## 2 Measurement of the Membrane Penetration: A Novel Procedure  
### Basic Principle  
For isotropic CD tests, the total volume change of the coarse granular specimen is composed of two parts: the volume deformation of the soil skeleton and the volume change caused by membrane penetration under the specific stress state conditions. In order to measure membrane penetration of coarse granular materials, this paper starts from the basic theory of soil deformation. Figure 1 shows the relation curves of volume change versus the confining pressure, which has been widely recognized in previous researches [6; 13]. The corresponding formula [26] may be expressed as:  
\[\Delta V_{T}(P)=\Delta V_{\mathrm{s}}(P)+\Delta V_{m}(P), \tag{1}\]  
where \(P\) is the confining pressure, \(\Delta V_{T}(P)\) [mL] is the total volume change under special consolidation confining pressures, \(\Delta V_{\mathrm{s}}(P)\) [mL] is the volume change of the soil skeleton which is the real volume change of the sample, and \(\Delta V_{m}(P)\) [mL] is the volume change caused by membrane penetration.  
The volume change of soil skeleton can be expressed as:  
\[\Delta V_{\mathrm{s}}(P)=\varepsilon_{\mathrm{s}}(P)V_{0} \tag{2}\]  
The membrane penetration can then be written as:  
\[\Delta V_{m}(P)=\varepsilon_{m}(P)A_{m} \tag{3}\]  
where \(V_{0}\) [cm\({}^{3}\)] is the initial volume of the specimen, and \(\varepsilon_{\mathrm{s}}(P)\) [cm\({}^{3}\)/cm\({}^{2}\)] and \(\varepsilon_{m}(P)\) [cm\({}^{3}\)/cm\({}^{2}\)] are the soil volumetric strain and the membrane penetration per unit membrane surface area under special confining pressures, respectively.\(A_{m}\) [cm\({}^{2}\)] is the specimen surface area covered by the membrane.  
Equation (1) can be rewritten as  
\[\begin{array}{rl}\Delta V_{T}(P)&=\frac{1}{4}\varepsilon_{s}(P)\pi D^{2}H+ \varepsilon_{m}(P)\pi DH\\ &=\frac{1}{4}\varepsilon_{s}(P)A_{m}D+\varepsilon_{m}(P)A_{m}\end{array} \tag{4}\]  
where \(D\) [mm] is the diameter of the specimen and \(H\) [mm] is the height of the specimen.  
In this paper, the proposed method to measure membrane penetration of coarse granular materials can be expressed as  
\[\frac{\Delta V_{T}(P)}{A_{m}}=\frac{1}{4}\varepsilon_{s}(P)D+\varepsilon_{m}(P) \tag{5}\]  
In Equation (5), different confining pressures are applied to the peripherals of multiple specimens to measure the total volume of drainage \(\Delta V_{T}(P)\) [mL], where \(\Delta V_{T}(P)/A_{m}\) [cm\({}^{3}\)/cm\({}^{2}\)] is set as the vertical coordinate, and \(D\) [mm] is the horizontal coordinate. The relationship between the specimen diameter and volume change can be established by extending the lines intersecting the vertical axis, and the intercept of the vertical coordinate axis is the membrane penetration per unit area, as shown in Figure 2.  
Compared with previous methods requiring the application of brass rod in the central of specimen, this method can effectively transmit the hydrostatic pressure onto the soil following the vertical direction. Therefore, when the top loading platen rests on the brass rod, the undesirable non-hydrostatic compression of the soil skeleton can be averted. The rigid inclusion also restricts the axial deformation of specimen, so that the unreasonable assumption of isotropic (\(\varepsilon_{v}=3\varepsilon_{a}\)) of granular materials under hydrostatic pressure which is underlain in previous methods is no longer necessary. Following this method, the membrane penetration of coarse granular materials can be derived theoretically with a series of isotropic CD tests on specimens of different diameters.  
Figure 1: Relation curves of volume change versus the confining pressure.