The formulation of the constitutive model including a set of parameters is here followed by the symmetry analysis, leading to invariance relations and master responses inherent to the model. In the context of rheology, this approach proves quite promising for dissipative material with a complex nonlinear constitutive law, including the state laws satisfied by the observable variables and the kinetics of internal variables traducing the evolution laws of the internal variables. The invariants associated to those symmetries can then be calculated and used to synthesize the material’s response into master responses reflecting the material’s response when the control parameters change. The proposed algorithm reflecting the entire methodology is summarized in the diagram of Fig. 1. The formulation of rheological models can be guided by the framework of the thermodynamics of irreversible processes, as explained in the sequel.

### Construction of the constitutive laws in the framework of TIP

In mechanics or rheology, constitutive laws for dissipative materials are generally written under the umbrella of the thermodynamics of irreversible processes (TIP) [42], relying on the energy density to describe the energetic part of the constitutive law and the dissipation potential for its dissipative part. Relying on Callen’s axiomatic, [43], and a generalization of De Donder’s thermodynamics, let the total internal energy \(E(\varvec{y},\varvec{z})\) characterizing the medium depend on a set of extensive variables \(\varvec{y}=(S,V\varepsilon ,N_k...)\), where *S*, \(V\varepsilon \), and \(N_k\) are respectively the entropy, the volume weighted deformation, the number of moles of the \(k^{th}\) species, further identified to a set of internal variables \(\varvec{z}\) describing the irreversible evolution of the microstructure (such as plasticity, viscoplasticity, and damage). The thermodynamical system is a finite volume *V* with boundary \(\partial V\) of a solid continuum body. The extensity property of *E* expresses as

$$\begin{aligned} E(\lambda \varvec{y},\lambda \varvec{z})=\lambda E(\varvec{y},\varvec{z}). \end{aligned}$$

(4.1)

Taking the derivative of previous equation with respect to \(\lambda \) at \(\lambda =1\), the Euler relation is obtained in the form of the Gibbs relation

$$\begin{aligned} E(\varvec{y},\varvec{z})=\varvec{Y}(\varvec{y},\varvec{z}).\varvec{y}-\varvec{A}(\varvec{y},\varvec{z}).\varvec{z} \end{aligned}$$

(4.2)

satisfied by the internal energy \(E(\varvec{y},\varvec{z})\), with \(\varvec{Y}=E_{,\varvec{y}}\) the intensive variables (vector) and \(\varvec{A}=-E_{,\varvec{z}}\) the generalized non equilibrium forces respectively. The intensive variables are dual of the extensive ones in a thermodynamical sense. For example, the second order Cauchy stress tensor \(\sigma \), is conjugated to the second order strain tensor \(\varepsilon \). The extensive variables are called control variables, since they are the arguments of an appropriate thermodynamical potential. Accounting for the Gibbs relation:

$$\begin{aligned} \frac{d E}{d t}(\varvec{y},\varvec{z})=\varvec{Y}(\varvec{y},\varvec{z}).\frac{d \varvec{y}}{d t}-\varvec{A}(\varvec{y},\varvec{z}).\frac{d \varvec{z}}{d t}, \end{aligned}$$

(4.3)

the Gibbs-Duhem relation results from the differentiation of Eq. (4.2), viz

$$\begin{aligned} \varvec{y}.\frac{d \varvec{Y}}{d t}-\varvec{z}.\frac{d \varvec{A}}{d t}=0. \end{aligned}$$

(4.4)

The constitutive law is specified in rate form from the total time derivation of the intensive variables \(\varvec{Y}\) and \(\varvec{A}\) as

$$\begin{aligned} \begin{array}{rcl} \dot{\varvec{Y}}(\varvec{y},\varvec{z})&{}=&{}\varvec{\varvec{a^u}}(\varvec{y},\varvec{z}).\dot{\varvec{y}}+\varvec{\varvec{b}}(\varvec{y},\varvec{z}).\dot{\varvec{z}}\\ -\dot{\varvec{A}}(\varvec{y},\varvec{z}) &{}=&{} \varvec{\varvec{b}}^T(\varvec{y},\varvec{z}).\dot{\varvec{y}}+\varvec{\varvec{g}}(\varvec{y},\varvec{z}).\dot{\varvec{z}} \end{array} \end{aligned}$$

(4.5)

with \(\varvec{\varvec{a^u}}(\varvec{y},\varvec{z})=E_{,\varvec{y}\varvec{y}}\) the Tisza matrix, \(\varvec{\varvec{b}}(\varvec{y},\varvec{z})=E_{,\varvec{z}\varvec{y}}\), \(\varvec{\varvec{g}}(\varvec{y},\varvec{z})=E_{,\varvec{z}\varvec{z}}\) the coupling and the dissipation matrices respectively, [44].

In the more general setting of the theory of irreversible process (abbreviated T.I.P.) or the thermodynamics with internal variables (abbreviated T.I.V.), the kinetic laws of the internal variables are written based on the setting up of the dissipation potential *D*, a positive and homogeneous function of degree *n* in the rate of the internal variables, [42], allowing to express the thermodynamic forces or affinities reflecting all dissipative variables \(\varvec{A}\) versus the dual internal variable \(\varvec{z}\) as

$$\begin{aligned} \varvec{A}{(\varvec{z})}=\frac{\partial D}{\partial \dot{\varvec{z}}}. \end{aligned}$$

(4.6)

Using previous relation results in a dissipation expressing as

$$\begin{aligned} \Phi =\varvec{A}.\dot{\varvec{z}}=\frac{\partial D (\varvec{z})}{\partial \dot{\varvec{z}}}=nD \ge 0 \end{aligned}$$

(4.7)

due to Euler’s identity for homogeneous functions of degree *n* (in classical T.I.P., *D* is taken as homogeneous of degree two). The previous relation can be inverted to give the rate of the internal variable versus the affinity, using the Legendre-Fenchel transform, involving the pseudo-potential of dissipation, defined as

$$\begin{aligned} D^*(\varvec{A})=\sup _{\varvec{A}}\,(\varvec{A}.\dot{\varvec{z}} - D(\dot{\varvec{z}})) \end{aligned}$$

(4.8)

where \(\varvec{A}\) is restricted to a convex set *K*, [42]. When \(D^*(\varvec{A})\) is differentiable, the evolution law of the internal variable \(\varvec{z}\) is given by

$$\begin{aligned} \dot{\varvec{z}}=\frac{\partial D^*(\varvec{A})}{\partial \varvec{A}}. \end{aligned}$$

(4.9)

Choosing as a specific model a quadratic and convex pseudo potential of dissipation \(D^*\), one obtains the following set of independent non linear kinetic relations (for a number of *N* processes)

$$\begin{aligned} \dot{z_k}=-\frac{z_k-z_k^r}{\tau _k}, \qquad k=1..N, \end{aligned}$$

(4.10)

considering a spectral decomposition of dissipative phenomena modelled by internal variables \(\dot{z_k}\). The variables \(z_k^r\) represent the value of the internal variable *z* at its relaxed equilibrium state, for the dissipation mode *k*, defined as the state for which the thermodynamic affinity \(A^r\) vanishes. The distribution of the relaxation times therein \(\tau _k\) is described from Prigogine’s fluctuation theorem, the contribution of each mode being proportional to the square root of the corresponding relaxation time [44].

A Lagrangian formulation of the previous dissipative constitutive laws is further constructed by the homotopy formula from the self-adjoint system of PDE’s (4.5), accounting for the Gibbs relation Gibbs-Duhem relations (4.4), see [45, 46]:

$$\begin{aligned} L=E_{,\varvec{y}}\cdot \dot{\varvec{y}}+ E_{,\varvec{z}}\cdot \dot{\varvec{z}}=\frac{d E(\varvec{y},\varvec{z})}{d t}. \end{aligned}$$

(4.11)

This thermodynamic Lagrangian incorporates the thermodynamic information related to the material, in terms of relations between the extensive control variables and the dual intensive thermodynamic forces, where the generalized coordinates \(\{\varvec{y},\varvec{z},\dot{\varvec{y}},\dot{\varvec{z}}\}\) correspond to the Lagrange variables of the considered spatially uniform system. The set of internal variables completes the set of control variables, ensuring the self-adjointness of the constitutive equations [45]. The kinetic information, viz the set of equations (4.10) is then incorporated into the previous thermodynamic Lagrangian via Lagrange multipliers \(\varvec{\lambda }=\{\lambda _k,k=1..N\}\), hence the augmented Lagrangian, \(L_{aug}\), accounting for both the thermodynamic and the kinetic information inherent to the material’s constitutive law, writes

$$\begin{aligned} l_{aug}={\dot{e}}+\sum _{k=1}^N \lambda _k \left( {\dot{z}}_k+\frac{z_k-z_k^r}{\tau _k}\right) , \end{aligned}$$

(4.12)

where the internal energy *E* has been replaced by the internal energy density *e*, which also satisfies the Maxwell conditions. The Lagrangian density \(l_{aug}\) in Eq. (4.12) is the volumetric density of a Lagrangian \(L_{aug}\), such that

$$\begin{aligned} \ L_{aug}=\int _{V} l_{aug} dV. \end{aligned}$$

(4.13)

The Lagrange multipliers in \(l_{aug}\) resemble thermodynamic affinities dual to the rate of internal variables, hence the kinetic part of the total Lagrangian thereabove resembles a dissipation potential. Observe that the incorporation of the kinetic laws Eq. (4.10) clearly breaks the symmetry (discrete symmetry) under time reversal \(t\rightarrow -t\), since the new Lagrangian contains an irreversible information. The set of Lagrange equations obtained from the stationarity condition of the action functional built from \( l_{aug}\)is completely equivalent to the constitutive equations (4.5) and the kinetic laws (4.10). The kinetic evolution equations Eq. (4.9) have symmetry properties depending on the specific form taken by the relaxation time therein.

### Symmetry analysis of the augmented Lagrangian

Relying on the formulated least action principle of the dissipative constitutive laws, we next exploit the associated variational symmetries to determine invariance properties of the set of constitutive equations (thermodynamic and kinetic informations), relying on the methodology presented in [45,46,47,48]. We focus in the sequel (but without lost of generality) on the specific energy \(e(\varepsilon ,s,z_k)\), depending on the strain \(\varepsilon \), the specific entropy *s*, and some specific internal variables \(z_k\), \(k=1..N\), accounting for internal dissipative phenomena (related to the microstructure).

The search for the variational symmetries of the Jacobi action built from the augmented Lagrangian amounts to find the infinitesimal generators on the (first order) jet space sustained by the control variables \(\{t,\varepsilon ,s,z_k\}\). As a concrete illustration, and without loss of generality, considering the time *t* as the sole independent variable and as the dependent variables the vector \(\varvec{y}={\varepsilon ,s,z_k}\), a generator of a symmetry group expresses as

$$\begin{aligned} \varvec{v}_{cont}=\xi \frac{\partial }{\partial t}+\phi ^\varepsilon \frac{\partial }{\partial \varepsilon }+\phi ^s \frac{\partial }{\partial s}+\phi ^{z_k} \frac{\partial }{\partial z_k} \end{aligned}$$

(4.14)

with the variations of the dependent variables given by

$$\begin{aligned} \delta \varepsilon =\mu \phi ^\varepsilon ~~;~~\delta s=\mu \phi ^s ~~;~~ \delta z_k=\mu \phi ^{z_k}, \end{aligned}$$

(4.15)

with \(\mu \) the group parameter, [5]. In Eq. (4.14), the subscript “cont” means that only components with respect to control variables are at first introduced. The variations given in Eq. (4.15) are responsible for the variations of intensive observable variables \(\delta \sigma \), \(\delta T\), and \(\delta A_k\), obtained from the second partial derivatives of the potential *e*. The symmetry group acting on the set of variables \(\{t,\varepsilon ,s,z_k\}\) is automatically extended to the enlarged set of variables \(\{t,\varepsilon ,s,z_k,\sigma ,T,A_k\}\): thereby, the total vector field, generator of the symmetries of the constitutive laws, is decomposed into the sum of the “control” vector field and the “observable” vector field:

$$\begin{aligned} \varvec{v}=\varvec{v}_{cont}+\varvec{v}_{obs} \quad \text {with} \quad \varvec{v}_{obs}=\phi ^{\sigma }\frac{\partial }{\partial \sigma }+\phi ^T \frac{\partial }{\partial T}+ \phi ^{A_k}\frac{\partial }{\partial {A_k}}. \end{aligned}$$

(4.16)

The components of the intensive variables of the observable and internal vector fields are given by the structure of the constitutive law (Eq. 4.5) for the elementary representative volume element (RVE):

$$\begin{aligned} \phi ^\sigma= & {} e_{,\varepsilon \varepsilon } \phi ^{\varepsilon }+e_{,s \varepsilon } \phi ^{s}+e_{,z_k \varepsilon } \phi ^{z_k} \end{aligned}$$

(4.17)

$$\begin{aligned} \phi ^T= & {} e_{,\varepsilon s} \phi ^{\varepsilon }+e_{,ss} \phi ^{s}+e_{,z_k s} \phi ^{z_k} \end{aligned}$$

(4.18)

$$\begin{aligned} \phi ^{A_i}= & {} e_{,\varepsilon z_i} \phi ^{\varepsilon }+e_{,s z_i} \phi ^{s}+e_{,z_k z_i} \phi ^{z_k}. \end{aligned}$$

(4.19)

The components of the vector field \(\varvec{v}_{obs}\) are thus completely determined by those of the vector field \(\varvec{v}_{cont}\). Let return now to a general situation, and decompose the Lagrangian of Eq. (4.12) into a thermodynamic and a kinetic contribution, as

$$\begin{aligned} l_{aug}=l_{thermo}+l_{kine} \end{aligned}$$

(4.20)

with

$$\begin{aligned} l_{thermo}={\dot{e}} \text{ and } l_{kine}= \sum _{k=1}^N \lambda _k \Big (\dot{z_k}+\frac{z_k-z_k^r}{\tau _k}\Big ). \end{aligned}$$

(4.21)

As shown previously, the invariance condition associated with the sole contribution \(l_{thermo}\) is automatically verified, see also [45]. The satisfaction of the Euler-Lagrange equations for \({\dot{e}}\) as a Lagrangian is fully equivalent to the Maxwell conditions for the second order partial derivatives of the potential function *e*. Thus, the invariance condition of the action integral simplifies to the differential condition involving the sole kinetic information (and kinetic Lagrangian)

$$\begin{aligned} \mathrm{pr}^{(1)} \varvec{v} l_{kine}+l_{kine} \mathrm{Div}\varvec{\xi }=0, \end{aligned}$$

(4.22)

accounting for the relation \(\mathrm{Div}\varvec{\xi }=D_t \varvec{\xi }=\dot{\varvec{\xi }}\).

The connection between the variational symmetry condition and the local symmetry of the field equations is next established. If \(\varvec{v}\) is a variational symmetry for some functional \(S=\int _t \int _V l dV dt\), it is also a symmetry of the corresponding Lagrange equations:

$$\begin{aligned} \mathrm{pr}^{(1)} \varvec{v} l+l \mathrm{Div}\varvec{\xi }=0 \quad \Rightarrow \quad \mathrm{pr}^{(1)} \varvec{v} \left( \varvec{E}(l)\right) =0, \end{aligned}$$

(4.23)

but the converse is generally not true: the set of all variational symmetries denoted by \(G_S\) is always included into the set of local symmetries, \(G_{\varvec{\Delta }}\), i.e. \(G_S \subset G_{\varvec{\Delta }}\). In the present case, we consider only variational symmetries along the optimal path where the constitutive laws are satisfied. Accounting for the expression of \(l_{kine}\), viz

$$\begin{aligned} l_{kine}=\sum _{k=1}^N \lambda _k \Big (\dot{z_k}+\frac{z_k-z_k^r}{\tau _k}\Big ), \end{aligned}$$

(4.24)

the condition \(l_{kine}=0\) is satisfied along the optimal path , hence the variational symmetry condition resumes to

$$\begin{aligned} \mathrm{pr}^{(1)} \varvec{v} l_{kine}=0 \text{ whenever } l_{kine}=0. \end{aligned}$$

(4.25)

Thanks to the linearity of the prolongation and doing a spectral decomposition of the dissipative mechanisms, the evolution of the internal variables are mutually independent, hence the symmetry conditions (4.25) may be decoupled, resulting in the new set of *N* independent symmetry conditions:

$$\begin{aligned} \mathrm{pr}^{(1)}\varvec{v}\Big (\dot{z_k}+\frac{z_k-z_k^r}{\tau _k}\Big )=0 \text{ whenever } \dot{z_k}+\frac{z_k-z_k^r}{\tau _k}=0, \end{aligned}$$

(4.26)

which is equivalent to the local symmetry condition.

The same condition holds for the more general evolution equation (4.9), thus allowing to extend the symmetry analysis to more complex dissipative phenomena. A classification of symmetries according to the form of the pseudo-potential of dissipation can also be done. Previous considerations entail that variational symmetries along the optimal path are fully equivalent to local symmetries directly computed from the constitutive laws.

The symmetries of the formulated augmented lagrangian formulation are those of the pseudo-potential of dissipation \(D^*(\varvec{A})\). The infinitesimal generator of the general invariance condition (4.25) is also split into the sum of a control and an observable vector field, with the components of the last contribution completely expressed from the components of the control vector; this implies searching for symmetries in a subset of the total jet space, since the state laws for the observable variables are automatically satisfied.

The presented general thermodynamically based material framework originates in pioneering works [49,50,51,52], and it covers a broad spectrum of rheological models in viscoelasticity, viscoplasticity, plasticity and also continuum damage mechanics.

In the next section, the invariance properties of the constitutive equations are exploited to set up a predictive methodology allowing to condense the materials response into equivalence principles. This has consequences on the experimental side, since the later enable to define equivalent experimental set up allowing a gain of time by selecting an optimal set of control variables (e.g. temperature, strain rate).

### Invariance properties of the constitutive equations

Considering for instance a general internal variable formulation of inelasticity for generalized standard materials, the symmetries of the constutive laws are implicitly reflected by the form taken by the thermodynamic potential (or the pseudo potential of dissipation introduced in Eq. (4.8)) and the dissipation function. The search for the Lie groups of constitutive laws is further used as a systematic tool for the construction of the so-called master curves, that condense the information related to the behavior of a material under varying experimental conditions. The experimental conditions are defined by the values taken by a set of control parameters such as temperature, strain rate. Hence, the knowledge of symmetry groups allows a prediction regarding the modification of the material’s response when these parameters vary. Thus, starting from a known set of constitutive equations, i.e. a known expression for *e*, \(z_k^r\) and \(\tau _k\), it is a priori possible to compute some symmetries of the behavior by applying the symmetry condition (4.26). Due to the equivalence of the local and variational symmetry conditions, a conservation law may be obtained using Noether’s theorem, viz \(\mathrm{Div}\varvec{P}=0\), with

$$\begin{aligned} P_i=\sum _{k=1}^q \sum _{j=1}^4 \xi _j u_{k,j}\frac{\partial l_{kine}}{\partial u_{k,i}}-\sum _{j=1}^q \phi _j \frac{\partial l_{kine}}{\partial u_{j,i}}-\xi _i l_{kine}, \end{aligned}$$

(4.27)

see e.g. [5], with \(l_{kine}\) the kinetic Lagrangian density, defined in the general thermodynamic setting exposed in the previous section as

$$\begin{aligned} l_{kine}=\sum _k \lambda _k \left( {\dot{z}}_k-\frac{\partial D^{*}(\varvec{A})}{\partial A_k}\right) \equiv 0. \end{aligned}$$

(4.28)

Recall that the dependent variables arguments of the kinetic Lagrangian, viz the variables \(u_{k}\), are internal variables. For a uniform elementary representative volume element, only the time derivative appears in previous conservation law expression; BVPs accounting for the spatial variation of the fields over spatial domains will be considered later on in this contribution.

### Application: construction of the master curve of a dry polyamid (PA66) and time temperature equivalence

The invariants build from the Lie symmetry analysis of the constitutive law will further be translated into master curves, as explained below. The constitutive model set up presently for the purpose of uniaxial tests involves the strain \(\epsilon \) as control variable, and the stress \(\sigma \) as the corresponding observable variable. The temperature *T*, dual of the entropy *s*, plays the role of a parameter. The internal variables \(z_k\) and the thermodynamic affinities \(A_k\) are not controlled; their values at the relaxed state are governed by the strain history. The exploitation of the general constitutive Eq. (4.5) and (4.10) delivers the following state law (see [53]):

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {{\dot{\sigma }}-E_u{\dot{\varepsilon }}+\sum _{k=1}^n b^1_k \frac{z_k-z_k^r}{\tau _k}=0}\\ \displaystyle {-{\dot{s}}+\alpha _u E_u {\dot{\varepsilon }}+\sum _{k=1}^n b^2_k \frac{z_k-z_k^r}{\tau _k}=0}\\ \displaystyle {-\dot{A_i}-b^1_i {\dot{\varepsilon }}+\sum _{k=1}^n g_{ik} \frac{z_k-z_k^r}{\tau _k}=0} \quad \quad i=1..n \end{array} \right. \end{aligned}$$

(4.29)

with \(E_u\) the constant instantaneous Young’s modulus, \(b^1_k\), \(\alpha _u\) (dilatation coefficients), \(b^2_k\), and \(g_{ik}\) being defined from the second order partial derivatives of the Helmholtz free energy, see [54]. The simplest model for the evolution of the internal variables is

$$\begin{aligned} z _k^r=c_k \varepsilon . \end{aligned}$$

(4.30)

If one starts from a particular expression of the kinetic laws, involving a temperature dependence of the relaxation times of Arrhenius kind, the following kinetic model may be written

$$\begin{aligned} {\dot{z}}_k+\frac{z_k-c_k \varepsilon }{\frac{h}{kT} \exp \left( \frac{\Delta H-T \Delta S_k}{RT} \right) }=0, \end{aligned}$$

(4.31)

where *h* is the Planck constant, *k* the Boltzmann constant, *R* the gas constant, \(c_k\), \(\Delta H\), \(\Delta S_k\) some material constants. The required symmetry condition expresses accordingly

$$\begin{aligned} \mathrm{pr}^{(1)}\varvec{v}\left( {\dot{z}}_k+\frac{z_k-c_k \varepsilon }{\frac{h}{kT} \exp \left( \frac{\Delta H-T \Delta S_k}{RT} \right) }\right) =0. \end{aligned}$$

(4.32)

The particular solution is obtained

$$\begin{aligned} \varvec{v}_0=\xi \frac{\partial }{\partial t}+\phi ^T\frac{\partial }{\partial T}+\phi ^{\sigma }\frac{\partial }{\partial \sigma }+\phi ^{s}\frac{\partial }{\partial s}+ \phi ^{A_k}\frac{\partial }{\partial A_k} \end{aligned}$$

(4.33)

with:

$$\begin{aligned} \xi =t;~~\phi ^T= & {} -\frac{R T^2}{R T+\Delta H};~~\phi ^{\sigma }=-\alpha _u E_u \phi ^T;\nonumber \\ \phi ^{A_k}= & {} -b^2_k;~ \phi ^T;~~\phi ^{\varepsilon }=0;~~\phi ^{z_k}=0. \end{aligned}$$

(4.34)

This solution is further interpreted as a mathematical formulation of the so-called time-temperature equivalence principle for polymers. The integration of the induced first order differential system leads to a one-parameter group of transformations, where the temperature is transformed as

$$\begin{aligned} {\bar{T}}= & {} \exp \left( L_W\left( \frac{\Delta H}{R}\exp \left( \frac{\mu T-T \ln (T^*)+\frac{\Delta H}{R}}{T}\right) \right) \right. \nonumber \\&\left. -\mu +\ln T^* -\frac{\Delta H}{RT}\right) \end{aligned}$$

(4.35)

with \(L_W(x)\) the Lambert function, and \(T^*=T/T_0\) with \(T_0=1\,\)K. The theoretical shift factor is obtained by inverting the previous implicit relation:

$$\begin{aligned} \frac{\mu (T,{\bar{T}})}{\ln 10}=\frac{\Delta H (T-{\bar{T}})}{R T {\bar{T}} \ln 10}+\frac{1}{\ln 10}\ln \frac{T}{{\bar{T}}}. \end{aligned}$$

(4.36)

This explicit expression of the group parameter \(\mu (T,{\bar{T}})\) highlights an invariance property satisfied by the secant modulus \(E_s(t,T)\), defined as the ratio of stress to strain:

$$\begin{aligned} E_s(t,T)=\frac{\sigma (t,T)}{\varepsilon (t,T)} =\frac{{\bar{\sigma }}({\bar{t}},{\bar{T}})}{{\bar{\varepsilon }}({\bar{t}},{\bar{T}})} =E_s({\bar{t}},{\bar{T}})=E_s(e^\mu t,{\bar{T}}) \end{aligned}$$

(4.37)

leading to the logarithmic relation between times:

$$\begin{aligned} \log {\bar{t}}=\log t +\frac{\mu }{\ln 10} \end{aligned}$$

(4.38)

This allows rewriting Eq. (4.37) under the form:

$$\begin{aligned} E_s(\log t,T)=E_s(\log t +\frac{\mu }{\ln 10 },{\bar{T}}). \end{aligned}$$

(4.39)

This relation links the two secant moduli \(E_s(\log {\bar{t}},{\bar{T}})\) and \(E_s(\log t,T)\) obtained at \({\bar{T}}\) and *T* by a translation by \(\frac{\mu }{\ln 10}\) on the time logarithmic scale, given explicitly in Eq. (4.36). The obtained invariance property is a theoretical formulation of the time-temperature equivalence principle. The latter was successfully confirmed by experimental data on various materials, and fitted with the empirical Williams-Landel-Ferry (WLF) expression (see e.g. [55,56,57]) or the Kohlrausch relation ([58,59,60]).

As an illustration of the time-temperature equivalence principle, and referring to the experimental validation of the predicted shift factor (DM-P5), let consider the data summarized in Fig. 2. The evolution of the secant modulus \(E_s(t,T)\) for isothermal tests on a polymer (polyamid 66, PA66) is plotted at different constant temperatures and for a given strain rate (\({\dot{\varepsilon }}=1.8 \times 10^{-4}\,\)s\(^{-1}\)). Every curve is parameterized by the temperature \({\bar{T}}\) (in the range 413K – 453K) and can be translated along the \(\log t\) axis with a horizontal shift factor \(a_{{\bar{T}} \rightarrow T}\) to coincide with a unique curve. This unique curve is called “master curve” at the reference temperature \(T=393K\), see Fig. 3.

The notion of equivalence transformation is exemplified below for the case of nonlinear wave propagation in fiber-reinforced materials, showing that the initial model including several parameters can be reduced to a model with no arbitrary parameters.