Secant

Material Percent neutralized Melt index at 190°C, dg/min Secant modulus, % extension, MPa Ultimate tensile strength, MPa Elongation at break, %... 

Method of False Position This variant is commenced by finding Xo andxi such th.atf(xo),f(x ) are of opposite signs. Then Oti = slope or secant hne joining [xo,f xo)] and [x, f x )] so that... 

In each following step Ot is the slope of the line joining [xk,fixi)] to the most recently determined point where/(x,) has the opposite sign from that of/(xi). This method is of first order. If one uses the most recently determined point (regardless of sign), the method is a secant method. 

The Wegstein method is a secant method applied to g x) — x — F x). Numerical Solution of Simultaneous Nonlinear Equations... 

Note When the elastic modulus is used, the elastic limit or proportional limit should be used with it in the formula. When the plastic modulus or Secant Modulus is used, it should be used with the corresponding yield strength. 

As indicated above, the stress-strain presentation of the data in isochronous curves is a format which is very familiar to engineers. Hence in design situations it is quite common to use these curves and obtain a secant modulus (see Section 1.4.1, Fig. 1.6) at an appropriate strain. Strictly speaking this will be different to the creep modulus or the relaxation modulus referred to above since the secant modulus relates to a situation where both stress and strain are changing. In practice the values are quite similar and as will be shown in the following sections, the values will coincide at equivalent values of strain and time. That is, a 2% secant modulus taken from a 1 year isochronous curve will be the same as a 1 year relaxation modulus taken from a 2% isometric curve. 

The only unknown on the right hand side is a value for modulus E. For the plastic this is time-dependent but a suitable value may be obtained by reference to the creep curves in Fig. 2.5. A section across these curves at the service life of 1 year gives the isochronous graph shown in Fig. 2.13. The maximum strain is recommended as 1.5% so a secant modulus may be taken at this value and is found to be 347 MN/m. This is then used in the above equation. 

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

Figure 4 Secant modulus versus percent conversion for Na and Ca salts of an ethylene/methacrylic acid ionomer.

SInaA Coiina A Tangent A Cotangent A Secant A Cosecant A... 

A chord of a circle (or sphere) is a line segment whose end points lie on the circle (or sphere). A line which intersects the circle (or sphere) in two points is a secant of the circle (or sphere). A diameter of a circle (or sphere) is a chord containing the center and a radius is a line segment from the center to a point on the circle (or sphere). 

Tangent-Secant Theorem— f given an angle with its vertex on a circle, formed by a secant ray and a tangent ray, then the measure of the angle is half the measure of the intercepted arc. 

Two-Secant Power Theorem— f given a circle C and an exterior point Q, let L be a secant line through Q, intersecting C at points R and S, and let be another secant line through Q, intersecting C at U and T, then... 

The functions versed sine, coversed sine, and exterior secant are defined as vers 0=1- cos 0 covers 0=1- sin 0 exsec 0 = sec 0 - 1... 

For the past century one successful approach is to plot a secant modulus that is at 1% strain or 0.85% of the initial tangent modulus and noting where they intersect the stress-strain curve (Fig. 2-2). However for many plastics, particularly the crystalline thermoplastics, this method is too restrictive. So in most practical applications the limiting strain is decided based on experience and/or in consultation between the designer and the plastic material manufacturer. Once the limiting strain is known, design methods based on its creep curves become rather straightforward (additional information to follow). 

Fig. 2-2 (a) Example of the modulus of elasticity determined on the initial straight portion of the stress-strain curve and secant modulus and (b) secant modulus for two different plastics that are 85% of the initial tangent modulus. 

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