Ex 1.
A well-known phenomenon of soft biological tissues is that they contract when they are explanted from the human body, which is also known as prestretch. A researcher identifies the mechanical properties of a piece of material after it is explanted, and finds that it follows the following relationship:
𝐹 = 𝐶(𝜆̃ − 1)
where 𝐶 represents the stiffness of the material and 𝜆̃ the stretch with respect to the contracted length of the tissue 𝐿𝑐. This means that 𝜆̃ = 𝐿/𝐿𝑐 where 𝐿 is the current length. A disadvantage is that this curve only represents the ex-vivo material behaviour, which may be different from the in-vivo behavior where the unloaded length equals 𝐿0 (with 𝐿0> 𝐿𝑐).
Determine the force–stretch behaviour for this material with respect to the in-vivo state (i.e. express the force 𝑭 in terms of 𝝀, where 𝝀 is defined with respect to 𝑳𝟎).
Ex 2.
Cartilage consists of different components, some of which are naturally exposed to extension and others have a tendency to be exposed to compression. This results from the fact that the length 𝐿𝑐 to which these components contract if they would be isolated from the cartilage differs between components.
Suppose that we can represent a piece of cartilage that is clamped at both ends by a model consisting of two parallel springs, where one spring represents the tissue components that are exposed to extension (with 𝐿1,𝑐= 0.9𝐿0 and stiffness 𝐶1), and the other represents the material in compression (with 𝐿2,𝑐= 1.2𝐿0 and stiffness 𝐶2).
(a) Determine the total force that the tissue is exerting on its surroundings at its original length 𝑳𝟎. (b) When 𝑪𝟏= 𝑪𝟐, will the tissue expand or contract when the clamps are removed?
Ex 3.
A problem in tissue engineering is compaction of the engineered construct. Cells in the construct apply forces to their environment, so a tissue that is grown in an incubator will shrink over time. Especially when biodegradable scaffolds are used, this can have a large impact, because the shape of the final construct is unclear and this influences the functionality. An experiment to measure the (very small) forces exerted by the cells in a construct is shown in the figure. A tissue is fixed to clamps A and B. The clamp B is attached with metal leaf springs to a rigid frame. The bending of the springs can be used to measure the force applied by the cells, and consequently the bending has to be incorporated in the analysis1.
This is done by modelling the complete system with two springs, mutually connected at point C. The tissue is modelled as a spring with stiffness 𝐶𝑡and rest length 𝐿𝑡. The force-measuring leaf springs are lumped into
1Vlimmeren, M. A. A. van, Driessen-Mol, A., Oomens, C. W. J. Baaijens, F. P. T. (2011) An in vitro model system to quantify stress generation, compaction and retraction in engineered heart valve tissue. Tissue Eng. Part C 17, 983–991
one single spring with stiffness 𝐶𝑠and length 𝐿𝑠. The relation between force F and extension ratio 𝐿/𝐿0 of a spring with stiffness 𝐶 is given by:
𝐹 = 𝐶(𝐿/𝐿0− 1)
The force exerted by the tissue is modelled as a concentrated force F in negative x-direction that is applied at point C.
Determine the displacement 𝒖𝑪of point C as a function of F, 𝑪𝒕, 𝑪𝒔, 𝑳𝒕, 𝑳𝒔.
