Initial modulus tangent

When 1 wt% of CNTs were added to the rubber the stress level for the nanocomposite material increased from 0.2839 to 0.56413 MPa. At 10 wt% of CNTs the stress value obtained reached 2.55 MPa which is nine times that of pure NR." Nanocomposites of NR reinforced with SiC nanoparticles and single-walled carbon nanotubes were synthesized and processed using toluene as the solvent." Both types of nanocomposite showed an increase in initial modulus with increasing filler content, with a maximum initial modulus at 1.5% filler content. The tangent modulus at 200% strain further confirms the enhancement of reinforcement by the nanofillers at 1.5% content. 

If the initial modulus and the incremental modulus at any time later in the creep test are the result of secondary bond stretching alone, then one would expect the incremental compliance (or modulus) to be the same as the initial modulus. However, the initial strain and compliance almost certainly include viscoelastic deformation for very short relaxation times. Once the sample has been in creep for 3700 hours, one would expect all of the short relaxation time processes to have taken place. Thus, the incremental compliance might include only bond stretching, giving a smaller value for the incremental compliance than the iiutial compliance and a larger incremental or tangent modulus than iiutial modulus (i.e. at t = 0, but just after loading). 

Initial Modulus versus Incremental (or Tangent) Modulus... 

To determine the tangent modulus, a straight line is drawn tangent to the curve the slope of which is reported. Typically, the steepest part of the eurve is selected this is invariably the initial slope, in which case it is reported as the initial modulus. When the tangent is taken at a specific strain level, this value must also be reported. To determine the chord modulus, two points on the eurve are selected and the slope of the line connecting them is calculated. When the initial point chosen is 0% strain, the value is reported as the secant modulus of the upper level, e.g., 1% or 2% secant modulus. In all modulus calculations it is assumed that deformation is homogeneous at low strains and that deformation... 

Figure 26 Schematic illustration of four of the principal methods used to ealeulate elastic modulus, (a) Initial modulus (b) tangent (1%) (c) chord, (1-2%) (d) secant (1%).

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. 

The flexural modulus is the ratio, within the elastic limit, of stress to corresponding strain. It is calculated by drawing a tangent to the steepest initial straight-line portion of the load-deflection curve and using an appropriate formula. 

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. 

In practice the computerized data reduction procedure consists of the series of steps listed in Table I. For both multimodal resonant frequency data and constant frequency dynamic taken at several different frequencies, the correct value of the reference temperature (Tq) is that which reduces the scatter of the modulus and loss tangent data to fit a smooth curve. The initially assumed value of T0 is taken as the temperature corresponding to the loss tangent maximum or Tg. If this value of T0 does not give satisfactory superposition, the assumed value of T0 is adjusted to a slightly higher value and the procedure is repeated until a satisfactory superposition is achieved. Previous... 

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)... 

The standard test methods, calculate the tensile modulus by drawing a tangent to the initial linear part of the stress-strain curve and calculating the slope of the line. In cases where no clearly defined linear portion exists, the secant modulus should be determined. 

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