In solid mechanics, the linear stability analysis of an elastic solution is studied using the method of incremental deformations superposed on finite deformations.[1] The method of incremental deformation can be used to solve static,[2] quasi-static [3] and time-dependent problems.[4] The governing equations of the motion are ones of the classical mechanics, such as the conservation of mass and the balance of linear and angular momentum, which provide the equilibrium configuration of the material. The main corresponding mathematical framework is described in the main Raymond Ogden's book Non-linear elastic deformations and in Biot's book Mechanics of incremental deformations,[5] which is a collection of his main papers.
Let
l{E}\inR3
l{B}0,l{B}a\inl{E}
{\bf\chi}
l{B}0
l{B}a
\chi
C1
l{B}0
l{B}a
{\bfx}={\bf\chi}({\bfX})
{\bfX}
Considering a hyperelastic material with an elastic strain energy density
W({\bfF})
{\bfS}
{\bfS}=
\partialW | |
\partial{\bfF |
For a quasi-static problem, without body forces, the equilibrium equation is
\begin{align} {\rmDiv}{\bfS}&=0 && Equilibrium,\\[3pt] \end{align}
where
{\rmDiv}
If the material is incompressible,[8] i.e. the volume of every subdomains does not change during the deformation, a Lagrangian multiplier[9] is typically introduced to enforce the internal isochoric constraint
\det{\bfF}=1
\begin{align} {\bfS}&=
\partialW | |
\partial{\bfF |
Let
\partiall{B}0
l{B}0
\partiall{B}a
l{B}a
\GammaD
\partiall{B}0
\GammaN
\partiall{B}0=\GammaD\cup\GammaN
{\bf
* | |
u} | |
0 |
\GammaD
{\bf
* | |
t} | |
0 |
\GammaN
\begin{align} {\bfu}(\bf{X})&={\bf
* | |
u} | |
0 |
&& {\rmon}\GammaD,\\[3pt] {\bfS}\rm ⋅ {\bfN}&={\bf
* | |
t} | |
0 |
&& {\rmon}\GammaN,\\[3pt] \end{align}
where
{\bfu}={\bfx}-{\bfX}=\chi({\bfX})-{\bfX}
{\bfN}
\partiall{B}0
The defined problem is called the boundary value problem (BVP). Hence, let
{\bfx}0=\chi0({\bfX})
W
\gamma
W
To improve this method, one has to superpose a small displacement
\delta{\bfx}
{\bfx}0
\bar{{\bfx}}={\bfx}0+\delta{\bfx}={\bfx}0+\chi1({\bfx}0)
where
\bar{{\bfx}}
\chi1({\bfx}0)
{\bfx}0
\deltal{B}a
In the following, the incremental variables are indicated by
\delta(\bullet)
\bar{\bullet}
The perturbed deformation gradient is given by:
\bar{{\bfF}}={\bfF}0+\delta{\bfF}=(I+\Gamma){\bfF}0
where
\Gamma={\rmgrad}\chi1({\bfx}0)
{\rmgrad}
The perturbed Piola stress is given by:
\bar{{\bfS}}={\bfS}0+\delta{\bfS}={\bfS}0+
\partial{\bfS | |
0}{\partial |
{\bf
0} | |
F}}| | |
{\bfF |
:\delta{\bfF}
where
:
\partial{\bfS | |
0}{\partial |
{\bf
0} | |
F}}| | |
{\bfF |
=l{A}1
\delta{\bfF}
{\bfS}
{\bfF}
W
\bar{{\bfS}}=
\partialW | |
\partial{\bfF |
If the material is incompressible, one gets
\delta{\bfS}=
\partial{\bfS | |
0}{\partial |
{\bfF}}\delta{\bfF}=l{A}1\delta{\bfF}+p({\bfF}0)-1\delta{\bfF}({\bfF}0)-1-\deltap({\bfF}0)-1,
where
\deltap
p
l{A}1
({\bfS},{\bfF})
It is useful to derive the push-forward of the perturbed Piola stress be defined as
\delta{\bfS}0={\bfF}0\delta{\bfS}=
1 | |
l{A} | |
0 |
\Gamma+p\Gamma-\deltap{\rmI},
1 | |
l{A} | |
0 |
1 | |
(l{A} | |
0) |
ijhk={\bf
0 | |
F} | |
i\gamma |
{\bf
0 | |
F} | |
h\beta |
1 | |
l{A} | |
\gammaj\betak |
Expanding the equilibrium equation around the basic solution, one gets
{\rmDiv}(\bar{\bfS})={\rmDiv}({\bfS}0+\delta{\bfS})={\rmDiv}({\bfS}0)+{\rmDiv}(\delta{\bfS})=0.
{\bfS}0
{\rmdiv}(\delta{\bfS}0)=0,
where
{\rmdiv}
The incremental incompressibility constraint reads
\det({\bfF}0+\delta{\bfF})=1.
Expanding this equation around the basic solution, as before, one gets
{\rmtr}(\Gamma)=0.
Let
\overline{\delta{\bfu}}
\overline{\delta{\bft}}
{\bf
* | |
u} | |
0 |
{\bf
* | |
t} | |
0 |
{ \begin{align} \delta{\bfu}({\bfx})&=\overline{\delta{\bfu}}&& {\bfx}\in{\GammaD
where
\delta{\bfu}=\chi1({\bfx}0)-{\bfx}
{\GammaD
The incremental equations
{\rmdiv}(\delta{\bfS}0)=0 \hbox{forcompressiblematerial}
{ \begin{align} \begin{cases} {\rmdiv}(\delta{\bfS}0)&=0\\ {\rmtr}(\Gamma)&=0 \end{cases} \hbox{forincompressiblematerial} \end{align} }
represent the incremental boundary value problem (BVP) and define a system of partial differential equations (PDEs).[13] The unknowns of the problem depend on the considered case. For the first one, such as the compressible case, there are three unknowns, such as the components of the incremental deformations
\delta{u}1({\bfx}),\delta{u}2({\bfx}),\delta{u}3({\bfx})
\chi1({\bfx})=\delta{u}1({\bfx}){\bfe}1+\delta{u}2({\bfx}){\bfe}2+\delta{u}3({\bfx}){\bfe}3
\deltap
p
The main difficulty to solve this problem is to transform the problem in a more suitable form for implementing an efficient and robusted numerical solution procedure.[14] The one used in this area is the Stroh formalism. It was originally developed by Stroh [15] for a steady state elastic problem and allows the set of four PDEs with the associated boundary conditions to be transformed into a set of ODEs of first order with initial conditions. The number of equations depends on the dimension of the space in which the problem is set. To do this, one has to apply variable separation and assume periodicity in a given direction depending on the considered situation.[16] In particular cases, the system can be rewritten in a compact form by using the Stroh formalism. Indeed, the shape of the system looks like
d | |
d{\rmx |
where
η
{\rmx}
{\bfG}
{\bfG}= {\begin{bmatrix} {\bfG}1&{\bfG}2\\ {\bfG}3&{\bfG}4 \end{bmatrix}, }
where each block is a matrix and its dimension depends on the dimension of the problem. Moreover, a crucial property of this approach is that
{\bfG}4=({\bf
* | |
G} | |
1) |
{\bfG}4
{\bfG}1
The Stroh formalism provides an optimal form to solve a great variety of elastic problems. Optimal means that one can construct an efficient numerical procedure to solve the incremental problem. By solving the incremental boundary value problem, one finds the relations[18] among the material and geometrical parameters of the problem and the perturbation modes by which the wave propagates in the material, i.e. what denotes the instability. Everything depends on
\gamma
By this analysis, in a graph perturbation mode vs control parameter, the minimum value of the perturbation mode represents the first mode at which one can see the onset of the instability. For instance, in the picture, the first value of the mode
kz
0.3
\gamma=0
kz=0