Lebesgue norms

Non-linear constitutive equations are developed for highly filled polymeric materials. These materials typically exhibit an irreversible stress softening called the "Mullins Effect." The development stems from attempting to mathematically model the failing microstructure of these composite materials in terms of a linear cumulative damage model. It is demonstrated that p order Lebesgue norms of the deformation history can be used to describe the state of damage in these materials and can also be used in the constitutive equations to characterize their time dependent response to strain distrubances. This method of analysis produces time dependent constitutive equations, yet they need not contain any internal viscosity contributions. This theory is applied to experimental data and shown to yield accurate stress predictions for a variety of strain inputs. Included in the development are analysis methods for proportional stress boundary valued problems for special cases of the non-linear constitutive equation. 

To accomplish these goals systematically, a brief discussion of constitutive linearity and non-linearity is introduced in Section 2. In Section 3 the "stress softening" or "Mullins Effect" [2-8] is analyzed from a simple mechanical failure model of the composite material s microstructure and is seen to contain this type of nonlinear behavior. Also the p order Lebesgue norms [13,14] of the strain history are presented as being excellent memory measures of the strain history to use in the constitutive equations. In Section 4 the development is extended to include the general three-dimensional constitutive equation for isotropic materials. 

Mathematically the quantity is called the p order Lebesgue norm [13,14] L. The Lp norm has properties that are worth noting and these are listed below. 

---