1 Introduction

It is known that the first version of the principle of Saint-Venant was conceived within a theoretical framework of a system of elliptical partial differential equations. Then, this principle has been adapted for classical elasticity in different versions to cover many different situations, see [1,2,3,4].

It is noted that an essential feature, common to all versions of the Saint-Venant principle, is that for a cylinder loaded on one end only, the strain energy stored in a portion of the body decays, if this portion is away from the loaded end.

A review of the various generalizations of the Saint-Venant principle can be found in papers [5, 6].

Even though this principle was “born” in the context of elliptical equations, it was extended to parabolic equations. The following studies are significant in this respect: [7,8,9,10,11]. In this case, the main concern of the authors was to obtain some spatial decay estimates with regard to the transient heat conduction.

Even in the case of hyperbolic equations, the Saint-Venant principle has enjoyed the interest of researchers, even though the number of studies in this case is not as high as in the case of the two other types of equations.

It is worth noting the contributions of studies [12,13,14,15] in which are investigated some decay estimates regarding the steady state solutions of the hyperbolic equations or some energy bounds in the case of transient solutions for the same type of equations.

In the last period of time, the Saint-Venant principle has been extended to continuous environments that take into account other effects such as the thermal effect or the effect of voids.

Thus, besides the versions specific to the elasticity theory, specific extensions appeared in the thermoelasticity theory, in the viscoelasticity theory or in the theory of porous media. The results obtained in the works [16, 17] are well known in this context.

After the theories for bodies with microstructure were developed, naturally, the principle was generalized to meet the needs of these media. Initiated by Eringen (see [18, 19]), the studies dedicated to microstructure environments have gained significant importance in the last decade of time.

A dipolar structure occupies a place between the theories that are dedicated to the microstructure. It is dedicated to avoid the known paradoxes: The energy equation is a parabolic type, and in the same time there is no elastic term in this equation. As such, under these conditions, the waves of heat will propagate with infinite speed. In short, the theory of dipolar bodies appears as a result of the inclusion, in the theory, of the idea of the unit cell. The dipolar structure models, for example, a molecule of a polymer, a crystallite of a polycrystal or a grain of a granular material. To highlight the importance of the theory of dipolar bodies, we will note the interest shown for this theory by many well-known researchers. In this regard, we evidentiate the papers by Mindlin [20], Green and Rivlin [21], Fried and Gurtin [22] and so on. Other approaches to dipolar bodies can be found elsewhere [23,24,25,26,27,28,29, 32,33,34,35,36]. In this, last line is the version of the Saint-Venant principle that we propose in our study. It is a generalization of this principle in order to cover the theory of elastic bodies with a dipolar structure.

The structure of our study is the following. In Sect. 2, we introduce the mixed initial boundary value problem for dipolar bodies in the context of thermoelastostatics. Then, we systematize the main notations, the equilibrium equations and the equation of energy, the initial data and the boundary data. In Sect. 3, we prove our main results.

2 Basic equations and conditions

We consider that an elastic dipolar body occupies an unbounded cylinder D which is only loaded at a plane end \(D_0\). In the main result of our study, we prove that in a region of the cylinder, which is situated at a certain distance from the loaded end, the internal energy is decreasing with regard to distance. If we denote with (x, y, z) the coordinates of an arbitrary point from the cylinder D, then we consider that the cylinder is situated in the \(z-\)axis direction and the plane end \(D_0\) is located at \(z=h_0, h_0>0\).

If we do not have to refer to the z-axis of the cylinder, then we would note the coordinates of the points in the body with \(x_1, x_2, x_3\), represented in a fixed system of rectangular Cartesian axes \(Ox_i,\;i=1,2,3\). For a simplified writing, we will use the notation x to replace the triplet \(\left( x_1,x_2,x_3\right) \). Apart from some specific situations, the functions that we will use in the following have as domain of definition the cylinder \(\bar{D}\times (0,\infty )\), where \(\bar{D}=D\cup \partial D\). In other words, these functions depend on (x, t), where x is the spatial variable and is used to designate the position of a material point and t is the time variable. When no confusion can occur, we will avoid explicitly writing the dependency of functions on the time variable or the spatial variables. If an index is repeated in a monomer, then we understand Einstein’s rule regarding summation. Also, we will use the Cartesian tensor and vector notations. The notation \(\dot{f}=\partial f/\partial t\), that is, a point placed overhead, is used to designate the differentiation of a function regarding its time variable t, while \(f_{,j}=\partial f/\partial x_j\) is the notation for the partial differentiation of a function with regard to its spatial variable \(x_j\).

In order to characterize the behavior of a body with a dipolar structure, we will use a displacement vector with components \(u_i\) and a tensor of dipolar displacement, \(\varphi \), whose components are denoted by \(\varphi _{ij}\).

The measures of the strain are the tensors \(\varepsilon _{ij}\), \(\gamma _{ij}\) and \(\chi _{ijk}\) which we introduce by means of the following geometric equations (see Eringen [19]):

$$\begin{aligned} \varepsilon _{ij}(v)=\frac{1}{2}\left( u_{i,j}+u_{j,i}\right) ,\quad \gamma _{ij}(v)=u_{j,i}-\varphi _{ij},\quad \chi _{ijk}(v)=\varphi _{jk,i}. \end{aligned}$$
(1)

In these equations, which are also called the strain–displacement equations, we denoted by v the whole displacement, namely \(v=\left( u_i,\varphi _{ij}\right) \).

In the following, we will consider only the context of a linear theory, and as such, it is normal to take into account an internal energy which is a quadratic form regarding all its variables. We will denote by \(\varPsi \) the density of the internal energy. We can develop in series, relative to its reference state, the internal energy density, so based on the principle of energy conservation, we can only retain the homogeneous terms of the second order. Therefore, we will use the following form of the internal energy, which corresponds to the global displacement v:

$$\begin{aligned} \varPsi (v)= & {} \frac{1}{2}C_{ijmn}\varepsilon _{ij}(v)\varepsilon _{mn}(v)+ G_{ijmn}\varepsilon _{ij}(v)\gamma _{mn}(v)+ F_{ijmnr}\varepsilon _{ij}(v)\chi _{mnr}(v)\nonumber \\&+\frac{1}{2}B_{ijmn}\gamma _{ij}(v)\gamma _{mn}(v) +D_{ijmnr}\gamma _{ij}(v)\chi _{mnr}(v) +\frac{1}{2}A_{ijkmnr}\chi _{ijk}(v)\chi _{mnr}(v). \end{aligned}$$
(2)

The constitutive equations will be introduced based on the measures of strain. As usual, the constitutive equations give the expressions of the tensors of the stress, that is, \(\tau _{ij}\), \(\sigma _{ij}\) and \(\mu _{ijk}\), in terms of the strain tensors:

$$\begin{aligned} \tau _{ij}(v)= & {} C_{ijmn} \varepsilon _{mn}(v)+G_{mnij} \gamma _{mn}(v)+F_{mnrij} \chi _{mnr}(v),\nonumber \\ \sigma _{ij}(v)= & {} G_{ijmn} \varepsilon _{mn}(v)+B_{ijmn} \gamma _{mn}(v)+D_{ijmnr} \chi _{mnr}(v),\nonumber \\ \mu _{ijk}(v)= & {} F_{ijkmn} \varepsilon _{mn}(v)+D_{mnijk} \gamma _{mn}(v)+A_{ijkmnr} \chi _{mnr}(v). \end{aligned}$$
(3)

In fact, the expressions of the tensors of the stress, \(\tau _{ij}\), \(\sigma _{ij}\) and \(\mu _{ijk}\) are derived from the internal energy \(\varPsi \) by using the next equations:

$$\begin{aligned} \tau _{ij}= \frac{\partial \varPsi }{\partial \varepsilon _{ij}},\quad \sigma _{ij}= \frac{\partial \varPsi }{\partial \gamma _{ij}},\quad \mu _{ijk}= \frac{\partial \varPsi }{\partial \chi _{ijk}}. \end{aligned}$$

The results of our study can be obtained if we impose to the energy density to be a quadratic form, positive definite, regarding its variables, namely to the tensors of the deformation \(\varepsilon _{ij}\), \(\gamma _{ij}\) and \(\chi _{ijk}\).

As usual, the elastic coefficients \(A_{ijkmnr}\), \(B_{ijmn}\), \(C_{ijmn}\), \(D_{ijmn}\), \(F_{ijkmn}\) and \(G_{ijkmn}\) used in Eqs. (2) and (3) are functions that characterize the elastic properties of the material. We will assume that functions depend only on the material points and are bounded functions on the domain D. In addition, based on Eq. (2) we can suppose that the above tensors of elasticity verify the following symmetry relations, on D:

$$\begin{aligned} C_{ijmn}=C_{mnij}=C_{jimn},\quad B_{ijmn}=B_{mnij},\quad A_{ijkmnr}=A_{mnrijk}. \end{aligned}$$
(4)

In order to obtain the relations (4), we have taken into account that the tensor \(\varepsilon _{ij}\) is symmetrical, considering Eq. (1).

In the absence of body forces and dipolar body forces, the equations of motion receive the following form:

$$\begin{aligned} \left[ \tau _{ij}(v)+\sigma _{ij}(v)\right] _{,j}= & {} \varrho \ddot{u}_i, \nonumber \\ \left[ \mu _{ijk}(v)\right] _{,i}+\sigma _{jk}(v)= & {} I_{js}\ddot{\varphi }_{ks}, \end{aligned}$$
(5)

which holds on D. By using the procedure of Toupin [4], we deduce that there exists the constant \(\mu _M\) for which the following inequality takes place on D:

$$\begin{aligned} \tau _{ij}(v)\tau _{ij}(v)+\sigma _{ij}(v)\sigma _{ij}(v)+\mu _{ijk}(v)\mu _{ijk}(v)\le 2\mu _M \varPsi (v). \end{aligned}$$
(6)

According to Gurtin [1], the constant \(\mu _M\) is the maximum elastic modulus. The authors of the study Mehrabadi et al. [30] presented certain maximum values for the moduli \(\mu _M\), in the case of different symmetries of elastic coefficients. These values are correlated with lower and upper bounds relative to the measure of the strain energy for different kinds of elastic materials.

In the regular points of the surface \(\partial D\), we can define the surface force tractions, which correspond to the displacement v, and which are given by

$$\begin{aligned} t_{i}(v)=\left( \tau _{ij}(v)+\sigma _{ij}(v)\right) n_j,\quad m_{jk}(v)=\left( \mu _{ijk}(v)\right) n_i. \end{aligned}$$
(7)

Since the normal is a unit vector, from (6) and (7) it is easy to deduce that

$$\begin{aligned} t_{i}(v)t_{i}(v)+m_{jk}(v)m_{jk}(v)\le 2\mu _M \varPsi (v). \end{aligned}$$
(8)

Furthermore, if we associate to the stress tensors the quantity

$$\begin{aligned} T(v)=\left( \tau _{ij}(v), \sigma _{ij}(v), \mu _{ijk}(v)\right) , \end{aligned}$$

then we can use as a magnitude of T(v), the size

$$\begin{aligned} \left| T(v)\right\| =\left[ \tau _{ij}(v)\tau _{ij}(v)+\sigma _{ij}(v)\sigma _{ij}(v)+ \mu _{ijk}(v)\mu _{ijk}(v)\right] ^{1/2}. \end{aligned}$$

In the following, we will need the strain energy, denoted by W(v), which corresponds to a smooth vector field v and is defined by means of the density of strain energy \(\varPsi (v)\) on D as follows:

$$\begin{aligned} W(v)= & {} \int _{D}\varPsi (v) dV= \int _{\varOmega } \left[ \frac{1}{2}C_{ijmn}\varepsilon _{ij}(v)\varepsilon _{mn}(v)\right. \nonumber \\&+G_{ijmn}\varepsilon _{ij}(v)\gamma _{mn}(v)+ F_{ijmnr}\varepsilon _{ij}(v)\chi _{mnr}(v)+ \frac{1}{2}B_{ijmn}\gamma _{ij}(v)\gamma _{mn}(v)\nonumber \\&\left. +D_{ijmnr}\gamma _{ij}(v)\chi _{mnr}(v) +\frac{1}{2}A_{ijkmnr}\chi _{ijk}(v)\chi _{mnr}(v) \right] \mathrm{d}V. \end{aligned}$$
(9)

3 Main results

In all that follows we will denote by D a prismatic bar of length h consisting of an elastic dipolar material, which is unstressed and has any materially uniform cross section. In the reference configuration, which is free of stress, we consider a fixed system of Cartesian coordinates so that the longitudinal bar axis is equal to the \(x_3\)-axis; the loaded end is located in the plane \(x_3 = 0\) , and in the bar for each point we have \(x_3\ge O\). We denote by \(\varSigma _0\) and \(\varSigma _h\) those cross sections in the bar which correspond to the planes \(x_3=0\) and \(x_3=h\), respectively. These sections and any other cross section in the bar are assumed to be some simply connected regular regions.

Other theoretical auxiliary results can be found in [31]. The lateral boundary of the bar will be denoted by \(\varSigma \) such that in fact our domain will be the cylinder delimited by the cross sections \(\varSigma _0\) and \(\varSigma _h\) and the lateral surface.

We will consider the following boundary conditions:

$$\begin{aligned} \left( \tau _{ij}(v)+\sigma _{ij}(v)\right) n_j= & {} t_{i}(v)=0,\quad \left( \mu _{ijk}(v)\right) n_i=m_{jk}(v)=0,\;\text{ on }\;L, \nonumber \\ \left( \tau _{ij}(v)+\sigma _{ij}(v)\right) n_j= & {} t_{i}(v)=t_i^{(0)},\quad \left( \mu _{ijk}(v)\right) n_i=m_{jk}(v)=m_{jk}^{(0)},\;\text{ on }\;\varSigma _{0},\nonumber \\ \left( \tau _{ij}(v)+\sigma _{ij}(v)\right) n_j= & {} t_{i}(v)=t_i^{(h)},\quad \left( \mu _{ijk}(v)\right) n_i=m_{jk}(v)=m_{jk}^{(h)},\;\text{ on }\;\varSigma _{h}, \end{aligned}$$
(10)

where the functions \(t_i^{(0)}\), \(t_i^{(h)}\), \(m_{jk}^{(0)}\) and \(m_{jk}^{(h)}\) are prescribed and satisfy some suitable smoothness hypotheses.

It is known fact that the displacement field \(v=\left( u_i,\varphi _{ij}\right) \) is a solution of the classical Saint-Venant’s problem if their components satisfy the equilibrium equations (5) and, also, the conditions to the limit (10).

We will use the following notation:

$$\begin{aligned} D_h=\left\{ \left( x_1,x_2,x_3\right) \in D: x_3>h\right\} , \end{aligned}$$

and denote by L the maximum value of \(x_3\) on D.

In order to obtain the main result of our study, we will need some auxiliary results.

For this purpose, we will consider the total energy associated with our domain \(D_h\), at a given moment \(t\in [0,\infty )\), denoted by \({\mathcal {E}}\) and defined by:

$$\begin{aligned} {\mathcal {E}}(h,t)= & {} \frac{1}{2}\int _{D_h}\left[ \varrho \dot{u}_i (v)\dot{u}_i(v)+I_{jk}\dot{\varphi }_{js}(v)\dot{\varphi }_{ks}(v)\right] \mathrm{d}V\nonumber \\&+\int _{D_h} \left[ \frac{1}{2}C_{ijmn}\varepsilon _{ij}(v)\varepsilon _{mn}(v)+ G_{ijmn}\varepsilon _{ij}(v)\gamma _{mn}(v)+ F_{ijmnr}\varepsilon _{ij}(v)\chi _{mnr}(v)\right. \nonumber \\&\left. +\frac{1}{2}B_{ijmn}\gamma _{ij}(v)\gamma _{mn}(v) +D_{ijmnr}\gamma _{ij}(v)\chi _{mnr}(v) +\frac{1}{2}A_{ijkmnr}\chi _{ijk}(v)\chi _{mnr}(v)\right] \mathrm{d}V. \end{aligned}$$
(11)

As it is known, the first integral in (11) is the kinetic energy associated with \(D_h\), at a given moment \(t\in [0,\infty )\).

We will need the following regularity conditions:

  1. (i)

    \(u_i, \dot{u}_i, \ddot{u}_i, u_{i,j}, \dot{u}_{i,j}\), \(\varphi _{jk}, \dot{\varphi }_{jk}, \ddot{\varphi }_{jk}, \varphi _{jk,i}, \dot{\varphi }_{jk,i}\) are continuous functions on \(\bar{D}\times [0,\infty )\);

  2. (ii)

    \(\varepsilon _{ij}, \gamma _{ij}\) and \(\chi _{ijk}\) are continuous tensors on \(\bar{D}\times [0,\infty )\);

  3. (iii)

    \(\tau _{ij}, \tau _{ji,j}, \sigma _{ij}, \sigma _{ji,j}, \mu _{ijk}\) and \(\mu _{ijk,i}\) are continuous tensors on \(\bar{D}\times [0,\infty )\);

According to Gurtin [3], we say that the ordered array \(\{u_i,\varphi _{jk}, \varepsilon _{ij}, \gamma _{ij}, \chi _{ijk}, \tau _{ij}, \sigma _{ij}, \mu _{ijk}\}\) is an admissible process if the geometric equations (1), the constitutive equations (3) and the equations of motion (5) are satisfied and, also, take place the conditions of regularity i)–iii).

In the following theorem, we deduce the first auxiliary result, that is, we obtain a bound for the total energy, in case of zero initial conditions:

$$\begin{aligned} u_i(x,t)= & {} 0,\quad \dot{u}_i(x,t)=0,\nonumber \\ \varphi _{jk}(x,0)= & {} 0,\quad \dot{\varphi }_{jk}(x,0)=0,\;\text{ on }\; D, \end{aligned}$$
(12)

and null surface forces traction

$$\begin{aligned} t_i(x,t)=0,\quad m_{jk}(x,t)=0,\; \text{ on }\;\left( \partial D{\setminus } \varSigma _0\right) \times [0,\infty ). \end{aligned}$$
(13)

Theorem 1

Consider an admissible process which corresponds to null body forces, to zero dipolar body forces, to homogeneous initial conditions (12) and to null surface forces traction (13).

Let \(\left( h_0, t_0\right) \) be a pair of constants fixed so that \(0\le h_0\le L\) and \(0\le t_0<\infty \). We have two situations.

  1. 1.

    If \(h\in \left[ 0,h_0\right] \) and \(t\in \left[ t_1, t_0-\sqrt{\frac{\varrho }{\mu _M}}\left( h-h_0\right) \right] \), then we have the estimate:

    $$\begin{aligned} {\mathcal {E}}(h,t)\ge {\mathcal {E}}(h_0,t_0). \end{aligned}$$
    (14)
  2. 2.

    If \(h\in \left[ h_0, L\right] \) and \(t\in \left[ t_2, t_0+\sqrt{\frac{\varrho }{\mu _M}}\left( h-h_0\right) \right] \), then we have the estimate:

    $$\begin{aligned} {\mathcal {E}}(h,t)\le {\mathcal {E}}(h_0,t_0). \end{aligned}$$
    (15)

    Here, we used the notations:

    $$\begin{aligned} t_1= & {} \max \left[ 0, t_0+\sqrt{\frac{\varrho }{\mu _M}}\left( h-h_0\right) \right] , \end{aligned}$$
    (16)
    $$\begin{aligned} t_2= & {} \max \left[ 0, t_0-\sqrt{\frac{\varrho }{\mu _M}}\left( h-h_0\right) \right] . \end{aligned}$$
    (17)

Proof

Recall that we have noted with \(\varSigma _h\) the intersection with D of the plane \(x_3=h\) and to facilitate the demonstration, we consider the function f(h, t) defined by:

$$\begin{aligned} f(h,t)=\int _0^t\int _{\varSigma _h}\left[ t_i(\tau )\dot{u}_i(\tau )+m_{jk}(\tau )\dot{\varphi }_{jk}(\tau )\right] \mathrm{d}A \mathrm{d}\tau . \end{aligned}$$
(18)

First, clearly, we have

$$\begin{aligned} \frac{\partial f(h,t)}{\partial t}=\int _{\varSigma _h}\left[ t_i(t)\dot{u}_i(t)+m_{jk}(t)\dot{\varphi }_{jk}(t)\right] \mathrm{d}A. \end{aligned}$$
(19)

Second, if we take into account Eqs. (1), (3) and (5) and considering the null initial conditions (12) and the homogeneous surface forces traction (13), for \(h, \bar{h}\in [0,L]\), \(\bar{h}\le h\), then we have

$$\begin{aligned} f(h,t)-f(\bar{h},t)= & {} \frac{1}{2} \int _{\bar{h}}^h \int _{\varSigma _l}\left[ \varrho \dot{u}_i (t)\dot{u}_i(t)+I_{jk}\dot{\varphi }_{js}(t)\dot{\varphi }_{ks}(t)\right] \mathrm{d}A \mathrm{d}l\nonumber \\&+\int _{\bar{h}}^h\int _{\varSigma _l} \left[ \frac{1}{2}C_{ijmn}\varepsilon _{ij}(t)\varepsilon _{mn}(t)+ G_{ijmn}\varepsilon _{ij}(t)\gamma _{mn}(t)+ F_{ijmnr}\varepsilon _{ij}(t)\chi _{mnr}(t)\right. \nonumber \\&\left. +\frac{1}{2}B_{ijmn}\gamma _{ij}(t)\gamma _{mn}(t) +D_{ijmnr}\gamma _{ij}(t)\chi _{mnr}(t) +\frac{1}{2}A_{ijkmnr}\chi _{ijk}(t)\chi _{mnr}(t)\right] \mathrm{d}A \mathrm{d}l,\nonumber \\ \end{aligned}$$
(20)

such that, by direct differentiation, we easily deduce the relation

$$\begin{aligned} \frac{\partial f(h,t)}{\partial h}= & {} \frac{1}{2} \int _{\varSigma _h}\left[ \varrho \dot{u}_i (t)\dot{u}_i(t)+I_{jk}\dot{\varphi }_{js}(t)\dot{\varphi }_{ks}(t)\right] \mathrm{d}A\nonumber \\&+\int _{\varSigma _h} \left[ \frac{1}{2}C_{ijmn}\varepsilon _{ij}(t)\varepsilon _{mn}(t)+ G_{ijmn}\varepsilon _{ij}(t)\gamma _{mn}(t)+ F_{ijmnr}\varepsilon _{ij}(t)\chi _{mnr}(t)\right. \nonumber \\&\left. +\frac{1}{2}B_{ijmn}\gamma _{ij}(t)\gamma _{mn}(t) +D_{ijmnr}\gamma _{ij}(t)\chi _{mnr}(t) +\frac{1}{2}A_{ijkmnr}\chi _{ijk}(t)\chi _{mnr}(t) \right] \mathrm{d}A. \end{aligned}$$
(21)

We now apply the inequality of the arithmetic–geometric mean and the inequality of Schwarz such that by taking into account the inequality (8), we can find an arbitrary positive constant \(\alpha \) so that

$$\begin{aligned} \left| \frac{\partial f(h,t)}{\partial h}\right|\le & {} \frac{1}{2} \int _{\varSigma _h}\alpha \rho \left[ \dot{u}_i (t)\dot{u}_i(t)+\dot{\varphi }_{jk}(t)\dot{\varphi }_{jk}(t)\right] \mathrm{d}A\nonumber \\&+\int _{\varSigma _h}\frac{1}{\alpha \rho } \left[ \frac{1}{2}C_{ijmn}\varepsilon _{ij}(t)\varepsilon _{mn}(t)+ G_{ijmn}\varepsilon _{ij}(t)\gamma _{mn}(t)+ F_{ijmnr}\varepsilon _{ij}(t)\chi _{mnr}(t)\right. \nonumber \\&\left. +\frac{1}{2}B_{ijmn}\gamma _{ij}(t)\gamma _{mn}(t) +D_{ijmnr}\gamma _{ij}(t)\chi _{mnr}(t) +\frac{1}{2}A_{ijkmnr}\chi _{ijk}(t)\chi _{mnr}(t)\right] \mathrm{d}A. \end{aligned}$$
(22)

If we choose \(\alpha =\left( \mu _M/\rho \right) ^{1/2}\) and consider the relation (19), we obtain the following relations:

$$\begin{aligned} \left( \frac{\mu _M}{\rho }\right) ^{1/2}\frac{\partial f}{\partial h}+\frac{\partial f}{\partial t}\le & {} 0, \nonumber \\ \left( \frac{\mu _M}{\rho }\right) ^{1/2}\frac{\partial f}{\partial h}-\frac{\partial f}{\partial t}\le & {} 0. \end{aligned}$$
(23)

Based on (21), we deduce that for all fixed \(t\in [0,\infty )\),

  1. (a)

    f(h, t) is an increasing function on [0, L], with respect to h.

    Taking into account that the pair of constants \(\left( h_0, t_0\right) \) is fixed so that \(0\le h_0\le L\) and \(0\le t_0<\infty \), based on (23)\(_1\) we deduce that:

  2. (b)

    \(f\left( h,t_0+\sqrt{\frac{\varrho }{\mu _M}}\left( h-h_0\right) \right) \) is an increasing function on [0, L), with respect to h;

  3. (c)

    \(f\left( h,t_0-\sqrt{\frac{\varrho }{\mu _M}}\left( h-h_0\right) \right) \) is an increasing function on [0, L), with respect to h;

Based on the above results (a), (b), (c), we can prove the estimates (14) and (15) of the theorem.

1. In the case \(h\in [0, h_0]\) and \(t\in [t_1, t_0]\), \(t_1\) defined in Eq. (16), we will use the notation \(h_1=h_0+\sqrt{\frac{\varrho }{\mu _M}}\left( t-t_0\right) \) and, clearly, we have \(h_1\in \left[ h, h_0\right] \). As such, based on the affirmation (a) we deduce

$$\begin{aligned} f(h,t)\ge f(h_1, t). \end{aligned}$$
(24)

Also, by using the case (b), we obtain the estimation:

$$\begin{aligned} f(h_1,t)\ge f(h_0, t_0), \end{aligned}$$
(25)

and then, from (24) and (25) we obtain:

$$\begin{aligned} f(h,t)\ge f(h_0, t_0). \end{aligned}$$
(26)

In the case \(h\in [0, h_0]\) and \(t\in [t_0, t_2]\), \(t_2\) defined in Eq. (17), we will use the notation \(h_2=h_0-\sqrt{\frac{\varrho }{\mu _M}}\left( t-t_0\right) \) and, clearly, we have \(h_2\in \left[ h, h_0\right] \). As such, based on the affirmation (a) we deduce

$$\begin{aligned} f(h,t)\ge f(h_2, t). \end{aligned}$$
(27)

Also, by using the case (c), we obtain the estimation:

$$\begin{aligned} f(h_2,t)\ge f(h_0, t_0). \end{aligned}$$
(28)

Clearly, from (27) and (28) we obtain, again, the estimation (26).

On the other hand, from the boundary conditions (13) we can easily deduce that for all \(t\ge 0\), we have

$$\begin{aligned} f(L,t)=0. \end{aligned}$$
(29)

As a consequence of this relation, from (11) and (20) we arrive to the conclusion that

$$\begin{aligned} f(h,t)={\mathcal {E}}(h,t), \end{aligned}$$
(30)

and, from this and (28) we obtain the estimate (14).

2. Using an analogous procedure, we will prove the second estimation of the theorem, namely (15).

We will also consider two situations.

First, if \(h\in \left[ h_0, L\right] \) and \(t\in \left[ t_2, t_0\right] \), \(t_2\) defined in Eq. (17), then we have \(h_2\in \left[ h, h_0\right] \), and as such, based on the affirmation (a) we deduce

$$\begin{aligned} f(h,t)\le f(h_2, t). \end{aligned}$$
(31)

Also, with the help of (c) we obtain the estimation:

$$\begin{aligned} f(h_2,t)\le f(h_0, t_0), \end{aligned}$$
(32)

and then, from (31) and (32) we obtain

$$\begin{aligned} f(h,t)\le f(h_0, t_0). \end{aligned}$$
(33)

In the case \(h\in \left[ h_0, L\right] \) and \(t\in [t_0, t_1]\), \(t_1\) defined in Eq. (16), we have \(h_1\in \left[ h_0, h\right] \). As such, based on the affirmation (a) we deduce

$$\begin{aligned} f(h,t)\le f(h_1, t). \end{aligned}$$
(34)

Also, by using the case (b), we obtain the estimation:

$$\begin{aligned} f(h_1,t)\le f(h_0, t_0). \end{aligned}$$
(35)

Clearly, from (34) and (35) we obtain, again, the estimation (33).

Finally, based on (30) and (33) we obtain the estimate (15) so that the proof of Theorem  1 is complete. \(\square \)

In the following theorem, we also obtain an estimate for the total energy.

Theorem 2

Consider an admissible process which corresponds to null body forces, to null dipolar body charges, to zero initial data (12) and to null surface forces traction (13).

Then for any \(t\ge 0\) and \(h\in \left[ \left( \mu _M/\rho \right) ^{1/2}t, L\right] \), we have:

$$\begin{aligned} {\mathcal {E}}(h,t)=0. \end{aligned}$$
(36)

Also, for any \(t\ge 0\) and h such that \(0\le h\le \left( \mu _M/\rho \right) ^{1/2}t\le L\), we have:

$$\begin{aligned}&\frac{1}{2}\int _0^t\int _{D_h}\left[ \varrho \dot{u}_i (v)\dot{u}_i(v)+I_{jk}\dot{\varphi }_{js}(v)\dot{\varphi }_{ks}(v)\right] \mathrm{d}V\mathrm{d}s \nonumber \\&\quad +\int _{D_h} \left[ \frac{1}{2}C_{ijmn}\varepsilon _{ij}(v)\varepsilon _{mn}(v)+ G_{ijmn}\varepsilon _{ij}(v)\gamma _{mn}(v)+ F_{ijmnr}\varepsilon _{ij}(v)\chi _{mnr}(v)\right. \nonumber \\&\quad \left. +\frac{1}{2}B_{ijmn}\gamma _{ij}(v)\gamma _{mn}(v) +D_{ijmnr}\gamma _{ij}(v)\chi _{mnr}(v) +\frac{1}{2}A_{ijkmnr}\chi _{ijk}(v)\chi _{mnr}(v)\right] \mathrm{d}V \mathrm{d}s\nonumber \\&\le \left( 1- \sqrt{\frac{\rho }{\mu _M}}\frac{h}{t}\right) \frac{1}{2}\int _0^t\int _{D_0}\left[ \varrho \dot{u}_i (v)\dot{u}_i(v)+I_{jk}\dot{\varphi }_{js}(v)\dot{\varphi }_{ks}(v)\right] \mathrm{d}V\mathrm{d}s\nonumber \\&\quad +\int _{D_0} \left[ \frac{1}{2}C_{ijmn}\varepsilon _{ij}(v)\varepsilon _{mn}(v)+ G_{ijmn}\varepsilon _{ij}(v)\gamma _{mn}(v)+ F_{ijmnr}\varepsilon _{ij}(v)\chi _{mnr}(v)\right. \nonumber \\&\quad \left. +\frac{1}{2}B_{ijmn}\gamma _{ij}(v)\gamma _{mn}(v) +D_{ijmnr}\gamma _{ij}(v)\chi _{mnr}(v) +\frac{1}{2}A_{ijkmnr}\chi _{ijk}(v)\chi _{mnr}(v)\right] \mathrm{d}V \mathrm{d}s. \end{aligned}$$
(37)

Proof

First, we specify that the estimate (37) can be rewritten in a shorter form:

$$\begin{aligned} \int _0^t {\mathcal {E}}(h,s)\mathrm{d}s\le \left( 1- \sqrt{\frac{\rho }{\mu _M}}\frac{h}{t}\right) \int _0^t {\mathcal {E}}(0,s)\mathrm{d}s,\;\text{ for }\;t\ge 0,\quad 0\le h\le \sqrt{\frac{\mu _M}{\rho }}t\le L. \end{aligned}$$
(38)

By direct calculations, from (11) we deduce

$$\begin{aligned} {\mathcal {E}}(0,0)=0. \end{aligned}$$
(39)

On the other hand, we can use the estimates (14) and (15) in the particular case \(h_0=0\), \(t_0=0\), so that for \(0\le \left( \mu _M/\rho \right) ^{1/2}t\le h\) we obtain:

$$\begin{aligned} {\mathcal {E}}(h,t)\le {\mathcal {E}}(0,0)=0, \end{aligned}$$
(40)

and, because the total energy is positive, we are led to Eq. (36).

In order to prove (37), or its equivalent form (38), we will use Eq. (36), already proven. To this aim, we will fix \(t_1\ge 0\) and \(h_1\) such that \(t_1\ge \left( \rho /\mu _M\right) ^{1/2} h_1\).

To simplify writing, we introduce the constant a by:

$$\begin{aligned} a=\left( \rho /\mu _M\right) ^{1/2} h_1. \end{aligned}$$

Based on (36), we have

$$\begin{aligned} \int _0^{t_1}{\mathcal {E}}(h_1,s)\mathrm{d}s=\int _0^{a}{\mathcal {E}}(h_1,s)\mathrm{d}s+\int _a^{t_1}{\mathcal {E}}(h_1,s)\mathrm{d}s= \int _a^{t_1}{\mathcal {E}}(h_1,s)\mathrm{d}s, \end{aligned}$$

and after we change the variable

$$\begin{aligned} s=\left( 1-a\frac{h_1}{t_1}\right) \tau + ah_1, \end{aligned}$$
(41)

we deduce

$$\begin{aligned}&\int _0^{t_1}{\mathcal {E}}(h_1,s)\mathrm{d}s= \int _a^{t_1}{\mathcal {E}}(h_1,s)\mathrm{d}s\nonumber \\&\quad =\left( 1-a\frac{h_1}{t_1}\right) \int _0^{t_1}{\mathcal {E}}\left( h_1, \left( 1-a\frac{h_1}{t_1}\right) \tau + ah_1\right) \mathrm{d}\tau . \end{aligned}$$
(42)

Now, we will use the estimate (14) (15) from Theorem 1, in the particular case \(h_0=h_1\) and \(t_0=s\), in which \(h_1\) is defined after Eq. (40) and s is defined in (41).

In this way, we obtain:

$$\begin{aligned} {\mathcal {E}}(0,\tau )\ge {\mathcal {E}}(h_1,s), \end{aligned}$$

so that based on (42) we obtain the desired estimate (37), and the proof of Theorem 2 is finished.

Our study will be complete if we manage to extend the estimate (37) so as to cover the boundary data on the end surface \(\varSigma _0\). \(\square \)

Theorem 3

Suppose the hypotheses of Theorem 2 are satisfied.

If we specify the boundary data on the end surface \(\varSigma _0\), then the estimate (37) is in a complete form.

Proof

To this end, we need to obtain some bounds for

$$\begin{aligned} \int _0^{t}{\mathcal {E}}(0,s)\mathrm{d}s, \end{aligned}$$

expressed as functions of end data.

Namely, in view of (13), we set

$$\begin{aligned} t_i(x,t)=s_i(x,t),\quad m_{jk}(x,t)=\mu _{jk}(x,t),\quad (x,t)\in \varSigma _0 \times [0,\infty ), \end{aligned}$$
(43)

in which the functions \(s_i\) and \(\mu _{jk}\) are prescribed and continuously differentiable with regard to the time variable. Using the procedure of [16], we can find a positive constant C, which depends only on the domain D, the surface \(\varSigma _0\) and the elastic coefficients so that we have the estimate:

$$\begin{aligned} \int _0^{t}{\mathcal {E}}(0,s)ds\le C\left\{ \int _0^t\left( \int _0^{\tau }\int _{\varSigma _0}\left[ \dot{s}_i(s)\dot{s}_i(s)+\dot{\mu }_{jk}(s)\dot{\mu }_{jk}(s)\right] \mathrm{d}A\mathrm{d}s\right) ^{1/2}\mathrm{d}\tau \right\} ^2. \end{aligned}$$

This last estimate can be used in the spatial decay relation (37) so that we can obtain a complete form of this estimate which can describe the spatial decay of end effects.

In the last result of our study, we verify that the previous estimates remain valid if \(L\rightarrow \infty \), in other words, if the previous analysis can cover the case of unbounded regions, regarding the \(x_3-\)axis direction. \(\square \)

Theorem 4

If the domain D is an unbounded region in direction of the \(x_3-\) axis, then the estimate (37) is still valid.

Proof

According to hypothesis, the maximum value of \(x_3\) on D is infinite, that is, \(L\rightarrow \infty \). As a consequence, for a positive, arbitrarily fixed t, we can replace (29) by the following relation:

$$\begin{aligned} \lim \limits _{h\rightarrow \infty }f(h,t)=0. \end{aligned}$$
(44)

Here is a brief argumentation for (44). First, in (26) we fix \(t_0=0\) so that for an arbitrary \(h_0\) we obtain the following inequality:

$$\begin{aligned} f(h,t)\ge f\left( h_0,0\right) ,\quad h\in \left[ 0,h_0\right] , \quad t\in [0, \left( \rho /\mu _M\right) ^{1/2}\left( h_0-h\right) , \end{aligned}$$
(45)

and, of course, \(\left( h_0,0\right) \). As such, for arbitrary fixed t, we obtain

$$\begin{aligned} \lim \limits _{h\rightarrow \infty }f(h,t)\ge 0. \end{aligned}$$
(46)

Second, for an arbitrary \(h_0\) and \(t_0=0\), (35) becomes

$$\begin{aligned} f(h,t)\le f(h_0,0)=0, \quad t\in [0, \left( \rho /\mu _M\right) ^{1/2}\left( h_0-h\right) ,\quad h_0\le h, \end{aligned}$$

from where

$$\begin{aligned} \lim \limits _{h\rightarrow \infty }f(h,t)\le 0. \end{aligned}$$
(47)

Clearly, from (47) and (46) we obtain (44).

Finally, from (20) and (44) we deduce that Eq. (30) is true also in the case of the unbounded domain, so that we can use the estimations of Theorems 1 and 2. In this way, the proof of the theorem is finished. \(\square \)

4 Conclusions

In fact, the Saint-Venant’s principle suggests to analyze the behavior of the steady state equilibrium, regarding the spatial variables, in a dipolar elastic body of the form of a prismatic bar which is subjected, only on a plane end, to some nonhomogeneous conditions to the limit. More precisely, we have shown that at a certain distance from the end which is loaded, any motion in the body vanishes. More specifically, our estimates in the three theorems assure that the strain energy stored in the cylinder in a portion located at a given distance d from the end which is loaded is decreasing exponentially relative to the respective distance d. But as noted from the beginning (see Toupin [4]), as an inherent weakness of the Saint-Venant’s principle, in order to find the optimum decay rate, we must consider some specific cross sections. Otherwise, the use of this principle is ineffective.