A teatcup liner, for use in a method of providing enhanced teat massage during a milking operation, includes a cross-sectional shape defining plural corner portions and plural side portions with a first side portion being a weak side and each remaining side portion being a strong side; with the weak side having a greater flexibility than each strong side such that during a collapse phase, the collapse of the weak side occurring first and being greater than the collapse of each strong side creates an asymmetric pressure distribution with a greater pressurized area resulting in a prolonged contact surface against the teat end across a first surface region of the cross-sectional shape opposite the weak side as compared to a remainder of the cross-sectional shape, the increased pressurized area across the first surface region providing an enhanced teat massage during the collapse phase.