# Nonholonomic Geometry of Viscoanelastic Media and Experimental Confirmation

Authors: Armando Ciancio; Carlo Cattani
Journal: Mathematical Problems in Engineering
 (2013)

---

## 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).


---

## Body

## 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 ɛαβ(1) (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]:
(1)dτ~αβdt+R(d)0(τ)τ~αβ=R(d)2(ɛ)d2ɛ~αβdt2+R(d)1(ɛ)dɛ~αβdt+R(d)0(ɛ)ɛ~αβ,
where τ~αβ and ɛ~αβ are the deviators of the stress (ταβ) and the strain (ɛαβ) tensors, respectively.The quantitiesR are given by
(2)R(d)0(τ)=a(1,1)ηs(1,1)=1σ≥0,R(d)0(ɛ)=a(0,0)(a(1,1)-a(0,0))ηs(1,1)≥0,R(d)1(ɛ)=a(0,0)+a(1,1)ηs(1,1)ηs(0,0)≥0,R(d)2(ɛ)=ηs(0,0)≥0,
in which σ is the relaxation time and a(1,1)≥a(0,0)≥0 are the state coefficients, while ηs(0,0) and ηs(1,1) are the phenomenological coefficients related to the following physical phenomena:
(3)a(0,0)⟹elasticity,a(1,1)⟹anelasticity,ηs(0,0)⟹viscosity,ηs(1,1)⟹fluidity.
In this paper we reconsider the theory from the point of view of the nonholonomic geometry.In Sections2 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 Section5 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(4)R27=R2·13+1={s,T,u,ταβ(eq),ɛαβ,ταβ(1),ɛαβ(1)},
where s is the specific entropy, T is the absolute temperature, u is the specific internal energy, ταβ(eq) is the symmetric equilibrium stress tensor, ɛαβ is the total symmetric strain tensor, and ταβ(1) is the symmetric affinity stress tensor conjugate to the anelastic tensor ɛαβ(1) (the symmetric tensorial thermodynamic internal variable).Letν be the specific volume related to the mass density ϱ=constant (homogeneous media) by νϱ=1. The Gibbs-Pfaff equation of viscoanelastic media is
(5)Tds=du-νταβ(eq)dɛαβ+νταβ(1)dɛαβ(1).
This equation defines in R27 the nonholonomic contact distribution of dimension 26 so that the highest dimension of integral manifold is 13.TheC∞ representation of this integral manifold (of maximum dimension) is usually given as follows:
(6)s=Φ(u,ɛαβ,ɛαβ(1)),T-1=∂Φ∂u(u,ɛαβ,ɛαβ(1)),ταβ(eq)=-ϱT∂Φ∂ɛαβ(u,ɛαβ,ɛαβ(1)),ταβ(1)=ϱT∂Φ∂ɛαβ(1)(u,ɛαβ,ɛαβ(1)).
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(7)T-1,T-1ϱ-1ταβ(eq),T-1ϱ-1ταβ(1)
are related by the equations of motion and the equations of state [8, 9].TheC∞-representation is a generic element in the 1-jet space converted into(8)(u,ɛαβ,ɛαβ(1),s(u,ɛαβ,ɛαβ(1)),∂s∂u,-∂s∂ɛαβ,∂s∂ɛαβ(1)).
In order to fix a representative for the specific entropy s=Φ(u,ɛαβ,ɛαβ(1)), we need a supplementary condition as follows.Theorem 1.
Ifs=Φ(u,ɛαβ,ɛαβ(1)) is an homogeneous function of order one, then
(9)Φ=uT-1ϱTταβ(eq)ɛαβ+1ϱTταβ(1)ɛαβ(1).Proof.
The condition of homogeneity of order one(10)Φ(ku,kɛαβ,kɛαβ(1))=kΦ(u,ɛαβ,ɛαβ(1))
gives the PDE
(11)u∂Φ∂u+ɛαβ∂Φ∂ɛαβ+ɛαβ(1)∂Φ∂ɛαβ(1)=Φ(u,ɛαβ,ɛαβ(1)).
Then the entropy (solution of this PDE) appears as the potential (9).Corollary 2.
If the specific entropys=Φ(u,ɛαβ,ɛαβ(1)) is an homogeneous function of order one, then(1)
the variabless,u,ɛαβ,ɛαβ(1) are conjugated to the intensive variables T,ταβ(eq),ταβ(1);(2)
the variabless,u,ɛαβ,ɛαβ(1) are not essential parameters (because they are not independent).Proof.
From the expression (9) we find
(12)Tds+sdT=du-1ϱ(ɛαβdταβ(eq)+ταβ(eq)dɛαβ)+1ϱ(ɛαβ(1)dταβ(1)+ταβ(1)dɛαβ(1)).(1)
Replacing the relation (5), we get
(13)sdT=-1ϱɛαβdταβ(eq)+1ϱɛαβ(1)dταβ(1),and the first statement is true.(2)
The foregoing relation shows that, for example,(14)ɛαβs,ɛαβ(1)sare essential parameters.


## 3. Specific Entropy via Least Squares Lagrangian

The most convenient way to fix a representativeΦ of the specific entropy s is to look at (5) as a partial derivative evolution equation and to build the least squares method Lagrangian L(15)2L=∥T-1-∂Φ∂u∥2+∥ταβ(eq)+ρT∂Φ∂ɛαβ∥2+∥ταβ(1)-ρT∂Φ∂ɛαβ(1)∥2
and the functional
(16)∫Ω‍L(u,ɛαβ,ɛαβ(1),Φ,Φu,Φɛαβ,Φɛαβ(1))dΩ,
where Φu=∂Φ/∂u, Φɛαβ=∂Φ/∂ɛαβ, and Φɛαβ(1)=∂Φ/∂ɛαβ(1).The extremals are solutions of the Euler-Lagrange PDE(17)∂L∂Φ-(Du∂L∂Φu+Dɛαβ∂L∂Φɛαβ+Dɛαβ(1)∂L∂Φɛαβ(1))=0,
where D* is the total derivative with respect to the variable *.In our case, we get(18)∂L∂Φ=0,∂L∂Φu=-(T-1-∂Φ∂u),∂L∂Φɛαβ=ρT(ταβ(eq)+ρT∂Φ∂Φɛαβ),∂L∂Φɛαβ(1)=-ρT(ταβ(1)-ρT∂Φ∂Φɛαβ(1)).
So that, by replacing the partial derivatives of L in (17), there follows the Laplace equation for the entropy
(19)∂2Φ∂u2+ρ2T2∂2Φ∂ɛαβ2+ρ2T2∂2Φ∂ɛαβ(1)2=0.
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
(20)T(t)s˙(t)=u˙(t)-νταβ(eq)(t)ɛ˙αβ(t)+νταβ(1)(t)ɛ˙αβ(1)(t).
In order to be a geodesic, this curve must minimize the energy functional
(21)J=12∫t0t1‍(T˙2(t)+s˙2(t)+u˙2(t)+δαγδβδτ˙αβ(eq)τ˙γδ(eq)+δαγδβδτ˙αβ(1)τ˙γδ(1)+δαγδβδε˙αβε˙γδ+δαγδβδε˙αβ(1)ε˙γδ(1)T˙2(t)+s˙2(t)+u˙2(t)+δαγδβδτ˙αβ(eq))dt.
In short, we must solve the problem
(22)minJsubjectto(20).
To solve this problem, we use the method of Lagrange multipliers. For this we defined the constrained Lagrangian
(23)L1=12(T˙2(t)+s˙2(t)+u˙2(t)+δαγδβδτ˙αβ(eq)(t)τ˙γδ(eq)(t)+δαγδβδτ˙αβ(1)(t)τ˙γδ(1)(t)+δαγδβδε˙αβ(t)ε˙γδ(t)+δαγδβδε˙αβ(1)(t)ε˙γδ(1)(t)T˙2(t)+s˙2(t)+u˙2(t)+δαγδβδτ˙αβ(eq))+p(T(t)s˙(t)-u˙(t)+νταβ(eq)(t)ε˙αβ(t)-νταβ(1)(t)ε˙αβ(1)(t)T(t)s˙(t)-u˙(t)+νταβ(eq)),
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)
(24)∂L1∂xi-Dt∂L1∂x˙i=0,
where xi are the generalized coordinates:
(25)x1=T,x2=s,x3=u,xαβ4=ταβ(eq),xαβ5=ɛαβ,xαβ6=ταβ(1),xαβ7=ɛαβ(1).
So that explicitly from (23) and (24) we have
(26)ps˙-T¨=0,ddt(s˙+pT)=0,ddt(u˙-p)=0,pνɛ˙αβ-ddtτ˙αβ(eq)=0,pνɛ˙αβ(1)+ddtτ˙αβ(1)=0,ddt(ɛ˙αβ(1)-pνταβ(1))=0,ddt(ɛ˙αβ+pνταβ(eq))=0.The equations (26) with the condition (20) are the differential equations of geodesics.Let(27)T0=T(0),T˙0=T˙(0),s0=s(0),s˙0=s˙(0),u0=u(0),u˙0=u˙(0),τ0αβ(eq)=ταβ(eq)(0),τ˙0αβ(eq)=τ˙αβ(eq)(0),τ0αβ(1)=ταβ(1)(0),τ˙0αβ(1)=τ˙αβ(1)(0),ɛ0αβ=ɛαβ(0),ɛ˙0αβ=ɛ˙αβ(0),ɛ0αβ(1)=ɛαβ(1)(0),ɛ˙0αβ(1)=ɛ˙αβ(1)(0)
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:
(28)u(t)=u˙0t+u0s(t)=T˙0pcospt+s˙0psinpt+(s0-T˙0p)T(t)=T˙0psinpt-s˙0pcospt+(T0+s˙0p)ταβ(eq)=[τ0αβ(eq)-1νp(ɛ˙0αβ+pντ0αβ(eq))]cosνpt+1νpτ˙0αβ(eq)sinνpt+1νp(ɛ˙0αβ+pντ0αβ(eq))ɛαβ=-[τ0αβ(eq)-1νp(ɛ˙0αβ+pντ0αβ(eq))]sinνpt-1νpτ˙0αβ(eq)cosνpt+(ɛ0αβ+1νpτ˙0αβ(eq))ταβ(1)=[τ0αβ(1)+1νp(ɛ˙0αβ(1)+pντ0αβ(1))]cosνpt+1νpτ˙0αβ(1)sinνpt-1νp(ɛ˙0αβ(1)+pντ0αβ(1))ɛαβ(1)=-[τ0αβ(1)+1νp(ɛ˙0αβ(1)+pντ0αβ(1))]sinνpt-1νpτ˙0αβ(1)cosνpt-(ɛ0αβ(1)+1νpτ˙0αβ(1)).Proof.
Let us first solve the simplest equation: from (26)3 we have
(29)u˙-p=u˙0⟹u=(u˙0+p)t+u0.
From (26)1 it is
(30)ddt(ps-T˙)=0⟹T˙=p(s-s0)+T˙0
and deriving (26)2(31)s¨+pT˙=0.
By replacing T˙ with the previous expression we get
(32)s¨+p[p(s-s0)+T˙0]=0,
that is
(33)s¨+p2s=(p2s0-pT˙0).
This is a linear nonhomogeneous second-order (harmonic) equation whose solution is
(34)s(t)=(s0-T˙0p)+T˙0pcospt+s˙0psinpt.
With this function, from the expression
(35)T˙=p(s-s0)+T˙0,
we can compute also T(t):
(36)T˙=p[((s0-T˙0p)+T˙0pcospt+s˙0psinpt)-s0]+T˙0;
that is,
(37)T˙=T˙0cospt+s˙0sinpt.
The solution is
(38)T(t)=T˙0psinpt-s˙0pcospt+(T0+s˙0p).
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:
(39)ɛαβ⟹ɛαβ(1),ταβ(eq)⟹-ταβ(1).
Thus from (26)7 it is
(40)ɛ˙αβ+pνταβ(eq)=ɛ˙0αβ+pντ0αβ(eq);
that is,
(41)ɛ˙αβ=-pνταβ(eq)+(ɛ˙0αβ+pντ0αβ(eq)).
If we put this expression in (26)4 we get
(42)pν[-pνταβ(eq)+(ɛ˙0αβ+pντ0αβ(eq))]-ddtτ˙αβ(eq)=0,
and by some manipulation we get the (vectorial) linear second order harmonic equation for ταβ(eq):
(43)ddtτ˙αβ(eq)+p2ν2ταβ(eq)=pν(ɛ˙0αβ+pντ0αβ(eq)).
The solution is
(44)ταβ(eq)=[τ0αβ(eq)-1νp(ɛ˙0αβ+pντ0αβ(eq))]cosνpt+1νpτ˙0αβ(eq)sinνpt+1νp(ɛ˙0αβ+pντ0αβ(eq)),
so that by integrating (41) and using the previous equation, we can easily get the expression for ɛαβ(45)ɛαβ=-[τ0αβ(eq)-1νp(ɛ˙0αβ+pντ0αβ(eq))]sinνpt-1νpτ˙0αβ(eq)cosνpt+(ɛ0αβ+1νpτ˙0αβ(eq)).
With similar computations we get also the last two equations of (28).The Lagrangian multiplierp 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
(46)limt→∞u(t)=+∞,limt→∞|T(t)|≤|T0-s˙0p|+|T˙0p|+|s˙0p|,limt→∞|s(t)|≤|s0-T˙0p|+|T˙0p|+|s˙0p|,
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 u(47)u3=Aei(kx1-ωt),u1=u3=0
being i2=-1 and k=k1+ik2 the complex wave number, so that
(48)vs=ωk1
is the phase velocity and k2 is connected with the attenuation of the waves.Asɛαβ=1/2(∂uα/∂xβ+∂uβ/∂xα), from (1) one obtains
(49)k1=ωϱ{B(ω)(1+D(ω)+1)}1/2,k1=ωϱ{B(ω)(1+D(ω)-1)}1/2,
where
(50)B(ω)=R(d)0(τ)R(d)0(ɛ)+ω2(R(d)1(ɛ)-R(d)0(τ)R(d)2(ɛ))(R(d)0(ɛ)-ω2R(d)2(ɛ))2+(ωR(d)1(ɛ))2,B(ω)=ω2(R(d)0(ɛ)-R(d)1(ɛ)R(d)0(τ)-ω2R(d)2(ɛ))2{R(d)0(τ)R(d)0(ɛ)-ω2(R(d)1(ɛ)-R(d)0(τ)R(d)2(ɛ))}2.
The complex shear velocity is [14]
(51)vc=1vs+k2iω=G1(ω)+iG2(ω)ϱ,
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:
(52)k1=ω(ϱG1/2)(1+(G2/G1)2+1)G11+(G2/G1)2,k2=ω(ϱG1/2)(1+(G2/G1)2-1)G11+(G2/G1)2.
Let us consider the range of high frequency ωH≤ω≤ωU and ωτ≫1 (of order 102) so that no relaxation phenomena occur (see Figure 1).By putting(53)G1=G1H1,001sG2=G2H1,001-s,
where G1H=G1(ωH) and G2H=G2(ωH), (52) becomes
(54)k1=ωϱ2(G1H)21.0012s+(G2H)2+G1H1.001s(G1H)21.0012s+(G2H)2,k2=ωϱ2(G1H)21.0012s+(G2H)2-G1H1.001s(G1H)21.0012s+(G2H)2.
For the Polyisobutilene we have [15] the characteristic values which are
(55)σ=10-7sec.,ωH=3.2·1014Hz,ωU=6·1014Hz,G1U=2.4·109Pa,G2U=2.75·104Pa,
and the graphics confirm (see Figure 2) with experimental data the validity of the model proposed for viscoanelastic phenomena in continuous media.k
1 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.
(a)
(b)


## 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) (Figure2).



---

*Source: 524718-2013-11-10.xml*