Mechanics of biological structures (prima parte) - Prof. Vena

MECHANICS OF BIOLOGICAL STRUCTURES

LECTURE 2 – 10-10-2016

The topics of this course are divided into two main groups: hard tissues and soft tissues. If we want to model the mechanical behavior of these tissues we need a specific mathematical framework, while if we want to model the mechanical behavior of soft tissues we need another mathematical framework. The first part of the course is on hard tissue. However there are some properties which are common to both soft and hard tissues.

In particular, one of the most important function of the biological tissues is that they always exhibit a sort of hierarchical architecture, this means that we can observe our biological tissue by using different magnification lens. If we use the magnification lens we see a specific structure and going on we can see other substructure at each level.

We start from the macroscopic level and then we go down and down with a magnification lens until the primer structure, characterized by the components of our material. The idea is that all tissues exhibit a hierarchical levels of microstructures and the geometrical arrangement of the constituents at each level and the mechanical properties of the single constituent at each level define the mechanical properties that we observe at the macroscopic level.

All biological tissues exhibit an anisotropic mechanical properties. This means that the stress-strain response of the material will depends on the direction of loading with the respect to the direction of the microstructure of the tissue that we have in our material.

They are also non linear. In general, the bone can be approximated as a linear material for a very small strain but in any case all the other biological materials (in particular case the soft biological materials) exhibit a non linear stress-strain relationship. This means that when we take a little piece of the biological tissue and we put it under a tensile test equipment, the relationship between forces and displacement is always non linear and therefore the relationship between stress and strain is non linear. Non linearity is again given by the hierarchical arrangement of the constituent at the microscopic levels.

Another important feature is the residual stress. This means that in these tissues there is a pre-existing state of stress even if no external loading are applied. The most important example is the vascular vessels. There is an internal pressure inside the vessels, but the idea is that if we have an artery with no blood pressure inside it, some stresses are acting into the material and we can visualize this stress taking a knife and cutting the arterial wall longitudinally.

This is not typical for any engineering structure. In fact if we take an engineering structure and we have no load applied, there is no stress and so no strain.

NO LOAD → NO STRESS → NO STRAIN

In the case of biological tissue, like in the case of vessel we have a σ≠0. Making a cut longitudinally we will see that the artery will naturally open forming an angle α and this angle is proportional to the internal stress.

All biological tissues also exhibit a time dependent behavior. This means that if we apply a static loading in terms of stresses, the strain in our material will change with time and the motivation of this is linked at least to two phenomena: the first is the viscoelasticity (typical for polymers). A viscoelastic material is a material that if is subjected to some stresses, the structure elongate itself, even if the stress is kept constant. The second phenomenon is related to the water that we have in all the biological tissues. And so when we apply a load on the material, the water inside is squeezed out of our sample and this needs some time (poroelasticity). These two phenomena driven the time dependent behavior of the biological tissues.

All living materials are able to change their mechanical behavior and their microstructures according to the loading that we are applying. This is called self adapting behavior. For example, a clear example of self adaption is the formation of an aneurism (an enlargement of the vessels) that is the result of the continuous change of the micro arrangement of constituents of the tissue. The self adaption behavior is good in same case, but it is not good for pathological case (like aneurism).

In the slide we can see two typical tissues. The one on the left is the tendon, while the one on the right is a ligament. In both picture we see fibers in a vertical direction that are microstructure elements and both these tissues are anisotropic tissues, so if we apply loads along fibers, we will have a stiffness response which is higher with the respect to the response we have if we apply the load in the direction that is perpendicular to the fibers. The fibers on the left are more straight, while the fibers on the right exhibit a wave structure. What does it mean? This microstructures arrangement makes the two tissues very different from the mechanical point of view, this means that the tendon has a higher stiffness than the ligament.

This difference in their behavior is linked to the microstructure of the fibers.

The first 3X3 matrix is responsible to the relation between the direct stress and direct strain. The last 3X3 matrix is responsible for the relationship between shear stress and shear strain. The other two 3X3 matrix are responsible for direct stress and shear strain and for shear stress and direct stain.

For this specific symmetry there are some terms which are different from 0 (coupling terms). This is due by the fact that when we have the elongation (direct strain), in addition, we have also a shear strain. So when we applied a direct stress, we get a shear strain.

However, this is a material difficult to characterize because 13 parameters are too much to determine in laboratory, so we have to determinate some simpler material.

Trigonal materials have 3 different planes of symmetry. They are characterized by 8 independent parameters, but again we have some terms different from 0.

Orthotropic materials have 3 different planes of symmetry and they are characterized by 9 independent parameters and it is one of the best candidates to characterize the bone properties. It is a material that can be thought as a crystal material with a structure where a, b, c have three different structures. C11, C22 and C33 are strictly related to the elastic modulus in the three directions. C12, C13 and C23 are again related to three different Poisson ratio. C44, C55 and C66 are three different shear stiffness for three different planes.

Tetragonal materials have three planes of symmetry and they are characterized by 6 independent parameters.

The transverse isotropy materials are characterized by 5 independent parameters. In the transverse isotropy materials we have one single direction which is the direction of the main direction material response. All directions perpendicular to the main directions do not exhibit difference in their response, so they exhibit the same E. C11-C12 is the shear stress in the transverse direction.

The cubic symmetry materials are characterized by 3 independent parameters. The elastic modulus in the 3 perpendicular directions are the same (C11, C11, C11). It is similar to the isotropic material. If we look the top left block of the material, the cubic and the isotropic material are not distinguishable but if we look at the shear stiffness, the cubic material exhibit a shear stiffness which is an independent parameter, while for the isotropic material the shear stiffness is dependent on C11, C12.

For the isotropic materials the shear stiffness is: _{G =} ^E_2(1+v), while for cubic symmetry materials G is an additional independent parameter.