In this lecture, we will cover the following:

• How normal strain can be expressed in terms of displacement for a 1D bar
• How to derive the displacement–strain relationship using an infinitesimal element
• How shear strain in a 2D element relates to displacement gradients
• How to summarise the full set of stress–strain and strain–displacement relationships for linear elastic materials
• Why these material models are important for the finite element method

In this lecture, we establish the fundamental relationship between strain and displacement. We begin with a bar in uniaxial tension and isolate an infinitesimally small element to examine how it both elongates and translates under load. By differentiating the displacement with respect to position, we show that normal strain in the x-direction is equal to the displacement gradient,

This provides the crucial link between geometric deformation (displacement) and strain, which measures local deformation.

We then extend this reasoning to a 2D element undergoing shear deformation. By examining the small angular distortions that arise from displacement components in both x and y directions, we show that shear strain is given by the sum of displacement gradients,

Finally, we consolidate the relationships from this and the previous lecture into a compact summary: normal and shear strains expressed in terms of displacement gradients, and corresponding stress–strain relationships expressed using Young’s modulus, shear modulus, and Poisson’s ratio. Together, these equations form the linear elastic material model that underpins finite element analysis, highlighting the importance of accurate constitutive modelling.

Next up:

In the next lecture, we extend these 2D relationships into three dimensions and introduce matrix notation, setting the stage for the material stiffness matrix used in finite element formulations.