Stress-strain polynomial

Figure 11-134. The stress-strain polynomial for Du Font s Zytel 101 nylon.

Marks, W. F. Stress/Strain Polynomials Their Use Could Result in More Accurate Design Calculations. Du Pont Engineering Magazine, pp. 14-15, winter 1986. 

These properties were determined at an extension rate of 5 inches/ minute with force applied parallel to the rubber grain direction. Youngs modulus at zero strain was found by taking the first derivative of polynomial equations fitted to the low extension portion of the stress-strain curves. 

The experimental data from the tensile stress-strain tests that generated the curve for the Zytel 101 nylon resin in Figure 11-134 are accurately modeled by the computer in the following polynomial ... 

Problem. What is the stress at this elongation The tensile stress-strain curve for the nylon just described according to the conditions given is shown in Figure 11-135. The polynomial determined by the computer for this curve is... 

Unfortunately, this high a degree of accuracy is not attainable in all stress-strain problems. Because polynomials are based on curves generated by using only one stress direction, they do not apply in problems involving two or more directions at the same time. 

Neither the r9 component of the rate-of-strain tensor nor the simple velocity gradient dvg/dr vanishes at the gas-liquid interface. This is expected for inviscid flow because viscous stress is not considered, even in the presence of a signiflcant velocity gradient. Once again, the leading term in the polynomial expansion for vg, given by (11-126), is used to approximate the tangential velocity component for flow of an incompressible fluid adjacent to a zero-shear interface ... 

An alternative isotropic strain energy density function which can predict the appropriate type of stress stiffening for blood vessels is an exponential where the arguments is a polynomial of the strain invariants. 

The material will have a given strength expressed as stress or strain, beyond which it fails. In order to postulate the failure, it is necessary to have a failure criterion with an associate theory to be able to effect a satisfactory design. Such theories include maximum stress, maximum strain, Tsai-Hill (based on deviatoric strain energy theory) and Tsai-Wu (based on interactive polynomial theory). The Tsai-Wu theory is the most commonly used. 

By not using the polynomial but published data the following approach should be adapted. Under the same conditions, the flexural modulus of the Zytel material, which differs from its tensile modulus, becomes 5,068 MPa (735,000 psi). Tlie flexural modulus E then is equal to the stress S divided by the strain e, or E = S/e. The stress is therefore... 

Another way to use polynomials, which requires trial-and-error computer entry, is to find the strain when the stress is either known or has been specified. As an example, in a final evaluation of the problem above, the designer may worry that the assembly conditions may not be ideal f6r field retrofits. The decision is then made to allow for a stress safety factor of two. Thus, the design stress becomes 15,247/2 = 7,624 psi. A sequence of tries then follows where... 

Using this approach instead of the polynomial, the result would be off by 0.074 percent of the strain (1.114 - 1.040). Even though the stress is not particularly high, this is a significant error of 7 percent. 

Whitney and Drzal [87] presented an analytical model to predict the stresses in a system consisting of a broken fiber surrounded by an unbounded matrix. The model (Fig. 8) is based on the superposition of the solutions to two axisymmetric problems, an exact far-field solution and an approximate transient solution. The approximate solution is based on the knowledge of the basic stress distribution near the end of the broken fiber, represented by a decaying exponential function multiplied by a polynomial. Equilibrium equations and the boundary conditions of classical theory of elasticity are exactly satisfied throughout the fiber and matrix, while compatibility of displacements is only partially satisfied. The far-field solution away from the broken fiber end satisfies all the equations of elasticity. The model also includes the effects of expansional, hygrothermal strains and considers orthotropic fibers of the transversely isotropic class. 

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