Incremental deformations
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 formula_1 be a three-dimensional Euclidean space. Let formula_2 be two regions occupied by the material in two different instants of time. Let formula_3 be the deformation which transforms the tissue from formula_4, i.e. the material/reference configuration, to the loaded configuration formula_5, i.e. current configuration. Let formula_6 be a formula_7-diffeomorphism from formula_4 to formula_5, with formula_10 being the current position vector, given as a function of the material position formula_11. The deformation gradient is given by formula_12 Considering a hyperelastic material  with an elastic strain energy density formula_13, the Piola-Kirchhoff stress tensor formula_14 is given by formula_15. For a quasi-static problem, without body forces, the equilibrium equation is where formula_17 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 formula_18. So that, the expression of the Piola stress tensor becomes Boundary conditions. Let formula_20 be the boundary of formula_4, the reference configuration, and formula_22, the boundary of formula_23, the current  configuration. One defines the subset formula_24 of formula_20 on which Dirichlet conditions are applied, while Neumann conditions hold on formula_26, such that formula_27. If formula_28 is the displacement vector to be assigned at the portion formula_24 and formula_30 is the traction vector to be assigned to the portion formula_26, the boundary conditions can be written in mixed-form, such as where formula_33 is the displacement and the vector formula_34 is the unit outward normal to formula_35. Basic solution. The defined problem is called the boundary value problem (BVP). Hence, let formula_36 be a solution of the BVP. Since formula_37 depends  nonlinearly on the deformation gradient, this solution is generally not unique, and it depends on geometrical and material parameters of the problem. So, one has to employ the method of incremental deformation in order to highlight the existence of an adjacent solution for a critical value of a dimensionless parameter, called control parameter formula_38 which "controls" the onset of the instability. This means that by increasing the value of this parameter, at a certain point new solutions appear. Hence, the selected basic solution is not anymore the stable one but it becomes unstable. In a physical way, at a certain  time the stored energy, such as the integral of the density formula_37 over all the domain of the basic solution is bigger than the one of the new solutions. To restore the equilibrium, the configuration of the material moves to another configuration which has lower energy. Method of incremental deformations superposed on finite deformations. To improve this method, one has to superpose a small displacement formula_40 on the finite deformation basic solution formula_41. So that: where formula_43 is the perturbed position and formula_44 maps the basic position vector formula_41 in the perturbed configuration formula_46. In the following, the incremental variables  are indicated by formula_47, while the perturbed ones are indicated by formula_48. Deformation gradient. The perturbed deformation gradient is given by: where formula_50, where formula_51 is the gradient operator with respect to the current configuration. Stresses. The perturbed Piola stress is given by: where formula_53 is the increment in formula_54 and formula_55 is called the elastic moduli associated to the pairs formula_56. It is useful to derive the push-forward of the perturbed Piola stress be defined as where formula_58 is also known as "the tensor of instantaneous moduli", whose components are: Incremental governing equations. Expanding the equilibrium equation around  the basic solution, one gets Since formula_61 is the solution to the equation at the zero-order, the incremental equation can be rewritten as where formula_63 is the divergence operator with respect to the actual configuration. The incremental incompressibility constraint reads Expanding this equation around the basic solution, as before, one gets Incremental boundary conditions. Let formula_66 and formula_67 be the prescribed increment of formula_68 and formula_69 respectively. Hence, the perturbed boundary condition are where formula_71 is the incremental displacement and formula_72. Solution of the incremental problem. The incremental equations formula_73 formula_74 represent the incremental boundary value problem (BVP) and  define a system of partial differential equations (PDEs). 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 formula_75, linked to the perturbed deformation by this relation formula_76. For the latter case, instead, one has to take into account also the increment formula_53 of the Lagrange multiplier formula_54, introduced to impose the isochoric constraint. 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. The one used in this area is the Stroh formalism. It was originally developed by Stroh 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. 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 formula_79 where formula_80 is the vector which contains all the unknowns of the problem, formula_81 is the only variable on which the rewritten problem depends and the matrix formula_82 is so-called "Stroh" matrix and it has the following form 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 formula_84, i.e. formula_85 is the hermitian matrix of formula_86. Conclusion and remark. 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 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 formula_38, the selected parameter denoted as the control one. 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 formula_88 in which the instability emerges is around formula_89 since the trivial solution formula_90 and formula_91 does not have to be considered.