Isothermal viscoelasticity and energy

Within the scope of the linear viscoelasticity theory, the change in the inner energy of a viscoelastic body is induced either by heat exchange or by a work performance. The first law of thermodynamics, balance equation of a closed system is mostly referred when the thermodynamic consistency of some rheological model is required. Accordingly, within the frame of the isothermal viscoelastic investigation we just distinguish between the stored and dissipated energy. And this is the issue that the paper is focused on. Subjected to a load, the one degree of freedom viscoelastic models’ behaviour is traced, together with the observation of the energy – total, stored and dissipated. Nevertheless, the only stored energy in viscoelastic model is potential energy. General considerations are applied on Maxwell model subjected to the standard both creep and relaxation tests.


Introduction
This work is a continuation of author's investigation in the field, namely [1][2][3]. It is focused to the thermodynamic aspects of viscoelastic (VE) models. The exploration is carried out within the framework of the linear viscoelasticity theory. Viscoelastic bodies, represented by corresponding models are studied and the energy flow while the VE bodies are loaded. The isothermal state of a system is supposed.

Viscoelastic models and related equations
Viscoelasticity theory is a part of rheology, material science studying material which are neither purely elastic solids nor purely viscous liquids. Linear viscoelasticity fruitfully exploits the Boltzmann superposition principle.

Fundamental viscoelastic elements
Each viscoelastic model involves two types of fundamental elements, Hookean elastic matter (H) characterised by linear dependence between stress and strain ( ) = ( ) (1) and Newton viscous fluid (N) characterised by linear dependence between stress and strain rate, see Fig.1.
with [ ] being elastic Young modulus and [ . ] Newton viscous coefficient.  In parallel connection, Fig.2 left, total deformation is the same as the deformations of both involved elements and total stress is equal to the sum of particular values of stress.
In serial connection, Fig.2 right, total deformation is equal to the sum of both involved elements' deformations and total stress is of the same magnitude as the particular values of stresses of both fundamental elements involved. [4] ( ) = H ( ) + N ( ) (6) When we couple (6), (7) with (1) and (2) and we eliminate all indexed variables, we get the stressstrain relation of the model, an implicit form of constitutive equation of Maxwell model: Both constitutive equations (5) and (8) are linear differential equations easily solvable with respect to either or .

Thermodynamics in rheology
Each rheological element, accordingly a VE model, governed by a constitutive relation endowed by a potential energy p ≥ 0, has to be thermodynamically consistent. Hence the dissipation rate of the model energy has to be nonnegative [5] ̇d ( ) =̇( ) −̇p( ) ≥ 0 with being the total work of deformation, d the dissipated energy and p the potential (stored) energy.

Stored and dissipated energy in VE models
The total work of deformation in VE models is split to stored and dissipated energy. Generally, the stored energy can be potential and kinetic, nevertheless during viscoelastic investigation we consider the types of load causing negligible inertial impact. So further, alongside the text, we identify the stored energy with potential one. It is important to recall that all potential energy is stored due to elastic element involved; and all energy spent to viscous element deformation is dissipated, changed to heat. Indeed, as we have isothermal state, all dissipated energy is lost energy as far as any other movement on VE model concerns. Now we trace the total deformation energy flow during a load process and its partition to the potential and dissipated component having in mind the constitutive equation of VE model which expresses the dependence of the reaction on the action. First we are interested in a total work of deformation. The stress power of VE model, i.e. the rate of deformation work which is performed per time and volume unit, is [8] ̇( ) = ( )( ). (9) Consequently, the deformation work is For the sake of expressing (9) and (10)  which are specific for specific load type for particular VE model. Then we can rewrite the total deformation energy rate and the total deformation energy itself in the form Remark: General rules for the derivation of time dependent material characteristics are elaborated in [3] and synoptically provided in [2].
The dissipation energy rate and the dissipation energy itself of a VE model can be written as: Thereafter, the potential (stored) energy rate and the potential energy itself is given as follows:

Energy of Maxwell model subjected to a load
There are two typical lab tests -creep test and relaxation test obviously executed in material science in order to mutual comparison of rheological material. That is why we deal with the load typical for these tests.
Following consideration is focused on the Maxwell model exclusively. We study the energy quantification of the model tied up with the mechanical response of the model subjected to both standard loading tests. Herein we use the thermodynamic consideration provided in Chapter 3. As mentioned above, Maxwell model (M)=(H) -(N) performed in Fig. 2 right, is governed by its constitutive relation (8).

Creep test on Maxwell modelconstant stress imposed
When a constant stress is applied at (M) and maintained in time: ( ) = * ; ∈ ( 0 , 1 ⟩, the responding deformation function is yielded as a solution of the simple differential equation Solution of (18) with the initial condition (0) = H (0) = * taken into account, is Let 0 = 0 be an initial time instant in which the constant stress * is suddenly imposed and being maintained in time. All absorbed energy is stored by the spring in that time instant due to immediate deformation of the spring. The dashpot initial displacement is zero. So

Relaxation test on Maxwell modelconstant strain imposed
A constant strain is immediately applied and maintained in time: ( ) = * ; ∈ ( 0 = 0, 1 ⟩. The responding stress function is given as a solution of the differential equation As soon as the model is step wisely loaded, the immediate deformation occurs. The initial deformation of (N) is zero. Later on, the deformation of (N) increases together with corresponding decrease of deformation of (H). This process (until persisting deformation load) continues until the elastic element deformation diminishes back to zero, e.g. all deformation of the viscoelastic material is transferred to the viscous element.
The stress function, i.e. the solution of (22) with the initial condition (0) = * taken into account is From (9) As physically expected and documented in the following computation, the dissipation energy increases together with the increase of the dashpot deformation; and the potential energy decreases down to zero adequately; as both, potential and dissipation energy in any time instant, are summed up to the total energy ( ) = p ( ) + d ( ) = We can easy see, that both stored (25) and dissipated (26) energy are summed up to the total (24) deformation energy.
Finally it is worth mentioning that forms (9) -(14) are general for Maxwell model, accordingly any loading function of time can be taken as the action -either ( ) or ( ) yielding the total, stored and dissipation energy functions matching with the mechanical response, reaction of the Maxwell model.

Conclusion
In this work the recent investigation of author is presented dealing with thermodynamic aspects of the viscoelastic bodies represented by their models. First a general consideration concerning the thermodynamics in rheology is provided together with the supporting formulas derivation. Then the Maxwell model is studied from the thermodynamic point of view. The resulting forms and graphical performance document the validity of the previous theoretical consideration.
Nowadays, the prediction of behaviour of materials under various types of load is of a great significance in material science. The theoretical evaluation of corresponding energy can help efficiently with the energy consumption prediction of new soft materials. This work is supported by grant VEGA 1/0456/17 and APVV-18-0052.