Internal state variable theory

In the next sections, the multiscale modeling methods are presented from the different disciplines perspectives. Clearly one could argue that overlaps occur, but the idea here is to present the multiscale methods from the paradigm from which they started. For example, the solid mechanics internal state variable theory includes mathematics, materials science, and numerical methods. However, it clearly started from a solid mechanics perspective and the starting point for mathematics, materials science, and numerical methods has led to other different multiscale methods. 

In terms of cyclic plasticity, Shenoy et al. [139, 140], Wang et al. [137], and McDowell [141] performed hierarchical multiscale modeling of Ni-based superalloys employing internal state variable theory. Fan et al. [142] performed a hierarchical multiscale modeling strategy for three length scales. 

In order to start the multiscale modeling, internal state variables were adopted to reflect void/crack nucleation, void growth, and void coalescence from the casting microstructural features (porosity and particles) under different temperatures, strain rates, and deformation paths [115, 116, 221, 283]. Furthermore, internal state variables were used to reflect the dislocation density evolution that affects the work hardening rate and, thus, stress state under different temperatures and strain rates [25, 283-285]. In order to determine the pertinent effects of the microstructural features to be admitted into the internal state variable theory, several different length scale analyses were performed. Once the pertinent microstructural features were determined and included in the macroscale internal state variable model, notch tests [216, 286] and control arm tests were performed to validate the model s precision. After the validation process, optimization studies were performed to reduce the weight of the control arm [287-289]. 

Now that the top-down internal state variable theory was established, the bottom-up simulations and experiments were required. At the atomic scale (nanometers), simulations were performed using Modified Embedded Atom Method, (MEAM) Baskes [176], potentials based upon interfacial atomistics of Baskes et al. [177] to determine the conditions when silicon fracture would occur versus silicon-interface debonding [156]. Atomistic simulations showed that a material with a pristine interface would incur interface debonding before silicon fracture. However, if a sufficient number of defects were present within the silicon, it would fracture before the interface would debond. Microstructural analysis of larger scale interrupted strain tests under tension revealed that both silicon fracture and debonding of the silicon-aluminum interface in the eutectic region would occur [290, 291]. 

M.F. Horstemeyer et al A multiscale analysis of fixed-end simple shear using molecular dynamics, crystal plasticity, and a macroscopic internal state variable theory. Modell. Sim. Mat. Sci. Eng. 11, 265-286 (2003)... 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Specific applications of the theory are not considered in this chapter. Only one example, that of small deformation classical plasticity, is worked out in Section 5.3. The set of internal state variables k is taken to be comprised of... 

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

The normality conditions (5.56) and (5.57) have essentially the same forms as those derived by Casey and Naghdi [1], [2], [3], but the interpretation is very different. In the present theory, it is clear that the inelastic strain rate e is always normal to the elastic limit surface in stress space. When applied to plasticity, e is the plastic strain rate, which may now be denoted e", and this is always normal to the elastic limit surface, which may now be called the yield surface. Naghdi et al. by contrast, took the internal state variables k to be comprised of the plastic strain e and a scalar hardening parameter k. In their theory, consequently, the plastic strain rate e , being contained in k in (5.57), is not itself normal to the yield surface. This confusion produces quite different results. 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

In Section 5.2 the set of internal state variables k was introduced. In the referential theory, a similar set of referential internal state variables K will be introduced in the same way without further physical identification at this stage. It will merely be assumed that each member of the set K is invariant under the coordinate transformation (A.50) representing a rigid rotation and translation of the coordinate frame. 

One effective hierarchical method for multiscale bridging is the use of thermodynamically constrained internal state variables (IS Vs) that can be physically based upon microstructure-property relations. It is a top-down approach, meaning the IS Vs exist at the macroscale but reach down to various subscales to receive pertinent information. The ISV theory owes much of its development to the state variable thermodynamics constructed by Helmholtz [4] and Maxwell [5]. The notion of ISV was introduced into thermodynamics by Onsager [6, 7] and was applied to continuum mechanics by Eckart [8, 9]. 

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. 

Kratochvfl, J., DUlon, O.W. Thermodynamics of elastic-plastic materials as a theory with internal state variables. J. Appl. Phys. 40(8), 3207-3218 (1969)... 

Lion, A., Liebl, C., et al. (2010). "Representation of the glass-transition in mechanical and thermal properties of glass-forming materials A three-dimensional theory based on thermodynamics with internal state variables." Journal of the Mechanics and Physics of Solids 58(9) 1338-1360. 

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... 

Let us now briefly summarize the essentials of MC-SCF theory. It is helpful to take an entirely different approach to the usual formulation of closed shell SCF theory. In fact, it proves to be useful to think of the MC-SCF process in a similar way to geometry optimization. Thus we shall view the orbital and Cl coefficient variables that occur in MC-SCF in the same way as the internal geometrical variables in a geometry optimization. Technically in order to do this we must assume that we have an orthogonal set of starting orbitals o and an orthogonal set of Cl vectors K>. The MC-SCF Cl expansion (for state K) is written as... 

The radius thus calculated from the theory of Smith and Symons does not correspond to any known property of halide ions. However, when the acceptable physical model of Franck and Platzman is combined with the concept of a variable radius, as proposed by Smith and Symons, both absolute value and environmental effects can be accounted for. This was done in the theory of Stein and Treinin (18, 19, 47), using an improved energetic cycle to obtain absolute values of r, the spectroscopically effective radius of the cavity containing the X ion. These values were then found to correspond to the known partial ionic radii in solution, as did values of dr/dT to values obtained from other experiments. The specific effects of temperature, solvents, and added salts could be used to differentiate between internal and such CTTS transitions where the electron interacts in the excited state strongly with the medium. These spectroscopic aspects of the theory were examined later in detail and compared with experiment by Treinin and his co-workers (3, 4, 32, 33, 42,48). 

Internal variables are introduced in relation (8.16) formally. However, the success of the theory depends on the proper choice of the internal variables for the considered case. Consideration of models usually helps to recognise which quantities describe the deviation of the system from its equilibrium state and which can be used as internal variables. A set of internal variables were identified in Chapter 2 for dilute polymer solutions and in Chapter 7 for polymer melts.1... 

One of the prominent features of polymeric liquids is the property to recover partially the pre-deformation state. Such behaviour is analogous to a rubber band snapping back when released after stretching. This is a consequence of the relaxation of macromolecular coils in the system every deformed macro-molecular coil tends to recover its pre-deformed equilibrium form. In the considered theory, the form and dimensions of the deformed macromolecular coil are connected with the internal variables which have to be considered when the tensor of recoverable strain is to be calculated. Further on, we shall consider the simplest case, when the form and dimensions of macromolecular coils are determined by the only internal tensor. In this case, the behaviour of the polymer liquid is considered to describe by one of the constitutive equations (9.48)-(9.49) or (9.58). 

Control is the theory that deals with the dynamic behavior of systems with inputs and outputs. In production engineering, control theory has been heavily applied in machines - especially in computerized numerical control (CNC) machine tools. In the basic principle, the external input to the system is called the reference. In production, it is usually selected as the desired position to be followed. The objective of control is to manipulate one or more variables of the system over a certain time such that the desired states, e.g., the outputs of the system, can follow the external reference input (trajectory). In CNC machine tools, the internal variable is the motor torque/force that can be manipulated so that the actual position can follow the external reference. 

The theories with internal variables provide detailed description of microstructure by introducing additional variables relevant to the microstructure of the system, and enlarge the domain of application of thermodynamics. Introduction of internal variables may lead a possibility to include the influence of microstructural effects into the description of a macroscopic phenomenon without changing of space and time scales. There are two types of internal variables internal degrees of freedom and internal variables of state. Internal variables of state must have no inertia and they produce no external work. Internal degrees of freedom, on the other hand, have both inertia and flux. 

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