Integral and Constitutive Equations

Appendix E Integral and Constitutive Equations Equation (E.13) was thus written as ... 

Combining the spatial Darcy and constitutive equations and integrating across the cake with limits of (0, L) on X and (0, Ap ) on p yields ... 

Two general methods for the development of single integral nonlinear constitutive equations that have been used are the rational (functional) thermodynamic approach and the state variable approach (or irreversible thermodynamic approach), each of which are described in a well-documented survey by K. Hutter (1977). In rational thermodynamics, the free energy is represented as a function of strain (or stress), temperature, etc, and then constitutive equations are formed by taking appropriate derivatives of the free energy. The state variable approach includes certain internal variables in order to represent the internal state of a material. Constitutive equations which describe the evolution of the internal state variables are included as a part of the theory. Onsager introduced the concept of internal variables in thermodynamics and this formalism was later used... 

The Rouse model gives the following integral-type constitutive equation (Doi and Edwards 1986)... 

Substituting the constitutive equation directly into the compatibility equation and interchanging the roles of integration and differentiation equation (5.13) becomes... 

Some of the integral or differential constitutive equations presented in this and the previous section have an exact equivalent in the other group. There are, however, equations in both groups that have no equivalent in the other category. 

The integrals in Equation (3.32) are found using a quadrature over the element domain The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection terms in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding)... 

The design of smart materials and adaptive stmctures has required the development of constitutive equations that describe the temperature, stress, strain, and percentage of martensite volume transformation of a shape-memory alloy. These equations can be integrated with similar constitutive equations for composite materials to make possible the quantitative design of stmctures having embedded sensors and actuators for vibration control. The constitutive equations for one-dimensional systems as well as a three-dimensional representation have been developed (7). 

If a motion is specified with satisfies the continuity condition, the velocity, strain, and density at each material particle are determined at each time t throughout the motion. Given the constitutive functions (e, k), c(e, k), b( , k), and a s,k) with suitable initial conditions, the constitutive equations (5.1), (5.4), and (5.11) may be integrated along the strain history of each material particle to determine its stress history. If the density, velocity, and stress histories are substituted into (5.32), the history of the body force at each particle may be calculated, which is required to sustain the motion. Any such motion is termed an admissible motion, although all admissible motions may not be attainable in practice. 

This full set of self-consistent equations is clearly very difficult to solve, even numerically. However, good approximations of closed integral type have been proposed. These essentially ignore the s-dependence of the survival and orientation functions, which makes them a physically appeaUng approach in the case of wormlike surfactants [71,72]. For ordinary monodisperse polymers the following approximate integral constitutive equation results ... 

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