure. A commonly used test method for measuring strength of brittle materials such as ceramics and glasses is the bend or flexure test. The test specimen can have a circular, square, or rectangular cross section and is uniform along the complete length. Such a specimen is much less expensive to fabricate than a tensile specimen. The test is conducted with the same kind of universal testing machine used for tensile and compressive strength measurements. The test specimen is supported at the ends and the load is applied either at the centre (threepoint loading) or at two positions (four-point loading), Fig. 6.. The longitudinal stresses in the specimens are tensile at their lower surfaces and compressive at their upper surfaces. The bend or flexural strength is defined as the maximum uniaxial tensile strength at failure and it is often referred to as the flexural strength or modulus of rupture (MOR). P P/2 P/2 L/2 P/2 L P/2 c = t/2 a P/2 L P/2 c = t/2 (a) Figure 6.: Two bend-test methods for brittle materials: (a) three-point bending; (b) four-point bending. The areas on the beams represent the bending moment diagrams. Note the region of constant maximum bending moment in (b); by contrast, the maximum bending moment occurs only at the centre of the specimen in (a). (b) Three-point testing is still an option, but four-point is preferred since more material is exposed to high stress. Also, because of the larger volume of material subjected to the same bending moment in Fig. 6.b, there is a higher probability that defects exist in this volume than in that in Fig. 6.a. Consequently, the four-point test gives a lower modulus of rupture than the three-point test. Figure 6.2 illustrates three-point bending arrangement which is capable of 80 bend angle for welded materials. 6.2 Three-Point Bend Test of Brittle Materials Considering a three point bend test of an elastic material, when the load P is applied at the mid-span of specimen in an x-y plane, stress distribution across the specimen thickness (t) is demonstrated in Fig. 6.3. The stress is essentially zero at the neutral axis N-N. Stresses in the y axis in the positive direction represent tensile stresses whereas stresses in the negative direction represent compressive stresses. Within the elastic range, brittle materials show a linear relationship of load and deflection where yielding occurs on a thin layer of the 4
Figure 6.2: Example of a weld plate bend tested under a three-point bend arrangement. specimen surface at the mid-span, Fig. 6.3a. This in turn leads to crack initiation which finally proceeds to specimen failure. Ductile materials however provide load-deflection curves which deviate from a linear relationship before failure takes place as opposed to those of brittle materials previously mentioned. Furthermore, it is also difficult to determine the beginning of yielding in this case. The stress distribution of a ductile material after yielding is given in Fig. 6.3b. Therefore, it can be seen that bend testing is not suitable for ductile materials due to difficulties in determining the yield point of the materials under bending and the obtained stress-strain curve in the elastic region may not be linear. The results obtained might not be validated. As a result, the bend test is therefore more appropriate for testing brittle materials whose stress-strain curves (Fig. 6.4) show its linear elastic behaviour just before the materials fail. P w x P/2 z y P/2 c c t/2 t/2 t/2 t/2 c c (a) Figure 6.3: Stress distributions in a rectangular bar when (a) elastically bended and (b) after yielding (b) 5
Figure 6.4: Flexural stress and flexural strain relationship. 6.2. Determination of flexural strength and elastic modulus For a beam in flexure, the maximum outer fibre stress from simple beam theory is: σ fb = M c I where M is the applied moment, c is the distance of the outer fibres (where the tensile force is acting) from the neutral axis, and I is the moment of inertia of the cross section about the neutral axis. It is assumed that the material behaviour is consistent with Hooke s law. For a rectangular test specimen I = bt 3 /2 and c = t/2, where b and t are the width and thickness of the specimen, respectively. With reference to 3-point a configurations, Fig. 6.4, the bending moment can be calculated as M = Replacing the values of M, c and I in eq.(6.) to obtain the flexural bending strength σ fb = P L 4 (6.) (6.2) 3PL 2bt 2 (6.3) From the experimental result, we can also obtain the elastic modulus of the material according to the linearelastic analysis. The deflection of the beam () from the centre as illustrated in Fig. 6.3 can be expressed as δ = PL 3 48 E I where I = wt 3 2. The elastic modulus (E B ) in the above equation can be calculated from the slope of the load-deflection curve (dp/d) in the linear region as follows (6.4) E B = L 3 48 I (dp dδ ) (6.5) E B = m L 3 (6.6) 4 w t 3 where m = (dp/d) is the slope of the tangent to the straight-line portion of the load-deflection beam. 6
6.3 The Weibull modulus The strength characterisation data of ceramics reported in terms of flexural bend strength or MOR is relatively inexpensive, straightforward and quick process. But it has severe limitation on the usability of MOR data: the measured strength varies significantly depending on the size of the specimen tested and whether it is loaded in three-point or four-point. Ceramics do not have uniquely defined failure strength. A given batch of ceramic specimens usually does not show a constant measured strength instead they show a range of strengths. The strength of ceramic is determined by a combination of two material parameters the toughness and the crack size. Since fracture toughness is not a variable, the variation in strength comes from a variation in the size of the largest defect (crack), Fig. 6.5, and the microstructure. There are lots of small defects, besides the largest ones, present in the material which are stressed but they do not reach their critical stress for propagation. Hence, it is necessary to use sample sets of ten or more specimens for even the simplest of goals such as determining an average strength for material ranking, material development, or materials specification purposes and a probabilistic approach is needed to interpret test data. Thus, it is also widely recognized that brittle material strengths are statistical in nature. Largest defect Figure 6.4: Flaw distribution in ceramic materials. When analysing the variation in the strength of ceramics, a particular function, due to the Swedish engineer Weibull, has been found to be useful. The simplest form of Weibull approach of characterising the flaw distribution is based on a weakest link model and it is analogous to the breaking of a length of chain. Failure occurs when the weakest link breaks. In a series of chains of a particular length, the weakest link in each length is of different strength and this controls the variation in strength. In ceramics, the links could represent small volumes of material containing a flaw and the weakest link is equivalent to the region with the largest flaw. Considering the two-parameter model, the probability of some length L of a link failing at a stress is P f = P s = exp { ( σ m ) } (6.7) σ 0 Where P f and P s are, respectively, the probability of failure and survival, 0 is a constant called the normalising stress, and m is a number, usually referred to as the Weibull modulus, which reflects the degree of variability in strength the higher the m is, the less variable is the strength. In this Weibull function, the Weibull modulus, m, defines the shape of the failure distribution curve. If m tends to be zero, the failure probability tends to become independent of the applied stress. If m =, the failure probability becomes a simple asymptotic exponential distribution. If m =, the distribution is a step function with P f = 0 when < 0, and P f = when > 0. P f P f m = m = 0 0 7
The Weibull modulus, m, also defines the width of the probability distribution, Fig. 6.2. If m is large, the distribution is narrow showing a small spread of failure strength, indicating high reliability of the material. If, on the other hand, m is small, the distribution is wide showing a large variation in the failure strength, indicating unreliability of the material. Poor ceramics have m in the range of 3 0, while good engineering ceramics have m in the range 0 40, and homogeneous metals and alloys have m well above 00. P f P f large m small m 0 0 A substantial number of test samples are required to determine accurate value for the Weibull slope m. Testing of only 0 samples can result in an error as high as ±40% in the m value. Over 80 samples are required to obtain 90% confidence in the m value. 6.3. Determining Weibull parameters from experimental data Taking natural logarithms in both sides of eq.(3.7), we get ln P s = ( σ m ) σ 0 (6.8) Taking natural logarithms again ln ( ln P s ) = m ln σ m ln σ 0 (6.9) ln { ln( P f )} = m ln σ m ln σ 0 (6.0) Thus plotting ln ( ln P s ) or ln ( ln ( P f ) as ordinate against ln as abscissa should give a straight line of the form y = mc + c. Here the gradient of the equation is the Weibull modulus m and the intercept to y-axis is the normalising stress 0 in the form of m ln 0. In the analysis of a number of strength data of a typical material through Weibull statistics, the plotting positions for abscissa are obvious but not the ordinate. The failure probability at a given stress is found by first ranking the failure stresses in order of strength. The failure probability of nth ranked sample from a total of N would be P f =( P s ) =n/(n+). The figure shown below is a Weibull plot of nine fracture strength data of abraded glass rods. The Weibull modulus of the sample is 4.76, indicating very poor reliability of the data. 8