A nipple cup liner having a longitudinal shape extending along a central longitudinal axis (X) and configured to be attached to an outer shell (4) of a nipple cup applied to an animal's teat during a milking operation The teat cup liner includes a barrel (2) that defines an internal space (21) that receives a teat during milking operation and has a length (L) along a longitudinal axis (X), When the part (2) is in a resting state, the part (2) has a cross-sectional shape that crosses the longitudinal axis (X). Each side part is connected to two of the corner parts (23), the first one of the side parts (24-1) is the weak side, and the remaining side parts (24- 2) is the strong side and the weak side has more flexibility than each strong side, milking operation During the crushing process, the weak side crushes towards the longitudinal axis (X) before each strong side crushes, and the weaker side displacement towards the longitudinal axis (X) moves towards the longitudinal axis (X), respectively. A pressure region greater than the corresponding crushing displacement on the strong side and greater than the remainder of the cross-sectional shape relative to the nipple end across the first surface region of the cross-sectional shape opposite the weak side Creates an asymmetric pressure distribution with respect to the teat end located in the barrel with an extended contact across the first surface region between the outer surface of the teat end and the adjacent inner surface of the teat cup liner The surface provides an increased pressure area across the first surface area and an extended contact surface provides an enhanced nipple massage during the crushing process. [Selection] Figure 5中央長手方向軸（Ｘ）に沿って延びる長手方向形状を有し、かつ搾乳操作時に動物の乳頭に付与される乳頭カップの外殻（４）に装着されるように構成された乳頭カップライナーであって、乳頭カップライナーが、搾乳操作時に乳頭を受ける内部空間（２１）を規定しかつ長手方向軸（Ｘ）に沿って長さ（Ｌ）を有する樽状部（２）を含み、樽状部（２）が、休止状態では、長手方向軸（Ｘ）を横切る横断面形状を有し、横断面形状が複数のコーナー部（２３）及び複数の側部（２４－１，２４－２）を規定し、各側部が前記コーナー部（２３）の二つに接続しており、側部の第一のもの（２４－１）が弱い側であり、それぞれの残りの側部
