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
$$ A = \epsilon  \frac{1}{2}(\partial _{x} u)^{2}. $$
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
$$ \epsilon = h(T)  l(T, T_{t}). $$
(12)
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
$$ \bigl(h(T)  l(T, T_{t}) \bigr)_{tt} = \partial _{xx}T. $$
(13)
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
$$ T'' + c^{2} l''\bigl(T, c\,T'\bigr) = c^{2} h''(T), $$
(14)
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:
$$ T'(\xi ), T''(\xi ) \to 0\quad \text{as } \xi \to \pm \infty . $$
(15)
Integrating (14) introduces an integration constant. Using boundary conditions (15), we find that the integration constant is zero due to
$$ l'\bigl(T, c T'\bigr) = l_{T}\bigl(T, c T'\bigr) T'  c l_{T'}\bigl(T, c T'\bigr) T''. $$
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
$$ \begin{aligned} l\bigl(T, c T'\bigr) &= \frac{1}{c^{2}} \biggl( \frac{T_{\infty}^{+} + T_{\infty}^{}}{2}  T \biggr) + \biggl(h(T)  \frac{h(T_{\infty}^{+}) + h(T_{\infty}^{})}{2} \biggr) \\ &\quad {}+ \frac{l(T_{\infty}^{+}, 0) + l(T_{\infty}^{}, 0)}{2}. \end{aligned} $$
(16)
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
$$ \epsilon = h(T)  \gamma (T) T_{t}. $$
(17)
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
$$ h(T)_{tt}  \bigl(\gamma (T) T_{t} \bigr)_{tt} = T_{xx}. $$
(18)
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
$$ h(T)_{tt}  \psi (T)_{ttt} = T_{xx}. $$
(19)
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
$$ c^{2} h''(T) +c^{3} \psi '''(T) = T''. $$
(20)
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
$$ c^{2} h'(T) + c^{3}\psi ''(T) = T' $$
holds. Integrating one more time we obtain
$$ c^{2} h(T) + c^{3} \psi '(T) = T + A_{2}, $$
(21)
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
$$ A_{2} = \frac{1}{2} \bigl(c^{2} \bigl(h \bigl(T_{\infty}^{+}\bigr) + h\bigl(T_{\infty}^{} \bigr) \bigr)\bigl(T_{ \infty}^{+} + T_{\infty}^{} \bigr) \bigr), $$
so that the wave speed c is found to satisfy
$$ c^{2} = \frac{T_{\infty}^{}  T_{\infty}^{+}}{ h(T_{\infty}^{})  h(T_{\infty}^{+})}. $$
(22)
Using the above value of \(A_{2}\) in (21) and writing \(\psi '(T) = \gamma (T) T'\) we obtain
$$ T'= \frac{1}{c^{3} \gamma (T)} \biggl[T  \frac{(T_{\infty}^{+} + T_{\infty}^{})}{2}  c^{2} \biggl(h(T)  \frac{h(T_{\infty}^{+}) + h(T_{\infty}^{})}{2} \biggr) \biggr]. $$
(23)
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).