Linear constitutive equation

Whilst obtaining this is the ultimate goal for many rheologists, in practice it is not possible to develop such an expression. However, our mechanical analogues do allow us to develop linear constitutive equations which allow us to relate the phenomena of linear viscoelastic measurements. For a spring the relationship is straightforward. When any form of shear strain is placed on the sample the shear stress responds instantly and is proportional to the strain. The constant of proportionality is the shear modulus... 

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. 

By adopting similar procedures used for linear equations by Biot (1954) and in consideration of generalized forces corresponding to the inner variables being zero, we have the non-linear constitutive equation of thermo-visco-elasticity (see Liu et al, 2001 for details) written for i =l, 2, 3... 

Without going into details we note also that sufficient conditions of minima of a may be discussed and further simpliflcations of these results using objectivity (3.124) and material symmetry may be obtained using Cauchy-Green tensors, Piola-Kirchhoff stress tensors, the known linearized constitutive equations of solids follow, e.g. Hook and Fourier laws with tensor (transport) coefficients which are reduced to scalars in isotropic solids (e.g. Cauchy law of deformation with Lame coefficients) [6, 7, 9, 13, 14]. 

Recall that the mixture invariance described in Sect. 4.4 means that all balances (4.95)-(4.102) remain valid with primed quantities defined by transformations (4.108)-(4.113). But here we proceed further with functions (4.222), (4.223) instead of (4.103), the Eqs.(4.108)-(4.112) permit the formulation of linear constitutive equations with primed quantities by constitutive principles analogously as in Sect. 4.5. Remaining parts of Sects. 4.5 and 4.6 will be done with analogous (but primed) definitions keeping the rule that the definitions themselves are mixture invariant (cf. above (4.114)). Procedure and description will be similar as in Sect. 4.4. Quantities or expressions which do not change by using (4.108)-(4.113) we denote as mixture invariant, e.g. quantities (4.113). For simplicity, we use the primes for mixture invariant quantities rather exceptionally. 

Using Auld s notation [3], the full tensor form of the linear constitutive equations with the strain and charge displacement as dependent on the stress and applied field may be written as... 

The derivation of ARRs from a bond graph of a hybrid system model that holds for all system modes is illustrated by means of a simple network with one switch and elements with a linear constitutive equation displayed in Fig. 4.1. 

In order to see how an incremental bond graph model for a bond graph element is obtained, a linear 1-port C-element with the nominal capacitance Cn is considered. In the following, an index n indicates a parameter or a variable of the non-faulty bond graph model with nominal parameters. In the case of a time constant parameter variation AC the linear constitutive equation of a 1-port C element in derivative causality takes the form... 

While the relation between the power variables of the non-thermal port may be linear, for the thermal port it is always nonlinear. If a linear constitutive equation... 

For small e, it follows from the linear constitutive equation (7.11) that tieis) approaches 3i (0). An example of risie) is shown in Fig. 7.20. 

As has been shown by Rosenberg [29], a linear state space equation in terms of Xj and the input vector u can be derived from the equations of the bond graph junction structure and the linear constitutive equations of the storage fields and the dissipative fields by eliminating Xd and other variables. The result is... 

The above properties are static physical properties which are determined with a linearly increasing applied force. Polymeric materials, including structural adhesives, have another important set of physical properties due to the fact that these materials behave in a manner that is not only elastic, but also viscoelastic in response to an applied stress. Viscous response may be treated by means of the linear constitutive equation formalism. Thus for a polymeric body, following FerryEq. (14) may be written ... 

Table 4.1 Thermodynamic potentials and linear constitutive equations...

Equations of this kind are called linear constitutive equations. The derivatives of the dependent variables with respect to the independent ones represent material coefficients. Recalling Eqs. (4.8), (4.9), and (4.10) they may also be written as partial second derivatives of the appropriate thermodynamic potential. Because of this second-derivative properly these coefficients are called second-order material coefficients. 

These are summarized in Table 4.1. The first colunm shows the three independent variables. In the second colunm the name of the associated thermocfynamic potential is given together with the defining Legendre transform. The third colunm shows the total differential of this potential, followed in the remaining columns by the corresponding linear constitutive equations. The material coefficients appearing in them and listed in Table 4.2 will be explained later. 

Book content is otganized in seven chapters and one Appendix. Chapter 1 is devoted to the fnndamental principles of piezoelectricity and its application including related histoiy of phenomenon discoveiy. A brief description of crystallography and tensor analysis needed for the piezoelectricity forms the content of Chap. 2. Covariant and contravariant formulation of tensor analysis is omitted in the new edition with respect to the old one. Chapter 3 is focused on the definition and basic properties of linear elastic properties of solids. Necessary thermodynamic description of piezoelectricity, definition of coupled field material coefficients and linear constitutive equations are discussed in Chap. 4. Piezoelectricity and its properties, tensor coefficients and their difierent possibilities, ferroelectricity, ferroics and physical models of it are given in Chap. 5. Chapter 6. is substantially enlarged in this new edition and it is focused especially on non-linear phenomena in electroelasticity. Chapter 7. has been also enlarged due to mary new materials and their properties which appeared since the last book edition in 1980. This chapter includes lot of helpful tables with the material data for the most today s applied materials. Finally, Appendix contains material tensor tables for the electromechanical coefficients listed in matrix form for reader s easy use and convenience. 

The linear constitutive equations of the two coupled fields read ... 

Which modifies (50) to the case of quasi-linear constitutive equation ... 

Determine the linear constitutive equations of a semi-permeable membrane. 

Since in the case of linear constitutive equations the X=l, and hence we get the symmetry relations as follows... 

Regarding the A=1 in case of linear constitutive equations, we receive the relations... 

This linear constitutive equation is based on the principle that the effects of sequential changes in strain are additive - ... 

In the case of small deformations (i.e. invoking geometrical linearization) Hooke s law for anisotropic elastic solids (i.e. the physically linearized constitutive equation) can be written in direct tensor notation as... 

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. 

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