# Three-point flexural test

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The three-point bending flexural test provides values for the modulus of elasticity in bending ${\displaystyle E_{f}}$, flexural stress ${\displaystyle \sigma _{f}}$, flexural strain ${\displaystyle \epsilon _{f}}$ and the flexural stress–strain response of the material. This test is performed on a universal testing machine (tensile testing machine or tensile tester) with a three-point or four-point bend fixture.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.

## Testing method

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. The sample is placed on two supporting pins a set distance apart.

Calculation of the flexural stress ${\displaystyle \sigma _{f}}$

${\displaystyle \sigma _{f}={\frac {3FL}{2bd^{2}}}}$ for a rectangular cross section
${\displaystyle \sigma _{f}={\frac {FL}{\pi R^{3}}}}$ for a circular cross section [1]

Calculation of the flexural strain ${\displaystyle \epsilon _{f}}$

${\displaystyle \epsilon _{f}={\frac {6Dd}{L^{2}}}}$

Calculation of flexural modulus ${\displaystyle E_{f}}$ [2]

${\displaystyle E_{f}={\frac {L^{3}m}{4bd^{3}}}}$

in these formulas the following parameters are used:

• ${\displaystyle \sigma _{f}}$ = Stress in outer fibers at midpoint, (MPa)
• ${\displaystyle \epsilon _{f}}$ = Strain in the outer surface, (mm/mm)
• ${\displaystyle E_{f}}$ = flexural Modulus of elasticity,(MPa)
• ${\displaystyle F}$ = load at a given point on the load deflection curve, (N)
• ${\displaystyle L}$ = Support span, (mm)
• ${\displaystyle b}$ = Width of test beam, (mm)
• ${\displaystyle d}$ = Depth or thickness of tested beam, (mm)
• ${\displaystyle D}$ = maximum deflection of the center of the beam, (mm)
• ${\displaystyle m}$ = The gradient (i.e., slope) of the initial straight-line portion of the load deflection

curve, (N/mm)

• ${\displaystyle R}$ = The radius of the beam, (mm)

## Fracture toughness testing

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 single edge notch bending specimen is [3]

{\displaystyle {\begin{aligned}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}+12.3\left({\frac {a}{W}}\right)^{5/2}\right.\\&\qquad \left.-21.2\left({\frac {a}{W}}\right)^{7/2}+21.8\left({\frac {a}{W}}\right)^{9/2}\right]\end{aligned}}}

where ${\displaystyle P}$ is the applied load, ${\displaystyle B}$ is the thickness of the specimen, ${\displaystyle a}$ is the crack length, and ${\displaystyle W}$ 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 ${\displaystyle K_{Ic}}$.

The ASTM D5045-14 [4] and E1290-08 [5] Standards suggests the relation

${\displaystyle K_{\rm {I}}={\cfrac {6P}{BW}}\,a^{1/2}\,Y}$

where

${\displaystyle Y={\cfrac {1.99-a/W\,(1-a/W)(2.15-3.93a/W+2.7(a/W)^{2})}{(1+2a/W)(1-a/W)^{3/2}}}\,.}$

The predicted values of ${\displaystyle K_{\rm {I}}}$ are nearly identical for the ASTM and Bower equations for crack lengths less than 0.6${\displaystyle W}$.

## Standards

• ISO 12135: Metallic materials. Unified method for the determination of quasi-static fracture toughness.
• ISO 12737: Metallic materials. Determination of plane-strain fracture toughness.
• ISO 178: Plastics—Determination of flexural properties.
• ASTM D790: Standard test methods for flexural properties of unreinforced and reinforced plastics and electrical insulating materials.
• ASTM E1290: Standard Test Method for Crack-Tip Opening Displacement (CTOD) Fracture Toughness Measurement.
• ASTM D7264: Standard Test Method for Flexural Properties of Polymer Matrix Composite Materials.
• ASTM D5045: Standard Test Methods for Plane-Strain Fracture Toughness and Strain Energy Release Rate of Plastic Materials.

## Related Research Articles

Young's modulus, or the Young modulus, is a mechanical property that measures the stiffness of a solid material. It defines the relationship between stress and strain in a material in the linear elasticity regime of a uniaxial deformation.

In materials science, deformation refers to any changes in the shape or size of an object due to

A fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displacement develops perpendicular to the surface of displacement, it is called a normal tensile crack or simply a crack; if a displacement develops tangentially to the surface of displacement, it is called a shear crack, slip band, or dislocation.

Compressive strength or compression strength is the capacity of a material or structure to withstand loads tending to reduce size, as opposed to tensile strength, which withstands loads tending to elongate. In other words, compressive strength resists compression, whereas tensile strength resists tension. In the study of strength of materials, tensile strength, compressive strength, and shear strength can be analyzed independently.

In materials science and metallurgy, toughness is the ability of a material to absorb energy and plastically deform without fracturing. One definition of material toughness is the amount of energy per unit volume that a material can absorb before rupturing. It is also defined as a material's resistance to fracture when stressed.

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.

Thermal shock is a type of rapidly transient mechanical load. By definition, it is a mechanical load caused by a rapid change of temperature of a certain point. It can be also extended to the case of a thermal gradient, which makes different parts of an object expand by different amounts. This differential expansion can be more directly understood in terms of strain, than in terms of stress, as it is shown in the following. At some point, this stress can exceed the tensile strength of the material, causing a crack to form. If nothing stops this crack from propagating through the material, it will cause the object's structure to fail.

Work hardening, also known as strain hardening, is the strengthening of a metal or polymer by plastic deformation. Work hardening may be desirable, undesirable, or inconsequential, depending on the context.

Delamination is a mode of failure where a material fractures into layers. A variety of materials including laminate composites and concrete can fail by delamination. Processing can create layers in materials such as steel formed by rolling and plastics and metals from 3D printing which can fail from layer separation. Also, surface coatings such as paints and films can delaminate from the coated substrate.

The stress intensity factor, , is used in fracture mechanics to predict the stress state near the tip of a crack or notch caused by a remote load or residual stresses. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials, and is a critical technique in the discipline of damage tolerance. The concept can also be applied to materials that exhibit small-scale yielding at a crack tip.

In materials science, fracture toughness is the critical stress intensity factor of a sharp crack where propagation of the crack becomes unlimited and rapid. The critical value of stress intensity factor in mode I loading measured under plane strain conditions is known as the plane strain fracture toughness, denoted . Fracture toughness is a quantitative way of expressing a material's resistance to crack propagation and standard values for a given material are generally available. Slow self-sustaining crack propagation known as stress corrosion cracking, can occur in a corrosive environment above the threshold and below . Limited crack extension can occur at lower values of stress intensity factor with fatigue crack growth.

Flexural strength, also known as modulus of rupture, or bend strength, or transverse rupture strength is a material property, defined as the stress in a material just before it yields in a flexure test. The transverse bending test is most frequently employed, in which a specimen having either a circular or rectangular cross-section is bent until fracture or yielding using a three point flexural test technique. The flexural strength represents the highest stress experienced within the material at its moment of yield. It is measured in terms of stress, here given the symbol .

The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material. The theoretical concept of J-integral was developed in 1967 by Cherepanov and in 1968 by Jim Rice independently, who showed that an energetic contour path integral was independent of the path around a crack.

In mechanics, the flexural modulus or bending modulus is an intensive property that is computed as the ratio of stress to strain in flexural deformation, or the tendency for a material to resist bending. It is determined from the slope of a stress-strain curve produced by a flexural test, and uses units of force per area.

Failure theory is the science of predicting the conditions under which solid materials fail under the action of external loads. The failure of a material is usually classified into brittle failure (fracture) or ductile failure (yield). Depending on the conditions most materials can fail in a brittle or ductile manner or both. However, for most practical situations, a material may be classified as either brittle or ductile. Though failure theory has been in development for over 200 years, its level of acceptability is yet to reach that of continuum mechanics.

According to the classical theories of elastic or plastic structures made from a material with non-random strength (ft), the nominal strength (σN) of a structure is independent of the structure size (D) when geometrically similar structures are considered. Any deviation from this property is called the size effect. For example, conventional strength of materials predicts that a large beam and a tiny beam will fail at the same stress if they are made of the same material. In the real world, because of size effects, a larger beam will fail at a lower stress than a smaller beam.

Crack tip opening displacement (CTOD) or is the distance between the opposite faces of a crack tip at the 90° intercept position. The position behind the crack tip at which the distance is measured is arbitrary but commonly used is the point where two 45° lines starting at the crack tip and intercepting the crack faces. The parameter is used in Fracture mechanics to characterise the loading on a crack and can be related to other crack tip loading parameters such as the stress intensity factor and the elastic-plastic J-integral.

The four-point bending flexural test provides values for the modulus of elasticity in bending , flexural stress , flexural strain and the flexural stress-strain response of the material. This test is very similar to the three-point bending flexural test. The major difference being that with the addition of a fourth bearing the portion of the beam between the two loading points is put under maximum stress, as opposed to only the material right under the central bearing in the case of three point bending.

A crack growth equation is used for calculating the size of a fatigue crack growing from cyclic loads. The growth of fatigue cracks can result in catastrophic failure, particularly in the case of aircraft. A crack growth equation can be used to ensure safety, both in the design phase and during operation, by predicting the size of cracks. In critical structure, loads can be recorded and used to predict the size of cracks to ensure maintenance or retirement occurs prior to any of the cracks failing.

## References

1. "Chapter 4 Mechanical Properties of Biomaterials". Biomaterials – The intersection of Biology and Material Science. New Jersey, United States: Pearson Prentice Hall. 2008. p. 152.
2. Zweben, C., W. S. Smith, and M. W. Wardle (1979), "Test methods for fiber tensile strength, composite flexural modulus, and properties of fabric-reinforced laminates", Composite Materials: Testing and Design (Fifth Conference), ASTM InternationalCS1 maint: multiple names: authors list (link)
3. Bower, A. F. (2009). Applied mechanics of solids. CRC Press.
4. ASTM D5045-14: Standard Test Methods for Plane-Strain Fracture Toughness and Strain Energy Release Rate of Plastic Materials, West Conshohocken, PA: ASTM International, 2014
5. E1290: Standard Test Method for Crack-Tip Opening Displacement (CTOD) Fracture Toughness Measurement, West Conshohocken, PA: ASTM International, 2008