## Flexural Stress Strain

Flexural stress-strain testing according to ASTM D 790 determines the load necessary to generate a given level of strain on a specimen typically using a three-point load (Fig. 2.6). Testing is performed at specified

! Modulus vs. specific gravity ratio for different materials follows a straight line

! Modulus vs. specific gravity ratio for different materials follows a straight line

Example of a three-point ASTM D 790 flexural test specimen

constant rate of crosshead movement based on material being tested. A solid plastic is usually at 0.05 in./min., foamed plastic at 0. 1 in./ min.,

Simple beam equations are used to determine the stresses on specimens at different levels of crosshead displacement. Using traditional beam equations and section properties in Fig. 2.5, the following relationships can be derived where Y is the deflection at the load point:

Bending stress where a = 3FL/2bh2 (2-10)

Bending or flexural modulus where E = FL3/4bh3Y (2-11)

Using these relationships, the flexural strength (also called the modulus of rupture) and the flexural modulus of elasticity can be determined. Table 2.1 provides examples of the flexural modulus of elasticity for polypropylene with nothing added (NEAT/Chapter 1) and reinforced with glass fibers and talc.

Polypropylene NEAT and filled flexural modulus of elasticity data

A flexural specimen is not in a state of uniform stress on the specimen. When a simply supported specimen is loaded, the side of the material opposite the loading undergoes the greatest tensile loading. The side of the material being loaded experiences compressive stress (Fig. 2.7). These stresses decrease linearly toward the center of the sample.

NEAT plastic 40wt°/o glass fiber 40wt°/o talc

180,000 psi (1,240 M Pa) 1,100,000 psi (7,600 MPa) 575,000 psi

Figure 2.7 Tensile-compressive loading occurs on a flexural specimen

Figure 2.7 Tensile-compressive loading occurs on a flexural specimen

Theoretically the center is a plane, called the neutral axis, that experiences no stress.

In the flexural test the tensile and compressive yield stresses of a plasdc may cause the stress distribution within the test specimen to become very asymmetric at high strain levels. This change causes the neutral axis to move from the center of the specimen toward the surface that is in compression. This effect, along with specimen anisotropy due to processing, may cause the shape of the stress-strain curve obtained in flexure to differ significandy from that of the normal S-S curve.

The S-S behavior of plastics in flexure generally follows that of tension and compression tests for either unreinforced or reinforced plastics. The flexural E tends to be the average between the tension and compression Es. The flexural yield point follows that observed in tension.

For the standard ASTM flexural strengths most plastics are higher than their ultimate tensile strengths, but may be either higher or lower than compressive strengths. Since most plastics exhibit some yielding or nonlinearity in their tensile S-S curve, there is a shift from triangular stress distribution toward rectangular distribution when the product is subject to bending. This behavior with plastics is similar to that when designing in steel and also for ultimate design strength in concrete. Shifts in the neutral axis resulting from differences in the yield strain and post-yield behavior in tension and compression usually affect the correlation between the modulus of rupture and the uniaxial strength results. The modulus of rupture reflects in part nonlinearities in stress distribution caused by plastification or viscoelastic nonlinearities in the cross-section.

Plastics such as short-fiber reinforced plastics with fairly linear stressstrain curves to failure usually display moduli of rupture values that are higher than the tensile strength obtained in uniaxial tests; wood behaves much the same way. Qualitatively, this can be explained from statistically considering flaws and fractures and the fracture energy available in flexural samples under a constant rate of deflection as compared to tensile samples under the same load conditions. These differences become less as the thickness of the bending specimen increases, as would be expected by examining statistical considerations.

The cantilever beam is another flexural test that is used to evaluate different plastics and structures such as beam designs. It is used in creep and fatigue testing and for conducting testing in different environments where the cantilever test specimen under load is exposed to chemicals, moisture, etc.

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