Stress-strain curves: linear and non-linear cases

Four types of behaviour

 (i) Linear elastic                 (ii) Non-linear elastic
  s |        /                       s |           _.-'
    |       /                           |        .-'
    |      /   loading and              |      .'    loading and
    |     /    unloading on             |    .'      unloading on
    |    /     the same line            |   /        the same curve
    |   /                               |  /
    |  /                                | /
    +-/------------------ e             +/-------------------- e

 (iii) Linear inelastic             (iv) Non-linear inelastic
  s |        /|                       s |           _.-'|
    |       / |                           |        .-'   |
    |      /  | unloading                 |      .'     /   unloading path
    |     /  /  parallel                  |    .'     .'    differs from the
    |    /  /                             |   /     .'      loading path
    |   /  /                              |  /    .'
    |  /  /                               | /   .'
    +-/--/----------- e                   +/--.'--------------- e
       ^ residual strain                       ^ residual strain

• (i) Linear elastic: stress proportional to strain, and on unloading the strain returns to zero along the same line (e.g. steel below yield).
• (ii) Non-linear elastic: the curve is curved but unloading follows the loading path with no residual strain (e.g. rubber).
• (iii) Linear inelastic: the loading line is straight, but on unloading a permanent strain remains (idealised plastic behaviour).
• (iv) Non-linear inelastic: curved loading path and permanent strain on unloading; this is the actual behaviour of concrete.

Reasons for non-linearity of concrete

1. Heterogeneous, three-phase structure. Aggregate is stiff and elastic, paste is softer and the interface is weak. Their different stiffness and strain capacity give a curved composite response.
2. Microcracks. Bond cracks already exist at the interface before load (from bleeding, shrinkage, temperature). Above 30% of $f_c$ they grow, and above 50 to 75% mortar cracks join them. Each new crack reduces stiffness, so the slope falls continuously.
3. Creep and time-dependent flow of gel. Part of the strain is viscous and not recovered, so unloading leaves residual strain.
4. Porosity and moisture. Water in the gel pores and capillaries moves under stress, adding to non-linear deformation.
5. Stress concentration at pores and voids leads to local yielding at stresses lower than the average strength.