Geology

Ø  If a rock sample is loaded, after removal of the load, the sample tends to recover its original shape and size, the rock is said to possess elastic properties.

Ø  If an external force, producing deformation does not exceed a certain limit; the deformation disappears with the removal of stress.

 The limit of deformation up to which the material is elastic is known as elastic limit.

Ø   The relationship between stress and strain can be explained with the help of a law known as Hook's Law, propose by Robert Hook in 1678.

Ø   This law is applicable, if the material is loaded below the elastic limit. (Figure 1)

Ø  Hook's law states, “The stress applied on material is directly proportional to the Strain produced.

 Mathematically

Where

σ = Applied stress

ε = Strain or deformation produced

E = Young’s modulus or Modulus of Elasticity

 It is the ratio of applied stress to the corresponding strain. It is denoted by E and mathematically represented as

Ø  For the validity of the above equation, it is necessary that the material should be isotropic and homogeneous.

Ø  Modulus of elasticity of rocks depends upon the rock type, its porosity, and grain size and water contents.

Ø  High value of modulus of elasticity indicates the good quality of rock having sound composition.

The ratio of shear stress to the corresponding shear strain is called the Shear or                             Rigidity Modulus, designated as 'G'.

It may also be calculated with the help of Young’s Modulus and Poisson’s ratio such as;

Where “v “is the Poisson’s Ratio.                                                   

Ø  It is the ratio between the radial strain to axial strain. (Figure.2)

Ø  t can also be defined as the ratio of lateral to longitudinal strain

Mathematically

Where

v = Poisson's ratio

E r = Radial strain

E a = Axial strain

Diameter increases as a result of decrease in length

Decrease in size of the core sample from its axis or longitudinally, there will be decrease in size and shortening in length.

Poisson’s Ratio may also be defined as “The ratio of the fractional transverse contraction to the fractional longitudinal of a body under tensile stress.”

Mathematically

Where

E trans = Strain at right angle to the direction of applied load

E long = Strain parallel to the direction of force

Ø  The material is elastic, if it returns exactly to its original shape on unloading, there is no residual strain

Ø  Materials that deform permanently are called inelastic.

Ø  Three commonly defined types of ideal elastic behavior (Figure 3)

a.       Perfectly elastic

b.      Elastic with hysterersis

c.       Linearly elastic

d.      Linear elastic perfectly plastic

e.       Linear elastic strain hardening

f.       Linear elastic strain softening

Ø  Stress-strain relationship follows a single curve regardless of shape.

Ø  For very stress, there is a unique level of strain.

Ø  Unloading path differs from the loading path.

Ø  Loaded-unloaded path is not only reversible but also a straight line

Ø  Many hard and dense rocks are linearly elastic until cracks start to form at stress level of perhaps 60-70% of rupture

Hard and porous rocks tend to behave elastically with Hysteresis at stress levels up to two-third of their compressive strength, after which they suffer permanent strain.

Ø  It is the elastic behavior linearly when the rock is loaded it causes to produce a linear strain.

Ø  Then a point comes when Strain continues even Stress is not increased further.

Ø  Now another point comes when the rock does not come back to original shape even stress has been removed.

Ø  This point shows rupture in rock.

Ø  It is the linear elastic behavior when a continuous stress causes a continuous strain until rupture occurs.

Ø  A point comes after rupture when a small increase in stress causes a large increase in strain. 

Ø  This point shows continuous rupture in rock.

Ø  It is the linear elastic behavior when a continuous stress causes a continuous strain until rupture occurs.

Ø  A point comes after rupture when strain continuously increases even applied stress is decreased.

All these elastic behaviors have been shown in following graphical diagrams;

Ø  When a sample is loaded and unloaded it normally recovers its shape and size, but sometimes a part of the deformation remains. This part of irreversible deformation is referred as the plastic behavior of the rock(Figure 4)

Ø  Plasticity is defined as a property of solid material to deform continuously and permanently with rupture.

Ø  At an ordinary temperature and pressure, rocks behave elastically, but in an environment at a high temperature and pressure, plastic deformation of the rock takes place.

Ø  In a plastic state permanent deformation may occur without rupture.

Ø  Brittle materials follow elastic properties and there is a collapse after elastic limit, but the ductile materials follow the plastic laws after crossing the elastic limit.                    

These are the tests which are performed on rock samples in laboratories to find the different parameters of strength.

For the determination of UCS, the following procedure is adopted

Ø  An axial load is applied to the rock sample till its failure.

Ø  The load acts in one direction only with no lateral pressure.

Ø  True compression failure in a rock can only occur through internal collapse of the rock structure due to compression of pores, thus resulting in grain fracture and movement along crystal boundaries.

UCS of various rocks varies depending upon their characteristics such as;

Ø  Most igneous rocks have porosity <1%, UCS>200 MPa

Ø  Sedimentary rocks with density  <2.3 g/m³ generally have UCS < 70 MPa

Ø  UCS increases with age in most sedimentary rocks due to increased lithification and reduced porosity.

Ø  Sedimentary rocks become stronger with age and tectonic stress.

We can compare between stronger and weaker rocks on basis of UCS and other characters giving in following table;

The common methods for the determination of unconfined compressive strength are given as:

a)      Compression test

b)      Point load test

c)      Schmidt rebound hammer test

Ø  Cylindrical sample of rock with a length of two to three times its diameter is cut from a core using a diamond saw, and the ends are ground flat and perpendicular axis using a lapping machine as shown in figure 5 and 6.

Ø  The sample is loaded in a compressive testing machine using spherical seating to ensure the load is applied axially.

Ø  Load is continuously increased until the sample is failed.

Ø  A calibrated pressure gauge is used to determine the peak load i.e. the load at which the rock fails.

Ø  At the time of failure, the reading is recorded from the gauge

Ø  The compressive strength of the rock is calculated using the following relation         

Where

σ_c = Unconfined Compressive strength

P = Applied load / stress

D = Diameter of the cylindrical

Ø  Core specimen or irregular rock fragments are loaded to failure between the conical plates of a portable lightweight tester as shown in figure 7.

Ø  Cylindrical sample of rock loaded across its diameter between two 60˚ steel points with tip radius of 5 mm.

Ø  Failure load (P) and platen separation (D) are measured.

Ø  The point load strength index “Is₍₅₀₎” is calculated by applying the correction to the uncorrected point load strength multiplying it by correction factor( F ).

UCS of cylindrical core samples with L/D ratio 2 to 1 can be calculated as;    

Ø  Originally designed to determine the surface hardness of concrete, give a quick approximately estimate of the rock strength.

Ø  The very portable type L hammer used in rock testing, contains a spring loaded plunger which is pressed against an outcrop surface or core specimen.

Ø  The rebound of the hammer, also called the Schmidt Rebound is noted.

Ø  A relationship has been developed between the numbers and the strength of the rock. The scale on the hammer is between 0-80. The height of the rebound is proportional to the hardness of the rock surface.

Ø  The value of the rebound is zero for very soft material.

Ø  The value of the rebound will be closer to 60 when the surface is hard and unbroken.

Ø  Rebound values correlates with UCS and decline significantly in fractured rocks.

Ø  Before going for the test, it should be noted that the surface to be tested must be clean, smooth and unaffected by the joints and fissures; otherwise unsatisfactory low values of the result would be recorded.

Ø  Wet surface will also give different values.

Ø  It is designated to function in horizontal surface, correction have to be made for the used at alternative angles.

Tensile strength of a material is defined as a maximum tensile stress, which a material is capable of bearing. It is maximum stress developed in a specimen in a tension test performed to rupture it.

Ø  Joints are mostly the product of tensile forces.

Ø  In a closely jointed rock mass, the overall tensile strength is zero.

Ø  It is important in designing roof & domes of the underground opening. A rock slab or beam subjected to bending also experience a tensile stress

Tensile strength = 1/10 (compressive strength)

Commonly used methods are

a)      Direct tensile test

b)      Point load test

c)      Brazilian test

d)     Bending test

Brazilian and bending test are indirect method

Ø  Rock sample of specific shape, having diameter (D) is stressed along its axis by a uniaxial tensile load as shown in figure in 7.

Ø  Tensile strength is calculated as the rupture (T) divided by the cross-sectional area (A).

Ø  To grip the sample from both ends, there should be a specific shape of the sample, which is difficult to prepare.

Ø  To grip cylindrical sample with fixing of material at both ends.

Ø  Due to above mentioned difficulties, various methods have been advised.

Ø  Reichmath (1963) described a method for the determination of tensile strength of a rock material as shown in figure 5.

Ø  Core sample is kept in such way that its axis is horizontal.

Ø  Compressive load is applied on the curved surface of specimen by a compression testing machine through small diameter hard steel rollers at right angles to the axis of specimen.

Ø  This load produces tensile stresses perpendicular to the axis of loading.

Ø  Tensile Strength is calculated as;              

Where

σ_t = Tensile Strength (Kg/cm²)

P = Failure Load (Kg)

D = Core Diameter