Editing Soft tissue (section)

== Mechanical characteristics ==
At small [[Deformation (mechanics)|strains]], elastin confers [[stiffness]] to the tissue and stores most of the [[strain energy]]. The collagen fibers are comparatively inextensible and are usually loose (wavy, crimped). With increasing tissue deformation the collagen is gradually stretched in the direction of deformation. When taut, these fibers produce a strong growth in tissue stiffness. The [[composite material|composite]] behavior is analogous to a [[nylon stocking]], whose rubber band does the role of elastin as the [[nylon]] does the role of collagen. In soft tissues, the collagen limits the deformation and protects the tissues from injury.

Human soft tissue is highly deformable, and its mechanical properties vary significantly from one person to another. Impact testing results showed that the stiffness and the damping resistance of a test subject's tissue are correlated with the mass, velocity, and size of the striking object. Such properties may be useful for forensics investigation when contusions were induced.<ref>{{cite journal | vauthors = Amar M, Alkhaledi K, Cochran D |title=Estimation of mechanical properties of soft tissue subjected to dynamic impact |journal=Journal of Engineering Research |date=2014 |volume=2 |issue=4 |doi=10.7603/s40632-014-0026-8|pages=87–101|doi-access=free }}</ref> When a solid object impacts a human soft tissue, the energy of the impact will be absorbed by the tissues to reduce the effect of the impact or the pain level; subjects with more soft tissue thickness tended to absorb the impacts with less aversion.<ref>{{cite book | vauthors = Alkhaledi K, Cochran D, Riley M, Stentz T, Bashford G, Meyer G | title = Proceedings of the 29th Annual European Conference on Cognitive Ergonomics | chapter = The psycophysical effects of physical impact to human soft tissue | date = August 2011 | pages = 269–270 | doi = 10.1145/2074712.2074774 | isbn = 9781450310291 | s2cid = 34428866 }}</ref>

[[Image:Pseudoelastic response (stress vs stretch ratio).png|300px|right|thumb|Graph of [[Stress (mechanics)#Euler-Cauchy's stress principle|lagrangian stress]] (T) versus [[Strain (materials science)|stretch ratio]] (λ) of a preconditioned soft tissue]]
Soft tissues have the potential to undergo large deformations and still return to the initial configuration when unloaded, i.e. they are [[hyperelastic material]]s, and their [[stress-strain curve]] is [[nonlinear]]. The soft tissues are also [[viscoelastic]], [[Incompressible flow|incompressible]] and usually [[anisotropic]]. Some viscoelastic properties observable in soft tissues are: [[Relaxation (physics)|relaxation]], [[Creep (deformation)|creep]] and [[hysteresis]].<ref name="Humphrey">{{cite journal | vauthors = Humphrey JD |title=Continuum biomechanics of soft biological tissues |journal=Proceedings of the Royal Society of London A |year=2003|volume=459 |pages=3–46|doi=10.1098/rspa.2002.1060|bibcode=2003RSPSA.459....3H|issue=2029|s2cid=108637580 }}</ref><ref name="Fung">{{cite book | vauthors = Fung YC |title=Biomechanics: Mechanical Properties of Living Tissues|publisher=Springer-Verlag |location=New York|year=1993|page= 568|isbn=0-387-97947-6}}</ref> In order to describe the mechanical response of soft tissues, several methods have been used. These methods include: hyperelastic macroscopic models based on strain energy, mathematical fits where nonlinear constitutive equations are used, and structurally based models where the response of a linear elastic material is modified by its geometric characteristics.<ref>{{cite journal | vauthors = Sherman VR, Yang W, Meyers MA | title = The materials science of collagen | journal = Journal of the Mechanical Behavior of Biomedical Materials | volume = 52 | pages = 22–50 | date = December 2015 | pmid = 26144973 | doi = 10.1016/j.jmbbm.2015.05.023 | doi-access = free }}</ref>

=== Pseudoelasticity ===
Even though soft tissues have viscoelastic properties, i.e. stress as function of strain rate, it can be approximated by a [[Hyperelastic material|hyperelastic]] model after '''precondition''' to a load pattern. After some cycles of loading and unloading the material, the mechanical response becomes independent of strain rate.
:<math>\mathbf{S}=\mathbf{S}(\mathbf{E},\dot{\mathbf{E}}) \quad\rightarrow\quad \mathbf{S}=\mathbf{S}(\mathbf{E})</math>

Despite the independence of strain rate, preconditioned soft tissues still present hysteresis, so the mechanical response can be modeled as hyperelastic with different material constants at loading and unloading. By this method the elasticity theory is used to model an inelastic material. Fung has called this model as '''pseudoelastic''' to point out that the material is not truly elastic.<ref name="Fung"/>

=== Residual stress ===
In physiological state soft tissues usually present [[residual stress]] that may be released when the tissue is [[Surgery|excised]]. [[Physiologists]] and [[histologists]] must be aware of this fact to avoid mistakes when analyzing excised tissues. This retraction usually causes a [[visual artifact]].<ref name="Fung"/>

=== Fung-elastic material ===
[[Yuan-Cheng Fung|Fung]] developed a [[constitutive equation]] for preconditioned soft tissues which is 
:<math>W = \frac{1}{2}\left[q + c\left( e^Q -1 \right) \right]</math>
with
:<math>q=a_{ijkl}E_{ij}E_{kl} \qquad Q=b_{ijkl}E_{ij}E_{kl}</math>
quadratic forms of [[strain (mechanics)|Green-Lagrange strains]] <math>E_{ij}</math> and <math>a_{ijkl}</math>, <math>b_{ijkl}</math> and <math>c</math> material constants.<ref name="Fung"/> <math>W</math> is the [[Strain energy density function|strain energy function]] per volume unit, which is the mechanical strain energy for a given temperature.

==== Isotropic simplification ====
The Fung-model, simplified with isotropic hypothesis (same mechanical properties in all directions). This written in respect of the principal stretches (<math>\lambda_i</math>):
:<math>W = \frac{1}{2}\left[a(\lambda_1^2 + \lambda_2^2 + \lambda_3^2 - 3) + b\left( e^{c(\lambda_1^2 + \lambda_2^2 + \lambda_3^2 - 3)} -1 \right) \right]</math> , 
where a, b and c are constants.

==== Simplification for small and big stretches ====
For small strains, the exponential term is very small, thus negligible.
:<math>W = \frac{1}{2}a_{ijkl}E_{ij}E_{kl}</math>

On the other hand, the linear term is negligible when the analysis rely only on big strains.
:<math>W = \frac{1}{2}c\left( e^{b_{ijkl}E_{ij}E_{kl}} -1 \right)</math>

=== Gent-elastic material ===
{{Further|Gent (hyperelastic model)}}
:<math>W = - \frac{\mu J_m}{2} \ln \left(1 - \left( \frac{\lambda_1^2 + \lambda_2^2 + \lambda_3^2 - 3}{J_m} \right) \right)</math>

where <math>\mu > 0</math> is the shear modulus for infinitesimal strains and <math>J_m > 0</math> is a stiffening parameter, associated with limiting chain extensibility.<ref>{{cite journal| vauthors = Gent AN |title=A new constitutive relation for rubber|journal=Rubber Chem. Technol.|year=1996|volume=69|pages=59–61|doi=10.5254/1.3538357}}</ref> This constitutive model cannot be stretched in uni-axial tension beyond a maximal stretch <math>J_m</math>, which is the positive root of

:<math>\lambda_m^2 + 2\lambda_m - J_m - 3 = 1 </math>