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.[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.[5] The main corresponding mathematical framework is described in the main Raymond Ogden's book Non-linear elastic deformations[1] and in Biot's book Mechanics of incremental deformations,[6] which is a collection of his main papers.

## Contents

* 1 Nonlinear elasticity
* 1.1 Kinematics and mechanics
* 1.2 Boundary conditions
* 1.3 Basic solution
* 2 Method of incremental deformations superposed on finite deformations
* 2.1 Deformation gradient
* 2.2 Stresses
* 2.3 Incremental governing equations
* 2.4 Incremental boundary conditions
* 3 Solution of the incremental problem
* 3.1 Conclusion and remark
* 4 See also
* 5 References

## Nonlinear elasticity

### Kinematics and mechanics

Scheme of the incremental deformation

Let ℰ∈ℝ3 be a three-dimensional Euclidean space. Let ℬ0,ℬa∈ℰ be two regions occupied by the material in two different instants of time. Let χ be the deformation which transforms the tissue from ℬ0, i.e. the material/reference configuration, to the loaded configuration ℬa, i.e. current configuration. Let χ be a C1-diffeomorphism[7] from ℬ0 to ℬa, with 𝐱=χ(𝐗) being the current position vector, given as a function of the material position 𝐗. The deformation gradient[5] is given by 𝐅=Grad𝐱=∂χ(𝐗)∂𝐗.

Considering a hyperelastic material with an elastic strain energy density[5] W(𝐅), the Piola-Kirchhoff stress tensor 𝐒 is given by 𝐒=∂W∂𝐅.

For a quasi-static problem, without body forces, the equilibrium equation is

Div𝐒=0Equilibrium,

where Div is the divergence[1] with respect to the material coordinates.

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⁡𝐅=1. So that, the expression of the Piola stress tensor becomes

𝐒=∂W∂𝐅−p𝐅−1Definition of the stress.

### Boundary conditions

Let ∂ℬ0 be the boundary of ℬ0, the reference configuration, and ∂ℬa, the boundary of ℬa, the current configuration.[1] One defines the subset ΓD of ∂ℬ0 on which Dirichlet conditions are applied, while Neumann conditions hold on ΓN, such that ∂ℬ0=ΓD∪ΓN. If 𝐮0* is the displacement vector to be assigned at the portion ΓD and 𝐭0* is the traction vector to be assigned to the portion ΓN, the boundary conditions can be written in mixed-form, such as

𝐮(𝐗)=𝐮0*onΓD,𝐒T⋅𝐍=𝐭0*onΓN,

where 𝐮=𝐱−𝐗=χ(𝐗)−𝐗 is the displacement and the vector 𝐍 is the unit outward normal to ∂ℬ0.

### Basic solution

The defined problem is called the boundary value problem (BVP). Hence, let 𝐱0=χ0(𝐗) be a solution of the BVP. Since W depends nonlinearly[10] 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 γ which "controls" the onset of the instability.[11] 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 W over all the domain of the basic solution is bigger than that of the new solutions. To restore the equilibrium, the configuration of the material moves to another configuration which has lower energy.[12]

## Method of incremental deformations superposed on finite deformations

To improve this method, one has to superpose a small displacement δ𝐱 on the finite deformation basic solution 𝐱0. So that:

𝐱¯=𝐱0+δ𝐱=𝐱0+χ1(𝐱0),

where 𝐱¯ is the perturbed position and χ1(𝐱0) maps the basic position vector 𝐱0 in the perturbed configuration δℬa.

In the following, the incremental variables are indicated by δ(∙), while the perturbed ones are indicated by ∙¯.[1]

### Deformation gradient

The perturbed deformation gradient is given by:

𝐅¯=𝐅0+δ𝐅=(𝐈+Γ)𝐅0,

where Γ=gradχ1(𝐱0), where grad is the gradient operator with respect to the current configuration.

### Stresses

The perturbed Piola stress is given by:

𝐒¯=𝐒0+δ𝐒=𝐒0+∂𝐒0∂𝐅|𝐅0:δ𝐅

where : denotes the contraction between two tensors, a fourth-order tensor ∂𝐒0∂𝐅|𝐅0=𝒜1 and a second-order tensor δ𝐅. Since 𝐒 depends on 𝐅 through W, its expression can be rewritten by emphasizing this dependence, such as

𝐒¯=∂W∂𝐅|𝐅0+∂W∂𝐅∂𝐅|𝐅0:δ𝐅.

If the material is incompressible, one gets

δ𝐒=∂𝐒0∂𝐅δ𝐅=𝒜1δ𝐅+p(𝐅0)−1δ𝐅(𝐅0)−1−δp(𝐅0)−1,

where δp is the increment in p and 𝒜1 is called the elastic moduli associated to the pairs (𝐒,𝐅).

It is useful to derive the push-forward of the perturbed Piola stress be defined as

δ𝐒0=𝐅0δ𝐒=𝒜01Γ+pΓ−δpI,

where 𝒜01 is also known as the tensor of instantaneous moduli, whose components are:

(𝒜01)ijhk=𝐅iγ0𝐅hβ0𝒜γjβk1.

### Incremental governing equations

Expanding the equilibrium equation around the basic solution, one gets

Div(𝐒¯)=Div(𝐒0+δ𝐒)=Div(𝐒0)+Div(δ𝐒)=0.

Since 𝐒0 is the solution to the equation at the zero-order, the incremental equation can be rewritten as

div(δ𝐒0)=0,

where div is the divergence operator with respect to the actual configuration.

The incremental incompressibility constraint reads

det⁡(𝐅0+δ𝐅)=1.

Expanding this equation around the basic solution, as before, one gets

tr(Γ)=0.

### Incremental boundary conditions

Let δ𝐮‾ and δ𝐭‾ be the prescribed increment of 𝐮0* and 𝐭0* respectively. Hence, the perturbed boundary condition are

δ𝐮(𝐱)=δ𝐮‾𝐱∈ΓDδ𝐒0T⋅𝐧=δ𝐭‾𝐱∈ΓN,

where δ𝐮=χ1(𝐱0)−𝐱 is the incremental displacement and ΓD∪ΓN=∂Ω.

## Solution of the incremental problem

The incremental equations

div(δ𝐒0)=0for compressible material
{div(δ𝐒0)=0tr(Γ)=0for incompressible material

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 δu1(𝐱),δu2(𝐱),δu3(𝐱), linked to the perturbed deformation by this relation χ1(𝐱)=δu1(𝐱)𝐞1+δu2(𝐱)𝐞2+δu3(𝐱)𝐞3. For the latter case, instead, one has to take into account also the increment δp of the Lagrange multiplier p, 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.[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.[15] Indeed, the shape of the system looks like

ddxη=𝐆η,

where η is the vector which contains all the unknowns of the problem, x is the only variable on which the rewritten problem depends and the matrix 𝐆 is so-called Stroh matrix and it has the following form

𝐆=[𝐆1𝐆2𝐆3𝐆4],

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 𝐆4=(𝐆1)*, i.e. 𝐆4 is the hermitian matrix of 𝐆1.[17]

### Conclusion and remark

Result of the linear stability analysis.

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 γ, 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 kz in which the instability emerges is around 0.3 since the trivial solution γ=0 and kz=0 does not have to be considered.

## See also

* Deformation (mechanics)
* Elastic instability
* Continuum mechanics

## References

1. ↑ 1.0 1.1 1.2 1.3 1.4 Ogden, R. W. (1997). Non-linear elastic deformations (Corr. ed.). Mineola, N.Y.: Dover. ISBN 978-0-486-69648-5.
2. ↑ Mora, Serge (2010). "Capillarity Driven Instability of a Soft Solid". Physical Review Letters 105 (21). doi:10.1103/PhysRevLett.105.214301. PMID 21231307.
3. ↑ Holzapfel, G. A.; Ogden, R. W. (31 March 2010). "Constitutive modelling of arteries". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466 (2118): 1551–1597. doi:10.1098/rspa.2010.0058.
4. ↑ Gower, A.L.; Destrade, M.; Ogden, R.W. (December 2013). "Counter-intuitive results in acousto-elasticity". Wave Motion 50 (8): 1218–1228. doi:10.1016/j.wavemoti.2013.03.007.
5. ↑ 5.0 5.1 5.2 Gurtin, Morton E. (1995). An introduction to continuum mechanics (6th [Dr.]. ed.). San Diego [u.a.]: Acad. Press. ISBN 978-0-12-309750-7.
6. ↑ Biot, M.A. (April 2009). "XLIII.". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 27 (183): 468–489. doi:10.1080/14786443908562246.
7. ↑ Rudin, Walter (1991). Functional analysis (2nd ed.). New York: McGraw-Hill. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi.
8. ↑ Horgan, C. O.; Murphy, J. G. (2018-03-01). "Magic angles for fibrous incompressible elastic materials" (in en). Proc. R. Soc. A 474 (2211). doi:10.1098/rspa.2017.0728. ISSN 1364-5021.
9. ↑ Bertsekas, Dimitri P. (1996). Constrained optimization and Lagrange multiplier methods. Belmont, Mass.: Athena Scientific. ISBN 978-1-886529--04-5.
10. ↑ Ball, John M. (December 1976). "Convexity conditions and existence theorems in nonlinear elasticity". Archive for Rational Mechanics and Analysis 63 (4): 337–403. doi:10.1007/BF00279992.
11. ↑ Levine, Howard A. (May 1974). "Instability and Nonexistence of Global Solutions to Nonlinear Wave Equations of the Form Pu tt = -Au + ℱ(u)". Transactions of the American Mathematical Society 192: 1–21. doi:10.2307/1996814.
12. ↑ Reid, L.D. Landau and E.M. Lifshitz; translated from the Russian by J.B. Sykes and W.H. (1986). Theory of elasticity (3rd English ed., rev. and enl. by E.M. Lifshitz, A.M. Kosevich, and L.P. Pitaevskii. ed.). Oxford [England]: Butterworth-Heinemann. ISBN 978-0-7506-2633-0.
13. ↑ Evans, Lawrence C. (2010). Partial differential equations (2nd ed.). Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-4974-3.
14. ↑ Quarteroni, Alfio (2014). Numerical models for differential problems (Second ed.). Milano: Springer Milan. ISBN 978-88-470-5522-3.
15. ↑ 15.0 15.1 Stroh, A. N. (April 1962). "Steady State Problems in Anisotropic Elasticity". Journal of Mathematics and Physics 41 (1–4): 77–103. doi:10.1002/sapm196241177.
16. ↑ Destrade, M.; Ogden, R.W.; Sgura, I.; Vergori, L. (April 2014). "Straightening wrinkles". Journal of the Mechanics and Physics of Solids 65: 1–11. doi:10.1016/j.jmps.2014.01.001.
17. ↑ Zhang, Fuzhen (2011). Matrix theory: basic results and techniques (2nd ed.). New York: Springer. ISBN 978-1-4614-1099-7. OCLC 756201359.
18. ↑ Ní Annaidh, Aisling; Bruyère, Karine; Destrade, Michel; Gilchrist, Michael D.; Maurini, Corrado; Otténio, Melanie; Saccomandi, Giuseppe (2012-03-17). "Automated Estimation of Collagen Fibre Dispersion in the Dermis and its Contribution to the Anisotropic Behaviour of Skin" (in en). Annals of Biomedical Engineering 40 (8): 1666–1678. doi:10.1007/s10439-012-0542-3. ISSN 0090-6964. PMID 22427196.

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