Four-Point Bending Test of Determining Stress-Strain Curves Asymmetric between Tension and Compression

Abstract

Keywords inverse problem · plasticity · four-point bending A method of determining both uniaxial tension and compression stressstrain curves from the result of a single four-point bending test was demonstrated. Stress-strain curves of magnesium showing tension-compression asymmetry due to twinning deformation and those of an S45C steel due to the Bauschinger effect were calculated. The Mayville-Finnie equation was modified slightly for this calculation. The calculation is sensitive to small change in the derivative of bending curve, revealing an aspect of inverse problem.An inverse problem in elasto-plastic bending of a material may be to determine uniaxial stress-strain (s-s) curve of the material. This classical subject was discussed by Nadai [1]. Later, Mayville and Finnie studied this problem, and derived a formula [2]. Despite the potential of the formula and also the usefulness of bending test, very few application has been found. Let us consider four-point bending of a prismatic bar (Fig.1). The bar is simply supported at points A and B, and a load P/2 is subjected at points C and D equally. Pure bending with the radius of curvature ρ occurs between C and D, where a constant bending moment of M = P d/2 is subjected. The bending strains of the outermost surfaces ϵ 1 (compression) and ϵ 2 (tension) are measured with two pieces of strain gauge. Accordingly, two curves of bending load vs. bending strain relation (P -ϵ curves) are obtained: ϵ 1 = ϵ 1 (P ) and ϵ 2 = ϵ 2 (P ). The bar has a constant rectangular cross section, where a local y coordinate is taken as in Fig.2. Yield stress in compression is Y 1 and in tension Y 2 . A neutral axis NN is located at y = y 0 . When s-s curves are asymmetric between tension and compression, the neutral axis is away from the centroid axis at y = h/2 while bending.A P -ϵ curve can be derived readily from the exact shape of s-s curve, if it is known. This is a forward problem in bending. For example, a material with an ideal elastic-perfect plastic s-s behavior illustrated in Fig.3(a) shows P-ϵ curves of Fig.3(b). The calculation was carried out by assuming that the modulus of elasticity E was 100 GPa, the yield stresses Y 1 in compression, 100MPa, Y 2 in tension, 200 MPa, and the dimensions, b=5 mm, h=1 mm, d=10 mm. Properties of P-ϵ curves listed below are common to any elasto-plastic(i) The linear relationship in elastic bending is given by(ii) When the compression side starts yielding, tension and compression curves branch off from the linear relationship at a point where the load is P 1 and the strain isTitle Suppressed Due to Excessive Length 3 (iii) When the tension side starts yielding, the two curves change the slopes at a point where the load is P 2 and the strain isThe inverse problem is to determine s-s curves from the result of bending.Without solving the problem, we can determine the values of E, Y 1 and Y 2 from P-ϵ curves. Let us start from the P-ϵ curves of Fig.3(b). The slope of linear relationship determines E as 100GPa from Eq.(1). The first and second ...

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