Internal Variable Approach

Abstract A fully coupled model of hygro-thermo-chemo-mechanical phenomena in concrete is presented. A mechanistic approach has been used to obtain the governing equations, by means of the hybrid mixture theory. The final equations are written in terms of the chosen primary and internal variables. The model takes into account coupling between hygral, thermal, chemical phenomena (hydration or dehydration), and material deformations, as well as changes of concrete properties, caused by these processes, e.g. porosity, permeability, stress-strain relation, etc. 

The behavior of complex dynamical systems can be analyzed and represented in a number of ways. Figure 1 represents one such approach, a constraint-response plot. A constraint, in this case [A], is any variable which the experimenter can control directly. A response, [X]ss in this case, is a measurable property of the system which depends upon the constraint values. The constraints are the external variables, e.g., the temperature of the bath surrounding the reactor or the reservoir concentrations, while the responses are the internal variables, e.g., the temperature or concentration of species in the reactor. The phase trajectory diagram of Fig. 4 is one type of response-response plot. Obviously, in a complex system, there will be several constraints and responses subject to independent (or coupled) variation. 

Next to metals, probably the synthetic polymer-based composites have been modeled most by hierarchical multiscale methods. Different multiscale formulations have been approached top-down internal state variable approaches, self-consistent (or homogenization) theories, and nanoscale quantum-molecular scale methods. 

The pressure p includes both the partial pressure of the gas of Brownian particles n(N +1 )T and the partial pressure of the carrier monomer liquid. We shall assume that the viscosity of the monomer liquid can be neglected. The variables xt k in equation (9.19) characterise the mean size and shape of the macromolecular coils in a deformed system. The other variables ut k are associated mainly with orientation of small rigid parts of macromolecules (Kuhn segments). As a consequence of the mesoscopic approach, the stress tensor (9.19) of a system is determined as a sum of the contributions of all the macromolecules, which in this case can be expressed by simple multiplication by the number of macromolecules n. The macroscopic internal variables x -k and u"k can be found as solutions of relaxation equations which have been established in Chapter 7. However, there are two distinctive cases, which have to be considered separately. 

Extended nonequilibrium thermodynamics theory is often applied to flowing polymer solutions. This theory includes relevant fluxes and additional independent variables in describing the flowing polymer solutions. Other contemporary thermodynamic approaches for this problem are GENERIC formalism, matrix method, and internal variables (Jou and Casas-Vazquez, 2001), which are summarized in the following sections. 

Classifying variables into fast or slow is a typical approach in chemical kinetics to apply the method of (quasi)stationary concentrations, which allows the initial set of differential equations to be largely reduced. In the chemically reactive systems near thermodynamic equihbrium, this means that the subsystem of the intermediates reaches (owing to quickly changing variables) the stationary state with the minimal rate of entropy production (the Rayleigh Onsager functional). In other words, the subsys tern of the intermediates becomes here a subsystem of internal variables. 

In the DARC/PELCO Structural method an internal variable pertaining to the ordered generation of an exhaustive topology is used. We show to what extent this variable combines some qualities of other structural variables, avoids some of their drawbacks and permits an approach to prediction reliability ... 

For flows where compressibility effects in a gas are important the use of the particle mass as internal coordinate may be advantageous because this quantity is conserved under pressure changes [11]. In this approach it is assumed that all the relevant internal variables can be derived from the particle mass, so the particle number distribution is described by the particle mass, position and time. Under these conditions, the dispersed phase flow fields are characterized by a single distribution function /(m, r,t) such that f m,r,t)drdm is the number of particles with mass between m and m+dm, at time t and within dr of position r. Notice that the use of particle diameter and particle mass as inner coordinates give rise to equivalent population balance formulations in the case of describing incompressible fluids. 

By using a very similar approach to the one outlined above for the PBE, it is possible to derive a GPBE for an NDF that includes particle velocity as an internal variable. We will denote this general NDF as n(t,x,, ) (i.e. without subscripts on n). The simplest GPBE (i.e. velocity without other internal coordinates) is known as the Boltzmann kinetic equation and was first derived in the context of gas theory (Chapman Cowling, 1961). The final form of the GPBE is... 

The difference in approach is self-evident. In the mechanistically based model, key internal and external variables are identified. Their variabilities are readily incorporated into the model to assess the overall variabihty in response. The contribution of each of the random variables on the variability in response may be readily assessed. Given the explicit functional dependence, when duly validated, it can be used to predict response beyond the range of the experimental data. The experi-entially based statistical model, on the other hand, represents a statistical fit to the data in which the key internal variables could not be identified. As such, it is incapable of capturing the functional dependence on these variables, and its usefulness is limited to the range of the experimental data. Because experimental (including measurement) errors are lumped into estimates of the fitting parameters and their variability, the quality of the subsequent reliabihty analyses may be overly conservative, or uncertain. A more detailed discussion of these approaches may be found in [7]. 

A fictive temperature 7 based approach firstly introduced by Tool [47] has been proved to be extremely successful in supplying the information about the free volume or the structure in the formulation of the free energy density. The fictive temperature 7/ is an internal variable to characterize the actual thermodynamic state during the glass transition, defined as the temperature at which the temporary nonequilibrium stmcture at T is in equilibrium [20]. It was assumed that the rate change of the fictive temperature is proportional to its deviation from the actual temperature and the proportionality factor depends on both T and Tf [48], as indicated in the evolution equation [47] ... 

The next step of importance is the description of inhomogeneous systems in terms of local equilibria. This process involves the division of the system to be described into subsystems, as discussed in Sect. 2.2.1, and shown in the upper half of Fig. 2.80. This is to be followed by the description of the system as a function of time as an additional variable. Following the system as a function of time allows the study of the kinetics of the processes seen when approaching equilibrium. Finally, it may be necessary to identify additional, internal variables to describe the nonequiUbrium states as a function of time. The kinetics of polymers may be sufficiently slow to decouple consecutive steps. For very slow responses, it may lead to arrested equilibria, such as seen in materials below their glass transitions as described in Sect. 2.4.4. The following Section makes use of the just summarized concepts in an attempt to achieve a depiction of the nonequilibrium state of macromolecules. 

The same methodology can be apvplied to the generic internal state variables approach to get the corresponding strain increment and tangent matrix. 

The two constitutive laws presented in this work were obtained considering an isotropic material at a reference state, i.e., a state without any influence of external parameters such as temperature. The constitutive parameters evolutions are instead taking into account within the numerical simulation and thus the constitutive laws have to be discretized using an Euler scheme. A case study using the strain based internal state variables approach has been presented. This study shows how creep/relaxation could influence the results of an industrial problem such as the "baking" of a carbonaceous ramming paste. 

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... 

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