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| [[File:Trois-points-p1040189.jpg|thumb|1940s flexural test machinery working on a sample of concrete]]
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| [[File:Three point flexural test.jpg|thumb| Test fixture on universal testing machine for three point flex test]]
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| {{Refimprove|date=March 2010}}
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| The '''three point bending [[flexure|flexural]] test''' provides values for the [[Flexural modulus|modulus of elasticity
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| in bending]] <math>E_f</math>, [[flexural stress]] <math>\sigma_f</math>, flexural strain <math>\epsilon_f</math> and the flexural stress-strain response of the material. The main advantage of a three point flexural test is the ease of the specimen preparation and testing. However, this method has also some disadvantages: the results of the testing method are sensitive to specimen and loading geometry and strain rate.
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| == Testing method ==
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| The [[test method]] for conducting the test usually involves a specified [[test fixture]] on a [[universal testing machine]]. Details of the test preparation, conditioning, and conduct affect the test results.
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| Calculation of the flexural stress <math>\sigma_f</math>
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| :<math>\sigma_f = \frac{3 F L}{2 b d^2}</math> for a rectangular cross section
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| :<math>\sigma_f = \frac{F L}{\pi R^3}</math> for a circular cross section<ref>{{cite book|title=Biomaterials - The intersection of Biology and Material Science|publisher=[[Pearson Prentice Hall Bioengineering]]|location=[[New Jersey]], [[United States]]|year=2008|chapter=Chapter 4 Mechanical Properties of Biomaterials|page=152}}</ref>
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| Calculation of the flexural strain <math>\epsilon_f</math>
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| :<math>\epsilon_f = \frac{6Dd}{L^2}</math>
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| Calculation of [[flexural modulus]] <math>E_f</math><ref>{{cite book|title=Salters Horners Advanced Physics for Edexcel AS Physics|publisher=[[Pearson Education]]|location=[[Essex]], [[United Kingdom]]|year=2008|chapter=Activity 20 - Bendy Wafer}}</ref><!-- Not included in the main students' book, but in an activity sheet. Instead of "m", the book has W (weight)/y (deflection at point of application of load). ISBN for students' book: ISBN 978-1-4058-9602-3 -->
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| :<math>E_f = \frac{L^3 m}{4 b d^3}</math>
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| in these formulas the following parameters are used:
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| * <math>\sigma_f</math> = Stress in outer fibers at midpoint, ([[MPa]])
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| * <math>\epsilon_f</math> = Strain in the outer surface, (mm/mm)
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| * <math>E_f</math> = flexural Modulus of elasticity,(MPa)
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| * <math>F</math> = load at a given point on the load deflection curve, ([[Newton (unit)|N]])
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| * <math>L</math> = Support span, (mm)
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| * <math>b</math> = Width of test beam, (mm)
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| * <math>d</math> = Depth of tested beam, (mm)
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| * <math>D</math> = maximum deflection of the center of the beam, (mm)
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| * <math>m</math> = The gradient (i.e., slope) of the initial straight-line portion of the load deflection
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| curve,(P/D), (N/mm)
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| * <math>R</math> = The radius of the beam, (mm)
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| == Fracture toughness testing ==
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| [[File:SingleEdgeNotchBending.svg|thumb|right|300px|Single edge notch bending specimen (also called three point bending specimen) for fracture toughness testing.]]
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| The [[fracture toughness]] of a specimen can also be determined using a three-point flexural test. The [[stress intensity factor]] at the crack tip of a [[three point flexural test|single edge notch bending specimen]] is<ref name=bower>{{cite book|title=Applied mechanics of solids|author=Bower, A. F.| year=2009| publisher=CRC Press.}}</ref>
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| :<math>
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| \begin{align}
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| K_{\rm I} & = \frac{4P}{B}\sqrt{\frac{\pi}{W}}\left[1.6\left(\frac{a}{W}\right)^{1/2} - 2.6\left(\frac{a}{W}\right)^{3/2}
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| + 12.3\left(\frac{a}{W}\right)^{5/2} \right.\\
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| & \qquad \left.- 21.2\left(\frac{a}{W}\right)^{7/2} + 21.8\left(\frac{a}{W}\right)^{9/2} \right]
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| \end{align}
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| </math>
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| where <math>P</math> is the applied load, <math>B</math> is the thickness of the specimen, <math>a</math> is the crack length, and
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| <math>W</math> is the width of the specimen. In a three-point bend test, a fatigue crack is created at the tip of the notch by cyclic loading. The length of the crack is measured. The specimen is then loaded monotonically. A plot of the load versus the crack opening displacement is used to determine the load at which the crack starts growing. This load is substituted into the above formula to find the fracture toughness <math>K_{Ic}</math>.
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| The ASTM E1290-08 Standard suggests the relation
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| :<math> | |
| K_{\rm I}= \cfrac{6P}{BW}\,a^{1/2}\,Y
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| </math>
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| where
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| :<math>
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| Y=\cfrac{1.99-a/W\,(1-a/W)(2.15-3.93a/W+2.7(a/W)^{2})}{(1+2a/W)(1-a/W)^{3/2}} \,.
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| </math>
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| The predicted values of <math>K_{\rm I}</math> are nearly identical for the ASTM and Bower equations for crack lengths less than 0.6<math>W</math>.
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| ==Standards==
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| * [[International Organization for Standardization|ISO]] 12135: Metallic materials. Unified method for the determination of quasi-static fracture toughness
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| * [[International Organization for Standardization|ISO]] 12737: Metallic materials. Determination of plane-strain fracture toughness
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| * [[ASTM]] D790: Standard test methods for flexural properties of unreinforced and reinforced plastics and electrical insulating materials
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| * [[International Organization for Standardization|ISO]] 178: Plastics—Determination of flexural properties
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| * ASTM E1290: Standard Test Method for Crack-Tip Opening Displacement (CTOD) Fracture Toughness Measurement.
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| ==See also==
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| *[[Bending]]
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| *[[Euler-Bernoulli beam equation]]
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| *[[Flexural strength]]
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| *[[List of area moments of inertia]]
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| *[[Second moment of area]]
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| == References ==
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| {{reflist}}
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| [[Category:Materials testing]]
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| [[Category:Mechanics]]
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