Where does stiffness collide?

The degree to which a body or material resists elastic deformation when a load is applied: a stiff structure moves little for a given force, and in the simplest case stiffness is the ratio of applied force to the resulting displacement, the spring constant. Ashby's 2011 Materials Selection in Mechanical Design keeps stiffness, which depends on a part's shape and on the elastic modulus of its material, apart from strength, which is the load a part can carry before it yields or breaks, and it treats the material property Young's modulus as the intrinsic quantity that a component's stiffness is partly built from. This referent is a mechanical relation between force and movement in a solid object, and it has nothing to do with a differential equation that is hard to integrate or with a joint that is hard to move after rest.[1]

Unit

The ratio of applied force to displacement, in newtons per metre for a linear spring

Recovery means

Elastic stiffness is fully recoverable: within the elastic limit the object returns to its shape once the load is removed

Measured by

Read from the slope of a force against displacement curve, or computed from a part's geometry and its material's elastic modulus

A property of a differential equation or system whose solution contains parts that change on very different time scales, so that a simple step by step explicit method must take impractically tiny steps to stay stable even after the fast parts have died away. Curtiss and Hirschfelder's 1952 Integration of Stiff Equations named this behaviour and showed why ordinary explicit schemes fail on such problems, motivating the implicit methods still used for them. This referent is a difficulty in computing the solution of equations, defined by a spread of decay rates rather than by any physical hardness, and it is unrelated to a solid object resisting a load or to a stiff limb, sharing only the word.[2]

Unit

A single system of equations, classed as stiff by the spread of its time scales, its stiffness ratio

Recovery means

Not applicable

Measured by

Diagnosed by the ratio of the largest to smallest eigenvalue magnitudes of the system, and by explicit methods failing to stay stable at reasonable step sizes

A reduced ease and range of movement in a joint, so that bending or straightening it feels difficult or restricted, classically worse after a period of rest and eased by moving, as in the morning stiffness reported in arthritis. Halls and colleagues' 2015 study Stiffness is more than just duration and severity found that people with rheumatoid arthritis live it as a felt resistance and difficulty in moving that is not captured fully by how long it lasts, and argued for measuring its impact more broadly. This referent is a bodily symptom experienced by a person, timed by how long it lasts and rated by how much it limits daily activity, and it is not the force to displacement ratio of a solid nor a hard to integrate equation.[3]

Unit

One person's joint stiffness, described by its duration, for example minutes of morning stiffness, and its severity

Recovery means

Often eases with gentle movement and warmth as the joint is used, though it may return after rest

Measured by

Assessed clinically by asking how long stiffness lasts after waking and how severely it limits movement, often on a patient rated scale