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# Multiaxial Temperature- and Time-Dependent Failure Model

- Monday, 12 February 2007

### This model should be applicable to a variety of materials.

A temperature- and time-dependent mathematical model predicts the conditions for failure of a material subjected to multiaxial stress. The model was initially applied to a filled epoxy below its glass-transition temperature, and is expected to be applicable to other materials, at least below their glass-transition temperatures. The model is justified simply by the fact that it closely approximates the experimentally observed failure behavior of this material: The multiaxiality of the model has been confirmed (see figure) and the model has been shown to be applicable at temperatures from —20 to 115 °F (–29 to 46 °C) and to predict tensile failures of constant-load and constant-load-rate specimens with failure times ranging from minutes to months.

The model is embodied in the following equation for the failure condition:

*AP*^{2}*J*_{2} + *BPI*_{1} = 1

where

*A*and*B*are parameters that define the shape of an ellipsoidal failure surface in multiaxial stress space;*P*is a scaling factor that accounts for the temperature and time dependences of the material;*J*_{2}is the second deviatoric stress invariant, given by

*I*_{1}is the first stress invariant, given by σ_{11}+ σ_{22}+ σ_{33};- the numerical subscripts denote Cartesian coordinate axes; and
- σ
_{ij}denotes the stress.

In the special case of constant *P*, this model is equivalent to a modified Drucker-Pager model, and to the Tsai-Wu failure model that is traditionally used in evaluating composite materials.

*A*(1.0 for this material) and

*B*(0.31754 for this material).

The value of the scale factor *P* is simply whatever value is needed to make a given failure surface pass through a known failure point for a given temperature and failure time. Hence, for example, once *A* and *B* are known, *P* as a function of time and temperature can be determined simply by solving the basic model equation for *P* and then inserting stress values from tensile or shear tests that involve known failure times and temperatures.

*This work was done by David Richardson, Michael McLennan, Gregory Anderson, David Macon, and Alicia Batista-Rodriguez of Thiokol Propulsion Corp. for Marshall Space Flight Center. For further information, please contact the company at (435) 863-3511. MFS-31750.*