Nonholonomic Geometry of Viscoanelastic Media and Experimental Confirmation

Abstract

A thermodynamical model for viscoanelastic media is analyzed using the nonholonomic geometry. A 27-dimensional manifold is introduced, and the differential equations for the geodetics are determined and analytically solved. It is shown that, in this manifold, the best specific entropy is a harmonic function. In the linear case the propagation of transverse acoustic waves is studied, and the theoretical results are compared with some experimental data from a polymeric material (polyisobutylene).

1. Introduction

From macroscopic point of view the most popular mathematical approaches to nonequilibrium thermodynamics are based both on the Caratheodory theory [1–3] which involves Pfaff equations and on the contact structure of thermodynamic state space [4–6]. If these equations are completely integrable, then the thermodynamics is called holonomic otherwise nonholonomic.

The theory of nonequilibrium holonomic thermodynamics for mechanical phenomena in continuous media was developed in the 90s (see e.g., [7] and references therein). The phenomenological equations were derived [8, 9] by introducing the tensorial internal variables 𝜀⁽¹⁾_𝛼𝛽 (𝛼,𝛽=1,2,3) which occur in the entropy production.

In particular, if one linearizes this theory by neglecting the cross effects among the irreversible phenomena (heat flow, mechanical viscosity, and anelastic deformations), the following rheological equations for distortional phenomena are obtained [10]: 𝑑𝜏_𝛼𝛽𝑑𝑡+𝑅^(𝜏)_(𝑑)0𝜏_𝛼𝛽=𝑅^(𝜀)_(𝑑)2𝑑²𝜀̃_𝛼𝛽𝑑𝑡²+𝑅^(𝜀)_(𝑑)1𝑑𝜀̃_𝛼𝛽𝑑𝑡+𝑅^(𝜀)_(𝑑)0𝜀̃_𝛼𝛽, (1) where 𝜏_𝛼𝛽 and 𝜀̃_𝛼𝛽 are the deviators of the stress (𝜏_𝛼𝛽) and the strain (𝜀_𝛼𝛽) tensors, respectively.

The quantities 𝑅 are given by 𝑅^(𝜏)_(𝑑)0=𝑎^(1,1)𝜂_𝑠^(1,1)=1𝜎𝑅≥0,_{(𝜀)(𝑑)0}=𝑎^(0,0)𝑎^(1,1)−𝑎^(0,0)𝜂_𝑠^(1,1)𝑅≥0,_{(𝜀)(𝑑)1}=𝑎^(0,0)+𝑎^(1,1)𝜂_𝑠^(1,1)𝜂_𝑠^(0,0)𝑅≥0,_{(𝜀)(𝑑)2}=𝜂_𝑠^(0,0)≥0, (2) in which 𝜎 is the relaxation time and 𝑎^(1,1)≥𝑎^(0,0)≥0 are the state coefficients, while 𝜂_𝑠^(0,0) and 𝜂_𝑠^(1,1) are the phenomenological coefficients related to the following physical phenomena: 𝑎^(0,0)𝑎⟹elasticity,_(1,1)𝜂⟹anelasticity,_𝑠(0,0)𝜂⟹viscosity,_𝑠(1,1)⟹fluidity. (3) In this paper we reconsider the theory from the point of view of the nonholonomic geometry.

In Sections 2 and 3 the analytic properties of the entropy are discussed, and in Section 4 the differential equations of geodetics in the space of state are obtanied.

Finally, in Section 5 we study the transverse waves and we will show the connection between complex numbers and the shear complex modulus. By applying these results to a polymeric material, as polyisobutylene we will also show that the expected results of the theoretical model are in agreement with the experimental data.

2. Gibbs-Pfaff Equation of Viscoanelastic Media

Let us define the space of states 𝑅²⁷=𝑅^2⋅13+1=𝑠,𝑇,𝑢,𝜏^(eq)_𝛼𝛽,𝜀_𝛼𝛽,𝜏⁽¹⁾_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽, (4) where 𝑠 is the specific entropy, 𝑇 is the absolute temperature, 𝑢 is the specific internal energy, 𝜏^(eq)_𝛼𝛽 is the symmetric equilibrium stress tensor, 𝜀_𝛼𝛽 is the total symmetric strain tensor, and 𝜏⁽¹⁾_𝛼𝛽 is the symmetric affinity stress tensor conjugate to the anelastic tensor 𝜀⁽¹⁾_𝛼𝛽 (the symmetric tensorial thermodynamic internal variable).

Let 𝜈 be the specific volume related to the mass density 𝜚=constant (homogeneous media) by 𝜈𝜚=1. The Gibbs-Pfaff equation of viscoanelastic media is 𝑇𝑑𝑠=𝑑𝑢−𝜈𝜏^(eq)_𝛼𝛽𝑑𝜀_𝛼𝛽+𝜈𝜏⁽¹⁾_𝛼𝛽𝑑𝜀⁽¹⁾_𝛼𝛽. (5) This equation defines in 𝑅²⁷ the nonholonomic contact distribution of dimension 26 so that the highest dimension of integral manifold is 13.

The 𝐶^∞ representation of this integral manifold (of maximum dimension) is usually given as follows: 𝑠=Φ𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽,𝑇⁻¹=𝜕Φ𝜕𝑢𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽,𝜏^(eq)_𝛼𝛽=−𝜚𝑇𝜕Φ𝜕𝜀_𝛼𝛽𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽,𝜏⁽¹⁾_𝛼𝛽=𝜚𝑇𝜕Φ𝜕𝜀⁽¹⁾_𝛼𝛽𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽. (6) This parametrization is based on the arbitrary 𝐶^∞-function Φ. So that we have a family of integral manifolds of dimension 13 indexed on the arbitrary function Φ.

The state parameters 𝑇⁻¹,𝑇⁻¹𝜚⁻¹𝜏^(eq)_𝛼𝛽,𝑇⁻¹𝜚⁻¹𝜏⁽¹⁾_𝛼𝛽 (7) are related by the equations of motion and the equations of state [8, 9].

The 𝐶^∞-representation is a generic element in the 1-jet space converted into𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽,𝑠𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽,𝜕𝑠𝜕𝑢,−𝜕𝑠𝜕𝜀_𝛼𝛽,𝜕𝑠𝜕𝜀⁽¹⁾_𝛼𝛽. (8) In order to fix a representative for the specific entropy 𝑠=Φ(𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽), we need a supplementary condition as follows.

Theorem 1. If 𝑠=Φ(𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽) is an homogeneous function of order one, then 𝑢Φ=𝑇−1𝜏𝜚𝑇^{(𝑒𝑞)𝛼𝛽}𝜀_𝛼𝛽+1𝜏𝜚𝑇^(1)𝛼𝛽𝜀⁽¹⁾_𝛼𝛽. (9)

Proof. The condition of homogeneity of order one Φ𝑘𝑢,𝑘𝜀_𝛼𝛽,𝑘𝜀⁽¹⁾_𝛼𝛽=𝑘Φ𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽 (10) gives the PDE 𝑢𝜕Φ𝜕𝑢+𝜀_𝛼𝛽𝜕Φ𝜕𝜀_𝛼𝛽+𝜀⁽¹⁾_𝛼𝛽𝜕Φ𝜕𝜀⁽¹⁾_𝛼𝛽=Φ𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽. (11) Then the entropy (solution of this PDE) appears as the potential (9).

Corollary 2. If the specific entropy 𝑠=Φ(𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽) is an homogeneous function of order one, then(1)the variables 𝑠,𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽 are conjugated to the intensive variables 𝑇,𝜏^(𝑒𝑞)_𝛼𝛽,𝜏⁽¹⁾_𝛼𝛽;(2)the variables 𝑠,𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽 are not essential parameters (because they are not independent).

Proof. From the expression (9) we find 1𝑇𝑑𝑠+𝑠𝑑𝑇=𝑑𝑢−𝜚𝜀_𝛼𝛽𝑑𝜏^(eq)_𝛼𝛽+𝜏^(eq)_𝛼𝛽𝑑𝜀_𝛼𝛽+1𝜚𝜀⁽¹⁾_𝛼𝛽𝑑𝜏⁽¹⁾_𝛼𝛽+𝜏⁽¹⁾_𝛼𝛽𝑑𝜀⁽¹⁾_𝛼𝛽. (12)(1) Replacing the relation (5), we get 1𝑠𝑑𝑇=−𝜚𝜀_𝛼𝛽𝑑𝜏^(eq)_𝛼𝛽+1𝜚𝜀⁽¹⁾_𝛼𝛽𝑑𝜏⁽¹⁾_𝛼𝛽, (13)and the first statement is true.(2) The foregoing relation shows that, for example, 𝜀_𝛼𝛽𝑠,𝜀⁽¹⁾_𝛼𝛽𝑠 (14)are essential parameters.

3. Specific Entropy via Least Squares Lagrangian

The most convenient way to fix a representative Φ of the specific entropy 𝑠 is to look at (5) as a partial derivative evolution equation and to build the least squares method Lagrangian 𝐿2𝐿=𝑇⁻¹−𝜕Φ𝜕𝑢²+𝜏^(eq)_𝛼𝛽+𝜌𝑇𝜕Φ𝜕𝜀_𝛼𝛽²+𝜏⁽¹⁾_𝛼𝛽−𝜌𝑇𝜕Φ𝜕𝜀⁽¹⁾_𝛼𝛽² (15) and the functional _Ω𝐿𝑢,𝜀_𝛼𝛽,𝜀⁽¹⁾_𝛼𝛽,Φ,Φ_𝑢,Φ_𝜀𝛼𝛽,Φ_𝜀⁽¹⁾_𝛼𝛽𝑑Ω, (16) where Φ_𝑢=𝜕Φ/𝜕𝑢, Φ_𝜀𝛼𝛽=𝜕Φ/𝜕𝜀_𝛼𝛽, and Φ_𝜀⁽¹⁾_𝛼𝛽=𝜕Φ/𝜕𝜀⁽¹⁾_𝛼𝛽.

The extremals are solutions of the Euler-Lagrange PDE 𝜕𝐿−𝐷𝜕Φ^𝑢𝜕𝐿𝜕Φ_𝑢+𝐷_𝜀𝛼𝛽𝜕𝐿𝜕Φ_𝜀𝛼𝛽+𝐷_𝜀⁽¹⁾_𝛼𝛽𝜕𝐿𝜕Φ_𝜀⁽¹⁾_𝛼𝛽=0, (17) where 𝐷_∗ is the total derivative with respect to the variable ∗.

In our case, we get 𝜕𝐿𝜕Φ=0,𝜕𝐿𝜕Φ_𝑢=−𝑇⁻¹−𝜕Φ,𝜕𝑢𝜕𝐿𝜕Φ_𝜀𝛼𝛽=𝜌𝑇𝜏^(eq)_𝛼𝛽+𝜌𝑇𝜕Φ𝜕Φ_𝜀𝛼𝛽,𝜕𝐿𝜕Φ_𝜀⁽¹⁾_𝛼𝛽=−𝜌𝑇𝜏⁽¹⁾_𝛼𝛽−𝜌𝑇𝜕Φ𝜕Φ_𝜀⁽¹⁾_𝛼𝛽. (18) So that, by replacing the partial derivatives of 𝐿 in (17), there follows the Laplace equation for the entropy 𝜕²Φ𝜕𝑢²+𝜌²𝑇²𝜕²Φ𝜕𝜀²_𝛼𝛽+𝜌²𝑇²𝜕²Φ𝜕𝜀²⁽¹⁾_𝛼𝛽=0. (19) Consequently, we have the following.

Theorem 3. The best entropy for the nonholonomic nonequilibrium thermodynamics is an harmonic function.

4. Geodesics

Any curve in the distribution (5) is described by 𝑇(𝑡)𝑠(̇𝑡)=𝑢̇(𝑡)−𝜈𝜏^(eq)_𝛼𝛽(𝑡)𝜀̇_𝛼𝛽(𝑡)+𝜈𝜏⁽¹⁾_𝛼𝛽(𝑡)𝜀̇⁽¹⁾_𝛼𝛽(𝑡). (20) In order to be a geodesic, this curve must minimize the energy functional 1𝐽=2^𝑡1_𝑡0𝑇̇²(𝑡)+𝑠²̇(𝑡)+𝑢̇²(𝑡)+𝛿_𝛼𝛾𝛿_𝛽𝛿𝜏̇^(eq)_𝛼𝛽𝜏̇^(eq)_𝛾𝛿+𝛿_𝛼𝛾𝛿_𝛽𝛿𝜏̇⁽¹⁾_𝛼𝛽𝜏̇⁽¹⁾_𝛾𝛿+𝛿_𝛼𝛾𝛿_𝛽𝛿𝜀̇_𝛼𝛽𝜀̇_𝛾𝛿+𝛿_𝛼𝛾𝛿_𝛽𝛿𝜀̇⁽¹⁾_𝛼𝛽𝜀̇⁽¹⁾_𝛾𝛿𝑑𝑡. (21) In short, we must solve the problem min𝐽subjectto(20). (22) To solve this problem, we use the method of Lagrange multipliers. For this we defined the constrained Lagrangian 𝐿₁=12𝑇̇²(𝑡)+𝑠²̇(𝑡)+𝑢̇²(𝑡)+𝛿_𝛼𝛾𝛿_𝛽𝛿𝜏̇^(eq)_𝛼𝛽(𝑡)𝜏̇^(eq)_𝛾𝛿(𝑡)+𝛿_𝛼𝛾𝛿_𝛽𝛿𝜏̇⁽¹⁾_𝛼𝛽(𝑡)𝜏̇⁽¹⁾_𝛾𝛿(𝑡)+𝛿_𝛼𝛾𝛿_𝛽𝛿𝜀̇_𝛼𝛽(𝑡)𝜀̇_𝛾𝛿(𝑡)+𝛿_𝛼𝛾𝛿_𝛽𝛿𝜀̇⁽¹⁾_𝛼𝛽(𝑡)𝜀̇⁽¹⁾_𝛾𝛿(𝑡)+𝑝𝑇(𝑡)𝑠(̇𝑡)−𝑢̇(𝑡)+𝜈𝜏^(eq)_𝛼𝛽(𝑡)𝜀̇_𝛼𝛽(𝑡)−𝜈𝜏⁽¹⁾_𝛼𝛽(𝑡)𝜀̇⁽¹⁾_𝛼𝛽(𝑡), (23) where 𝑝 is the Lagrangian multiplier.

Theorem 4. The geodesics of the nonholonomic viscoanelastic distribution are solutions of the Euler-Lagrange ODEs for the Lagrangian (23) 𝜕𝐿₁𝜕𝑥^𝑖−𝐷_𝑡𝜕𝐿₁𝜕𝑥̇^𝑖=0, (24) where 𝑥^𝑖 are the generalized coordinates: 𝑥¹=𝑇,𝑥²=𝑠,𝑥³𝑥=𝑢,_4𝛼𝛽=𝜏^(𝑒𝑞)_𝛼𝛽,𝑥⁵_𝛼𝛽=𝜀_𝛼𝛽,𝑥⁶_𝛼𝛽=𝜏⁽¹⁾_𝛼𝛽,𝑥⁷_𝛼𝛽=𝜀⁽¹⁾_𝛼𝛽. (25) So that explicitly from (23) and (24) we have 𝑝̇̈𝑑𝑠−𝑇=0,̇𝑑𝑑𝑡𝑠+𝑝𝑇=0,̇𝜀̇𝑑𝑡𝑢−𝑝=0,𝑝𝜈_𝛼𝛽−𝑑𝜏̇𝑑𝑡^{(𝑒𝑞)𝛼𝛽}=0,𝑝𝜈𝜀̇⁽¹⁾_𝛼𝛽+𝑑𝜏̇𝑑𝑡^(1)𝛼𝛽𝑑=0,𝜀̇𝑑𝑡^(1)𝛼𝛽−𝑝𝜈𝜏⁽¹⁾_𝛼𝛽𝑑=0,𝜀̇𝑑𝑡^𝛼𝛽+𝑝𝜈𝜏^(𝑒𝑞)_𝛼𝛽=0. (26)

The equations (26) with the condition (20) are the differential equations of geodesics.

Let 𝑇₀=𝑇(0),𝑇̇₀=𝑇̇(0),𝑠₀(0),𝑠̇=𝑠₀=𝑠(̇0),𝑢₀=𝑢(0),𝑢̇₀=𝑢̇(0),𝜏₀^(eq)_𝛼𝛽=𝜏^(eq)_𝛼𝛽(0),𝜏̇₀^(eq)_𝛼𝛽=𝜏̇^(eq)_𝛼𝛽(0),𝜏₀⁽¹⁾_𝛼𝛽=𝜏⁽¹⁾_𝛼𝛽(0),𝜏̇₀⁽¹⁾_𝛼𝛽=𝜏̇⁽¹⁾_𝛼𝛽(0),𝜀_0𝛼𝛽=𝜀_𝛼𝛽(0),𝜀̇_0𝛼𝛽=𝜀̇_𝛼𝛽(0),𝜀₀⁽¹⁾_𝛼𝛽=𝜀⁽¹⁾_𝛼𝛽(0),𝜀̇₀⁽¹⁾_𝛼𝛽=𝜀̇⁽¹⁾_𝛼𝛽(0) (27) be the given (constant) initial values.

By some explicit computation we can easily show that the following.

Theorem 5. The geodesics of the nonholonomic viscoanelastic distribution, as solution of the Cauchy problem (26) and (27), are the family of curves: 𝑢(𝑡)=𝑢̇₀𝑡+𝑢₀𝑠(t)=𝑇̇₀𝑝𝑠̇cos𝑝𝑡+⁰𝑝sin𝑝𝑡+𝑠₀−𝑇̇₀𝑝𝑇(𝑡)=𝑇̇₀𝑝𝑠̇sin𝑝𝑡−⁰𝑝cos𝑝𝑡+𝑇₀+𝑠₀̇𝑝𝜏^(𝑒𝑞)_𝛼𝛽=𝜏₀^(𝑒𝑞)_𝛼𝛽−1𝜀̇𝜈𝑝⁰_𝛼𝛽+𝑝𝜈𝜏₀^(𝑒𝑞)_𝛼𝛽+1cos𝜈𝑝𝑡𝜏̇𝜈𝑝^0(𝑒𝑞)_𝛼𝛽1sin𝜈𝑝𝑡+𝜀̇𝜈𝑝⁰_𝛼𝛽+𝑝𝜈𝜏₀^(𝑒𝑞)_𝛼𝛽𝜀_𝛼𝛽=−𝜏₀^(𝑒𝑞)_𝛼𝛽−1𝜀̇𝜈𝑝⁰_𝛼𝛽+𝑝𝜈𝜏₀^(𝑒𝑞)_𝛼𝛽−1sin𝜈𝑝𝑡𝜏̇𝜈𝑝^0(𝑒𝑞)_𝛼𝛽cos𝜈𝑝𝑡+𝜀_0𝛼𝛽+1𝜏̇𝜈𝑝^0(𝑒𝑞)_𝛼𝛽𝜏⁽¹⁾_𝛼𝛽=𝜏₀⁽¹⁾_𝛼𝛽+1𝜀̇𝜈𝑝⁰⁽¹⁾_𝛼𝛽+𝑝𝜈𝜏₀⁽¹⁾_𝛼𝛽+1cos𝜈𝑝𝑡𝜏̇𝜈𝑝⁰⁽¹⁾_𝛼𝛽1sin𝜈𝑝𝑡−𝜀̇𝜈𝑝⁰⁽¹⁾_𝛼𝛽+𝑝𝜈𝜏₀⁽¹⁾_𝛼𝛽𝜀⁽¹⁾_𝛼𝛽=−𝜏₀⁽¹⁾_𝛼𝛽+1𝜀̇𝜈𝑝⁰⁽¹⁾_𝛼𝛽+𝑝𝜈𝜏₀⁽¹⁾_𝛼𝛽−1sin𝜈𝑝𝑡𝜏̇𝜈𝑝⁰⁽¹⁾_𝛼𝛽cos𝜈𝑝𝑡−𝜀₀⁽¹⁾_𝛼𝛽+1𝜏̇𝜈𝑝⁰⁽¹⁾_𝛼𝛽. (28)

Proof. Let us first solve the simplest equation: from (26)₃ we have 𝑢̇−𝑝=𝑢̇₀⟹𝑢=𝑢̇₀+𝑝𝑡+𝑢₀. (29) From (26)₁ it is 𝑑̇𝑇̇𝑑𝑡𝑝𝑠−=0⟹𝑇=𝑝𝑠−𝑠₀+𝑇̇₀ (30) and deriving (26)₂𝑠̈+𝑝𝑇̇=0. (31) By replacing 𝑇̇ with the previous expression we get 𝑠̈+𝑝𝑝𝑠−𝑠₀+𝑇̇₀=0, (32) that is 𝑠̈+𝑝²𝑠=𝑝²𝑠₀−𝑝𝑇̇₀. (33) This is a linear nonhomogeneous second-order (harmonic) equation whose solution is 𝑠(𝑡)=𝑠₀−𝑇̇₀𝑝+𝑇̇₀𝑝𝑠̇cos𝑝𝑡+⁰𝑝sin𝑝𝑡. (34) With this function, from the expression 𝑇̇=𝑝𝑠−𝑠₀+𝑇̇₀, (35) we can compute also 𝑇(𝑡): 𝑇̇=𝑝𝑠₀−𝑇̇₀𝑝+𝑇̇₀𝑝𝑠̇cos𝑝𝑡+⁰𝑝sin𝑝𝑡−𝑠₀+𝑇̇₀; (36) that is, 𝑇̇=𝑇̇₀cos𝑝𝑡+𝑠₀̇sin𝑝𝑡. (37) The solution is 𝑇(𝑡)=𝑇̇₀𝑝𝑠̇sin𝑝𝑡−⁰𝑝cos𝑝𝑡+𝑇₀+𝑠₀̇𝑝. (38) Concerning (26)_4,5,6,7 we can notice that it is enough to solve (26)_4,7 since their solutions are formally equal to the solutions of (26)_5,6 since these equations coincide with (26)_4,7 apart from the substitutions: 𝜀_𝛼𝛽⟹𝜀⁽¹⁾_𝛼𝛽,𝜏^(eq)_𝛼𝛽⟹−𝜏⁽¹⁾_𝛼𝛽. (39) Thus from (26)₇ it is 𝜀̇_𝛼𝛽+𝑝𝜈𝜏^(eq)_𝛼𝛽=𝜀̇_0𝛼𝛽+𝑝𝜈𝜏₀^(eq)_𝛼𝛽; (40) that is, 𝜀̇_𝛼𝛽=−𝑝𝜈𝜏^(eq)_𝛼𝛽+𝜀̇_0𝛼𝛽+𝑝𝜈𝜏₀^(eq)_𝛼𝛽. (41) If we put this expression in (26)₄ we get 𝑝𝜈−𝑝𝜈𝜏^(eq)_𝛼𝛽+𝜀̇_0𝛼𝛽+𝑝𝜈𝜏₀^(eq)_𝛼𝛽−𝑑𝜏̇𝑑𝑡^(eq)𝛼𝛽=0, (42) and by some manipulation we get the (vectorial) linear second order harmonic equation for 𝜏^(eq)_𝛼𝛽: 𝑑𝜏̇𝑑𝑡^(eq)𝛼𝛽+𝑝²𝜈²𝜏^(eq)_𝛼𝛽=𝑝𝜈𝜀̇_0𝛼𝛽+𝑝𝜈𝜏₀^(eq)_𝛼𝛽. (43) The solution is 𝜏^(eq)_𝛼𝛽=𝜏₀^(eq)_𝛼𝛽−1𝜀̇𝜈𝑝⁰_𝛼𝛽+𝑝𝜈𝜏₀^(eq)_𝛼𝛽+1cos𝜈𝑝𝑡𝜏̇𝜈𝑝^0(eq)_𝛼𝛽1sin𝜈𝑝𝑡+𝜀̇𝜈𝑝⁰_𝛼𝛽+𝑝𝜈𝜏₀^(eq)_𝛼𝛽, (44) so that by integrating (41) and using the previous equation, we can easily get the expression for 𝜀_𝛼𝛽𝜀_𝛼𝛽=−𝜏₀^(eq)_𝛼𝛽−1𝜀̇𝜈𝑝⁰_𝛼𝛽+𝑝𝜈𝜏₀^(eq)_𝛼𝛽−1sin𝜈𝑝𝑡𝜏̇𝜈𝑝^0(eq)_𝛼𝛽cos𝜈𝑝𝑡+𝜀_0𝛼𝛽+1𝜏̇𝜈𝑝^0(eq)_𝛼𝛽. (45) With similar computations we get also the last two equations of (28).

The Lagrangian multiplier 𝑝 is obtained by inserting the functions (28) and derivatives into (20).

It should be noticed that the the projection of the geodesics (28) into different planes gives rise to well known curves. For instance, in the plane ⟨𝑇,𝑠⟩ (28) are the parametric equations of a cycloid. Moreover, by assuming that all the initial values are positive, the asymptotic limits give lim_𝑡→∞𝑢(𝑡)=+∞,lim_𝑡→∞|𝑇(𝑡)|≤𝑇₀−𝑠₀̇𝑝+𝑇̇₀𝑝+𝑠₀̇𝑝,lim_𝑡→∞|𝑠(𝑡)|≤𝑠₀−𝑇̇₀𝑝+𝑇̇₀𝑝+𝑠₀̇𝑝, (46) which are in agreement with physical consideration especially for the entropy 𝑠 which is upper bounded. Analogously we have similar bounded asymptotic limits for the vectorial functions.

5. Experimental Approach to the Linear Response Theory

In a previous paper [11] by the application of the linear response theory [12–15] numerical values of (1) were considered, and the results were compared with experimental data.

In this section we study another aspect of the transversal waves propagation in viscoanelastic media, and we apply the theoretical results to a polymeric material as the polyisobutylene.

We consider, with respect to the Cartesian orthogonal axes (𝑥₁,𝑥₂,𝑥₃), the following displacement 𝐮𝑢₃=𝐴𝑒^{𝑖(𝑘𝑥1−𝜔𝑡)},𝑢₁=𝑢₃=0 (47) being 𝑖²=−1 and 𝑘=𝑘₁+𝑖𝑘₂ the complex wave number, so that 𝑣_𝑠=𝜔𝑘₁ (48) is the phase velocity and 𝑘₂ is connected with the attenuation of the waves.

As 𝜀_𝛼𝛽=1/2(𝜕𝑢_𝛼/𝜕𝑥_𝛽+𝜕𝑢_𝛽/𝜕𝑥_𝛼), from (1) one obtains 𝑘₁=𝜔√𝜚𝐵(𝜔)1+𝐷(𝜔)+1^1/2,𝑘₁=𝜔√𝜚𝐵(𝜔)1+𝐷(𝜔)−1^1/2, (49) where 𝐵(𝜔)=𝑅^(𝜏)_(𝑑)0𝑅^(𝜀)_(𝑑)0+𝜔²𝑅^(𝜀)_(𝑑)1−𝑅^(𝜏)_(𝑑)0𝑅^(𝜀)_(𝑑)2𝑅_(𝑑)0^(𝜀)−𝜔²𝑅_(𝑑)2^(𝜀)2+𝜔𝑅_(𝑑)1^(𝜀)2,𝐵(𝜔)=𝜔²𝑅_(𝑑)0^(𝜀)−𝑅_(𝑑)1^(𝜀)𝑅_(𝑑)0^(𝜏)−𝜔²𝑅_(𝑑)2^(𝜀)2𝑅_(𝑑)0^(𝜏)𝑅_(𝑑)0^(𝜀)−𝜔²𝑅_(𝑑)1^(𝜀)−𝑅_(𝑑)0^(𝜏)𝑅_(𝑑)2^(𝜀)2. (50) The complex shear velocity is [14] 𝑣_𝑐=1𝑣_𝑠+𝑘₂=𝐺𝑖𝜔¹(𝜔)+𝑖𝐺₂(𝜔)𝜚, (51) where 0≤𝑠≤257, 𝐺₁(𝜔) (storage modulus), and 𝐺₂(𝜔) (loss modulus) are, respectively, linked with the nondissipative and dissipative phenomena, and their experimental curves are plotted in Figure 1.

Figure 1 

Generic storage and loss moduli.

From (47) to (50), the following relations are obtained: 𝑘₁=𝜔𝜚𝐺₁/21+𝐺₂/𝐺₁²+1𝐺₁1+𝐺₂/𝐺₁²,𝑘₂=𝜔𝜚𝐺₁/21+𝐺₂/𝐺₁²−1𝐺₁1+𝐺₂/𝐺₁². (52) Let us consider the range of high frequency 𝜔_𝐻≤𝜔≤𝜔_𝑈 and 𝜔𝜏≫1 (of order 10²) so that no relaxation phenomena occur (see Figure 1).

By putting 𝐺₁=𝐺_1𝐻1,001^𝑠𝐺₂=𝐺_2𝐻1,001^−𝑠, (53) where 𝐺_1𝐻=𝐺₁(𝜔_𝐻) and 𝐺_2𝐻=𝐺₂(𝜔_𝐻), (52) becomes 𝑘₁=𝜔𝜚2𝐺_1𝐻²1.001^2𝑠+𝐺_2𝐻²+𝐺_1𝐻1.001^𝑠𝐺_1𝐻²1.001^2𝑠+𝐺_2𝐻²,𝑘₂=𝜔𝜚2𝐺_1𝐻²1.001^2𝑠+𝐺_2𝐻²−𝐺_1𝐻1.001^𝑠𝐺_1𝐻²1.001^2𝑠+𝐺_2𝐻². (54) For the Polyisobutilene we have [15] the characteristic values which are 𝜎=10⁻⁷sec.,𝜔_𝐻=3.2⋅10¹⁴𝜔Hz,_𝑈=6⋅10¹⁴𝐺Hz,_1𝑈=2.4⋅10⁹Pa,𝐺_2𝑈=2.75⋅10⁴Pa, (55) and the graphics confirm (see Figure 2) with experimental data the validity of the model proposed for viscoanelastic phenomena in continuous media.

(a)

(b)

Figure 2 

𝑘1 and 𝑘2 for Polyisobutilene: (M.w.106g/mol; 𝑇0=273K) the experimental curves (in black) and the theoretical curves (in red) obtained by our model.

6. Conclusions

From the viewpoint of nonholonomic irreversible thermodynamics, it is shown that the best specific entropy is an harmonic function in a 27-dimensional manifold. The differential equations of geodetics are obtained and the corresponding curves are explicitly computed. In the linearized theory it is shown that the theoretical results are in agreement with the experimental data in the case of polymeric material (Polyisobutilene) (Figure 2).

Conflict of Interests

The authors declare that there is no conflicts of interests regarding the publication of this paper.