**Shrinkage Index (Is)**

Shrinkage Index is the range of consistency in which soil is in a semi-solid state of consistency.

It is defined as numerical difference of plastic limit & shrinkage limit.

Is=Wp-Ws.

**Shrinkage Ratio (R or SR)**

Shrinkage Ratio is defined ratio of decrease in volume of soil expressed as a percentage of its dry volume to the corresponding change in its water content above shrinkage limit.

SR or R=**\(\frac{\frac{V_1-V_2}{V_d}\times 100}{w_1-w_2}\).**

If water content is reduced up to shrinkage limit, w2=ws, V2=Vd.

SR or R =\(\frac{\frac{V_1-V_d}{V_d}\times 100}{w_1-w_s}\).

w1-w2=\(\frac{\text{change in mass of water}}{\text{mass of solids}}\).

w1-w2=\(\frac{(V_1-V_d)×ρ_w}{M_s}\).

w1-ws=\(\frac{(V_1-V_d)×ρ_w}{M_d}\).

\(R=\frac{{\frac{V_1-V_d}{V_d}\times 100}}{\frac{(V_1-V_d)×ρ_w}{M_d}}\)\(R=\frac{Md}{Vd·ρ_w}\)

\(R=\frac{ρ_d}{ρ_w}\)=Gd

R= Mass specific gravity of soil in its dry state.

Note

ws=\(\frac{ρ_w}{ρ_d}\)–\(\frac{ρ_w}{ρ_s}\).

ws=\(\frac{1}{R}\)–\(\frac{1}{G}\). Very Important

**Volumetric shrinkage (Vs)**

Volumetric shrinkage is defined decrease in the volume of soil expressed as the percentage of its dry volume when the water content is reduced from its given value up to the shrinkage limit.

Vs=\(\frac{V1-Vd}{Vd}×100\).

R =\(\frac{\frac{V_1-V_2}{V_d}\times 100}{w_1-w_2}\).

If w2=ws ⇒ V2=Vd.

R =\(\frac{\frac{V_1-V_d}{V_d}\times 100}{w_1-w_s}\).

Vs= (w1-ws)R.

**Linear Shrinkage (Ls)**

Linear Shrinkage is defined as decrease in one dimension of soil expressed as a percentage of its original dimension, when water content is reduced from its given value upto shrinkage limit.

In case of soil, liner dimension is considered as thickness ( as area of soil is infinite).

Ls= 100{1-\([\frac{100}{100+Vs}]^{\frac{1}{3}}\)}

**Plasticity Index (Ip)**

Plasticity Index is range of consistency in which soil exhibits plastic properties or behaves like plastic materials.

It is defined numerical difference of liquid limit and plastic limit.

\(Ip = w_L-w_p\).

Liquid limit, Plastic limit, and Plasticity index are all dependent upon type & amount of clay mineral. (In Plasticity Index significance of amount in more.)

**Note-**

- For Sand, Gravel plastic limit is not attained hence they are reported as non plastic soil.
- It plastic limit of soil comes equal or greater than liquid limit, plasticity index is reported as zero.
- When a non- plastic soil like sand(Ip1) is mixed with plastic soil like clay (Ip2) then plasticity index of mix is given by

Ip mix=\(\frac{Ip1·x1+Ip2·x2}{x1+x2}\).