On the decay of exponential type for the solutions in a dipolar elastic body

Abstract

Our study is concerned with a generalization of the principle of Saint-Venant in the context of the theory of elastic bodies with a dipolar structure. It is an extension of the form of this principle proposed by Toupin in the case of classical linear elasticity. We consider a straight cylinder which is occupied by an dipolar elastic body that is only loaded at one end and our main finding proves that the internal energy in a portion of this cylinder, that is located at a given distance regarding the end that is loaded, is exponentially decreasing, regarding the respective distance.

Keywords:

• Thermoelastostatics
• bodies with dipolar structure
• the principle of Saint-Venant
• exponential decay

1. Introduction

In classical linear elasticity there are a lot of versions of Saint-Venant's principle, most of them have been discussed by Gurtin in his monograph [1]. In this regard, we recall the papers Edelstein [2], Knowles [3], Toupin [4] and Boley [5]. Later the principle was extended for different contexts, such as for composite plane elasticity, for different kinds of anisotropic or isotropic bodies, for the Laplace's equation, for three-dimensional problems, and even for nonlinear problems. Many of these generalizations have been summarized by Horgan in his review paper [6]. See also [7]. A further comprehensive reference is Knowles [8]. The Saint-Venant principle is a proper concept for elliptical equations. An important property for any form of the principle of Saint-Venant is the following. It is considered a homogeneous elastic material which occupy a prismatic bar having an arbitrary length and cross-section and is loaded on one end only by a self-equilibrated force. In these circumstances the elastic energy $E\left(d\right)$ stored in a portion of the body which is located at the distance d from the loaded end satisfies the inequality (1) $E\left(d\right)\le E\left(0\right)\exp [-(d-l\left)/s_c\right(l\left)\right].$(1) This inequality is addressed in detail in the paper [9].

Here l is the length of a slice of the prismatic bar and $s_c\left(l\right)$ can be defined by means of the elastic moduli of the dipolar body (the minimum and the maximum moduli) and the smallest nonzero characteristic frequency of the free vibration of this slice. In some papers $s_c\left(l\right)$ is called the characteristic decay length. However, Ericksen [10] has proven that in some specific conditions, $s_c\left(l\right)$ does not depend on the minimum elastic modulus, as such, it depends only on the maximum of elastic modulus and is not depending on the minim elastic modulus.

Based on an idea given by Toupin [4] and Knowles [8], the principle of Saint-Venant is useful to evaluate the strain energy decay in the cylinder, away from its end which is loaded.

Although at first this principle appeared as a feature for the partial differential equations of elliptical type, naturally it has been raised the issue of some extensions of the principle in order to apply it to parabolic partial differential equations. In this context, we note the studies by Chirita [11], Horgan et al. [9], and Nunziato [12].

The principle was not used at first in the context of hyperbolic equations, but not so long after, more and more studies with Saint Venant type results were published for the description of elastic wave propagation. In this respect, the studies by Chirita [13], Iesan and Quintanilla [14], Flavin and Knops [15], Flavin et al. [16], are significant.

As the theories of different media with microstructure become more and more important After the theories for bodies with microstructure were developed, the problem of the extension of this principle was naturally raised to meet the requirements of these bodies. The works on bodies with microstructure were initiated by Eringen (see, for instance, [17,18]) and in the last decade of time, it has greatly increased the number of studies dedicated to microstructure.

Among the theories that aim to microstructure, special importance is given to the theory of dipolar elastic media. All these theories make the next paradox disappear: the equation energy does not have elastic terms in its structure and this partial-derivative equation is of parabolic type. As such, because of these contradictions, the propagation of the heat waves take place with an infinite speed.

It can be said that the elastic bodies with a dipolar structure appeared by including it in a certain theory of the idea of the unit cell. This unit cell models a molecule of a polymer, a crystallite of a polycrystal or a grain of a granular material. If we analyse the significance given by some important researchers to solids with a structure of the dipolar type, we deduce that the theory of elasticity of dipolar bodies is one of the most important modern theories of microstructure environments. In this sense, it is enough to list a few studies, such as Mindlin [19], Green and Rivlin [20], or Fried and Gurtin [21], which have a good recognition among researchers. Then the number of approaches of these media with dipolar structure has increased considerably [22–32]. Our study is organized as follows. In Section 2 we introduce the basic equations and conditions which define the mixed initial-boundary values problem in the context of the elastostatics of dipolar materials. Also, here we introduce the important notations, the equations in the case of equilibrium and the equation of energy, the relations to the limit and the initial restrictions. In the first part of Section 3 we prove some auxiliary results and then we present the main results of our work.

2. Basic equations and conditions

Let us consider a finite spatial region Ω which is occupied by an elastic material with a dipolar structure. We assume that Ω is a regular domain from the Euclidean three dimensional space $R^3$. We will denote by $\mathrm{\partial }\mathrm{\Omega }$ the border surface of the domain Ω and we will use the unit outward normal of the surface $\mathrm{\partial }\mathrm{\Omega }$, having $n_i$ as components. We assume that $\mathrm{\partial }\mathrm{\Omega }$ is a smooth surface and an arbitrary point P in the domain Ω is identifiable through three coordinates $x_1,x_2,x_3$. For simplicity of writing, instead of the triplet $(x_1,x_2,x_3)$, the notation x is used, that is $x=(x_1,x_2,x_3)$. The functions that we use in what follows are assumed be dependent on the spatial variable x and t, the time variable, namely, $f=f(x,t)$, and the domain of definition for these functions is the cylinder $\overline{\mathrm{\Omega }}\times (0,\mathrm{\infty })$, with $\overline{\mathrm{\Omega }}=\mathrm{\Omega }\cup \mathrm{\partial }\mathrm{\Omega }$. In some situations, we avoid writing dependencies on some functions of their spatial or temporal variables, if there is no possibility of confusion. We used the Einstein rule of summation with regards to repeating subscripts and adopt the tensor and vector notation, usually by bold.

A superposed dot is used for the differentiation regarding the variable t, that is, we will use the notation $\dot{f}=\mathrm{\partial }f/\mathrm{\partial }t$. Also, in order to designates the partial derivative of a function f regarding a spatial coordinate $x_j$ we used a subscript preceded by a comma, that is, $f_{,j}=\mathrm{\partial }f/\mathrm{\partial }{x}_j$.

We will use the vector of the displacement, denoted $\mathbf{u}$, with the components $u_i$, and a tensor of the dipolar dispacement, denoted by $\mathit{\phi }$, and having the components ${\phi }_{ij}$ in order the describe the evolution of the the dipolar elastic body.

To characterize the strain measures, we used the tensors denoted by $e_{ij}$, ${\gamma }_{ij}$ and ${\chi }_{ijk}$ defined by the following relations: (2) $\begin{array}{rl}{e}_{ij}\left(w\right)& =\frac{1}{2}\left(u_{i,j}+u_{j,i}\right),{\nu }_{ij}\left(w\right)=u_{j,i}-{\phi }_{ij},\\ {\chi }_{ijk}\left(w\right)& ={\phi }_{jk,i},\end{array}$(2) which are called the strain-displacement relations (see, for instance, Eringen [17]).

Here we used the notation w to designate the global the displacement, that is, w has the components $w=(u_i,{\phi }_{ij})$.

It is known that the relation which give the strain tensors by means of the displacement are also called the geometric equations.

Now, we can introduce the tensors of the stress, denoted by $t_{ij}$, ${\sigma }_{ij}$ and ${\eta }_{ijk}$, by means of the constitutive equations, because these equations give the stress tensors as functions of the strain: Having the measures of strain, (3) $\begin{array}{rl}{t}_{ij}\left(w\right)& =C_{ijmn}{e}_{mn}\left(w\right)+G_{mnij}{\nu }_{mn}\left(w\right)+F_{mnrij}{\chi }_{mnr}\left(w\right),\\ {\sigma }_{ij}\left(w\right)& =G_{ijmn}{e}_{mn}\left(w\right)+B_{ijmn}{\nu }_{mn}\left(w\right)+D_{ijmnr}{\chi }_{mnr}\left(w\right),\\ {\eta }_{ijk}\left(w\right)& =F_{ijkmn}{e}_{mn}\left(w\right)+D_{mnijk}{\nu }_{mn}\left(w\right)+A_{ijkmnr}{\chi }_{mnr}\left(w\right),\end{array}$(3) because the strain tensors are already introduced.

In what follows we consider only the linear theory, that is why we should consider that the internal energy, denoted by Ψ, is a quadratic form, with regards to all constitutive variables. It will be used in the following form (4) $\begin{array}{rl}\mathrm{\Psi }\left(w\right)& =\frac{1}{2}{C}_{ijmn}{e}_{ij}\left(w\right)e_{mn}\left(w\right)+G_{ijmn}{e}_{ij}\left(w\right){\nu }_{mn}\left(w\right)\\ & +F_{ijmnr}{e}_{ij}\left(w\right){\chi }_{mnr}\left(w\right)+\\ & +\frac{1}{2}{B}_{ijmn}{\nu }_{ij}\left(w\right){\nu }_{mn}\left(w\right)+D_{ijmnr}{\nu }_{ij}\left(w\right){\chi }_{mnr}\left(w\right)\\ & +\frac{1}{2}{A}_{ijkmnr}{\chi }_{ijk}\left(w\right){\chi }_{mnr}\left(w\right),\end{array}$(4) where we evidentiated that the internal energy corresponds to the displacement v.

To this aim, the internal energy is developed in series, relative to the initial reference, and only linear and square terms are retained and then it used the principle of conservation of energy.

In fact, the tensors of the stress, $t_{ij}$, ${\sigma }_{ij}$ and ${\eta }_{ijk}$, from (3(3) $\begin{array}{rl}{t}_{ij}\left(w\right)& =C_{ijmn}{e}_{mn}\left(w\right)+G_{mnij}{\nu }_{mn}\left(w\right)+F_{mnrij}{\chi }_{mnr}\left(w\right),\\ {\sigma }_{ij}\left(w\right)& =G_{ijmn}{e}_{mn}\left(w\right)+B_{ijmn}{\nu }_{mn}\left(w\right)+D_{ijmnr}{\chi }_{mnr}\left(w\right),\\ {\eta }_{ijk}\left(w\right)& =F_{ijkmn}{e}_{mn}\left(w\right)+D_{mnijk}{\nu }_{mn}\left(w\right)+A_{ijkmnr}{\chi }_{mnr}\left(w\right),\end{array}$(3) ) are deduced by using the internal energy Ψ in the following relations: $t_{ij}=\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}},{\sigma }_{ij}=\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}},{\eta }_{ijk}=\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}.$ Our following results can be deduced, by assuming that to the density of the internal energy is a quadratic positive definite functional, with respect to the tensors of strain $e_{ij}$, ${\nu }_{ij}$ and ${\chi }_{ijk}$.

As usual, the elastic coefficients $A_{ijkmnr}$, $B_{ijmn}$, $C_{ijmn}$, $D_{ijmn}$, $F_{ijkmn}$, and $G_{ijkmn}$ from Equation (4(4) $\begin{array}{rl}\mathrm{\Psi }\left(w\right)& =\frac{1}{2}{C}_{ijmn}{e}_{ij}\left(w\right)e_{mn}\left(w\right)+G_{ijmn}{e}_{ij}\left(w\right){\nu }_{mn}\left(w\right)\\ & +F_{ijmnr}{e}_{ij}\left(w\right){\chi }_{mnr}\left(w\right)+\\ & +\frac{1}{2}{B}_{ijmn}{\nu }_{ij}\left(w\right){\nu }_{mn}\left(w\right)+D_{ijmnr}{\nu }_{ij}\left(w\right){\chi }_{mnr}\left(w\right)\\ & +\frac{1}{2}{A}_{ijkmnr}{\chi }_{ijk}\left(w\right){\chi }_{mnr}\left(w\right),\end{array}$(4) ) characterize the elastic properties of the material and these are supposed be bounded functions defined on Ω, depending only on the spatial variables, not on time variable.

Also, based on the fact that the strain tensor $e_{ij}$ is a symmetrical tensor, it is easy to obtain the elasticity tensors obey to the following symmetry relations: Ω: (5) $\begin{array}{rl}{C}_{ijmn}& =C_{mnij}=C_{jimn},B_{ijmn}=B_{mnij},\\ A_{ijkmnr}& =A_{mnrijk}.\end{array}$(5) In case the body forces and dipolar body forces are absent, the equilibrium equations can be write in the next form: (6) $\begin{array}{rl}{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}& =0,\\ {\left({\eta }_{ijk}\left(w\right)\right)}_{,i}+{\sigma }_{jk}\left(w\right)& =0,\end{array}$(6) that are satisfied on whole domain Ω. With the help of Toupin procedure [4], we can find a positive constant ${\mu }_M$ so that on Ω we have the inequality: (7) $t_{ij}\left(w\right)t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right){\sigma }_{ij}\left(w\right)+{\eta }_{ijk}\left(w\right){\eta }_{ijk}\left(w\right)\le 2{\mu }_M\mathrm{\Psi }\left(w\right).$(7) In the work [1] Gurtin shown that the constant ${\mu }_M$ is the maximum value of the elastic modulus. In the paper by Mehrabadi et al. [33] are presented certain estimations of the maximum moduli ${\mu }_M$, in correlation with different types of symmetries for the elastic coefficients. These estimations are in correspondence with lower bound and upper bound of the energy of strain for many kinds of elastic media.

Now, we introduce some surface tractions, corresponding to the global displacement w, having the expressions: (8) $t_i\left(w\right)=\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j,m_{jk}\left(w\right)=\left({\eta }_{ijk}\left(w\right)\right)n_i,$(8) in all regular points of $\mathrm{\partial }\mathrm{\Omega }$.

Taking into account that normal vector is an unitary vector, we can get from (8(8) $t_i\left(w\right)=\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j,m_{jk}\left(w\right)=\left({\eta }_{ijk}\left(w\right)\right)n_i,$(8) ) and (7(7) $t_{ij}\left(w\right)t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right){\sigma }_{ij}\left(w\right)+{\eta }_{ijk}\left(w\right){\eta }_{ijk}\left(w\right)\le 2{\mu }_M\mathrm{\Psi }\left(w\right).$(7) ) the following estimation: (9) $t_i\left(w\right)t_i\left(w\right)+m_{jk}\left(w\right)m_{jk}\left(w\right)\le 2{\mu }_M\mathrm{\Psi }\left(w\right).$(9) If we consider, for the tensors of stress, the association $T\left(w\right)=\left(t_{ij}\left(w\right),{\sigma }_{ij}\left(w\right),{\eta }_{ijk}\left(w\right)\right),$ then a magnitude of $T\left(w\right)$ can be used the size: $\left|T\left(w\right)\right.∥={\left[t_{ij}\left(w\right)t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right){\sigma }_{ij}\left(w\right)+{\eta }_{ijk}\left(w\right){\eta }_{ijk}\left(w\right)\right]}^{1/2}.$ In the following we will use the energy of strain, corresponding to the smooth vector w, which is denoted by $S\left(w\right)$ and is defined on Ω by: (10) $S\left(w\right)={\int }_{\mathrm{\Omega }}\mathrm{\Psi }\left(w\right)\mathrm{d}V.$(10) We intend to equip the set of uniform vector fields of displacements v, defined on the domain Ω, with the structure of a real vector space. Moreover, the structure can be extended to a Hilbert space, introducing a scalar product that generates a norm.

Thus, if we take into account the expression (4(4) $\begin{array}{rl}\mathrm{\Psi }\left(w\right)& =\frac{1}{2}{C}_{ijmn}{e}_{ij}\left(w\right)e_{mn}\left(w\right)+G_{ijmn}{e}_{ij}\left(w\right){\nu }_{mn}\left(w\right)\\ & +F_{ijmnr}{e}_{ij}\left(w\right){\chi }_{mnr}\left(w\right)+\\ & +\frac{1}{2}{B}_{ijmn}{\nu }_{ij}\left(w\right){\nu }_{mn}\left(w\right)+D_{ijmnr}{\nu }_{ij}\left(w\right){\chi }_{mnr}\left(w\right)\\ & +\frac{1}{2}{A}_{ijkmnr}{\chi }_{ijk}\left(w\right){\chi }_{mnr}\left(w\right),\end{array}$(4) ) of the deformation energy density $\mathrm{\Psi }\left(w\right)$, by using the functional $W\left(w\right)$, we can build the next bilinear form as follows: (11) $\begin{array}{rl}E(v,w)& =\frac{1}{2}{\int }_{\mathrm{\Omega }}\left\{C_{ijmn}{e}_{ij}\left(v\right)e_{mn}\left(w\right)+G_{ijmn}\right.\\ & \times \left[e_{ij}\left(v\right){\nu }_{mn}\left(w\right)+e_{ij}\left(w\right){\nu }_{mn}\left(v\right)\right]+\\ & +F_{ijmnr}\left[e_{ij}\left(v\right){\chi }_{mnr}\left(w\right)+e_{ij}\left(w\right){\chi }_{mnr}\left(v\right)\right]\\ & +B_{ijmn}{\nu }_{ij}\left(v\right){\nu }_{mn}\left(w\right)\\ & +D_{ijmnr}\left[{\nu }_{ij}\left(v\right){\chi }_{mnr}\left(w\right)+{\nu }_{ij}\left(w\right){\chi }_{mnr}\left(v\right)\right]\\ & \left.+A_{ijkmnr}{\chi }_{ijk}\left(v\right){\chi }_{mnr}\left(w\right)\right\}\mathrm{d}V.\end{array}$(11) The displacement v is as above, that is, $v=(u_i,{\phi }_{ij})$ and w is another uniform vector field of displacements, namely $w=(v_i,{\psi }_{ij})$.

With the help of the bilinear form $E(v,w)$ we define the inner product (12) $⟨v,w⟩=2E(v,w),$(12) from where we obtain the norm $\parallel v\parallel =\sqrt{<v,v>}$.

According to Iesan and Quintanilla [34] (see also [35]), in the case that the displacements v and w satisfy the equations of equilibrium (6(6) $\begin{array}{rl}{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}& =0,\\ {\left({\eta }_{ijk}\left(w\right)\right)}_{,i}+{\sigma }_{jk}\left(w\right)& =0,\end{array}$(6) ), then taking into account the relations (8(8) $t_i\left(w\right)=\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j,m_{jk}\left(w\right)=\left({\eta }_{ijk}\left(w\right)\right)n_i,$(8) ) and by using the divergence theorem, we can rewrite the inner product in the form (13) $⟨v,w⟩={\int }_{\mathrm{\partial }\mathrm{\Omega }}\left[v_i{t}_i\left(w\right)+{\psi }_{jk}{m}_{jk}\left(w\right)\right]\mathrm{d}A,$(13) and the following reciprocity relation holds true: (14) $\begin{array}{rl}& {\int }_{\mathrm{\partial }\mathrm{\Omega }}\left[v_i{t}_i\left(w\right)+{\psi }_{jk}{m}_{jk}\left(w\right)\right]\mathrm{d}A\\ & ={\int }_{\mathrm{\partial }\mathrm{\Omega }}\left[u_i{t}_i\left(w\right)+{\phi }_{jk}{m}_{jk}\left(w\right)\right]\mathrm{d}A.\end{array}$(14)

3. Main results

In the following we will consider as domain Ω a bar of prismatic form, having the length h, which consists of an dipolar elastic media. We will suppose that the bar has a materially cross-section which is uniform at each point and the bar is unstressed. The axis of the bar is the $x_3$-axis from a system of Cartesian coordinates and in any point of the bar we have $x_3\ge O$. The plane $x_3=0$ is the loaded end of the bar. The cross-sections in the prismatic bar corresponding to the planes $x_3=0$ and $x_3=h$ are denoted by $S_1$ and $S_2$. Any cross-section in the prismatic bar is a regular region which is supposed be simply connected.

For other theoretical considerations see [36]. If we denote by L the lateral surface of the prismatic, the the domain Ω is the cylinder consisting of the lateral surface L and the cross-sections $S_1$ and $S_2$.

To complete the mixed problem we will add the next boundary data (15) $\begin{array}{rl}& \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}L,\\ & \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=t_i^{\left(\alpha \right)},\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=m_{jk}^{\left(\alpha \right)},\text{on}{S}_{\alpha },\end{array}$(15) where $t_i^{\left(\alpha \right)}$ and $m_{jk}^{\left(\alpha \right)}$, $\alpha =1,2$, are given functions which satisfy certain usual hypotheses of regularity.

If the components of the displacement field $w=(u_i,{\phi }_{ij})$ satisfy the equations of equilibrium (6(6) $\begin{array}{rl}{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}& =0,\\ {\left({\eta }_{ijk}\left(w\right)\right)}_{,i}+{\sigma }_{jk}\left(w\right)& =0,\end{array}$(6) ) and the boundary conditions (15(15) $\begin{array}{rl}& \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}L,\\ & \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=t_i^{\left(\alpha \right)},\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=m_{jk}^{\left(\alpha \right)},\text{on}{S}_{\alpha },\end{array}$(15) ), we say that w is a solution of the classical Saint-Venant's problem.

It is also known (see, for instance, Iesan and Quintanilla [34]) that the solution of the Saint-Venant's problem is determined until a rigid motion and exists, if and only if, the following conditions occur (16) $\begin{array}{rl}& {\int }_{S_1}{t}_i^{\left(1\right)}\mathrm{d}A+{\int }_{S_2}{t}_i^{\left(2\right)}\mathrm{d}A=0,\\ & {\int }_{S_1}{e}_{ijk}\left[x_j{t}_k^{\left(1\right)}+m_{jk}^{\left(1\right)}\right]\mathrm{d}A\\ & +{\int }_{S_2}{e}_{ijk}\left[x_j{t}_k^{\left(2\right)}+m_{jk}^{\left(2\right)}\right]\mathrm{d}A=0,\end{array}$(16) where $e_{ijk}$ is the alternating symbol.

If we consider the so-called relaxed problem of Saint-Venant, then instead of the boundary conditions (15(15) $\begin{array}{rl}& \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}L,\\ & \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=t_i^{\left(\alpha \right)},\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=m_{jk}^{\left(\alpha \right)},\text{on}{S}_{\alpha },\end{array}$(15) ), the following boundary conditions are required (17) $\begin{array}{rl}& \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}L,\\ & R_i\left(w\right)={\mathcal{R}}_i,M_{jk}\left(w\right)={\mathcal{M}}_{jk},\text{on}{S}_1,\end{array}$(17) where ${\mathcal{R}}_i$ and ${\mathcal{M}}_{jk}$ are prescribed sizes and are the components of the resultant force and resultant moment, respectively. The components of moments are calculated for the tractions acting on the surface $S_1$ and are considered relative to the origin.

Consider $v_1$ a solution of the problem of Saint-Venant and $v_2$ a solution of the relaxed Saint-Venant's problem. Then the diference $v=v_1-v_2$ is a solution of the equilibrium equations (6(6) $\begin{array}{rl}{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}& =0,\\ {\left({\eta }_{ijk}\left(w\right)\right)}_{,i}+{\sigma }_{jk}\left(w\right)& =0,\end{array}$(6) ) on the domain Ω which satisfies the next boundary data: (18) $\begin{array}{rl}& \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}L,\\ & {\int }_{S_{\alpha }}{t}_i\left(w\right)\mathrm{d}A=0,{\int }_{S_{\alpha }}{e}_{ijk}\left[x_j{t}_k\left(w\right)+m_{jk}\left(w\right)\right]\\ & \mathrm{d}A=0,\alpha =1,2.\end{array}$(18) Consider the following eigenvalue problem (19) $\begin{array}{rl}& -{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}=\lambda u_i,\\ & -{\left({\eta }_{ijk}\left(w\right)\right)}_{,i}-{\sigma }_{jk}\left(w\right)=\lambda {\phi }_{jk}\text{in}V,\\ & \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}{S}_V,\end{array}$(19) where V is the domain from the slice of the bar (a cylinder) having the length l, which is located between the planes $x_3=s$ and $x_3=s+l$. The slice is taken normal to the generators of the cylinder and its total boundary surface $S_V$ is free of traction.

We will denote by ${\lambda }_0\left(l\right)$ the smallest nonzero eigenvalue of the eigenvalue problem (19(19) $\begin{array}{rl}& -{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}=\lambda u_i,\\ & -{\left({\eta }_{ijk}\left(w\right)\right)}_{,i}-{\sigma }_{jk}\left(w\right)=\lambda {\phi }_{jk}\text{in}V,\\ & \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}{S}_V,\end{array}$(19) ). Furthermore, we will use the notation $s_c\left(l\right)=2\sqrt{\frac{{\lambda }_M}{{\lambda }_0\left(l\right)}},$ which is called the characteristic decay length. Here ${\lambda }_M$ is the supremum of the eigenvalues of the operatorial equation $\mathrm{\Psi }\left(w\right)=\lambda w$. Recall that $S_{\alpha }$ is the notation for a generic cross-section in the prismatic bar which is located in the plane $x_3=\alpha$.

In the following theorem we prove the above mentioned extension of the Saint-Venant's principle.

Theorem 3.1

Consider that the prismatic body consists of a dipolar homogeneous material, has materially uniform cross-sections, and the strain energy density is a positive definite quadratic form. Furthermore, assume that the cylinder is loaded on the section $S_0$ by a system of forces which is self-equilibrated. If the displacement field v is a solution of the equilibrium equations (6(6) $\begin{array}{rl}{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}& =0,\\ {\left({\eta }_{ijk}\left(w\right)\right)}_{,i}+{\sigma }_{jk}\left(w\right)& =0,\end{array}$(6) ) and satisfies the boundary conditions (18(18) $\begin{array}{rl}& \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}L,\\ & {\int }_{S_{\alpha }}{t}_i\left(w\right)\mathrm{d}A=0,{\int }_{S_{\alpha }}{e}_{ijk}\left[x_j{t}_k\left(w\right)+m_{jk}\left(w\right)\right]\\ & \mathrm{d}A=0,\alpha =1,2.\end{array}$(18) ) then we have (20) $\frac{W\left(s\right)}{W\left(0\right)}\le {\mathrm{e}}^{(l-s)/s_c\left(l\right)}.$(20)

Proof.

We take into account that quadratic form Ψ is homogeneous with respect to its constitutive variables. If we use the known Euler's theorem regarding the homogeneous functions (see, for instance [34]), from (10(10) $S\left(w\right)={\int }_{\mathrm{\Omega }}\mathrm{\Psi }\left(w\right)\mathrm{d}V.$(10) ) we deduce (21) $\begin{array}{rl}W\left(s\right)& =\frac{1}{2}{\int }_{x_3\ge s}\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}{e}_{ij}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}{\nu }_{ij}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\chi }_{ijk}\right)\mathrm{d}V=\\ & =\frac{1}{2}{\int }_{x_3\ge s}\left[\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}{u}_{j,i}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}\left(u_{j,i}-{\phi }_{ij}\right)+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\phi }_{jk,i}\right]\mathrm{d}V\\ & =\frac{1}{2}{\int }_{L_s}\left[n_i\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}\right)u_j+n_i\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\phi }_{jk}\right]\mathrm{d}A\\ & =-\frac{1}{2}{\int }_{L_s}\left[\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{3j}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{3j}}\right)u_j+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}{\phi }_{jk}\right]\mathrm{d}A.\end{array}$(21) In the deduction of Equation (21(21) $\begin{array}{rl}W\left(s\right)& =\frac{1}{2}{\int }_{x_3\ge s}\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}{e}_{ij}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}{\nu }_{ij}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\chi }_{ijk}\right)\mathrm{d}V=\\ & =\frac{1}{2}{\int }_{x_3\ge s}\left[\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}{u}_{j,i}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}\left(u_{j,i}-{\phi }_{ij}\right)+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\phi }_{jk,i}\right]\mathrm{d}V\\ & =\frac{1}{2}{\int }_{L_s}\left[n_i\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}\right)u_j+n_i\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\phi }_{jk}\right]\mathrm{d}A\\ & =-\frac{1}{2}{\int }_{L_s}\left[\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{3j}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{3j}}\right)u_j+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}{\phi }_{jk}\right]\mathrm{d}A.\end{array}$(21) ) we used the geometric equations (1(1) $E\left(d\right)\le E\left(0\right)\exp [-(d-l\left)/s_c\right(l\left)\right].$(1) ), the divergence theorem and the fact that the unit outward normal satisfies the conditions $n_i=-{\delta }_{3i},\text{on}{L}_s.$ We can use the known Schwarz inequality and then the classical arithmetic-geometric inequality so that it is easy to prove the following elementary inequality (22) $2{\int }_{\mathrm{\Omega }}ab\mathrm{d}V\le x{\int }_{\mathrm{\Omega }}{a}^2\mathrm{d}V+\frac{1}{x}{\int }_{\mathrm{\Omega }}{b}^2\mathrm{d}V,$(22) which takes place for any x>0.

Inspired by inequality (22(22) $2{\int }_{\mathrm{\Omega }}ab\mathrm{d}V\le x{\int }_{\mathrm{\Omega }}{a}^2\mathrm{d}V+\frac{1}{x}{\int }_{\mathrm{\Omega }}{b}^2\mathrm{d}V,$(22) ), we will obtain some estimates for the last integral from (22(22) $2{\int }_{\mathrm{\Omega }}ab\mathrm{d}V\le x{\int }_{\mathrm{\Omega }}{a}^2\mathrm{d}V+\frac{1}{x}{\int }_{\mathrm{\Omega }}{b}^2\mathrm{d}V,$(22) ). Thus, considering the first part of this integral, we are led to (23) $\begin{array}{rl}& -\frac{1}{2}{\int }_{L_s}\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{3j}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{3j}}\right)u_j\mathrm{d}A\\ & \le \frac{1}{4}\left[x_1{\int }_{L_s}\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{3j}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{3j}}\right)\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{3j}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{3j}}\right)\mathrm{d}A\right.\\ & \left.+\frac{1}{x_1}{\int }_{L_s}{u}_j{u}_j\mathrm{d}A\right].\end{array}$(23) Similarly, regarding the second part of the last integral from (22(22) $2{\int }_{\mathrm{\Omega }}ab\mathrm{d}V\le x{\int }_{\mathrm{\Omega }}{a}^2\mathrm{d}V+\frac{1}{x}{\int }_{\mathrm{\Omega }}{b}^2\mathrm{d}V,$(22) ), considering (21(21) $\begin{array}{rl}W\left(s\right)& =\frac{1}{2}{\int }_{x_3\ge s}\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}{e}_{ij}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}{\nu }_{ij}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\chi }_{ijk}\right)\mathrm{d}V=\\ & =\frac{1}{2}{\int }_{x_3\ge s}\left[\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}{u}_{j,i}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}\left(u_{j,i}-{\phi }_{ij}\right)+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\phi }_{jk,i}\right]\mathrm{d}V\\ & =\frac{1}{2}{\int }_{L_s}\left[n_i\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}\right)u_j+n_i\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\phi }_{jk}\right]\mathrm{d}A\\ & =-\frac{1}{2}{\int }_{L_s}\left[\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{3j}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{3j}}\right)u_j+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}{\phi }_{jk}\right]\mathrm{d}A.\end{array}$(21) ) we deduce (24) $\begin{array}{rl}& -\frac{1}{2}{\int }_{L_s}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}{u}_j\mathrm{d}A\\ & \le \frac{1}{4}\left(x_2{\int }_{L_s}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}\mathrm{d}A+\frac{1}{x_2}{\int }_{L_s}{\phi }_{jk}{\phi }_{jk}\mathrm{d}A\right).\end{array}$(24) If we set $x_1=x_2=\alpha$, then taking into account the inequalities (23(23) $\begin{array}{rl}& -\frac{1}{2}{\int }_{L_s}\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{3j}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{3j}}\right)u_j\mathrm{d}A\\ & \le \frac{1}{4}\left[x_1{\int }_{L_s}\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{3j}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{3j}}\right)\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{3j}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{3j}}\right)\mathrm{d}A\right.\\ & \left.+\frac{1}{x_1}{\int }_{L_s}{u}_j{u}_j\mathrm{d}A\right].\end{array}$(23) ) and (24(24) $\begin{array}{rl}& -\frac{1}{2}{\int }_{L_s}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}{u}_j\mathrm{d}A\\ & \le \frac{1}{4}\left(x_2{\int }_{L_s}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}\mathrm{d}A+\frac{1}{x_2}{\int }_{L_s}{\phi }_{jk}{\phi }_{jk}\mathrm{d}A\right).\end{array}$(24) ), we obtain from (21(21) $\begin{array}{rl}W\left(s\right)& =\frac{1}{2}{\int }_{x_3\ge s}\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}{e}_{ij}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}{\nu }_{ij}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\chi }_{ijk}\right)\mathrm{d}V=\\ & =\frac{1}{2}{\int }_{x_3\ge s}\left[\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}{u}_{j,i}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}\left(u_{j,i}-{\phi }_{ij}\right)+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\phi }_{jk,i}\right]\mathrm{d}V\\ & =\frac{1}{2}{\int }_{L_s}\left[n_i\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}\right)u_j+n_i\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}{\phi }_{jk}\right]\mathrm{d}A\\ & =-\frac{1}{2}{\int }_{L_s}\left[\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{3j}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{3j}}\right)u_j+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}{\phi }_{jk}\right]\mathrm{d}A.\end{array}$(21) ) the following expression: (25) $\begin{array}{rl}W\left(s\right)& \le \frac{1}{4}\left[\alpha {\int }_{C_s}\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}\right)\mathrm{d}A\right.\\ & \left.+\frac{1}{\alpha }{\int }_{C_s}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}A\right].\end{array}$(25) To simplify writing, we denote the ordered array $(e_{ij},{\nu }_{ij},{\chi }_{ijk})$ by E and then we can write Ψ in the form (26) $\mathrm{\Psi }=\frac{1}{2}E\cdot TE,$(26) where T is a linear transformation.

From (25(25) $\begin{array}{rl}W\left(s\right)& \le \frac{1}{4}\left[\alpha {\int }_{C_s}\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}\right)\mathrm{d}A\right.\\ & \left.+\frac{1}{\alpha }{\int }_{C_s}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}A\right].\end{array}$(25) ), considering that Ψ is a positive definite quadratic form, we deduce (27) $\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }E}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }E}=TE\cdot TE=E\cdot T^{2}E\le {\lambda }_{M}E\cdot TE=2{\lambda }_M\mathrm{\Psi },$(27) where we denote by ${\lambda }_M$ the maximum eigenvalue of T.

If we take into account the inequality (27(27) $\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }E}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }E}=TE\cdot TE=E\cdot T^{2}E\le {\lambda }_{M}E\cdot TE=2{\lambda }_M\mathrm{\Psi },$(27) ), we are led from (25(25) $\begin{array}{rl}W\left(s\right)& \le \frac{1}{4}\left[\alpha {\int }_{C_s}\left(\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{e}_{ij}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\nu }_{ij}}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{ijk}}\right)\mathrm{d}A\right.\\ & \left.+\frac{1}{\alpha }{\int }_{C_s}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}A\right].\end{array}$(25) ) to: (28) $W\left(s\right)\le \frac{1}{4}\left[\alpha {\int }_{C_s}2{\lambda }_M\mathrm{\Psi }\mathrm{d}A+\frac{1}{\alpha }{\int }_{C_s}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}A\right].$(28) For some l>0, let us consider the notation $L_{s,l}=\left\{x=\left(x_1,x_2,x_3\right):x\in \mathrm{\Omega },s\le x_3\le s+l\right\},$ that is, $L_{s,l}$ is a part of the cylinder between the planes $x_3=s$ and $x_3=s+l$.

Now, an integration of the inequality (28(28) $W\left(s\right)\le \frac{1}{4}\left[\alpha {\int }_{C_s}2{\lambda }_M\mathrm{\Psi }\mathrm{d}A+\frac{1}{\alpha }{\int }_{C_s}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}A\right].$(28) ), relative to to time variable, over the interval $[s,s+l]$, gives us the relation (29) $V(s,l)\le \frac{\alpha {\lambda }_M}{2l}{\int }_{L_{s,l}}\mathrm{\Psi }\mathrm{d}V+\frac{1}{4\alpha l}{\int }_{L_{s,l}}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}V,$(29) where we used the notation (30) $V(s,l)=\frac{1}{l}{\int }_s^{s+l}W\left(r\right)\mathrm{d}r.$(30) Now, we intend to find a bound for the second integral from the right-hand side of Equation (24(24) $\begin{array}{rl}& -\frac{1}{2}{\int }_{L_s}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}{u}_j\mathrm{d}A\\ & \le \frac{1}{4}\left(x_2{\int }_{L_s}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }{\chi }_{3jk}}\mathrm{d}A+\frac{1}{x_2}{\int }_{L_s}{\phi }_{jk}{\phi }_{jk}\mathrm{d}A\right).\end{array}$(24) ). More specifically, we will obtain an upper bound for this integral which is an expression of the first integral from (29(29) $V(s,l)\le \frac{\alpha {\lambda }_M}{2l}{\int }_{L_{s,l}}\mathrm{\Psi }\mathrm{d}V+\frac{1}{4\alpha l}{\int }_{L_{s,l}}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}V,$(29) ).

As is known, for a rigid movement, the internal energy density is null, that is, $\mathrm{\Psi }=0$. As such, we have that zero is the smallest eigenvalue. To avoid the possibility of zero eigenvalue, or, in other words, to avoid the rigid displacement, we take smooth fields $u_i$ and ${\phi }_{ij}$ that satisfy (see [4]) (31) ${\int }_{L_{s,l}}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}V\ne 0.$(31) Let us consider the eigenvalue problem (19(19) $\begin{array}{rl}& -{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}=\lambda u_i,\\ & -{\left({\eta }_{ijk}\left(w\right)\right)}_{,i}-{\sigma }_{jk}\left(w\right)=\lambda {\phi }_{jk}\text{in}V,\\ & \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}{S}_V,\end{array}$(19) ) with $L_{s,l}$ instead of V. We multiply (19(19) $\begin{array}{rl}& -{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}=\lambda u_i,\\ & -{\left({\eta }_{ijk}\left(w\right)\right)}_{,i}-{\sigma }_{jk}\left(w\right)=\lambda {\phi }_{jk}\text{in}V,\\ & \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}{S}_V,\end{array}$(19) )${}_1$ by $u_i$ and (19(19) $\begin{array}{rl}& -{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}=\lambda u_i,\\ & -{\left({\eta }_{ijk}\left(w\right)\right)}_{,i}-{\sigma }_{jk}\left(w\right)=\lambda {\phi }_{jk}\text{in}V,\\ & \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}{S}_V,\end{array}$(19) )${}_2$ by ${\phi }_{jk}$. By adding the resulting equations and integrating over $L_{s,l}$, we obtain a relation to which we apply the divergence theorem and the null boundary conditions (19(19) $\begin{array}{rl}& -{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}=\lambda u_i,\\ & -{\left({\eta }_{ijk}\left(w\right)\right)}_{,i}-{\sigma }_{jk}\left(w\right)=\lambda {\phi }_{jk}\text{in}V,\\ & \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}{S}_V,\end{array}$(19) )${}_3$ and (19(19) $\begin{array}{rl}& -{\left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)}_{,j}=\lambda u_i,\\ & -{\left({\eta }_{ijk}\left(w\right)\right)}_{,i}-{\sigma }_{jk}\left(w\right)=\lambda {\phi }_{jk}\text{in}V,\\ & \left(t_{ij}\left(w\right)+{\sigma }_{ij}\left(w\right)\right)n_j=t_i\left(w\right)=0,\\ & \left({\eta }_{ijk}\left(w\right)\right)n_i=m_{jk}\left(w\right)=0,\text{on}{S}_V,\end{array}$(19) )${}_4$. In this way, we deduce that the lowest nonzero eigenvalue ${\lambda }_0\left(l\right)$ satisfies the inequality: (32) ${\lambda }_0\left(l\right)\le \frac{{\int }_{L_{s,l}}\mathrm{\Psi }\mathrm{d}V}{{\int }_{L_{s,l}}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}V},$(32) which is well-defined based on (31(31) ${\int }_{L_{s,l}}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}V\ne 0.$(31) ).

If we take into account (31(31) ${\int }_{L_{s,l}}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}V\ne 0.$(31) ), from (29(29) $V(s,l)\le \frac{\alpha {\lambda }_M}{2l}{\int }_{L_{s,l}}\mathrm{\Psi }\mathrm{d}V+\frac{1}{4\alpha l}{\int }_{L_{s,l}}\left(u_j{u}_j+{\phi }_{jk}{\phi }_{jk}\right)\mathrm{d}V,$(29) ) we deduce (33) $V(s,l)\le \frac{1}{l}\left(\frac{1}{2}\alpha {\lambda }_M+\frac{2}{{\lambda }_0\alpha }\right){\int }_{L_{s,l}}\mathrm{\Psi }\mathrm{d}V.$(33) If we choose $\alpha =\frac{2}{\sqrt{{\lambda }_0{\lambda }_M}},$ then the term $1/2\alpha {\lambda }_M+2/\left({\lambda }_0\alpha \right)$ from (33(33) $V(s,l)\le \frac{1}{l}\left(\frac{1}{2}\alpha {\lambda }_M+\frac{2}{{\lambda }_0\alpha }\right){\int }_{L_{s,l}}\mathrm{\Psi }\mathrm{d}V.$(33) ) takes the minimum value, denoted by $s_c\left(l\right)$, namely $s_c\left(l\right)=2\sqrt{\frac{{\lambda }_M}{{\lambda }_0}},$ and (33(33) $V(s,l)\le \frac{1}{l}\left(\frac{1}{2}\alpha {\lambda }_M+\frac{2}{{\lambda }_0\alpha }\right){\int }_{L_{s,l}}\mathrm{\Psi }\mathrm{d}V.$(33) ) becomes (34) $V(s,l)\le \frac{s_c\left(l\right)}{l}{\int }_{L_{s,l}}\mathrm{\Psi }\mathrm{d}V.$(34) Now, we differentiate in (30(30) $V(s,l)=\frac{1}{l}{\int }_s^{s+l}W\left(r\right)\mathrm{d}r.$(30) ) with respect to s such that we obtain $\frac{\mathrm{d}V(s,l)}{\mathrm{d}s}=\frac{1}{l}\left[W\right(s+l)-W(s\left)\right]=-\frac{1}{l}{\int }_{L_{s,l}}\mathrm{\Psi }\mathrm{d}V,$ and this relation together with (34(34) $V(s,l)\le \frac{s_c\left(l\right)}{l}{\int }_{L_{s,l}}\mathrm{\Psi }\mathrm{d}V.$(34) ) leads to (35) $s_c\left(l\right)\frac{\mathrm{d}V(s,l)}{\mathrm{d}s}+V(s,l)\le 0.$(35) Taking into account that $W\left(s\right)$ is a nonincreasing function of s, we deduce that (36) $W(s+l)\le V(s,l)\le W\left(s\right).$(36) If we integrate in (35(35) $s_c\left(l\right)\frac{\mathrm{d}V(s,l)}{\mathrm{d}s}+V(s,l)\le 0.$(35) ) and then use (36(36) $W(s+l)\le V(s,l)\le W\left(s\right).$(36) ) we are led to the inequality (37) $W(s_2+l)\le W\left(s_2\right){\mathrm{e}}^{(s_1-s_2)/s_c\left(l\right)}.$(37) Finally, if in (37(37) $W(s_2+l)\le W\left(s_2\right){\mathrm{e}}^{(s_1-s_2)/s_c\left(l\right)}.$(37) ) we choose $s_2=s-l$ and $s_1=0$, then we get the desired estimate (20(20) $\frac{W\left(s\right)}{W\left(0\right)}\le {\mathrm{e}}^{(l-s)/s_c\left(l\right)}.$(20) ) which concludes the proof of the theorem.

4. Conclusions

From the above, it can be deduced that the principle of Saint-Venant proposes an estimate of the behaviour of the steady state in a elastic material, with dipolar structure, which ‘fills’ a bar of prismatic form. For this bar we have not null conditions only at one end of it $x_3=0$. Specifically, any motion in the media vanishes, at a concrete distance d from the loaded end. According to the above inequality (20(20) $\frac{W\left(s\right)}{W\left(0\right)}\le {\mathrm{e}}^{(l-s)/s_c\left(l\right)}.$(20) ) we can deduce that the energy of strain which is stored in the prismatic bar, in an area situated at a specific distance d from the plane $x_3=0$ has a decrease of exponential type With regards to the distance d. In accordance with our initial remarks and taking into account the clarifications of Toupin from the paper [4], a weak point of the Saint-Venant principle must be specified. The use of this principle is efficient, in the sense of obtaining of an optimal rate of decay, only in the case when we consider certain cross-sections.

Disclosure statement

No potential conflict of interest was reported by the author(s).