A crack that still carries load
In concrete, soil, and other quasi-brittle solids the tip is not a mathematical point. Dugdale, Barenblatt, and Hillerborg replace the singular field with a traction–separation law on a process zone of finite length. The tensile strength ft is finite; the area under the law is the fracture energy Gf. Linear elasticity is recovered only if that zone is small compared with the crack. In a saturated porous solid the same surface carries an effective traction tc = td − α p n: water pressure is continuous, its normal gradient is not.
Traction–separation
Stress along the crack line
Gf = ∫ t dδ · ℓch = E Gf / ft²
Navy is the cohesive traction. Orange is the LEFM 1/√r field with the same far-field K. Lower Gf shortens the process zone and can produce snap-back under load control. In a saturated porous solid the same law sits on a Heaviside jump in displacement; water pressure is continuous, its normal gradient is not.