The elastic problem is governed by the field equations (
10), (
15) and (
23), with the boundary conditions (
11), (
19) and (
17). They can be rewritten in matrix form, both in the field:
$$\begin{aligned} \begin{aligned}&\varvec{\varepsilon }=\varvec{u}'+\varvec{\Gamma }\varvec{u},\\&\varvec{\tilde{\sigma }}'-\varvec{\varvec{\Gamma }}^{T}\varvec{\tilde{\sigma }}-\left( \mathring{N}\varvec{\Gamma }\varvec{u}\right) '+\mathring{N}\left( \varvec{\Gamma }^{T}\varvec{u}'+\varvec{\Omega }\varvec{u}\right) =\mathbf {0},\\&\varvec{\tilde{\sigma }}=\left( \varvec{C}^{0}+\mathring{N}\hat{\varvec{C}}\right) \varvec{\varepsilon }, \end{aligned} \end{aligned}$$
(31)
and at the boundary:
$$\begin{aligned} \begin{aligned}\varvec{A}_{u}\varvec{u}_{A}&=\mathbf {0},\\ \varvec{A}_{\sigma }\varvec{\tilde{\sigma }}_{A}&=\varvec{K}_{s}\varvec{A}_{\sigma }\varvec{u}_{A},\\ \varvec{\tilde{\sigma }}_{B}&=\mathring{N}_{B}\varvec{\Gamma }\varvec{u}_{B}. \end{aligned} \end{aligned}$$
(32)
Here:
\(\varvec{u}:=\left( u,v,w,\theta _{x},\theta _{y},\theta _{z}\right) ^{T}\)is the displacement column vector;
\(\varvec{\varvec{\Gamma },\,\Omega }\), as
\(\varvec{A}_{u},\varvec{A}_{\sigma },\) are Boolean matrices:
$$\begin{aligned} \begin{aligned}&\varvec{\Gamma }:=\left[ \begin{array}{cccccc} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} -1\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 1 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \end{array}\right] ,\qquad \varvec{\Omega }:=\left[ \begin{array}{cccccc} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 1 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 1 \end{array}\right] ,\\ \\&\varvec{A}_{u}:=\left[ \begin{array}{cccccc} 0 \quad &{} 1 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 1 \quad &{} 0 \quad &{} 0 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 1 \quad &{} 0 \quad &{} 0 \end{array}\right] ,\qquad \varvec{A}_{\sigma }:=\left[ \begin{array}{cccccc} 1 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 1 \quad &{} 0\\ 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 0 \quad &{} 1 \end{array}\right] , \end{aligned} \end{aligned}$$
(33)
and
\(\varvec{K}_{s}\) is the soil stiffness matrix:
$$\begin{aligned} \varvec{K}_{s}:=\left[ \begin{array}{ccc} k_{x} \quad &{} 0 \quad &{} 0\\ 0 \quad &{} k_{y} \quad &{} -k_{yz}\\ 0 \quad &{} -k_{yz} \quad &{} k_{z} \end{array}\right] . \end{aligned}$$
(34)
By expressing equilibrium in terms of displacements, the elastic problem is governed by the following set of ordinary differential equations:
$$\begin{aligned} \varvec{D}_{2}\left( \mathring{N}\right) \varvec{u}''+\varvec{D}_{1}\left( \mathring{N},\mathring{N}'\right) \varvec{u}'+\varvec{D}_{0}\left( \mathring{N},\mathring{N}'\right) \varvec{u}=\varvec{0}, \end{aligned}$$
(35)
with the boundary conditions:
$$\begin{aligned} \begin{aligned}&\varvec{A}_{u}\varvec{u}_{A}=\mathbf {0},\\&\varvec{B}_{H1}\left( \mathring{N}_{H}\right) \varvec{u}'_{H}+\varvec{B}_{H0}\left( \mathring{N}_{H}\right) \varvec{u}{}_{H}=\varvec{0},\qquad H=A,B. \end{aligned} \end{aligned}$$
(36)
In the previous equations, matrices
\(\varvec{D}\) and
\(\varvec{B}\) linearly depend on prestress and, possibly, on its gradient, according to:
$$\begin{aligned} \begin{aligned}\varvec{D}_{2}\left( \mathring{N}\right)&:=\varvec{D}_{2}^{0}+\mathring{N}\varvec{\hat{D}}_{2},\\ \varvec{D}_{i}\left( \mathring{N},\mathring{N}'\right)&:=\varvec{D}_{i}^{0}+\mathring{N}\varvec{\hat{D}}_{i}+\mathring{N}'\check{\varvec{D}}_{i},&i=0,1,\\ \varvec{B}_{Hj}\left( \mathring{N}_{H}\right)&:=\varvec{B}_{Hj}^{0}+\mathring{N}_{H}\varvec{\hat{B}}_{Hj},&j=0,1,\quad H=A,B, \end{aligned} \end{aligned}$$
(37)
where the following positions hold, in the field:
and at the boundary:
$$\begin{aligned} \begin{aligned}\varvec{B}_{A1}^{0}&:=\varvec{A_{\sigma }}\varvec{C}^{0},&\varvec{\hat{B}}_{A1}&:=\varvec{A_{\sigma }}\hat{\varvec{C}},\\ \varvec{B}_{A0}^{0}&:=\varvec{A_{\sigma }}\varvec{C}^{0}\varvec{\Gamma }-\varvec{K}_{s}\varvec{A}_{\sigma },&\varvec{\hat{B}}_{A0}&:=\varvec{A_{\sigma }}\hat{\varvec{C}}\varvec{\Gamma },\\ \varvec{B}_{B1}^{0}&:=\varvec{C}^{0},&\varvec{\hat{B}}_{B1}&:=\hat{\varvec{C}},\\ \varvec{B}_{B0}^{0}&:=\varvec{C}^{0}\varvec{\Gamma },&\varvec{\hat{B}}_{B0}&:=\hat{\varvec{C}}\varvec{\Gamma }-\varvec{\Gamma }. \end{aligned} \end{aligned}$$
(39)
It is observed that, while the matrix
\(\varvec{D}_{2}\left( \mathring{N}\right) \) is symmetric, the matrices
\(\varvec{D}_{1}\left( \mathring{N},\mathring{N}'\right) \) and
\(\varvec{D}_{0}\left( \mathring{N},\mathring{N}'\right) \) calls for the following discussion: (a) in the case in which
\(\mathring{N}'=0\) (i.e., the self-weight is absent),
\(\varvec{D}_{1}\left( \mathring{N},0\right) \) is skew-symmetric and
\(\varvec{D}_{0}\left( \mathring{N},0\right) \) is symmetric; (b) if
\(\mathring{N}'\ne 0\) (self-weigh does exist), then
\(\varvec{D}_{1}\left( \mathring{N},\mathring{N}'\right) \) and
\(\varvec{D}_{0}\left( \mathring{N},\mathring{N}'\right) \) are neither symmetric nor skew-symmetric. In spite of this occurrence, it is easy to check that the differential equations (
35) (as well the boundary conditions (
36)) are
self-adjoint, since the following Green Identity holds:
$$\begin{aligned} \begin{aligned}&\intop _{0}^{\ell }\varvec{v}^{T}\left( \varvec{D}_{2}\varvec{u}''+\varvec{D}_{1}\varvec{u}'+\varvec{D}_{0}\varvec{u}\right) \mathrm {d}s=\intop _{0}^{\ell }\varvec{u}^{T}\left( \varvec{D}_{2}\varvec{v}''+\varvec{D}_{1}\varvec{v}'+\varvec{D}_{0}\varvec{v}\right) \mathrm {d}s+\text {boundary conditions,}\end{aligned} \end{aligned}$$
(40)
consistently with the conservative nature of forces and constitutive law.