In solid mechanics, the linear stability analysis of an elastic solution is studied using the method of incremental deformations superposed on finite deformations. The method of incremental deformation can be used to solve static, quasi-static and time-dependent problems. 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, which is a collection of his main papers.
Nonlinear elasticity
Kinematics and mechanics
Let E ∈ R 3 {\displaystyle {\mathcal {E}}\in \mathbb {R} ^{3}} be a three-dimensional Euclidean space. Let B 0 , B a ∈ E {\displaystyle {\mathcal {B}}_{0},{\mathcal {B}}_{a}\in {\mathcal {E}}} be two regions occupied by the material in two different instants of time. Let χ {\displaystyle {\bf {\chi }}} be the deformation which transforms the tissue from B 0 {\displaystyle {\mathcal {B}}_{0}} , i.e. the material/reference configuration, to the loaded configuration B a {\displaystyle {\mathcal {B}}_{a}} , i.e. current configuration. Let χ {\displaystyle \chi } be a C 1 {\displaystyle C^{1}} -diffeomorphism from B 0 {\displaystyle {\mathcal {B}}_{0}} to B a {\displaystyle {\mathcal {B}}_{a}} , with x = χ ( X ) {\displaystyle {\bf {x}}={\bf {\chi }}({\bf {X}})} being the current position vector, given as a function of the material position X {\displaystyle {\bf {X}}} . The deformation gradient is given by
F = G r a d x = ∂ χ ( X ) ∂ X . {\displaystyle {\bf {F}}={\rm {Grad}}\,{\bf {x}}={\frac {\partial {\bf {\chi }}({\bf {X)}}}{\partial {\bf {X}}}}.}
Considering a hyperelastic material with an elastic strain energy density W ( F ) {\displaystyle W({\bf {F}})} , the Piola-Kirchhoff stress tensor S {\displaystyle {\bf {S}}} is given by S = ∂ W ∂ F {\displaystyle {\bf {S}}={\frac {\partial W}{\partial \,{\bf {F}}}}} . For a quasi-static problem, without body forces, the equilibrium equation is
D i v S = 0 Equilibrium , {\displaystyle {\begin{aligned}{\rm {Div}}\,{\bf {S}}&=0&&\qquad {\text{Equilibrium}},\\[3pt]\end{aligned}}}
where D i v {\displaystyle {\rm {Div}}} is the divergence with respect to the material coordinates. If the material is incompressible, i.e. the volume of every subdomains does not change during the deformation, a Lagrangian multiplier is typically introduced to enforce the internal isochoric constraint det F = 1 {\displaystyle \det {\bf {F}}=1} . So that, the expression of the Piola stress tensor becomes
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