# Material Shape Parameter (n)

In order to calculate the plasticity reduction factor given in Michael C.Y. Niu's book a constant (n) is being multiplied which is called the material shape parameter, does anyone know the meaning of material shape parameter. Thank you in advance!

Edit: The plasticity reduction factor formula is an empirical formula given in Michael Niu's Book, Airframe Stress Analysis, Chapter 12, Pg - 456

• Welcome to Aviation.SE. Does this question have any particular relation to aviation , as opposed to general material sciences and mechanical engineering?
– Ralph J
Commented Dec 8, 2023 at 18:36
• @RalphJ That is a good question, I would assume it would be related to aeronautical engineering which comes under the umbrella of aviation. This question in particular was meant to have cutouts in the fuselage skin panels. Commented Dec 11, 2023 at 14:07

Aircraft make use of many different materials, especially metallic alloys. In contrast to many other applications, weight is a critical parameter, and in order to reduce aircraft weight, accurate material models are needed to predict failure. Since aircraft consist mostly of thin-walled structures, besides the pure material strength, stability failure modes such as buckling or crippling are often critical. For stability calculations, the stiffness of the material is critical. One of the plasticity models most commonly used in aviation is Ramberg-Osgood, as given in the following equation:

$$\frac{E\cdot \epsilon}{\sigma_{0.7}} = \frac{\sigma}{\sigma_{0.7}} + \left(\frac{\sigma}{\sigma_{0.7}}\right)^n$$

E is the modulus of elasticity, $$\epsilon$$ the strain, $$\sigma$$ the stress, $$\sigma_{0.7}$$ the secant yield stress (where a line starting at the origin and with a 0.7E slope crosses the stress-strain curve of the material), and $$n$$ the material shape parameter.

To calculate the material shape parameter, usually the formula

$$n = 1 + \frac{\ln({17/7})}{\ln{(\sigma_{0.7}/\sigma_{0.85})}}$$

is used. Notice also that a single material can have different shape parameters $$n$$ depending on the testing conditions (strain rate, temperature,...)

In this figure you can see the effect of the parameter n on the stress-strain curve (the figure has been created with E=1 and $$\sigma_{0.7}=1$$). The higher n is, the closer is the behavior of the material to a perfect plastic material. With a low n, the stress-strain curve is closer to an ideal elastic material.

While this question could be treated as "material science and mechanical engineering", as mentioned in one of the comments, the development of these material models has been driven mainly by the needs of the aviation and space industries, and the primary users of these models are aerospace engineers.