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Konstantin Volokh's blog

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Mechanics of Soft Materials: Lecture Notes

I finished the grad course on Mechanics of Soft Materials. It took 14 weeks with 2 academic hours per week and it covered the following topics: 1 Tensors 2 Kinematics 3 Balance laws 4 Isotropic elasticity 5 Anisotropic elasticity 6 Viscoelasticity 7 Chemo-mechanical coupling 8 Electro-mechanical coupling.

I attach the class notes and I will be grateful for the remarks, corrections, and criticism from iMechanicians.

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Journal Club Theme of September 2009: Elasticity and Failure

Let us consider interaction of two atoms/molecules/particles. The reference distance between them corresponds to zero interaction force and zero stored energy. The interaction passes three stages with the increase of the distance. At the first stage the force increases proportionally to the increasing distance: the linear stage. At the second stage the force-distance relationship deviates from the linear proportionality: the nonlinear stage. At the third stage the force drops with the increasing distance: the separation or failure stage.

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Nice and low-cost introductory texts

There are two very nice companion texts on continuum mechanics and nonlinear elasticity printed by Dover recently: "Continuum mechanics" by Spencer and "An introduction to the theory of elasticity" by Atkin and Fox. Great and fairly affordable reading!

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Mechanics of growth and rupture of abdominal aortic aneurysm

We present a coupled mathematical model of growth and failure of the abdominal aortic aneurysm (AAA). The failure portion of the model is based on the constitutive theory of softening hyperelasticity where the classical hyperelastic law is enhanced with a new constant indicating the maximum energy that an infinitesimal material volume can accumulate without failure. The new constant controls material failure and it can be interpreted as the average energy of molecular bonds from the microstructural standpoint.

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Prediction of arterial failure

We enhanced a bi-layer fiber-matrix microstructural arterial model with softening and analyzed the artery inflation under the internal pressure. Numerical simulations lead to the following three findings. Firstly, it is found that the fiber strength dominates the strength of the media layer. Secondly, it is found that the strength of the media layer dominates the overall arterial strength and plays the crucial role in the load-bearing capacity of arteries. Thirdly, it is found that residual stresses can increase the overall arterial strength significantly. The pre-existing compression in arteries delays the onset of rupture like the pre-existing compression in the pre-stressed concrete delays the crack opening.

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Equilibrium equations for large deformations

Lagrangian or referential equilibrium equations for materials undergoing large deformations are of interest in the developing fields of mechanics of soft biomaterials and nanomechanics. The main feature of these equations is the necessity to deal with the First Piola-Kirchhoff, or nominal, stress tensor which is a two-point tensor referring simultaneously to the reference and current configurations.

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Griffith controversy

Using the Griffith energy method for analysis of cavitation under hydrostatic tension we conclude that the critical tension tends to infinity when the cavity radius approaches zero (IJSS, 2006, doi: 10.1016/j.ijsolstr.2006.12.022). The conclusion is physically meaningless, of course. Moreover, if we assume that the failure process occurs at the edge of the cavity then the critical tension should be length-independent for small but finite cavities while the Griffith analysis always exhibits length-dependence. The main Griffith idea - introduction of the surface energy - is controversial because it sets up the characteristic length, say, surface energy over volume energy. By no means is this approach in peace with the length-independent classical continuum mechanics.

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Prediction of femoral head collapse in osteonecrosis

OSTEONECROSIS is the death of bone that results in the collapse of the bony structure, leading to joint pain, bone destruction, and loss of function. Destruction of the bone frequently is severe enough to require joint replacement surgery. Osteonecrosis is a common disorder and accounts for 10% or more of the 500,000 total joint replacement procedures performed annually in the United States. Approximately 75% of patients with osteonecrosis are between 30 and 60 years of age.

From the point of view of mechanics, osteonecrosis means deterioration of mechanical properties of the bone. Decrease of the magnitude of the elastic modulus of the bone leads to its inability to bear the external load and culminates in bone damage and fracturing. For a couple of decades the engineers were trying to estimate the critical stress-strain state of the femoral head using the available data on the osteonecrotic bone properties, finite element analysis based on 3D elasticity, and Von Mises stress as a criticality condition. The fact that the cortical shell of the femoral head is significantly stiffer than the underlying cancellous bone did not attract much attention yet. However, from the solid mechanics point of view the difference in the stiffness of the cortical and cancellous parts of the femoral head under both normal and necrotic conditions is important. This difference allows for considering the femoral head as an elastic cortical shell on an elastic cancellous foundation. This, in its turn, suggests the buckling of the cortical shell as a possible starting point of the overall head collapse. The purpose of the study, described here, was to assess the cortical shell buckling scenario as a possible mechanism of the femoral head collapse at the various stages of osteonecrosis.

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Why fingerprints are different

A possible explanation of the variety of fingerprints comes from the consideration of the mechanics of tissue growth. Formation of fingerprints can be a result of the surface buckling of the growing skin. Remarkably, the surface bifurcation enjoys infinite multiplicity. The latter can be a reason for the variety of fingerprints. Tissue morphogenesis with the surface buckling mechanism and the growth theory underlying this mechanism are presented in the attached notes.

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