# Triaxial shear test

A triaxial shear test is a common method to measure the mechanical properties of many deformable solids, especially soil (e.g., sand, clay) and rock, and other granular materials or powders. There are several variations on the test.[1][2][3][4]

In a triaxial shear test, stress is applied to a sample of the material being tested in a way which results in stresses along one axis being different from the stresses in perpendicular directions. This is typically achieved by placing the sample between two parallel platens which apply stress in one (usually vertical) direction, and applying fluid pressure to the specimen to apply stress in the perpendicular directions. (Testing apparatus which allows application of different levels of stress in each of three orthogonal directions are discussed below, under "True Triaxial test".)

The application of different compressive stresses in the test apparatus causes shear stress to develop in the sample; the loads can be increased and deflections monitored until failure of the sample. During the test, the surrounding fluid is pressurized, and the stress on the platens is increased until the material in the cylinder fails and forms sliding regions within itself, known as shear bands. The geometry of the shearing in a triaxial test typically causes the sample to become shorter while bulging out along the sides. The stress on the platen is then reduced and the water pressure pushes the sides back in, causing the sample to grow taller again. This cycle is usually repeated several times while collecting stress and strain data about the sample. During the test the pore pressures of fluids (e.g., water, oil) or gasses in the sample may be measured using Bishop's pore pressure apparatus.

From the triaxial test data, it is possible to extract fundamental material parameters about the sample, including its angle of shearing resistance, apparent cohesion, and dilatancy angle. These parameters are then used in computer models to predict how the material will behave in a larger-scale engineering application. An example would be to predict the stability of the soil on a slope, whether the slope will collapse or whether the soil will support the shear stresses of the slope and remain in place. Triaxial tests are used along with other tests to make such engineering predictions.

During the shearing, a granular material will typically have a net gain or loss of volume. If it had originally been in a dense state, then it typically gains volume, a characteristic known as Reynolds' dilatancy. If it had originally been in a very loose state, then contraction may occur before the shearing begins or in conjunction with the shearing.

Sometimes, testing of cohesive samples is done with no confining pressure, in an unconfined compression test. This requires much simpler and less expensive apparatus and sample preparation, though the applicability is limited to samples that the sides won't crumble when exposed, and the confining stress being lower than the in-situ stress gives results which may be overly conservative. The compression test performed for concrete strength testing is essentially the same test, on apparatus designed for the larger samples and higher loads typical of concrete testing.

## Test Execution

For soil samples, the specimen is contained in a cylindrical latex sleeve with a flat, circular metal plate or platen closing off the top and bottom ends. This cylinder is placed into a bath of a hydraulic fluid to provide pressure along the sides of the cylinder. The top platen can then be mechanically driven up or down along the axis of the cylinder to squeeze the material. The distance that the upper platen travels is measured as a function of the force required to move it, as the pressure of the surrounding water is carefully controlled. The net change in volume of the material can also be measured by how much water moves in or out of the surrounding bath, but is typically measured - when the sample is saturated with water - by measuring the amount of water that flows into or out of the sample's pores.

### Rock

For testing of high-strength rock, the sleeve may be a thin metal sheeting rather than latex. Triaxial testing on strong rock is fairly seldom done because the high forces and pressures required to break a rock sample require costly and cumbersome testing equipment.

### Effective Stress

The effective stress on the sample can be measured by using a porous surface on one platen, and measuring the pressure of the fluid (usually water) during the test, then calculating the effective stress from the total stress and pore pressure.

### Triaxial test to determine the shear strength of a discontinuity

The triaxial test can be used to determine the shear strength of a discontinuity. A homogeneous and isotropic sample fails due to shear stresses in the sample. If a sample with a discontinuity is orientated such that the discontinuity is about parallel to the plane in which maximum shear stress will be developed during the test, the sample will fail due to shear displacement along the discontinuity, and hence, the shear strength of a discontinuity can be calculated.[5]

## Types of Triaxial Tests

There are several variations of the triaxial test:

### Consolidated Drained (CD)

In a 'consolidated drained' test the sample is consolidated and sheared in compression slowly to allow pore pressures built up by the shearing to dissipate. The rate of axial deformation is kept constant, i.e., strain is controlled. The idea is that the test allows the sample and the pore pressures to fully consolidate (i.e., adjust) to the surrounding stresses. The test may take a long time to allow the sample to adjust, in particular low permeability samples need a long time to drain and adjust strain to stress levels.

### Consolidated Undrained (CU)

In a 'consolidated undrained' test the sample is not allowed to drain. The shear characteristics are measured under undrained conditions and the sample is assumed to be fully saturated. Measuring the pore pressures in the sample (sometimes called CUpp) allows approximating the consolidated-drained strength. Shear speed is often calculated based on the rate of consolidation under a specific confining pressure (whilst saturated). Confining pressures can vary anywhere from 1 psi to 100 psi or greater, sometimes requiring special load cells capable of handling higher pressures.

### Unconsolidated Undrained (UU)

In an 'unconsolidated undrained' test the loads are applied quickly, and the sample is not allowed to consolidate during the test. The sample is compressed at a constant rate (strain-controlled).

### True Triaxial Test

Triaxial testing systems have been developed to allow independent control of the stress in three perpendicular directions. This allows investigation of stress paths not capable of being generated in axisymmetric triaxial test machines, which can be useful in studies of cemented sands and anisotropic soils. The test cell is cubical, and there are six separate plates applying pressure to the specimen, with LVDTs reading movement of each plate.[6] Pressure in the third direction can be applied using hydrostatic pressure in the test chamber, requiring only 4 stress application assemblies. The apparatus is significantly more complex than for axisymmetric triaxial tests, and is therefore less commonly used.

### Free end condition in triaxial testing

The Danish triaxial in action

Triaxial tests of classical construction had been criticized for their nonuniform stress and strain field imposed within the specimen during larger deformation amplitudes.[7] The highly localized discontinuity within a shear zone is caused by combination of rough end plates and specimen height.

To test specimens during larger deformation amplitude, "new" [8] and "improved"[9] version of the triaxial apparatus were made. Both the "new" and the "improved" triaxial follow the same principle - sample height is reduced down to one diameter height and friction with the end plates is canceled.

The classical apparatus uses rough end plates - the whole surface of the piston head is made up of rough, porous filter. In upgraded apparatuses the tough end plates are replaced with smooth, polished glass, with a small filter at the center. This configuration allows a specimen to slide / expand horizontally while sliding along the polished glass. Thus, the contact zone between sample and the end plates does not buildup unnecessary shear friction, and a linear / isotropic stress field within the specimen is sustained.

Due to extremely uniform, near isotropic stress field - isotropic yielding takes place. During isotropic yielding volumetric strain is isotopically distributed within the specimen, this improves measurement of volumetric response during CD tests and pore water pressure during CU loading. Also, isotropic yielding makes the specimen expand radially in uniform manner, as it is compressed axially. The walls of a cylindrical specimen remain straight and vertical even during large strain amplitudes (50% strain amplitude was documented by Vardoulakis (1980), using "improved" triaxial, on non saturated sand). This is in contrast with classical setup, where the specimen forms a bugle in the center, while keeping a constant radius at the contact with the end plates.

Post-liquefaction testing. The fine sand specimen was liquefied during CU cycles and recovered with CD cycles many times. The wrinkles formed due to extreme volume change imposed by iterating between CU liquefaction and draining. In liquefied state sample become soft enough to imprint thin latex. During CD cycles - stiff enough to preserve the imprinted pattern. No bulging or shear rupture is present despite numerous instances of pure plastic yielding.

The "new" apparatus has been upgraded to "the Danish triaxial" by L.B.Ibsen.[10] The Danish triaxial can be used for testing all soil types. It provides improved measurements of volumetric response - as during isotropic yielding, volumetric strain is distributed isotopically within the specimen. Isotropic volume change is especially important for CU testing, as cavitation of pore water sets the limit of undrained sand strength.[11] Measurement precision is improved by taking measurements near the specimen. The load cell is submerged and in direct contact with the upped pressure head of the specimen. Deformation transducers are attached directly to the piston heads as well. Control of the apparatus is highly automated, thus cyclic loading can be applied with great efficiency and precision.

The combination of high automation, improved sample durability and large deformation compatibility expands the scope of triaxial testing. The Danish triaxial can yield CD and CU sand specimens into plasticity without forming a shear rupture or bulging. A sample can be tested for yielding multiple times in a single, continuous loading sequence. Samples can even be liquefied to a large strain amplitude, then crushed to CU failure. CU tests can be allowed to transition into CD state, and cyclic tested in CD mode to observe post liquefaction recovery of stiffness and strength.[12] This allows to control the specimens to a very high degree, and observe sand response patterns which are not accessible using classical triaxial testing methods.

## Test standards

The list is not complete; only the main standards are included. For a more extensive listing, please refer to the websites of ASTM International (USA), British Standards (UK), International Organization for Standardization (ISO), or local organisations for standards.

• ASTM D7181-11: Standard Test Method for Consolidated Drained Triaxial Compression Test for Soils[13]
• ASTM D4767-11 (2011): Standard Test Method for Consolidated Undrained Triaxial Compression Test for Cohesive Soils[14]
• ASTM D2850-03a (2007): Standard Test Method for Unconsolidated-Undrained Triaxial Compression Test on Cohesive Soils[15]
• BS 1377-8:1990 Part 8: Shear strength tests (effective stress)Triaxial Compression Test[16]
• ISO/TS 17892-8:2004 Geotechnical investigation and testing—Laboratory testing of soil—Part 8: Unconsolidated undrained triaxial test[17]
• ISO/TS 17892-9:2004 Geotechnical investigation and testing—Laboratory testing of soil—Part 9: Consolidated triaxial compression tests on water-saturated soils[18]

## References

1. ^ Bardet, J.-P. (1997). Experimental Soil Mechanics. Prentice Hall. ISBN 978-0-13-374935-9.
2. ^ Head, K.H. (1998). Effective Stress Tests, Volume 3, Manual of Soil Laboratory Testing, (2nd ed.). John Wiley & Sons. ISBN 978-0-471-97795-7.
3. ^ Holtz, R.D.; Kovacs, W.D. (1981). An Introduction to Geotechnical Engineering. Prentice-Hall, Inc. ISBN 0-13-484394-0.
4. ^ Price, D.G. (2009). De Freitas, M.H., ed. Engineering Geology: Principles and Practice. Springer. p. 450. ISBN 3-540-29249-7.
5. ^ Goodman, R.E. (1989). Introduction to Rock Mechanics. Wiley; 2 edition. p. 576. ISBN 978-0-471-81200-5.
6. ^ Reddy, K.R.; Saxena, S.K.; Budiman, J.S. (June 1992). "Development of A True Triaxial Testing Apparatus" (pdf). Geotechnical Testing Journal. ASTM. 15 (2): 89–105.
7. ^ ROWE, P W, Barden, L, "IMPORTANCE OF FREE ENDS IN TRIAXIAL TESTING" Journal of Soil Mechanics & Foundations, Volume: 90
8. ^
9. ^ Vardoulakis, I. (1979). "Bifurcation analysis of the triaxial test on sand samples". Acta Mechanica. 32: 35. doi:10.1007/BF01176132.
10. ^ Ibsen, L.B. (1994). "The stable state in cyclic triaxial testing on sand". Soil Dynamics and Earthquake Engineering. 13: 63. doi:10.1016/0267-7261(94)90042-6.
12. ^ https://www.onepetro.org/conference-paper/ISOPE-I-15-114
13. ^ ASTM D7181 (2011). Standard Test Method for Consolidated Drained Triaxial Compression Test for Soils). ASTM International, West Conshohocken, PA, 2003.
14. ^ ASTM D4767-11 (2011). Standard Test Method for Consolidated Undrained Triaxial Compression Test for Cohesive Soils. ASTM International, West Conshohocken, PA, 2003. doi:10.1520/D4767-11.
15. ^ ASTM D2850 - 03a (2007). Standard Test Method for Unconsolidated-Undrained Triaxial Compression Test on Cohesive Soils. ASTM International, West Conshohocken, PA, 2003. doi:10.1520/D2850-03AR07.
16. ^ BS 1377-1 (1990). Methods of test for soils for civil engineering purposes. General requirements and sample preparation. BSI. ISBN 0-580-17692-4.
17. ^ ISO/TS 17892-8:2004 (2007). Geotechnical investigation and testing - Laboratory testing of soil - Part 8: Unconsolidated undrained triaxial test. International Organization for Standardization. p. 24.
18. ^ ISO/TS 17892-9:2004 (2007). Geotechnical investigation and testing -- Laboratory testing of soil -- Part 9: Consolidated triaxial compression tests on water-saturated soils. International Organization for Standardization. p. 30.

Borehole

A borehole is a narrow shaft bored in the ground, either vertically or horizontally. A borehole may be constructed for many different purposes, including the extraction of water, other liquids (such as petroleum) or gases (such as natural gas), as part of a geotechnical investigation, environmental site assessment, mineral exploration, temperature measurement, as a pilot hole for installing piers or underground utilities, for geothermal installations, or for underground storage of unwanted substances, e.g. in carbon capture and storage.

Clay

Clay is a finely-grained natural rock or soil material that combines one or more clay minerals with possible traces of quartz (SiO2), metal oxides (Al2O3 , MgO etc.) and organic matter. Geologic clay deposits are mostly composed of phyllosilicate minerals containing variable amounts of water trapped in the mineral structure. Clays are plastic due to particle size and geometry as well as water content, and become hard, brittle and non–plastic upon drying or firing. Depending on the soil's content in which it is found, clay can appear in various colours from white to dull grey or brown to deep orange-red.

Although many naturally occurring deposits include both silts and clay, clays are distinguished from other fine-grained soils by differences in size and mineralogy. Silts, which are fine-grained soils that do not include clay minerals, tend to have larger particle sizes than clays. There is, however, some overlap in particle size and other physical properties. The distinction between silt and clay varies by discipline. Geologists and soil scientists usually consider the separation to occur at a particle size of 2 µm (clays being finer than silts), sedimentologists often use 4–5 μm, and colloid chemists use 1 μm. Geotechnical engineers distinguish between silts and clays based on the plasticity properties of the soil, as measured by the soils' Atterberg limits. ISO 14688 grades clay particles as being smaller than 2 μm and silt particles as being larger.

Mixtures of sand, silt and less than 40% clay are called loam. Loam makes good soil and is used as a building material.

Direct shear test

A direct shear test is a laboratory or field test used by geotechnical engineers to measure the shear strength properties of soil or rock material, or of discontinuities in soil or rock masses.

The U.S. and U.K. standards defining how the test should be performed are ASTM D 3080, AASHTO T236 and BS 1377-7:1990, respectively. For rock the test is generally restricted to rock with (very) low shear strength. The test is, however, standard practice to establish the shear strength properties of discontinuities in rock.

The test is performed on three or four specimens from a relatively undisturbed soil sample. A specimen is placed in a shear box which has two stacked rings to hold the sample; the contact between the two rings is at approximately the mid-height of the sample. A confining stress is applied vertically to the specimen, and the upper ring is pulled laterally until the sample fails, or through a specified strain. The load applied and the strain induced is recorded at frequent intervals to determine a stress–strain curve for each confining stress. Several specimens are tested at varying confining stresses to determine the shear strength parameters, the soil cohesion (c) and the angle of internal friction, commonly known as friction angle (${\displaystyle \phi }$). The results of the tests on each specimen are plotted on a graph with the peak (or residual) stress on the y-axis and the confining stress on the x-axis. The y-intercept of the curve which fits the test results is the cohesion, and the slope of the line or curve is the friction angle.

Direct shear tests can be performed under several conditions. The sample is normally saturated before the test is run, but can be run at the in-situ moisture content. The rate of strain can be varied to create a test of undrained or drained conditions, depending whether the strain is applied slowly enough for water in the sample to prevent pore-water pressure buildup. Direct shear test machine is required to perform the test. The test using the direct shear machine determinates the consolidated drained shear strength of a soil material in direct shear.

The advantages of the direct shear test over other shear tests are the simplicity of setup and equipment used, and the ability to test under differing saturation, drainage, and consolidation conditions. These advantages have to be weighed against the difficulty of measuring pore-water pressure when testing in undrained conditions, and possible spuriously high results from forcing the failure plane to occur in a specific location.

The test equipment and procedures are slightly different for test on discontinuities.

Exploration geophysics

Exploration geophysics is an applied branch of geophysics, which uses physical methods, such as seismic, gravitational, magnetic, electrical and electromagnetic at the surface of the Earth to measure the physical properties of the subsurface, along with the anomalies in those properties. It is most often used to detect or infer the presence and position of economically useful geological deposits, such as ore minerals; fossil fuels and other hydrocarbons; geothermal reservoirs; and groundwater reservoirs.

Exploration geophysics can be used to directly detect the target style of mineralization, via measuring its physical properties directly. For example, one may measure the density contrasts between the dense iron ore and the lighter silicate host rock, or one may measure the electrical conductivity contrast between conductive sulfide minerals and the resistive silicate host rock.

Gravel

Gravel is a loose aggregation of rock fragments. Gravel is classified by particle size range and includes size classes from granule- to boulder-sized fragments. In the Udden-Wentworth scale gravel is categorized into granular gravel (2 to 4 mm or 0.079 to 0.157 in) and pebble gravel (4 to 64 mm or 0.2 to 2.5 in). ISO 14688 grades gravels as fine, medium, and coarse with ranges 2 mm to 6.3 mm to 20 mm to 63 mm. One cubic metre of gravel typically weighs about 1,800 kg (or a cubic yard weighs about 3,000 pounds).

Gravel is an important commercial product, with a number of applications. Many roadways are surfaced with gravel, especially in rural areas where there is little traffic. Globally, far more roads are surfaced with gravel than with concrete or tarmac; Russia alone has over 400,000 km (250,000 mi) of gravel roads. Both sand and small gravel are also important for the manufacture of concrete.

Natchez silt loam

In 1988, the Professional Soil Classifiers Association of Mississippi selected Natchez silt loam soil to represent the soil resources of the State. These soils exist on 171,559 acres (0.56% of state) of landscape in Mississippi.

Particle technology

Particle technology is the "science and technology related to the handling and processing of particles and powders." This applies to the production, handling, modification, and use of a wide variety of particulate materials, both wet or dry, in sizes ranging from nanometers to centimeters; its scope spans a range of industries to include chemical, petrochemical, agricultural, food, pharmaceuticals, mineral processing, civil engineering, advanced materials, energy, and the environment.

Response spectrum

A response spectrum is a plot of the peak or steady-state response (displacement, velocity or acceleration) of a series of oscillators of varying natural frequency, that are forced into motion by the same base vibration or shock. The resulting plot can then be used to pick off the response of any linear system, given its natural frequency of oscillation. One such use is in assessing the peak response of buildings to earthquakes. The science of strong ground motion may use some values from the ground response spectrum (calculated from recordings of surface ground motion from seismographs) for correlation with seismic damage.

If the input used in calculating a response spectrum is steady-state periodic, then the steady-state result is recorded. Damping must be present, or else the response will be infinite. For transient input (such as seismic ground motion), the peak response is reported. Some level of damping is generally assumed, but a value will be obtained even with no damping.

Response spectra can also be used in assessing the response of linear systems with multiple modes of oscillation (multi-degree of freedom systems), although they are only accurate for low levels of damping. Modal analysis is performed to identify the modes, and the response in that mode can be picked from the response spectrum. These peak responses are then combined to estimate a total response. A typical combination method is the square root of the sum of the squares (SRSS) if the modal frequencies are not close. The result is typically different from that which would be calculated directly from an input, since phase information is lost in the process of generating the response spectrum.

The main limitation of response spectra is that they are only universally applicable for linear systems. Response spectra can be generated for non-linear systems, but are only applicable to systems with the same non-linearity, although attempts have been made to develop non-linear seismic design spectra with wider structural application. The results of this cannot be directly combined for multi-mode response.

Seismoelectrical method

The seismoelectrical method (which is different from the electroseismic physical principle) is based on the generation of electromagnetic fields in soils and rocks by seismic waves. This technique is still under development and in the future it may have applications like detecting and characterizing fluids in the underground by their electrical properties, among others, usually related to fluids (porosity, transmissivity, physical properties).

Shear band

A shear band (or, more generally, a 'strain localization') is a narrow zone of intense shearing strain, usually of plastic nature, developing during severe deformation of ductile materials.

As an example, a soil (overconsolidated silty-clay) specimen is shown in Fig. 1, after an axialsymmetric compression test. Initially the sample was cylindrical in shape and, since symmetry was tried to be preserved during the test, the cylindrical shape was maintained for a while during the test and the deformation was homogeneous, but at extreme loading two X-shaped shear bands had formed and the subsequent deformation was strongly localized (see also the sketch on the right of Fig. 1).

Shear strength test

Soil shear strength tests are used to determine the load on soil. Some of these tests are:

direct shear test

triaxial shear test

vane shear test

unconfined compression test

Shear stress

A shear stress, often denoted by τ (Greek: tau), is the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section of the material. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.

Shear stress arises from shear forces, which are pairs of equal and opposing forces acting on opposite sides of an object.

Shearing (physics)

Shearing in continuum mechanics refers to the occurrence of a shear strain, which is a deformation of a material substance in which parallel internal surfaces slide past one another. It is induced by a shear stress in the material. Shear strain is distinguished from volumetric strain, the change in a material's volume in response to stress.

Silt

Silt is granular material of a size between sand and clay, whose mineral origin is quartz and feldspar. Silt may occur as a soil (often mixed with sand or clay) or as sediment mixed in suspension with water (also known as a suspended load) and soil in a body of water such as a river. It may also exist as soil deposited at the bottom of a water body, like mudflows from landslides. Silt has a moderate specific area with a typically non-sticky, plastic feel. Silt usually has a floury feel when dry, and a slippery feel when wet. Silt can be visually observed with a hand lens, exhibiting a sparkly appearance. It also can be felt by the tongue as granular when placed on the front teeth (even when mixed with clay particles).

Specific storage

In the field of hydrogeology, storage properties are physical properties that characterize the capacity of an aquifer to release groundwater. These properties are Storativity (S), specific storage (Ss) and specific yield (Sy).

They are often determined using some combination of field tests (e.g., aquifer tests) and laboratory tests on aquifer material samples. Recently, these properties have been also determined using remote sensing data derived from Interferometric synthetic-aperture radar.

Thixotropy

Thixotropy is a time-dependent shear thinning property. Certain gels or fluids that are thick, or viscous, under static conditions will flow (become thin, less viscous) over time when shaken, agitated, sheared or otherwise stressed (time dependent viscosity). They then take a fixed time to return to a more viscous state.

Some non-Newtonian pseudoplastic fluids show a time-dependent change in viscosity; the longer the fluid undergoes shear stress, the lower its viscosity. A thixotropic fluid is a fluid which takes a finite time to attain equilibrium viscosity when introduced to a steep change in shear rate. Some thixotropic fluids return to a gel state almost instantly, such as ketchup, and are called pseudoplastic fluids. Others such as yogurt take much longer and can become nearly solid. Many gels and colloids are thixotropic materials, exhibiting a stable form at rest but becoming fluid when agitated. Thixotropy arises because particles or structured solutes require time to organize. An excellent overview of thixotropy has been provided by Mewis and Wagner.Some fluids are anti-thixotropic: constant shear stress for a time causes an increase in viscosity or even solidification. Fluids which exhibit this property are sometimes called rheopectic. Anti-thixotropic fluids are less well documented than thixotropic fluids.

Tilt test (geotechnical engineering)

In geomechanics, a tilt test is a simple test to estimate the shear strength parameters of a discontinuity. Two pieces of rock containing a discontinuity are held in hand or mounted in test equipment with the discontinuity horizontal. The sample is slowly tilted until the top block moves. The angle with the horizontal at onset of movement is called the tilt-angle.

The size of the specimen is limited to 10–20 cm for hand-held tests, while machine-operated tilt test equipment may handle up to meter-sized samples. In the field, the angle can be determined most easily with an inclinometer as present in most geological or structural compasses.

Trench

A trench is a type of excavation or depression in the ground that is generally deeper than it is wide (as opposed to a wider gully, or ditch), and narrow compared with its length (as opposed to a simple hole).In geology, trenches are created as a result of erosion by rivers or by geological movement of tectonic plates. In the civil engineering field, trenches are often created to install underground infrastructure or utilities (such as gas mains, water mains or telephone lines), or later to access these installations. Trenches have also often been dug for military defensive purposes. In archaeology, the "trench method" is used for searching and excavating ancient ruins or to dig into strata of sedimented material.

Void ratio

The void ratio of a mixture is the ratio of the volume of voids to volume of solids.

It is a dimensionless quantity in materials science, and is closely related to porosity as follows:

${\displaystyle e={\frac {V_{V}}{V_{S}}}={\frac {V_{V}}{V_{T}-V_{V}}}={\frac {\phi }{1-\phi }}}$

and

${\displaystyle \phi ={\frac {V_{V}}{V_{T}}}={\frac {V_{V}}{V_{S}+V_{V}}}={\frac {e}{1+e}}}$

where ${\displaystyle e}$ is void ratio, ${\displaystyle \phi }$ is porosity, VV is the volume of void-space (such as fluids), VS is the volume of solids, and VT is the total or bulk volume. This figure is relevant in composites, in mining (particular with regard to the properties of tailings), and in soil science. In geotechnical engineering, it is considered as one of the state variables of soils and represented by the symbol e.

Note that in geotechnical engineering, the symbol ${\displaystyle \phi }$ usually represents the angle of shearing resistance, a shear strength (soil) parameter. Because of this, the equation is usually rewritten using ${\displaystyle n}$ for porosity:

${\displaystyle e={\frac {V_{V}}{V_{S}}}={\frac {V_{V}}{V_{T}-V_{V}}}={\frac {n}{1-n}}}$

and

${\displaystyle n={\frac {V_{V}}{V_{T}}}={\frac {V_{V}}{V_{S}+V_{V}}}={\frac {e}{1+e}}}$

where ${\displaystyle e}$ is void ratio, ${\displaystyle n}$ is porosity, VV is the volume of void-space (air and water), VS is the volume of solids, and VT is the total or bulk volume.

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