# Atterberg limits

The Atterberg limits are a basic measure of the critical water contents of a fine-grained soil: its shrinkage limit, plastic limit, and liquid limit.

Depending on its water content, a soil may appear in one of four states: solid, semi-solid, plastic and liquid. In each state, the consistency and behavior of a soil is different and consequently so are its engineering properties. Thus, the boundary between each state can be defined based on a change in the soil's behavior. The Atterberg limits can be used to distinguish between silt and clay, and to distinguish between different types of silts and clays.

These limits were created by Albert Atterberg, a Swedish agriculturist.[1] They were later refined by Arthur Casagrande.

Distinctions in soil are used in assessing the soils that are to have structures built on them. Soils when wet retain water, and some expand in volume. The amount of expansion is related to the ability of the soil to take in water and its structural make-up (the type of atoms present). These tests are mainly used on clayey or silty soils since these are the soils that expand and shrink due to moisture content. Clays and silts react with the water and thus change sizes and have varying shear strengths. Thus these tests are used widely in the preliminary stages of designing any structure to ensure that the soil will have the correct amount of shear strength and not too much change in volume as it expands and shrinks with different moisture contents.

## Laboratory tests

### Shrinkage limit

The shrinkage limit (SL) is the water content where further loss of moisture will not result in any more volume reduction.[2] The test to determine the shrinkage limit is ASTM International D4943. The shrinkage limit is much less commonly used than the liquid and plastic limits.

### Plastic limit

The Plastic Limit (PL) is determined by rolling out a thread of the fine portion of a soil on a flat, non-porous surface. The procedure is defined in ASTM Standard D 4318. If the soil is at a moisture content where its behavior is plastic, this thread will retain its shape down to a very narrow diameter. The sample can then be remolded and the test repeated. As the moisture content falls due to evaporation, the thread will begin to break apart at larger diameters.

The plastic limit is defined as the moisture content where the thread breaks apart at a diameter of 3.2 mm (about 1/8 inch). A soil is considered non-plastic if a thread cannot be rolled out down to 3.2 mm at any moisture possible.

### Liquid limit

Casagrande cup in action

The liquid limit (LL) is conceptually defined as the water content at which the behavior of a clayey soil changes from plastic to liquid. However, the transition from plastic to liquid behavior is gradual over a range of water contents, and the shear strength of the soil is not actually zero at the liquid limit. The precise definition of the liquid limit is based on standard test procedures described below.

The original liquid limit test of Atterberg's involved mixing a pat of clay in a round-bottomed porcelain bowl of 10–12 cm diameter. A groove was cut through the pat of clay with a spatula, and the bowl was then struck many times against the palm of one hand. Casagrande subsequently standardized the apparatus and the procedures to make the measurement more repeatable. Soil is placed into the metal cup portion of the device and a groove is made down its center with a standardized tool of 2 millimetres (0.079 in) width. The cup is repeatedly dropped 10 mm onto a hard rubber base at a rate of 120 blows per minute, during which the groove closes up gradually as a result of the impact. The number of blows for the groove to close is recorded. The moisture content at which it takes 25 drops of the cup to cause the groove to close over a distance of 12.7 millimetres (0.50 in) is defined as the liquid limit. The test is normally run at several moisture contents, and the moisture content which requires 25 blows to close the groove is interpolated from the test results. The liquid limit test is defined by ASTM standard test method D 4318.[3] The test method also allows running the test at one moisture content where 20 to 30 blows are required to close the groove; then a correction factor is applied to obtain the liquid limit from the moisture content.[4]

Another method for measuring the liquid limit is the fall cone test, also called the cone penetrometer test. It is based on the measurement of penetration into the soil of a standardized cone of specific mass. Although the Casagrande test is widely used across North America, the fall cone test is much more prevalent in Europe due to being less dependent on the operator in determining the Liquid Limit.[5]

## Derived limits

The values of these limits are used in a number of ways. There is also a close relationship between the limits and properties of a soil such as compressibility, permeability, and strength. This is thought to be very useful because as limit determination is relatively simple, it is more difficult to determine these other properties. Thus the Atterberg limits are not only used to identify the soil's classification, but it allows for the use of empirical correlations for some other engineering properties.

### Plasticity index

The plasticity index (PI) is a measure of the plasticity of a soil. The plasticity index is the size of the range of water contents where the soil exhibits plastic properties. The PI is the difference between the liquid limit and the plastic limit (PI = LL-PL). Soils with a high PI tend to be clay, those with a lower PI tend to be silt, and those with a PI of 0 (non-plastic) tend to have little or no silt or clay.

Soil descriptions based on PI:[6]

• (0)- Nonplastic
• (<7) - Slightly plastic
• (7-17) - Medium plastic
• >17 - Highly plastic

### Liquidity index

The liquidity index (LI) is used for scaling the natural water content of a soil sample to the limits. It can be calculated as a ratio of difference between natural water content, plastic limit, and liquid limit: LI=(W-PL)/(LL-PL) where W is the natural water content.

### Consistency index

The consistency index (CI) indicates the consistency (firmness) of a soil. It is calculated as CI = (LL-W)/(LL-PL), where W is the existing water content. Soil at the liquid limit will have a consistency index of 0, while soil at the plastic limit will have a consistency index of 1.

### Activity

The activity (A) of a soil is the PI divided by the percent of clay-sized particles (less than 2 μm) present. Different types of clays have different specific surface areas which controls how much wetting is required to move a soil from one phase to another such as across the liquid limit or the plastic limit. From the activity, one can predict the dominant clay type present in a soil sample. High activity signifies large volume change when wetted and large shrinkage when dried. Soils with high activity are very reactive chemically.

Normally the activity of clay is between 0.75 and 1.25, and in this range clay is called normal. It is assumed that the plasticity index is approximately equal to the clay fraction (A = 1). When A is less than 0.75, it is considered inactive. When it is greater than 1.25, it is considered active.

## Notes

1. ^ "Brief history of Swedish Soil Mechanics". Archived from the original on 2007-03-25. Retrieved 2007-01-15.
2. ^
3. ^ "ASTM D4318 - 10 Standard Test Methods for Liquid Limit, Plastic Limit, and Plasticity Index of Soils". ASTM. 2010. Retrieved 2011-02-18.
4. ^
5. ^ BS 1377 part 2
6. ^ Sowers, 1979

## References

• Soil Physical Properties--Mechanics
• Seed, H.B. (1967). "Fundamental Aspects of the Atterberg Limits". Journal of Soil Mechanics and Foundations Div., 92(SM4), Retrieved from http://trid.trb.org/view.aspx?id=38900
• Das, B. M. (2006). Principles of geotechnical engineering. Stamford,CT: Thomson Learning College.
• Sowers, 1979. Introductory Soil Mechanics and Foundations: Geotechnical Engineering, 4th Ed., Macmillan, New York. (as referenced in Coduto, 1999. Geotechnical Engineering: Principles and Practices. Prentice Hall. New Jersey.)
Albert Atterberg

Albert Mauritz Atterberg (19 March 1846 – 4 April 1916) was a Swedish chemist and agricultural scientist who created the Atterberg limits, which are commonly referred to by geotechnical engineers and engineering geologists today. In Sweden he is equally known for creating the Atterberg grainsize scale, which remains the one in use.

Atterberg received his Ph.D. in chemistry from Uppsala University in 1872 and then stayed there as a lecturer in analytical chemistry until 1877, during which time he traveled across Sweden and abroad to study the latest developments in organic chemistry. He then went on to become the principal of the Chemical Station and Seed Control Institute at Kalmar, publishing numerous papers on agricultural research dealing with the classification of varieties of oats and corn between 1891 and 1900.

It was towards the age of fifty-four that Atterberg, while continuing his work on chemistry, began to focus his efforts on the classification and plasticity of soils, for which he is most remembered. Atterberg was apparently the first to suggest the limit <0.002 mm as a classification for clay particles. He found that plasticity to be a particular characteristic of clay and as a result of his investigations arrived at the consistency limits which bear his name today. He also conducted studies aiming to identify the specific minerals that give a clayey soil its plastic nature.

Atterberg’s work on soil classification gained formal recognition from the International Society of Soil Science in a Berlin Conference in 1913. Two year later a U.S. Bureau of Standards report stated that Atterberg's method was "as simple a one as could be devised, and...it is well that we should become familiar with it." The U.S. Bureau of Chemistry and Soils adopted it in 1937.

The importance of Atterberg’s work has never been fully realized in his own field of agricultural science, nor in other subjects concerned with clays, such as ceramics. Its introduction to the field of geotechnical engineering was due to Karl Terzaghi, who came to realise its importance at a relatively early stage of his research. Terzaghi’s assistant, Arthur Casagrande, standardized the tests in his paper in 1932 and the procedures have been followed worldwide ever since.

He was the uncle of the composer Kurt Atterberg.

Atterberg

Atterberg may refer to:

Albert Atterberg (1846–1916), Swedish chemist and agricultural scientist

Kurt Atterberg (1887–1974), Swedish composer and engineer

Atterberg limits

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.

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.

Geotechnical investigation

Geotechnical investigations are performed by geotechnical engineers or engineering geologists to obtain information on the physical properties of soil earthworks and foundations for proposed structures and for repair of distress to earthworks and structures caused by subsurface conditions. This type of investigation is called a site investigation. Additionally, geotechnical investigations are also used to measure the thermal resistivity of soils or backfill materials required for underground transmission lines, oil and gas pipelines, radioactive waste disposal, and solar thermal storage facilities. A geotechnical investigation will include surface exploration and subsurface exploration of a site. Sometimes, geophysical methods are used to obtain data about sites. Subsurface exploration usually involves soil sampling and laboratory tests of the soil samples retrieved.

Surface exploration can include geologic mapping, geophysical methods, and photogrammetry, or it can be as simple as a geotechnical professional walking around on the site to observe the physical conditions at the site.

To obtain information about the soil conditions below the surface, some form of subsurface exploration is required. Methods of observing the soils below the surface, obtaining samples, and determining physical properties of the soils and rocks include test pits, trenching (particularly for locating faults and slide planes), boring, and in situ tests.

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.

Plasticity (physics)

In physics and materials science, plasticity describes the deformation of a (solid) material undergoing non-reversible changes of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself. In engineering, the transition from elastic behavior to plastic behavior is called yield.

Plastic deformation is observed in most materials, particularly metals, soils, rocks, concrete, foams, bone and skin. However, the physical mechanisms that cause plastic deformation can vary widely. At a crystalline scale, plasticity in metals is usually a consequence of dislocations. Such defects are relatively rare in most crystalline materials, but are numerous in some and part of their crystal structure; in such cases, plastic crystallinity can result. In brittle materials such as rock, concrete and bone, plasticity is caused predominantly by slip at microcracks. In cellular materials such as liquid foams or biological tissues, plasticity is mainly a consequence of bubble or cell rearrangements, notably T1 processes.

For many ductile metals, tensile loading applied to a sample will cause it to behave in an elastic manner. Each increment of load is accompanied by a proportional increment in extension. When the load is removed, the piece returns to its original size. However, once the load exceeds a threshold – the yield strength – the extension increases more rapidly than in the elastic region; now when the load is removed, some degree of extension will remain.

Elastic deformation, however, is an approximation and its quality depends on the time frame considered and loading speed. If, as indicated in the graph opposite, the deformation includes elastic deformation, it is also often referred to as "elasto-plastic deformation" or "elastic-plastic deformation".

Perfect plasticity is a property of materials to undergo irreversible deformation without any increase in stresses or loads. Plastic materials with hardening necessitate increasingly higher stresses to result in further plastic deformation. Generally, plastic deformation is also dependent on the deformation speed, i.e. higher stresses usually have to be applied to increase the rate of deformation. Such materials are said to deform visco-plastically.

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.

Sand

Sand is a granular material composed of finely divided rock and mineral particles. It is defined by size, being finer than gravel and coarser than silt. Sand can also refer to a textural class of soil or soil type; i.e., a soil containing more than 85 percent sand-sized particles by mass.The composition of sand varies, depending on the local rock sources and conditions, but the most common constituent of sand in inland continental settings and non-tropical coastal settings is silica (silicon dioxide, or SiO2), usually in the form of quartz. The second most common type of sand is calcium carbonate, for example, aragonite, which has mostly been created, over the past half billion years, by various forms of life, like coral and shellfish. For example, it is the primary form of sand apparent in areas where reefs have dominated the ecosystem for millions of years like the Caribbean.

Sand is a non-renewable resource over human timescales, and sand suitable for making concrete is in high demand.

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).

Soil classification

Soil classification deals with the systematic categorization of soils based on distinguishing characteristics as well as criteria that dictate choices in use.

Soil liquefaction

Soil liquefaction occurs when a saturated or partially saturated soil substantially loses strength and stiffness in response to an applied stress such as shaking during an earthquake or other sudden change in stress condition, in which material that is ordinarily a solid behaves like a liquid.

In soil mechanics, the term "liquefied" was first used by Allen Hazen in reference to the 1918 failure of the Calaveras Dam in California. He described the mechanism of flow liquefaction of the embankment dam as:

If the pressure of the water in the pores is great enough to carry all the load, it will have the effect of holding the particles apart and of producing a condition that is practically equivalent to that of quicksand… the initial movement of some part of the material might result in accumulating pressure, first on one point, and then on another, successively, as the early points of concentration were liquefied.

The phenomenon is most often observed in saturated, loose (low density or uncompacted), sandy soils. This is because a loose sand has a tendency to compress when a load is applied. Dense sands by contrast tend to expand in volume or 'dilate'. If the soil is saturated by water, a condition that often exists when the soil is below the water table or sea level, then water fills the gaps between soil grains ('pore spaces'). In response to soil compressing, the water pressure increases and the water attempts to flow out from the soil to zones of low pressure (usually upward towards the ground surface). However, if the loading is rapidly applied and large enough, or is repeated many times (e.g. earthquake shaking, storm wave loading) such that the water does not flow out before the next cycle of load is applied, the water pressures may build to the extent that it exceeds the force (contact stresses) between the grains of soil that keep them in contact. These contacts between grains are the means by which the weight from buildings and overlying soil layers is transferred from the ground surface to layers of soil or rock at greater depths. This loss of soil structure causes it to lose its strength (the ability to transfer shear stress), and it may be observed to flow like a liquid (hence 'liquefaction').

Although the effects of liquefaction have been long understood, engineers took more notice after the 1964 Niigata earthquake and 1964 Alaska earthquake. It was a major factor in the destruction in San Francisco's Marina District during the 1989 Loma Prieta earthquake, and in Port of Kobe during the 1995 Great Hanshin earthquake. More recently liquefaction was largely responsible for extensive damage to residential properties in the eastern suburbs and satellite townships of Christchurch, New Zealand during the 2010 Canterbury earthquake and more extensively again following the Christchurch earthquakes that followed in early and mid-2011. On 28th September 2018, an earthquake of 7.5 magnitude hit the Central Sulawesi province of Indonesia. Resulting soil liquefaction buried the suburb of Balaroa and Petobo village in 3 meters deep mud. The government of Indonesia is considering designating the two neighborhoods of Balaroa and Petobo, that have been totally buried under mud- as mass graves.The building codes in many countries require engineers to consider the effects of soil liquefaction in the design of new buildings and infrastructure such as bridges, embankment dams and retaining structures.

Soil mechanics

Soil mechanics is a branch of soil physics and applied mechanics that describes the behavior of soils. It differs from fluid mechanics and solid mechanics in the sense that soils consist of a heterogeneous mixture of fluids (usually air and water) and particles (usually clay, silt, sand, and gravel) but soil may also contain organic solids and other matter. Along with rock mechanics, soil mechanics provides the theoretical basis for analysis in geotechnical engineering, a subdiscipline of civil engineering, and engineering geology, a subdiscipline of geology. Soil mechanics is used to analyze the deformations of and flow of fluids within natural and man-made structures that are supported on or made of soil, or structures that are buried in soils. Example applications are building and bridge foundations, retaining walls, dams, and buried pipeline systems. Principles of soil mechanics are also used in related disciplines such as engineering geology, geophysical engineering, coastal engineering, agricultural engineering, hydrology and soil physics.

This article describes the genesis and composition of soil, the distinction between pore water pressure and inter-granular effective stress, capillary action of fluids in the soil pore spaces, soil classification, seepage and permeability, time dependent change of volume due to squeezing water out of tiny pore spaces, also known as consolidation, shear strength and stiffness of soils. The shear strength of soils is primarily derived from friction between the particles and interlocking, which are very sensitive to the effective stress. The article concludes with some examples of applications of the principles of soil mechanics such as slope stability, lateral earth pressure on retaining walls, and bearing capacity of foundations.

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.

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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