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Encyclopedia > Engineering strain
Continuum mechanics
General
Solid mechanics
Fluid mechanics
This article is about the deformation of materials. For other meanings, see strain.

In any branch of science dealing with materials and their behaviour, strain is the geometrical expression of deformation caused by the action of stress on a physical body. Strain is calculated by first assuming a change between two body states: the beginning state and the final state. Then the difference in placement of two points in this body in those two states expresses the numerical value of strain. Strain therefore expresses itself as a change in size and/or shape. Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Stress is the internal distribution of force per unit area that balances and reacts to external loads applied to a body. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... The law of conservation of mass/matter, also known as Law of Mass Conservation (or the Lomonosov-Lavoisier law), states that the mass of a closed system of substances will remain constant, regardless of the processes acting inside the system. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... Solid mechanics is the branch of physics and mathematics that concern the behavior of solid matter under external actions (e. ... In jewelry, a solid gold piece is the alternative to gold-filled or gold-plated jewelry. ... Elasticity is a branch of physics which studies the properties of elastic materials. ... For other uses, see Plasticity. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ... Figure 1: Rectangular specimen subject to compression, with Poissons ratio circa 0. ... Rheology is the study of the deformation and flow of matter. ... Fluid mechanics is the subdiscipline of continuum mechanics that studies fluids, that is, liquids and gases. ... A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ... Fluid statics (also called hydrostatics) is the science of fluids at rest, and is a sub-field within fluid mechanics. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations that describe the motion of fluid substances such as liquids and gases. ... The related Category:Units of viscosity has been nominated for deletion, merging, or renaming. ... A Newtonian fluid (named for Isaac Newton) is a fluid that flows like water—its shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. ... A non-Newtonian fluid is a fluid in which the viscosity changes with the applied strain rate. ... In engineering mechanics, deformation is a change in shape due to an applied force. ... Look up strain in Wiktionary, the free dictionary. ... Stress is the internal distribution of force per unit area that balances and reacts to external loads applied to a body. ...


If strain is equal over all parts of a body, it is referred to as homogeneous strain; otherwise, it is inhomogeneous strain. In its most general form, the strain is a symmetric tensor. The strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation: the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms εij = 1/2 γij (i...


In the case of geological action of the earth, if the release of stress through strain in rocks is sufficiently large, earthquakes may occur. Geology (from Greek γη- (ge-, the earth) and λογος (logos, word, reason)) is the science and study of the Earth, its composition, structure, physical properties, history, and the processes that shape it. ... An earthquake is a phenomenon that results from the sudden release of stored energy in the Earths crust that creates seismic waves. ...

Contents

Quantifying strain

Given that strain results in the deformation of a body, it can be measured by calculating the change in length of a line or by the change in angle between two lines (where these lines are theoretical constructs within the deformed body). The change in length of a line is termed the stretch, absolute strain, or extension, and may be written as delta ell. Then the (relative) strain, ε, is given by An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ... In the uniaxial tensile test commonly carried out to determine some properties of engineering materials, a small testpiece is stretched from an initial, undeformed length to a current, deformed length . ...

varepsilon = frac {delta ell}{ell_0}

where

varepsilon is strain in measured direction
ell_o is the original length of the material.
ell is the current length of the material.

The extension (delta ell) is positive if the material has gained length (in tension) and negative if it has reduced length (in compression). Because ell_o is always positive, the sign of the strain is always the same as the sign of the extension. Tension is a reaction force applied by a stretched string (rope or a similar object) on the objects which stretch it. ... Physical compression is the result of the subjection of a material to compressive stress, resulting in reduction of volume. ...


Strain has no units of measure because in the formula the units of length are cancelled. Dimensions of metres/metre or inches/inch are sometimes used for convenience, but generally units are left off and the strain sometimes is given as a percentage. The metre, or meter (U.S.), is a measure of length. ... An inch (plural: inches; symbol or abbreviation: in or, sometimes, ″ - a double prime) is the name of a unit of length in a number of different systems, including English units, Imperial units, and United States customary units. ... The percent sign A percentage is a way of expressing numbers as fractions of 100 and is often denoted using the percent sign, %. For example, 45. ...


Linear axial strain at single point

In the case of measuring strain in the selected point of the body, it is expressed as a strain where the distance ell between two points approaches zero:

varepsilon = mathop {lim_{ell to 0}} frac {{delta} {ell} } {ell}

where

varepsilon is strain in measured direction
{{delta} {ell} } is the length difference for current length ell.
ell is the current length of the material, which approaches zero.


Therefore linear strain is defined as change of distance in the close proximity of selected point.


The general case of linear strain

For the body of any shape, subjected to any deformation the values of strain will be different depending on the spatial direction of measurement. Considering the linear deformation in the point A placed at the start of coordinate system and a second point B placed along the x axis, which due to deformation has moved to the point B' the linear strain will be expressed as:

varepsilon_x = mathop {lim_{B to A}}{{|AB'|-|AB|} over {|AB|}}

Doing similar calculations for axes y and z respective values of εy and εz can be obtained. For any given displacement field overrightarrow u (the values of displacement vectors for all points in the body) the linear strain can be written as: A displacement field is an assignment of displacement vectors for all points in a body that is displaced from one state to another. ...

varepsilon_x = {{partial u_x} over {partial x}} ; varepsilon_y = {{partial u_y} over {partial y}} ; varepsilon_z = {{partial u_z} over {partial z}}

where

varepsilon_i is strain in direction along axis i
{{partial u_i} over {partial i}} is a differential of overrightarrow u at any point in the direction along axis i

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...

Shear strain

Similarly the angular change at any point between two lines crossing this point in a body can be measured as a shear (or shape) strain. Shear strain γ is the limit of ratio of angular difference between any two lines in a body before and after deformation, assuming that the lines lengths are approaching zero. Given a displacement field overrightarrow u like above, the shear strain can be written as follows: An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ... Shear strain is the components of a strain at a point that produce changes in shape of a body (distortion) without a volumetric change. ... A displacement field is an assignment of displacement vectors for all points in a body that is displaced from one state to another. ...

gamma_{xy} = {{partial u_x} over {partial y}} + {{partial u_y} over {partial x}} ; gamma_{yz} = {{partial u_y} over {partial z}} + {{partial u_z} over {partial y}} ; gamma_{xz} = {{partial u_x} over {partial z}} + {{partial u_z} over {partial x}}

Volumetric strain

Although linear strain ε and shear strain γ completely defines the state of deformation of a body, it is also possible to measure other characteristic strain values, like for example volumetric strain, which measures the ratio of change of body's volume. The definition of volumetric strain at selected point is: In engineering mechanics, deformation is a change in shape due to an applied force. ...

vartheta = lim_{V^{(0)} to 0}{V - V^{(0)} over {V^{(0)}}}

where

vartheta is volumetric strain
V(0) is initial volume
V is final volume

For cartesian coordinate system following expression is always true: Fig. ...

vartheta = varepsilon_x + varepsilon_y + varepsilon_z

where

vartheta is volumetric strain
varepsilon_x , varepsilon_y , varepsilon_z are strains along x, y and z axis

The strain tensor

Main article: Strain tensor

Using above notation for linear and shear strain it is possible to express strain as a strain tensor: The strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation: the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms εij = 1/2 γij (i...

varepsilon_{ij} = {1 over 2} left({nabla_i u_j + nabla_j u_i}right)

using indicial notation or using vector notation: In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ... For information on vectors as a mathematical object see vector (spatial). ...

varepsilon = {1 over 2} ( vec{nabla}vec{u} + (vec{nabla}vec{u})^T)

Comparing traditional notation with tensor notation following is obtained for cartesian coordinate system: Fig. ...

varepsilon_{ij}= left[{begin{matrix} {varepsilon _x } & frac {gamma _{xy} } {2} & frac {gamma _{xz} } {2} frac {gamma _{yx} } {2} & {varepsilon _y } & frac {gamma _{yz} } {2} frac {gamma _{zx} } {2} & frac {gamma _{zy} } {2} & {varepsilon _z } end{matrix}}right]

Then volumetric strain equals:

vartheta = varepsilon_{ij}g^{ij}

where gij is a contravariant metric tensor (using tensor notation: vartheta = tr(varepsilon))


Principal strains in two dimensions

Because the strain tensor is a real symmetric matrix, by singular value decomposition it can be represented as a set of orthogonal eigenvectors, directions along which there is no shear, only stretching or compression. In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...


Assuming the two dimensional strain tensor given as:

varepsilon_{ij}= left[{begin{matrix} {varepsilon _x } & {frac {gamma _{xy}} {2}} {frac {gamma _{xy}} {2}} & {varepsilon _y } end{matrix}}right]

Then principal strains varepsilon _1, varepsilon _2 are equal to:

varepsilon _1 = frac {varepsilon _x + varepsilon _ y}{2} + sqrt{ left( frac {varepsilon _x - varepsilon _y}{2} right)^2 + left( frac{gamma _{xy}} {2}right)^2 }
varepsilon _2 = frac {varepsilon _x + varepsilon _ y}{2} - sqrt{ left( frac {varepsilon _x - varepsilon _y}{2} right)^2 + left( frac{gamma _{xy}} {2}right)^2 }

The case of large deformations

Above reasoning assumes that body is subject to small deformations. It must be rememberred that with increasing deformation the linear strain error increases. For large deformations the strain tensor can be written as:

varepsilon_{ij} = {1 over 2}({g_{ij}-g_{ij}^{(0)}})

where


gij is the metric tensor of body after deformation


gij(0) is metric tensor of the undeformed body


Engineering strain vs. true strain

In the definition of linear strain (known technically as engineering strain), strains cannot be totaled. Imagine that a body is deformed twice, first by delta ell_1 and then by delta ell_2 (cumulative deformation). The final strain

epsilon = frac{delta ell_1 + delta ell_2}{ell_0}

is slightly different from the sum of the strains:

epsilon_1 = frac{delta ell_1}{ell_0}

and

epsilon_2 = frac{delta ell_2}{ell_0 + delta ell_1}

As long as delta ell_1 ll ell_0, it is possible to write:

epsilon_2 simeq frac{delta ell_2}{ell_0}

and thus

epsilon simeq epsilon_1 ; + epsilon_2 ;

True strain (aka natural strain and logarithmic strain), however, can be totaled. This is defined by:

exp(epsilon _{T}) = frac{ell_f}{ell_0}

and thus

epsilon _{T}= ln left (frac{ell_f}{ell_0} right )

The engineering strain formula is the series expansion of the true strain formula. As the degree of the taylor series rises, it approaches the correct function. ...


See also

Stress is the internal distribution of force per unit area that balances and reacts to external loads applied to a body. ... Typical foil strain gauge. ... The strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation: the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms εij = 1/2 γij (i... A stress-strain curve is a graph derived from measuring load (stress - σ) versus extension (strain - ε) for a sample of a material. ... In the uniaxial tensile test commonly carried out to determine some properties of engineering materials, a small testpiece is stretched from an initial, undeformed length to a current, deformed length . ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ... Figure 1: Rectangular specimen subject to compression, with Poissons ratio circa 0. ...

External links

  • Strain types listed: engineering, true, logarithmic and lagrange
v • d • e
General subfields within physics

Classical mechanics · Electromagnetism · Thermodynamics · General relativity · Quantum mechanics Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time and explaining them using mathematics. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Electromagnetism is the physics of the electromagnetic field; a field encompassing all of space which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... Fig. ...


Particle physics · Thermal physics · Condensed matter physics · Atomic, molecular, and optical physics Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Thermal physics is the combined study of thermodynamics, statistical mechanics, and kinetic theory. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Atomic, molecular, and optical physics is the study of matter-matter and light-matter interactions on the scale of single atoms or structures containing a few atoms. ...

Strain can also show that the figure is under lots and lot s of stress. Often times it can stretch a material out, occasionaly to the point of breakage.


  Results from FactBites:
 
OMEGA ENGINEERING - Strain Gage Technical Data (1135 words)
Several types of strain gages depend on the proportional variance of electrical resistance to strain: the piezoresistive or semi-conductor gage, the carbon-resistive gage, the bonded metallic wire, and foil resistance gages.
The resistance of a strain gage is defined as the electrical resistance measured between the two metal ribbons or contact areas intended for the connection of measurement cables.
The strain sensitivity k of a strain gage is the proportionality factor between the relative change of the resistance.
Strain - Wikipedia, the free encyclopedia (379 words)
In any branch of science dealing with materials and their behaviour, strain is the geometrical expression of deformation caused by the action of stress on a physical body.
Strain in the Earth resulting from stresses across faults results in motion over the fault surface and a combination of brittle and ductile deformation of the surrounding rocks.
Given that strain results in the deformation of a body, it can be measured by calculating the change in length of a line or by the change in angle between two lines (where these lines are theoretical constructs within the deformed body).
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