SOIL MECHANICS 191 of soil at its Plastic Limit is approximately 150 kPa (i.e., the strength of soil changes by about 100 times as the water content changes from the Liquid Limit to the Plastic Limit) Peak Strength Soils whose initial states are dense of critical have a peak strength before they reach a critical state, and they dilate during drained shear, as shown earlier in Figure The peak strengths vary with effective normal stress and specific volume, as shown in Figure 11 Samples which reach their peak states at the same specific volume have peak strengths on an envelope shown in Figure 11A The envelope is often approximated by a straight line, shown in Figure 11A given by ẳ c0p ỵ s0p tan f0p ẵ21 The peak friction angle, f0p , is a material parameter and, from Figure 11A f0p < f0c The cohesion intercept, c0p , is not a material parameter and its value depends on the specific volume Moreover c0p is not the strength at zero effective normal stress, as this must be zero for an uncemented granular material The linear approximation for peak strength given by eqn [21] is applicable only within the range for which data are available Figure 11B shows additional data at smaller normal effective stresses; there the envelope is now distinctly curved and passes through the origin The curved peak failure envelope, shown in Figure 11C, can be represented by a power law of the form ¼ As b ½22 where b is a material parameter and A depends on the specific volume From analyses of the stresses and strains in the soil block, shown in Figure 7A, peak shear strength is given by ¼ s0 tan f0c ỵ cị ẵ23 At the critical state, c ¼ and t0c is given by eqn [18] At the peak state, the angle of dilation is at a maximum The maximum rate of dilation is governed by the state parameter so the peak strength increases as the initial state moves away from the critical state line Equations [21, 22 and 23] are alternative theories for the peak strength of soils They all contain a combination of material parameters and state dependent parameters Equations [22 and 23] correctly give zero strength at zero effective stress Equation [21] is most commonly applied in practice Stiffness of Soil Figure 5A shows non-linear isotropic unloading and reloading behaviour Similar non-linear behaviour occurs during shearing, as shown in Figure 12A The tangent shear modulus G0 is the gradient of the stress-strain curve given by G0 ẳ dt dg ẵ24 At the start of shearing near the origin the shear modulus is G0o and at failure the shear modulus is zero Figure 11 Peak strength Figure 12 Stiffness and shear modulus