## 1 Introduction  
The triaxial test is a common method to obtain the stress-strain behavior of coarse granular materials in the laboratory [1]. When preparing coarse granular materials for triaxial tests, test specimen is typically covered by rubber membrane to prevent direct contact with water in the pressure chamber, so that boundary stress and control drainage conditions can be readily applied. However, due to the unevenness of the specimen, the confining pressure will cause the flexible rubber membrane to be pushed into the peripheral voids of the specimen, which is called membrane penetration [1]. This situation can be found in both conventional triaxial tests and true triaxial tests. By ignoring the impact of rubber membrane penetration, the real volume change of the specimen may be overestimated in the triaxial consolidated-drained test. Furthermore, in the undrained test, the total volume of the saturated specimen remains unchanged. When applying the confining pressure, positive pore pressure is generated in the specimen, the membrane will be withdrawn out of the peripheral voids of the soil specimen, and the effective confining pressure decreases; then, the membrane penetration will decrease with increasing pore water pressure. This leads to a certain amount of water in the soil skeleton being drained, invalidating the assumed testing conditions of a constant volume without drainage [2]. Consequently, the undrained test will become a drained test with decreasing measured pore pressure increment \(\Delta\)u, resulting in an unconservative overestimation of the cyclic strength of the soil in the cyclic triaxial test. Thus, the membrane penetration effect cannot be ignored, and the measurement error caused by the penetration effect must be corrected or reduced [2]. [PERSON] and [PERSON] [3] first recognized membrane penetration as a source of revised volume change and divided the total volume change into two parts: the volume change of the soil skeleton and the rubber membrane penetration volume. [PERSON] and [PERSON] assumed that the specimen behaved isotropically under hydrostatic loading, and the volumetric strains were calculated to be three times equivalent of the axial strains [4]. However, the strain of sand is anisotropic under hydrostatic loading, the radial strains will be larger than the axial strains, and the assumption overestimates the penetration volume change of the membrane, as explained by [PERSON] and [PERSON], [PERSON] and [PERSON], [PERSON] and [PERSON] [5; 6; 7]. To address the membrane penetration issue, researchers usually study membrane penetration based on theoretical derivation. [PERSON] et al. proposed a method for evaluating the membrane penetration volume change. The cylindrical rigid brass rods of different diameters were placed into the center of soil specimens, then the total volume change under different confining pressures was measured [8]. A linear relationship between the total volume change and the diameter of the brass rods can therefore be plotted. Since brass rods are relatively incompressible, the membrane penetration is equal to the total volume change when the diameter of the brass rods and the specimen diameter are the same. [PERSON] and [PERSON] [9] stated the following disadvantages of the method proposed by [PERSON] et al. (1) The relationship between the total volume change and diameter of the brass rod are not linear; and (2) when the top loading platen rests on the brass rod, the vertical stress on the soil will be lower than the applied hydrostatic stress, resulting in nonhydrostatic compression of the soil skeleton. A similar procedure was adapted by [PERSON] and [PERSON]. To ensure that the specimen was subjected to hydrostatic stress, the conventional rigid top platen was replaced with an annular flexible and lubricated top plate. [PERSON] et al. [10] adapted the above procedure to study membrane penetration, and found a good linear relationship between the total volume change and diameter of the brass rod when the confining stress was between 35 and 600 kPa.