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Stressratetype strainlimiting models for solids resulting from implicit constitutive theory
Advances in Continuous and Discrete Models volume 2023, Article number: 6 (2023)
Abstract
The main objective of this work is twofold. First, we investigate the stressratetype implicit constitutive relations for solids within the context of strainlimiting theory of material response. The relations we study are models for generalisations of elastic bodies whose strain depends on the stress and the stress rate. Secondly, we obtain travellingwave solutions for some special cases that are nonlinear in the stress. These are the first notion of solutions available in the literature for this type of models describing stressratetype materials.
1 Introduction
The usage of complex materials in technology forces mathematicians to model material response in such a way that it is not only general enough to explain observed phenomena in experiments, but also as simple as possible from the point of view of mathematical analysis. Due to the complexity in their material response as well as the presence of the vast spectrum of possibilities for modelling them, viscoelastic materials have attracted a serious amount of attention in recent years. In accordance with the requirements of applications, ratetype viscoelastic models that include information about the current values of the kinematical quantities are much more favourable over the integraltype models where one has to keep track of the history of these quantities. The same understanding is valid both for fluids and solids. From the modelling point of view, on the other hand, the independent variables differ depending on whether viscoelastic fluids or solids are modelled. In this manuscript, we aim to model stressratetype solids obeying the requirements of strainlimiting theory.
It all started when Rajagopal [1] introduced a new perspective for modelling elasticmaterial response starting with implicit constitutive relations. The approach attracted a lot of attention immediately due to the fact that it was possible to explain some experimentally observed phenomena where, after linearisation of the strain under the assumption of smallness of the deformation gradient, it was possible to obtain a nonlinear relationship between the linearised strain and the stress, which is not possible to obtain in classical elasticity. In those models, no restrictions are put on the state of the stress and hence they are called strainlimiting models. From a classical mechanical point of view, they are more appropriate for using the stress as a primitive variable supporting the reasoning that the stress is the cause of the deformation but not the effect.
Many studies exist on implicit constitutive modelling and strainlimiting response of materials including elastic and viscoelastic solids and fluids (we refer the reader to the review article by Şengül [2] and references therein both for mathematical treatments and experimental applications). In this context, mathematical analysis for the viscoelastic response of solids with strainratetype constitutive relations have been investigated in articles such as [3–5], and [6]. More recently, the threedimensional problem with periodic and Dirichlet boundary data have been studied by Bulíček, et al. in [7] and [8]. From an application point of view for such strainratetype models one can refer to experimental studies such as [9] where the mechanical behaviour of various types of silk and spider silk are investigated. Recently, in [10], Erbay and Şengül aimed to investigate models resulting from implicit constitutive relations depending on the stress and the stress rate. Following the approach introduced for fluids by Rajagopal and Srinivasa [11], they introduced a stressratetype model for solids, and showed that it is thermodynamically consistent. Such ratetype models are suitable for giving explanations to some experimental observations on various materials such as dragline silk (see e.g. [12]) where, following the approach of Holzapfel and Simo [13], a stresstype internal variable that obeys an evolution rule is introduced. As also noted in [6], due to the inherent causal relation modelling of such phenomena by defining the stress as a primitive variable instead of the strain is therefore more appropriate, which is one of the motivations to study such mathematical models.
The aim of this note is to develop a general stressratetype model for the response of solids within the context of strainlimiting theory using a thermodynamical point of view. We start by analysing the general nonlinear model and search for heteroclinic travellingwave solutions, which reduces the equation of motion to an ordinary differential equation. Due to having analytical techniques available in the theory of firstorder ordinary differential equations only in the cases where the derivative term is explicitly expressed as a function of other variables, we pass onto a special case that is linear in the stress rate. We look at travellingwave profiles for some special cases of nonlinearities chosen to be consistent with the restrictions we obtain during analysis.
2 Derivation of the model
We provide a brief introduction of the mathematical background covering fundamental equations that form the basis of continuum mechanics, and the formulation of implicit constitutive theory. Similar reviews can also be found in [2, 10, 14, 15].
Let u be the displacement of the body at the position \(\mathbf{x} \in \mathbb{R}^{3}\) of a particle at time t with respect to the undeformed reference configuration X, that is \(\mathbf{u} = \mathbf{x}\mathbf{X}\). The motion is expressed by the deformation map \(\chi (\mathbf{X})=\mathbf{x}\). The deformation gradient tensor F and the velocity field v are defined as \(\mathbf{F} = \frac{\partial \mathbf{x} }{\partial \mathbf{X}}\) and \(\mathbf{v}=\dot{\mathbf{x}}\); from these the definitions for linearised strain tensor, ϵ, and the symmetric part of the velocity gradient, D, are obtained as
The Almasi–Hamel strain tensor A and Cauchy–Green stretch tensor B, which is defined as \(\mathbf{B} = \mathbf{F} \mathbf{F}^{T}\), are related to ϵ as follows;
The strainlimiting theory of elastic solids is based on the assumption that
in some appropriate norm with \(\delta \ll 1\). This immediately implies that ignoring the higherorder terms in the definition of A we can approximate it by ϵ. As a result, by (2), we can replace the Cauchy–Green tensor B with \(\mathbf{I} + 2 \boldsymbol{\epsilon}\) in the constitutive relation for the response of the material.
This approach has been adopted in several studies investigating the response of elastic solids (see e.g. [16]) starting from the general implicit relations of the form \(\mathcal{F}(\mathbf{T}, \mathbf{B})=0\). Recently, models for viscoelastic materials have been investigated by Rajagopal and Saccomandi [3] and Erbay and Şengül [4]. In these studies, the response of material is strainrate type and hence the general constitutive relation to start with is given by \(\mathcal{F}(\mathbf{T}, \mathbf{B}, \mathbf{D}) = 0\). In this work, on the other hand, we are interested in the stressratetype viscoelastic response and hence the implicit constitutive relation we would like to consider is given by
Even though there is a vast literature on ratetype models for material response in the classical setting, there is no systematic way of generating new models that satisfy the requirements of thermodynamics. In [17], Prusa and Rajagopal considered general constitutive relations for fluids between the histories of the Cauchy stress and the relative deformation gradient, and showed that fluids defined through such models are in fact generalisations of ratetype fluids. This was a big step towards understanding new material phenomena by posing new models that are mechanically meaningful and mathematically tractable at the same time. Later, in [18], Rajagopal investigated anisotropy for simple materials where the class of response relations under considerations was implicit relations between the history of the stress, the history of the density, and the history of the deformation gradient. More recently, Erbay and Şengül [10] showed that by considering (4), it is possible to obtain a thermodynamically consistent model for stressratetype viscoelastic solids within the context of strainlimiting theory. This is a big step towards understanding new material phenomena by posing new models that are mechanically meaningful and mathematically tractable at the same time. In this work, we would like to generalise the model introduced in [10] as well as show the existence of travellingwave solutions under certain conditions on the nonlinearities, which is the first notion of solutions investigated in this context. More precisely, we would like to consider the constitutive relation
As noted by Şengül in [2] (see also [10]) under the assumption of isotropy, from (5) one obtains
with the scalar functions \(\alpha _{i}\) (\(i = 0, \ldots , 8\)) depending on the invariants
We would like to follow the approach of smallness of the displacement gradient as in (3). However, due to having the stress rate rather than the Cauchy–Green tensor in (4), in this case we need to make the following assumptions;
Using (7), we are able to replace B with \(\mathbf{I} + 2 \boldsymbol{\epsilon}\) in the constitutive relation (5).
3 Thermodynamical analysis
In the work of Erbay and Şengül [10], thermodynamical analysis of a model obtained from (5) under the assumptions of (7) was carried out using a Gibbs potential formulation. It was shown that the model
where h is the derivative of the complementary free energy satisfying \(h(0) = 0\) and κ is a nonnegative constant, is thermodynamically consistent. This meant that a model with (8) as the constitutive relation satisfies the first and second laws of thermodynamics in the form of the Clausius–Duhem inequality. The key idea was that a formulation based on the complementary free energy (or equivalently Gibbs free energy) was used instead of the classical formulation based on the Helmholtz energy function so that it was not necessary to introduce strainlike quantities. As a result, the thermodynamic potentials depend on the stress and the stress rate.
Without loss of generality, assuming that the density changes due to the propagating waves are neglected, similar arguments to those in [10] lead to
where \(A = A(T, \dot{T})\) is the Almasi–Hamel strain tensor and \(\phi _{c}\) is the complementary energy, which can be shown to be independent of the stress rate, that is, \(\phi _{c} = \phi _{c}(T)\). This inequality suggests that A consists of two parts, one that is equal to \(\partial _{T}\phi _{C}\), and the other, say \(l(T, \dot{T})\), which satisfies
Since l, in general, depends nonlinearly on T and Ṫ, it is difficult to draw a conclusion from (10). However, this is not the case when l depends linearly on the stress rate. Therefore, we will consider the case when
where \(h(T) = \partial _{T} \phi _{c}\) for some scalar function γ. In this case, one can conclude from the thermodynamic inequality (9) that \(\gamma (T) \geq 0\).
4 Travellingwave solutions
In onespace dimension, the linearised strain can be written as \(\epsilon = \partial _{x} u(x, t)\), where u is the displacement function and \(\partial _{x}\) stands for partial differentiation with respect to x. As expressed in [10], owing to (2), the Almansi–Hamel strain \(A(x, t)\) can be rewritten as
With the smallstrain assumption, we are able to replace A by ϵ in the equations derived above. Furthermore, in this case, the difference between the quantities measured in the reference and current configurations disappears and the material time derivatives become partial derivatives with respect to time. As a result of separating the Almansi–Hamel strain tensor into elastic and dissipative parts as in the previous section, we consider
Using the equation of motion given as \(u_{tt} = \partial _{x} T\) as well as the fact that in onespace dimension the linearised strain is \(\epsilon = \partial _{x}u\), we obtain
Passing to the travellingwave variable \(\xi = x  ct\) so that \(T = T(\xi )\) and \(T_{t} = c\,T'\), from (13) we obtain the ordinary differential equation given as
where ′ stands for differentiation with respect to ξ. We are considering heteroclinic travelling waves taking two constant values at minus and plus infinity, say \(T_{\infty}^{}\) and \(T_{\infty}^{+}\), respectively. Therefore, the boundary conditions we require are the following:
Integrating (14) introduces an integration constant. Using boundary conditions (15), we find that the integration constant is zero due to
Integrating once more with respect to ξ, and using \(T_{\infty}^{}\) and \(T_{\infty}^{+}\) as the two constant states at infinities in order to find the integration constant explicitly, we obtain
Clearly, (16) is a firstorder ordinary differential equation that is nonlinear in its highestorder derivative term. Therefore, in order to be able to proceed with the solution, it is more convenient to look at the case when the nonlinearity is linear in the highestorder derivative term (see also (11)). This means we should have started with the constitutive relation given by
Here, the stress dependence is nonlinear, whereas the stressrate dependence is linear. Once again, using the fact that in the onedimensional setting, \(\epsilon = u_{x}\) together with the equation of motion \(u_{tt} = T_{x}\), we obtain the partial differential equation
Looking at the second term on the lefthand side, we can define a function \(\psi (T)\) such that \(\psi (T)_{t}= \gamma (T) T_{t}\) so that we can rewrite (18) as
Defining ψ in this manner requires \(\psi _{T}(T) = \gamma (T)\), which we will use later. Looking at the travellingwave solutions of the form \(T = T(\xi )\) with \(\xi = x  ct\) again, we obtain
Integrating (20) once and using (15) as before to find the corresponding integration constant, say \(A_{1}\), we find that \(A_{1} = 0\) so that
holds. Integrating one more time we obtain
where \(A_{2}\) is the integration constant of this step. Now, using (15) again, we obtain that \(c^{2} h(T_{\infty}^{\pm}) = T_{\infty}^{\pm} + A_{2}\) and adding these equalities we find
so that the wave speed c is found to satisfy
Using the above value of \(A_{2}\) in (21) and writing \(\psi '(T) = \gamma (T) T'\) we obtain
Equation (23) is a firstorder ordinary differential equation and it possesses solutions whose implicit form can be found by a single integration. In order to be able to find solutions explicitly, we need to consider the restrictions we might need on the nonlinear functions γ and h. First, it is easy to check that the equilibrium points are \(T = T_{\infty}^{\pm}\). Moreover, considering the constitutive relation (17), when \(\gamma (T)= 0\) there is no stressrate part, which means we must be in the elastic case. Therefore, in this case, we cannot obtain heteroclinic travellingwave solutions. Similarly, when \(h(T)\) is linear in T, that is, \(h(T) = h'(0) T\) with \(h'(0) \neq 0\), we can only obtain a constant solution.
We can summarise our findings as follows:
Theorem 1
Assume that \(h \in C^{2}\), \(\psi \in C^{3}\), and the following conditions hold;

(a)
\(\psi \colon \mathbb{R} \to \mathbb{R}\) is a strictly increasing function of its variable;

(b)
\(h\colon \mathbb{R} \to \mathbb{R}\) is such that \(h(z) \neq h'(0) T\) with \(h'(0)\neq 0\);

(c)
\(T(z) \to T_{\infty}^{+}\) as \(z\to \infty \), and \(T(z) \to T_{\infty}^{}\) as \(z \to  \infty \).
Then, equation (18) possesses heteroclinic, travellingwave solutions \(T(\xi ) \in \mathcal{S}\) with the travellingwave variable \(\xi = x  c t\). The two constant states of the waves are given by \(T_{\infty}^{+}\) and \(T_{\infty}^{}\), which are also the equilibrium states of the equation, and c is the wave speed given by (22).
5 Travellingwave profiles for some special cases
Since heteroclinic travelling waves propagate from one constant state to another when \(c^{2}>0\), using (22) we must have either \(T_{\infty}^{} > T_{\infty}^{+}\) and \(h(T_{\infty}^{}) > h(T_{\infty}^{+})\), or \(T_{\infty}^{} < T_{\infty}^{+}\) and \(h(T_{\infty}^{}) < h(T_{\infty}^{+})\). Without loss of generality, we can take the first case and choose
In this case, we find that \(c^{2} = 1/h(1)\) so that (23) reduces to
Assuming that \(h(0) =0\), this equation reduces to
Looking at this equation and using (24) as equilibrium points, we can deduce that \(\gamma (0)\neq 0\) and \(\gamma (1) \neq 0\). Finally, one can show that the fact that \(T(\xi )\) is a solution implies that \(T(\xi +p)\) is also a solution, for any p. Hence, without loss of generality, we can choose \(T(0) = 1/2\).
To find the solution \(T(\xi )\) one can rewrite (26) to obtain
and integrating with respect to the travellingwave variable ξ we obtain
where the integration constant \(A_{3}\) depends on the integrated form of the expression on the left. Letting
one sees from (28) that
Taking all the restrictions we obtained for γ and h into account, in order to illustrate the travellingwave profiles, we will first consider when
The readers are referred to Appendix A for the calculations leading to the solution given in the implicit form as
which is plotted in Fig. 1.
These choices of nonlinearities clearly satisfy the restrictions required in Theorem 1 and hence they are important.
Next, we consider the case when
By keeping h the same and changing only the function γ, we aim to see how dramatically the travellingwave profile is effected. Moreover, in this case, \(\gamma > 0\) is satisfied only for a certain \(\mathcal{S}\) rather than the whole real line as in the previous case. For this case, we obtain the solution given implicitly as
for which the calculations are available in Appendix B and the travellingwave profile is given in Fig. 2.
Next, we consider the case when
This choice of functions is due to checking whether it is possible for the nonlinearity γ to be a transcendental function. It turns out that it is possible to have the existence of a travellingwave solution in this case. Here, we obtain
for which the travellingwave profile is shown in Fig. 3. The calculations can be found in Appendix C.
Our final choice for functions h and γ are
This choice of γ makes the linearised strain dependent on the arctangent function, which is also the case in the strainrate response of materials analysed in [6]. It is clear that a similar travellingwave profile is obtained here, in the stressrate model as well. We refer to Appendix D for the derivation of the solution that is implicitly expressed as
and is plotted in Fig. 4.
6 Conclusions
In this work, we introduce a stressratetype viscoelastic constitutive relation and analyse it from the point of view of thermodynamical consistency as well as the mathematical analysis of the corresponding heteroclinic travellingwave solutions. Our findings are important owing to the fact that this is the first time a stressratetype model for material response is shown to admit some notion of solutions. There are still a number of related open problems concerning the corresponding partial differential equation (18). This equation is different from classical equations resulting from mechanical theories in two respects, the first one being that the inertia term is nonlinear, and the second one that the unknown is not the displacement or the deformation, but the stress. As a result, there is no available way to approach the problem using standard techniques. In the case of the strainrate modelling, Erbay, Erkip and Şengül [19] obtained the localintime existence of solutions and Şengül [20] obtained global solutions under some restrictions of the nonlinearities for a slightly related partial differential equation by converting h and hence eliminating the nonlinearity on the inertia term as a result of a series change of variables. It would be favourable to attack these equations directly rather than trying to put them in classical form. However, such a mathematical tool does not currently exist. It is also worth mentioning that the partial differential equation obtained in the strainrate case and equation (18) have completely different behaviours in terms of their stabilities, as shown in [10] in the very simple case of linear h. Therefore, it is expected that a brand new approach should be adopted for (18). Hopefully, such a mathematical tool will be introduced as a result of careful investigations of experimental results on material behaviour.
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References
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First author’s email: dumanemre@sabanciuniv.edu.
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Appendices
Appendix A: The case of quadratics
Here, we consider the case when \(h(T) = T^{2}\), \(\gamma (T)=T^{2}+1\). From equation (22) we can conclude that \(c=1\). Inserting these into (28) we obtain
which implies
and hence \(T2 \log (T1)+\log (T)=\xi +A_{3}\). Inserting \(T(\xi =0)=1/2\) we are able to find the integration constant as \(A_{3} = \log (2)\frac{1}{2}\).
Appendix B: The case of quadratic and quartic
Here, we consider the case when \(h(T) = T^{2}\), \(\gamma (T)= (T\frac{1}{2} )^{4}\). From equation (22), we have \(c=1\) again. Inserting these into (28) we obtain
implying
Now, for brevity of calculation we define \(k:=T^{2}T\). It is important to note that this is not a change of variables and that the integration variable is still dT. We obtain
This gives
As a result we obtain
Inserting \(T(\xi =0)=1/2\) we are able to find the integration constant as \(A_{3} = \frac{1}{6}\).
Appendix C: The case of quadratic and logarithm
In this case we consider \(h(T) = T^{2}\), \(\gamma (T)=\log (2+T )\). From equation (22), we have \(c=1\) again. Inserting these into (28) we obtain
which is equivalent to
This gives
Defining \(k:=2+T\), the lefthand side becomes \(\int \frac{\log k}{k3}\,dk +\int \frac{\log k}{k2}\,dk \). Since both integrals are of the form \(\int \frac{\log x}{x  a} \,dx \), we evaluate this expression to insert the value for a later. We can also write the same expression by defining \(y:=xa\) as follows:
Now, we can define \(z=\frac{y}{a}\) to obtain \(\log a \log y + \int \frac{\log (z+1 )}{z} \,dz \). Using the polylogarithm identity
Inserting the values in place we obtain
Inserting \(T(\xi =0)=1/2\) we are able to find the integration constant as \(A_{3} = \operatorname{Li}_{2} (\frac{1}{4} )+\operatorname{Li}_{2} ( \frac{1}{6} )\log ^{2}(2)\log (3)(\log (2)+i \pi )\).
Appendix D: The case of quadratic and arctangent
We consider \(h(T) = T^{2}\), \(\gamma (T)=\arctan (T )+1\). From equation (22) we obtain \(c=1\). Inserting these into (28) it follows that
We use the complex logarithm form of the inverse tangent function to obtain
which can be rewritten as
Calculating the integrals, we obtain
As a result, we obtain
With some simplifications we have
Inserting \(T(0)=1/2\) we are able to find the integration constant as
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Duman, E., Şengül, Y. Stressratetype strainlimiting models for solids resulting from implicit constitutive theory. Adv Cont Discr Mod 2023, 6 (2023). https://doi.org/10.1186/s1366202303751x
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DOI: https://doi.org/10.1186/s1366202303751x
Keywords
 Stressratetype models
 Strainlimiting theory
 Implicit constitutive relation