  Lie groupoids and their associated algebroids arise naturally in the study of
the constitutive properties of continuous media. Thus, Continuum Mechanics and
Differential Geometry illuminate each other in a mutual entanglement of theory
and applications. Given any material property, such as the elastic energy or an
index of refraction, affected by the state of deformation of the material body,
one can automatically associate to it a groupoid. Under conditions of
differentiability, this material groupoid is a Lie groupoid. Its associated Lie
algebroid plays an important role in the determination of the existence of
material defects, such as dislocations. This paper presents a rather intuitive
treatment of these ideas.
