Box 1 Justification of Terzaghi’s Principle

Giuseppe Gambolati; Pietro Teatini

Box 1 Justification of Terzaghi’s Principle

To justify Equation 3, namely that Σ A_i<< 1, we can provide the following rough calculation. Assume the solid grains are spherical with radius r. According to Hertz’s theory (Hertz, 1881), the contact area A’ of two spheres pressed by the force P reads:

$\displaystyle A^{\prime }=1.23\pi \left (0.5\frac{Pr}{E_{r}} \right )$

(Box 1-1)

with E_r the sphere Young modulus (which reflects the stiffness of a solid as the ratio of its tensile stress and axial strain, ML^-1T^-2). Let’s take a representative porous medium depth (250 m) and spheres radius equal to, r = 0.5 mm, and assume full saturation. Considering the buoyant force exerted by water the weight P of a grain column of height h is:

$\displaystyle P=\frac{h}{2r}\gamma ^{\prime }\frac{4}{3}\pi r^{2}$

(Box 1-2)

where:

γ′

=

specific weight of the spheres minus the upward buoyant force (ML⁻²T⁻²)

Using Equation Box 1‑2, A′ above becomes Equation Box 1‑3:

$\displaystyle A^{\prime }=1.23\pi r^{2}\left ( \frac{\pi h\gamma ^{\prime}}{3E_{r}} \right )^{2/3}$

(Box 1-3)

Setting h = 250 m, γ′ = 1.7×10⁴ N/m³ (N is a Newton, the SI unit of force [MLT^-2], and 1 N is equal to 1 kg m s^-2) and E_r = 1×10¹¹ N/m² (corresponding to a volumetric grain compressibility c_b,r= 0.16×10^‑¹⁰ m²/N and a grain Poisson ratio v_r= 0.25, being:

$\displaystyle c_{b,r}=3\frac{1-2\nu _{r}}{E_{r}}$

we obtain:

$\displaystyle A^{\prime }=1.23\pi (0.5\ \textup{mm})^{2}\left ( \frac{\pi }{3}\frac{250\ \textup{m}\times 1.7\times 10^{-4}\frac{\textup{N}}{\textup{m}^{3}}}{1\times 10^{11}\frac{\textup{N}}{\textup{m}^{2}}} \right )^{2/3}$ $\displaystyle \cong 0.00121\ \textup{mm}^{2}$

that is, equal to 0.121 percent of the horizontal projection area of the spheres (equal to 1 mm²). Hence the assumption that Σ A_i<< 1 is fully warranted.

Return to where text linked to Box 1

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