Modulus tangent

Fig. 8. Tangent modulus (T), tenacity ( ), and elongation (A ) as a function of take-up velocity for nylon-6. To convert N /tex to g/den (gpd), multiply by...

Fig. 11. Effect of spinning speed on properties of nylon-6 fibers, where (T) is the tangent modulus (° ), the tenacity and (A ), the elongation. To convert...

External-pressure failure of shells can result from overstress at one extreme or n om elastic instability at the other or at some intermediate loading. The code provides the solution for most shells by using a number of charts. One chart is used for cylinders where the shell diameter-to-thickness ratio and the length-to-diameter ratio are the variables. The rest of the charts depic t curves relating the geometry of cyhnders and spheres to allowable stress by cui ves which are determined from the modulus of elasticity, tangent modulus, and yield strength at temperatures for various materials or classes of materials. The text of this subsection explains how the allowable stress is determined from the charts for cylinders, spheres, and hemispherical, ellipsoidal, torispherical, and conical heads. 

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. 

From a plot of this data Initial tangent modulus =... 

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. 

Secant modulus The secant modulus is the ratio of stress to the corresponding strain at any specific point on the stress-strain curve. As shown in Fig. 2-2(a), the secant modulus is the slope of the line joining the origin and a selected point C on the stress-strain curve this could represent a vertical line at the usual 1 % strain. The secant modulus line is plotted from the initial tangent modulus and where it intersects the stress-strain curve. The plotted line location is also based on the angle used in relation to the initial tangent line from the ab-... 

For materials that deviate from the proportionality law even well below the elastic limit, the slope of the tangent to the stress-strain curve at a low stress level is taken as the tensile modulus. When the stress-strain curve displays no proportionality at any stress level, the secant modulus is employed instead of the tensile modulus (Fig. 2-2). The secant modulus is the ratio of stress to corresponding strain, usually at 1% strain or 85% from the initial tangent modulus. 

Tahir and Birley24 have considered the tangent modulus at intervals on the stress relaxation curve and compared it to the modulus of the initial loading curve to derive what they termed the Modulus Enhancement Factor, hence adding to the information which can be gained from a stress relaxation test. 

Fiber type 90 flexural strength (MPa) 90 flexural strain (%) Tangent modulus of elasticity (GPa)... 

At stress, erxrs, the crack density remains essentially constant. It has thus been assumed that there is no additional stress transfer between the fibers and the matrix. In this case, the tangent modulus is given by 14... 

In practice, the tangent modulus is usually found to be smaller than predicted by Eqn. (58). Two factors are involved changes in the sliding stress and fiber failure. At high fiber stresses, the Poisson condition of the fiber reduces the radial stress, arr. Consequently, whenever the sliding stress can be represented by Eqn. (1), r decreases as the stress increases. The associated tangent modulus at fixed crack spacing is1... 

As the UTS is approached, significant fiber failures occur, which further reduce the tangent modulus. The basic stress-strain relationship is,64... 

Tangent modulus of elastcitv, ASTM D790-71. c Area under flexural stress-deflection curve. 

Fig, 40. Tangent modulus O and failure stress A A as a function of temperature, open symbols... 

Chemical system and specimen (1) Foam density (g/cc) (2) Water content after soaking %) (3) Compressive strength (psi) (4) E, initial tangent modulus of elasticity (psi) (5) Curing time (days)... 

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