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# Probabilistic Multi-Factor Interaction Model for Complex Material Behavior

- Created: Monday, 01 March 2010

### Complex material behavior is represented by a single equation of product form.

Complex material behavior is represented by a single equation of product form to account for interaction among the various factors. The factors are selected by the physics of the problem and the environment that the model is to represent. For example, different factors will be required for each to represent temperature, moisture, erosion, corrosion, etc. It is important that the equation represent the physics of the behavior in its entirety accurately.

The Multi-Factor Interaction Model (MFIM) is used to evaluate the divot weight (foam weight ejected) from the external launch tanks. The multi-factor has sufficient degrees of freedom to evaluate a large number of factors that may contribute to the divot ejection. It also accommodates all interactions by its product form. Each factor has an exponent that satisfies only two points — the initial and final points. The exponent describes a monotonic path from the initial condition to the final. The exponent values are selected so that the described path makes sense in the absence of experimental data. In the present investigation, the data used were obtained by testing simulated specimens in launching conditions. Results show that the MFIM is an effective method of describing the divot weight ejected under the conditions investigated.

The problem lies in how to represent the
divot weight with a single equation. A unique
solution to this problem is a multi-factor
equation of product form. Each factor is of
the following form (1 – *x _{i}*/

*x*)

_{f}*, where*

^{e}i*x*

_{i}is the initial value, usually at ambient conditions,

*x*the final value, and ei the exponent that makes the curve represented unimodal that meets the initial and final values. The exponents are either evaluated by test data or by technical judgment. A minor disadvantage may be the selection of exponents in the absence of any empirical data. This form has been used successfully in describing the foam ejected in simulated space environmental conditions. Seven factors were required to represent the ejected foam. The exponents were evaluated by least squares method from experimental data.

_{f}The equation is used and it can represent multiple factors in other problems as well; for example, evaluation of fatigue life, creep life, fracture toughness, and structural fracture, as well as optimization functions. The software is rather simplistic. Required inputs are initial value, final value, and an exponent for each factor. The number of factors is open-ended. The value is updated as each factor is evaluated. If a factor goes to zero, the previous value is used in the evaluation.

*This work was done by Galib H. Abumeri
and Christos C. Chamis of Glenn Research
Center.
*

*Inquiries concerning rights for the commercial use of this invention should be addressed
to NASA Glenn Research Center, Innovative
Partnerships Office, Attn: Steve Fedor, Mail
Stop 4–8, 21000 Brookpark Road, Cleveland,
Ohio 44135. Refer to LEW-18450-1.*