Constitutive equation integral form

This latter relation constitutes the integral form of Poisson s equation of electrostatics. 

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... 

Goddard (27) expressed the notion of the simple fluid constitutive equation in a co-rotational integral series. The integral series expansion had been used in the co-deformational frame by Green and Rivlin (28) and Coleman and Noll (29). The co-rotational expansion takes the form ... 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

One may use the linear viscoelastic data as a pure rheological characterization, and relate the viscoelastic parameters to some processing or final properties of the material inder study. Furthermore, linear viscoelasticity and nonlinear viscoelasticity are not different fields that would be disconnected in most cases, a linear viscoelastic function (relaxation fimction, memory function or distribution of relaxation times) is used as the kernel of non linear constitutive equations, either of the differential or integral form. That means that if we could define a general nonlinear constitutive equation that would work for all flexible chains, the knowledge of a single linear viscoelastic function would lead to all rheological properties. 

Additional complexity can he brought to the constitutive equation in its integral form. Indeed, the idea of rubber elasticity that is inherent to the Lodge model has been generalized by Kayes, Bernstein, Kearsley and Zapas [20-23] in a large class of constitutive equations. In a perfect body, the strain energy W may be linked to strain and stress by ... 

These constitutive equations differ in their mathematical form the Wagner equation is an integral equation whereas the Phan Thien Tanner model is a differential one. 

It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the Phan-Thien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory munerical results were also obtained with multi-mode integral constitutive equations using a spectnun of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [19]. 

The integral form is not the only possible form in which stress-strain relationships can be written. The constitutive equation for an isotropic material whose response is sensitive to the derivatives of stress and strain can be written as (5)... 

However expansions of the type of Eq. (20.4) would not be able to explain such phenomena as stress relaxation ( 10) or the overshoot phenomenon in stress growth ( 11). It might prove fruitful to explore alternate methods of obtaining constitutive equations which would not be of the form of Eq. (20.4), i.e. stress = function of y and its derivatives. After all, time derivatives of stress could appear [as in Eq. (4.23)], or stress could be given as an integral over the strain history. This question of the connection between dumbbell models and continuum mechanics has been studied extensively in a series of papers by Giesekus (3/). 

Just as Eqs. (6.58) and (6.59) were obtained in Chapter 6, the integral forms of Eq. (7.53) can be obtained. Then, with Eq. (7.49), the integral form of the constitutive equation for the Rouse chain model is given by (with u replaced by p)... 

Equation 1 is the continuum equilibrium condition, Eq. 2 are the constitutive equations, which specify the material behavior, and Eq. 3 are the kinematics, the displacement-deformation conditions. The forming machine defines the boundary conditions for this set of partial differential equations. It can be seen without going into details that those equations are essentially grade two in the displacements, which means that boundary conditions are in the displacements or in the first derivative of displacements. Eirst derivatives of displacements are in all constitutive equations connected to stress at least due to elasticity, which is common for all materials. Erom these two types of press machines can be derived, namely, path-driven machines, where boundary conditions for the displacements are prescribed, and force-driven machines, where stress boundary conditions are prescribed, which are integrated to the press force. Erom this it follows also that despite the possibilities of servo presses to operate under different modes of the drive, path, force, or energy, nothing really new is added, because the drive can only introduce either boundary condition at a time. 

The stress constitutive equation can also be formulated as an integral over the history of the deformation. The most common form used for simulations is... 

Other Constitutive Modei Descriptions. The above work describes a relatively simple way to think of nonlinear viscoelasticity, viz, as a sort of time-dependent elasticity. In solid polymers, it is important to consider compressibility issues that do not exist for the viscoelastic fluids discussed earlier. In this penultimate section of the article, other approaches to nonlinear viscoelasticity are discussed, hopefully not abandoning all simplicity. The development of nonlinear viscoelastic constitutive equations is a very sophisticated field that we will not even attempt to survey completely. One reason is that the most general constitutive equations that are of the multiple integral forms are cumbersome to use in practical applications. Also, the experimental task required to obtain the material parameters for the general constitutive models is fairly daunting. In addition, computationally, these can be difficult to handle, or are very CPU-time intensive. In the next sections, a class of single-integral nonlinear constitutive laws that are referred to as reduced time or material clock-type models is disscused. Where there has been some evaluation of the models, these are examined as well. 

The applicability of the CONNFFESSIT (Calculation Of Non-Newtonian Flows Finite Elements and Stochastic Simulation Technique) in its present form is limited to the solution of fluid mechanical problems of incompressible fluids under isothermal conditions. The method is based on a combination of traditional continuum-mechanical schemes for the integration of the mass- and momentum-conservation equations and a simulational approach to the constitutive equation. 

Equation (26) is what is known as a rate-type constitutive equation, and it gives the extra stress in implicit form. Integral equations, on the other hand, are explicit in the stress, and the simplest of these can be derived based on the Boltzmann superposition principle. According to this principle, the stress in a material at any time can be obtained by adding stress tbatjndivjduallv result from earh Vio... 

There are numerous other constitutive equations of both differential and integral type for polymer melts, and some do a better job of matching data from a variety of experiments than does the PTT equation. The overall structure of the differential equations is usually of the form employed here The total stress is a sum of individual stress modes, each associated with one term in the linear viscoelastic spectrum, and there is an invariant derivative similar in structure to the one in the PTT equation, but with different quadratic nonlinearities in t and Vv. The Giesekus model, for example, which is also widely used, has the following form ... 

Beyond the few categories discussed above, it is difficult to work with the ISF in its full form. However, by either assuming slow flows or small deformations, it is possible to obtain approximate expressions in the form of expansions. When the ISF is expanded for slow flows one obtains the order fluids we discussed earlier. When we expand for small deformations, the ISF yields integral relations which account for the fading memory. At the 0th order we again have the hydrostatic pressure, but now at the 1st order we get the constitutive equation for linear viscoelasticity, viz. 

Derail et al. [10,11] used a non-linear integral constitutive equation of the KBKZ [20] type, which, for uniaxial extension, has the form... 

Wagner [14,15] has provided a method for the prediction of normal stress difference from shear viscosity using a strain-dependent single-integral constitutive equation of the (Berstein-Kearsley-Zapas) 6KZ type. In an sq>propriately modified form, it can be written [13] as follows ... 

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