Tensors and tissue behaviour

Tensor order tells us how many indices describe a quantity. A scalar gives a magnitude with units. Temperature and mass are examples. It is a zero-order tensor. Body mass is scalar. Weight as a gravitational force has a direction and is a vector.

A vector adds direction to magnitude. Force is a first-order tensor, with components along the x, y and z axes. Add forces as vectors to find the resultant. Ground reaction force is the force the ground exerts on a paw. It contributes to the resultant; it is not another name for every resultant force.

Stress and strain are second-order tensors. In three dimensions, each uses a 3×3 matrix. For stress, one index identifies the surface orientation and the other gives the direction of action. Normal stresses act perpendicular to a surface; shear stresses act along it. Stress has units of force per area. Strain describes relative deformation. Both vary from point to point within a loaded tissue.

Bone is anisotropic: its response changes with loading direction. This reflects its microarchitecture, including aligned osteons and trabeculae. It is distinct from the effect of the bone’s overall shape.

An ideal elastic material has a unique load–deformation relationship and a rate-independent response. One-to-one does not mean a 1:1 numerical ratio or necessarily a straight line. A straight proportional relationship is a separate property: linearity.

Tendon, ligament and cartilage are viscoelastic. Their response depends on loading rate and time. Viscosity describes resistance to fluid flow; elasticity describes recoverable deformation of a solid. These water-rich tissues combine both behaviours. In a canine meniscus, slow loading lets fluid redistribute and the tissue is more compliant. Fast loading gives fluid less time to move, so its apparent stiffness rises.

References

Johnston & Tobias, Veterinary Surgery: Small Animal, 2nd ed., Ch39, Basic Concepts and Stress. NIST, SI Units—Mass: https://www.nist.gov/pml/owm/si-units-mass