Experimental studies on the influence of intermediate principal stress and inclination on the mechanical behaviour of angular sands

Abstract

A comprehensive experimental study has been made on angular sand to investigate various aspects of mechanical behavior. A hollow cylinder torsion testing apparatus is used in this program to apply a range of stress conditions on this angular quartzitic fine sand under monotonic drained shear. The effect of the magnitude and inclination of the principal stresses on an element of sand is studied through these experiments. This magnitude and inclination of the principal stresses are presented as an “ensemble measure of fabric in sands”. This ensemble measure of fabric in the sands evolves through the shearing process, and reaches the final state, which indeed has a unique fabric. The sand shows significant variation in strength with changing inclination of the principal stresses. The locus of the final stress state in principal stress space is also mapped from these series of experiments. Additional aspects of non-coaxiality, a benchmarking exercise with a few constitutive models is presented here. This experimental approach albeit indirect shows that a unique state which is dependent on the fabric, density and confining stress exists. This suite of experiments provides a well-controlled data set for a clear understanding on the mechanical behavior of sands.

Appendix

$$\begin{aligned} \overline{\sigma _z }= & {} \frac{W}{\pi \left( {r_o^2 -r_i^2 } \right) }+\frac{\left( {p_o r_o^2 -p_i r_i^2 } \right) }{\left( {r_o^2 -r_i^2 } \right) } \end{aligned}$$

(7)

$$\begin{aligned} \overline{\sigma _r }= & {} \frac{\left( {p_o r_o +p_i r_i } \right) }{\left( {r_o +r_i } \right) } \end{aligned}$$

(8)

$$\begin{aligned} \overline{\sigma _\theta }= & {} \frac{\left( {p_o r_o -p_i r_i } \right) }{\left( {r_o -r_i } \right) } \end{aligned}$$

(9)

$$\begin{aligned} \overline{\tau _{\theta z} }= & {} \frac{3.M_T }{2.\pi \left( {r_o^3 -r_i^3 } \right) }\end{aligned}$$

(10)

$$\begin{aligned} \overline{\varepsilon _z }= & {} \frac{\Delta H}{H} \end{aligned}$$

(11)

$$\begin{aligned} \overline{\varepsilon _r }= & {} -\frac{\left( {u_o -u_i } \right) }{\left( {r_o -r_i } \right) } \end{aligned}$$

(12)

$$\begin{aligned} \overline{\varepsilon _\theta }= & {} -\frac{\left( {u_o +u_i } \right) }{\left( {r_o +r_i } \right) } \end{aligned}$$

(13)

$$\begin{aligned} \overline{\gamma _{_{\theta z}} }= & {} -\frac{2\beta \left( {r_o^3 -r_i^3 } \right) }{3H\left( {r_0^2 -r_i^2 } \right) } \end{aligned}$$

(14)

$$\begin{aligned} \sigma _1= & {} \frac{\sigma _z +\sigma _\theta }{2}+\sqrt{\left( {\left( {\frac{\sigma _z -\sigma _\theta }{2}} \right) ^{2}+\left( {\tau _{\theta z} } \right) ^{2}} \right) } \end{aligned}$$

(15)

$$\begin{aligned} \sigma _3= & {} \frac{\sigma _z +\sigma _\theta }{2}\sqrt{\left( {\left( {\frac{\sigma _z -\sigma _\theta }{2}} \right) ^{2}+\left( {\tau _{\theta z} } \right) ^{2}} \right) } \end{aligned}$$

(16)

$$\begin{aligned} p^{\prime }= & {} \frac{\sigma _1^\prime +\sigma _2^\prime +\sigma _3^\prime }{3} \end{aligned}$$

(17)

$$\begin{aligned} q= & {} \frac{1}{\sqrt{2}}\left[ {\left( {\sigma _1^\prime -\sigma _2^\prime } \right) ^{2}+\left( {\sigma _2^\prime -\sigma _3^\prime } \right) ^{2}+\left( {\sigma _3^\prime -\sigma _1^\prime } \right) ^{2}} \right] ^{1/2} \nonumber \\\end{aligned}$$

(18)

$$\begin{aligned} \tau _{oct}= & {} \frac{1}{3}\left[ {\left( {\sigma _1^\prime -\sigma _2^\prime } \right) ^{2}+\left( {\sigma _2^\prime -\sigma _3^\prime } \right) ^{2}+\left( {\sigma _3^\prime -\sigma _1^\prime } \right) ^{2}} \right] ^{1/2} \end{aligned}$$

(19)

$$\begin{aligned} J_2= & {} \frac{1}{6}\left[ {\left( {\sigma _1^\prime -\sigma _2^\prime } \right) ^{2}+\left( {\sigma _2^\prime -\sigma _3^\prime } \right) ^{2}+\left( {\sigma _3^\prime -\sigma _1^\prime } \right) ^{2}} \right] \end{aligned}$$

(20)

$$\begin{aligned}&J_3 = \frac{1}{27}\Big [ \left( {\sigma _1 -\sigma _2 } \right) ^{2}\left( \sigma _1 +\sigma _2 -2.\sigma _3 \right) \nonumber \\&\qquad \quad +\left( {\sigma _2 -\sigma _3 } \right) ^{2}\left( {\sigma _2 +\sigma _3 -2.\sigma _1 } \right) \nonumber \\&\qquad \quad +\left( {\sigma _3 -\sigma _1 } \right) ^{2}\left( {\sigma _3 +\sigma _1 -2.\sigma _2 } \right) \Big ] \end{aligned}$$

(21)

$$\begin{aligned}&\quad q = \sqrt{3.J_2} \quad \left( {\hbox {for}\;\hbox {TX-C}\;\hbox {and}\;\hbox {TX-E}\;\;q=\sigma _1 -\sigma _3 } \right) \end{aligned}$$

(22)

$$\begin{aligned}&\quad \gamma _{oct} = \frac{2}{3}\left[ {\left( {\varepsilon _1 -\varepsilon _2 } \right) ^{2}+\left( {\varepsilon _2 -\varepsilon _3 } \right) ^{2}+\left( {\varepsilon _3 -\varepsilon _1 } \right) ^{2}} \right] ^{1/2} \nonumber \\\end{aligned}$$

(23)

$$\begin{aligned}&\quad D_p = \frac{{\left( {\varepsilon _{v,j+1} -\varepsilon _{v,j-1} } \right) -\left( {p_{j+1}^\prime -p_{j-1}^\prime } \right) }/K}{{\left( {\varepsilon _{q,j+1} -\varepsilon _{q,j-1} } \right) -\left( {q_{j+1} -q_{j-1} } \right) }/{3G}} \end{aligned}$$

(24)

$$\begin{aligned}&\quad {\eta } = \frac{q}{p^{\prime }} \end{aligned}$$

(25)

$$\begin{aligned}&\quad \sin \;3\theta = -\frac{3\sqrt{3}}{2}\cdot \frac{J_3 }{\left( {J_2 } \right) ^{3/2}} \end{aligned}$$

(26)

where ${\uptheta }$ varies from

$$\begin{aligned}&-\frac{\pi }{6}\le \theta \le \frac{\pi }{6} \end{aligned}$$

(27)

$$\begin{aligned}&\quad \hbox {Relative}\;\hbox {density},\;R.D = \frac{e_{\max } -e_{nat}}{e_{\max } -e_{\min }} \end{aligned}$$

(28)

$$\begin{aligned}&\quad Sin\varphi = \frac{\left( {\sigma _1^\prime -\sigma _3^\prime } \right) }{\sigma _1^\prime +\sigma _3^{\prime }} \end{aligned}$$

(29)

where ${\upvarphi }$ is the friction angle at the final stress state

where,

• ${\upsigma }_{\mathrm{z}}$ is the axial stress

• ${\upsigma }_{{\uptheta }}$ is the tangential stress

• ${\upsigma }_{\mathrm{r}}$ is the radial stress

• ${\uptau }_{\uptheta \mathrm{z}}$ is the shear stress along ${\uptheta }\hbox {z}$ plane

• $\hbox {r}_{\mathrm{o}}$ is the external radius of the specimen

• $\hbox {r}_{\mathrm{i}}$ is the internal radius of the specimen

• $\hbox {u}_{\mathrm{o}}$ is the external radial displacement

• $\hbox {u}_{\mathrm{i}}$ is the internal radial displacement

• H is the initial height of the specimen

• $\Delta \hbox {H}$ is the change in axial displacement

• $\hbox {J}_{2}$ is the second deviatoric stress invariant

• ${\upeta }$ is the stress ratio

• ${\uptau }_{\mathrm{oct}},\, {\upgamma }_{\mathrm{oct}}$ is the octahedral shear stress and strain

• ${\uptheta }$ is the Lode angle

• $\hbox {e}_{\max }$—maximum void ratio the material can achieve

• $\hbox {e}_{\min }$—minimum void ratio the material can achieve

• $\hbox {e}_{\mathrm{nat}}$—void ratio at which the specimen is prepared

Pressure ratio at the critical/final state obtained from the series #1 and series #2 experiments. This ratio was used to identify the level of stress non-uniformity in specimen. Guitterez et al. [27] and Nakata et al. [28] have recommended a range $0.75 \, < \, \hbox {P}_{\mathrm{i}}/\hbox {P}_{\mathrm{o}}< \, 1.3$. This range of pressures was maintained in the tests performed here. However, at the critical/final state the specimen with b = 0, ${\upalpha }\,=\,75^{\circ },\, 90^{\circ }$ and b = 1, ${\upalpha }\,=\,0^{\circ }$ violate this condition. A note to this effect has been made in the figures.