# Stiffness 1.01 When I began my MPhil/PhD it came with the promise of lots of related blog content. Well, here it begins… Broadly speaking, I’m looking into certain aspects of lower limb stiffness. Hopefully this first piece will serve as a neat and compact little introduction to the topic and set the foundations for some more in-depth posts that will be coming in the not too distant future.

What is stiffness?

Stiffness describes the deformation of an object in response to load. It’s a term from physics based on Hooke’s Law.

What is Hooke’s Law?

F = kx

Hooke’s Law states that “the force (F) required to deform a material is related to a proportionality constant (k) and the distance (x) that the material is deformed” (Butler et al, 2003).

How can we calculate stiffness?

We calculate stiffness by determining the proportionality constant (k; also called the spring constant). Assuming that an object obeys Hooke’s Law, its change in length will be a direct function of the forces acting upon it. We can therefore rearrange the above equation to determine k:

k = ΔFx (where ΔF is change in force and Δx is change in length)

So, if you know how much force is produced for a given length change then you can calculate stiffness.

What determines ‘limb stiffness’?

We can determine stiffness at various physiologic levels:

• Overall limb stiffness (i.e. leg stiffness)
• Single joint stiffness (i.e. ankle stiffness)
• Muscle tendon unit stiffness (i.e. medial gastrocnemius and Achilles acting together)
• Individual tissue stiffness (i.e. Achilles tendon)
• Individual fibre stiffness (i.e. single muscle fibre)

Each level of stiffness is a product of the subfactors from the levels below. Importantly, this means that changes in stiffness at lower levels aren’t always reflected higher up. For instance, an increase in Achilles tendon stiffness may not necessarily lead to an increase in leg stiffness.

How is stiffness modelled during athletic movement?

To model stiffness we have to make the assumption that the leg acts as a simple spring to support the mass of the body; we call this an ideal spring-mass system. Biomechanists designate three types of stiffness which can be calculated, the type used will be dependent on the task and system level to be analysed.

• Vertical stiffness (kvert) – used to determine limb stiffness during vertical tasks (i.e. jumping and hopping in place)
• Leg stiffness (kleg) –  used to determine limb stiffness during horizontal (i.e. running, jumping and bounding) as well as vertical tasks
• Torsional stiffness (kjoint) – used to determine joint stiffness (important as these forces are now acting rotationally as opposed to linearly)

All of these are based on the Hookean premise that we can calculate k, our spring constant and stiffness value, if we know how much force is produced for a given change in length.

Are there problems with this approach?

To use our spring-mass model we must accept that out ‘spring’ is massless, moves only in one direction, and has a stiffness that is independent of time, length, or velocity. We effectively have to discount the role of the nervous system, muscle reflexes and anything going on at the other end of our spring. There are problems with it, as there will be with any reductionist approach, but it’s the best we have moment.