Definitions

# Critical state soil mechanics

Critical State Soil Mechanics is the area of Soil Mechanics that encompasses the conceptual models that represent the mechanical behavior of saturated remolded soils based on the Critical State concept. The Critical State concept is an idealization of the observed behavior of saturated remoulded clays in triaxial compression tests, and it is assumed to apply to undisturbed soils. It states that soils and other granular materials, if continuously distorted (sheared) until they flow as a frictional fluid, will come into a well-defined critical state. At the onset of the critical state, shear distortions $varepsilon_s$ occur without any further changes in mean effective stress $p\text{'}$, deviatoric stress $q$ (or yield stress, $sigma_y$, in uniaxial tension according to the von Mises yielding criterion), or specific volume $nu$:
$frac\left\{partial p\text{'}\right\}\left\{partial varepsilon_s\right\}=frac\left\{partial q\right\}\left\{partial varepsilon_s\right\}=frac\left\{partial nu\right\}\left\{partial varepsilon_s\right\}=0$
where,
$nu=1+e$
$p\text{'}=frac\left\{1\right\}\left\{3\right\}\left(sigma_1\text{'}+sigma_2\text{'}+sigma_3\text{'}\right)$
$q= sqrt\left\{frac\left\{\left(sigma_1\text{'} - sigma_2\text{'}\right)^2 + \left(sigma_2\text{'} - sigma_3\text{'}\right)^2 + \left(sigma_1\text{'} - sigma_3\text{'}\right)^2\right\}\left\{2\right\}\right\}$
However, for triaxial conditions $sigma_2\text{'}=sigma_3\text{'}$. Thus,
$p\text{'}=frac\left\{1\right\}\left\{3\right\}\left(sigma_1\text{'}+2sigma_3\text{'}\right)$
$q=\left(sigma_1\text{'}-sigma_3\text{'}\right)$

All critical states, for a given soil, form a unique line called the Critical State Line (CSL) defined by the following equations in the space $\left(p\text{'}, q, v\right)$:

$q=Mp\text{'}$
$nu=Gamma-lambda ln\left(p\text{'}\right)$
where $M$, $Gamma$, and $lambda$ are soil constants. The first equation determines the magnitude of the deviatoric stress $q$ needed to keep the soil flowing continuously as the product of a frictional constant $M$ (capital $mu$) and the mean effective stress $p\text{'}$. The second equation states that the specific volume $nu$ occupied by unit volume of flowing particles will decrease as the logarithm of the mean effective stress increases.

## History

In an attempt to advance soil testing techniques, Kenneth Harry Roscoe of Cambridge University, in the late forties and early fifties, developed a simple shear apparatus in which his successive student attempted to study the changes in conditions in the shear zone both in sand and in clay soils. In 1958 a study of the yielding of soil based on some Cambridge data of the simple shear apparatus tests, and on much more extensive data of triaxial tests at Imperial College, London, led to the publication of the critical state concept . Roscoe obtained his undergraduate degree in mechanical engineering and his experiences trying to create tunnels to escape when held as a prisoner of war by the Nazis during WWII introduced him to soil mechanics. Subsequent to this 1958 paper, concepts of plasticity were introduced by Schofield and published later in a classic text book . Schofield was taught at Cambridge by a Prof. Baker, a structural engineer who was a strong believer in designing structures that would fail "plastically". Prof. Baker's theories strongly influenced Schofield's thinking on soil shear.

## Original Cam-Clay Model

The Original Cam-Clay model is based on the the assumption that the soil is isotropic, elasto-plastic, deforms as a continuum, and it is not affected by creep.