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应变strain


应变在力学中定义为一微小材料元素承受应力时所产生的单位长度变形量。因此是一个无量纲的物理量。公式记为




其中是应变,是材料元素的长度,是承受应力的变形量。

在直杆模型中,除了长度方向由长度改变量除以原长而得“线形变”,另外还定义了压缩时以截面边长(或直径)改变量除以原边长(或直径)而得的“横向应变”。对大多数材料,横向应变的绝对值约为线应变的绝对值的三分之一至四分之一。二者之比的绝对值称作“泊松系数”。


应力概念 物体由于外因(受力、湿度、温度场变化等)而变形时,在物体内各部分之间产生相互作用的内力,以抵抗这种外因的作用,并力图使物体从变形后的位置回复到变形前的位置。在所考察的截面某一点单位面积上的内力称为应力。同截面垂直的称为正应力或法向应力,同截面相切的称为剪应力或切应力。应力会随着外力的增加而增长,对于某一种材料,应力的增长是有限度的,超过这一限度,材料就要破坏。对某种材料来说,应力可能达到的这个限度称为该种材料的极限应力。极限应力值要通过材料的力学试验来测定。将测定的极限应力作适当降低,规定出材料能安全工作的应力最大值,这就是许用应力。材料要想安全使用,在使用时其内的应力应低于它的极限应力,否则材料就会在使用时发生破坏。 工程构件,大多数情形下,内力并非均匀分布,通常“ 破坏”或“ 失效”往往从内力集度最大处开始,因此,有必要区别并定义应力概念。


有些材料在工作时,其所受的外力不随时间而变化,这时其内部的应力大小不变,称为静应力;还有一些材料,其所受的外力随时间呈周期性变化,这时内部的应力也随时间呈周期性变化,称为交变应力。材料在交变应力作用下发生的破坏称为疲劳破坏。通常材料承受的交变应力远小于其静载下的强度极限时,破坏就可能发生。另外材料会由于截面尺寸改变而引起应力的局部增大,这种现象称为应力集中。对于组织均匀的脆性材料,应力集中将大大降低构件的强度,这在构件的设计时应特别注意。


应变概念

机械零件和构件等物体内任一点(单元体)因外力作用引起的形状和尺寸的相对改变。与点的正应力和切应力(见应力)相对应,应变分为线应变和角应变。受力零件和构件上的每一点都可取一个微小的正六面体,称为单元体。单元体任一边的线长度的相对改变称为线应变或正应变;单元体任意两边所夹直角的改变称为角应变或切应变,以弧度来度量。线应变和角应变是度量零件内一点处变形程度的两个几何量。零件变形后,单元体体积的改变与原单元体体积之比,称为体积应变。线应变、角应变和体积应变都是无量纲的量。当单元体各个面上的切应力都等于零,而只有正应力作用时,称该单元体为主单元体,它的各个面称为主平面,各主平面交线的方向称为主方向。沿主方向的线应变称为主应变。当外力卸除后,物体内部产生的应变能够全部恢复到原来状态的,称为弹性应变;如只能部分地恢复到原来状态,其残留下来的那一部分称为塑性应变。


A 线应变[1]

  在直角坐标中所取单元体为正六面体时,三条相互垂直的棱边的长度在变形前后的改变量与原长之比,定义为线应变,用ε表示。一点在x、y、z方向的线应变分别为εx、εy、εz。线应变以伸长为正,缩短为负。


  B 切应变

  单元体的两条相互垂直的棱边,在变形后的直角改变量,定义为角应变或切应变,用γ表示。一点在x-y方向、y-z方向、z-x方向的切应变,分别为γxy、γyz、γzx。切应变以直角减少为正,反之为负。


  C 一点的应变状态

  一点的应变分量εx、εy、εz、γxy、γyz、γzx已知时,在该点处任意方向的线应变,以及通过该点任意两线段间的直角改变量,都可根据应变分量的坐标变换公式求出。该点的应变状态也就确定。

  表示一点应变状态的9个应变分量εx、εy、εz、γxy、γyx、γyz、γzy、γzx、γxz组成的应变张量,即

  式中 右边的张量中的切应变用εxy、εxz、---表示,适用于使用张量的附标标号的表示法;

  左边张量中的切应变用γxy、γxz、---表示,是工程习惯表示法。

  二者概念相同,大小相差一倍。应变张量也是二阶对称量,其中切应变分量εxy=εyx,...。

http://en.wikipedia.org/wiki/Strain_(materials_science)#Strain

Deformation (mechanics)

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It has been suggested that Deformation (engineering) be merged into this article or section. (Discuss) Proposed since September 2008.

Deformation in continuum mechanics is the transformation of a body from a referenceconfiguration to a current configuration.[1] A configuration is a set containing the positions of all particles of the body. Contrary to the common definition of deformation, which implies distortion or change in shape, the continuum mechanics definition includes rigid body motions where shape changes do not take place ([1] footnote 4, p. 48).

The cause of a deformation is not pertinent to the definition of the term. However, it is usually assumed that a deformation is caused by external loads,[2] body forces (such as gravity or electromagnetic forces), or temperature changes within the body.

Strain is a description of deformation in terms of relative displacement of particles in the body.

Different equivalent choices may be made for the expression of a strain field depending on whether it is defined in the initial or in the final placement and on whether the metric tensor or its dual is considered.

In a continuous body, a deformation field results from a stress field induced by applied forces or is due to changes in the temperature field inside the body. The relation between stresses and induced strains is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials. Deformations which are recovered after the stress field has been removed are called elastic deformations. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations remain even after stresses have been removed. One type of irreversible deformation is plastic deformation, which occurs in material bodies after stresses have attained a certain threshold value known as the elastic limit or yield stress, and are the result of slip, or dislocationmechanisms at the atomic level. Another type of irreversible deformation is viscous deformation, which is the irreversible part of viscoelastic deformation.

In the case of elastic deformations, the response function linking strain to the deforming stress is the compliance tensor of the material.

Contents[hide]· 1 Straino 1.1 Strain measures§ 1.1.1 Engineering strain§ 1.1.2 Stretch ratio§ 1.1.3 True strain§ 1.1.4 Green strain§ 1.1.5 Almansi straino 1.2 Normal straino 1.3 Shear straino 1.4 Metric tensor· 2 Description of deformationo 2.1 Affine deformationo 2.2 Rigid body motion· 3 Displacemento 3.1 Displacement gradient tensor· 4 Examples of deformationso 4.1 Plane deformation§ 4.1.1 Isochoric plane deformation§ 4.1.2 Simple shear· 5 See also· 6 References· 7 Further reading  [show]Laws [show]Solid mechanics [show]Fluid mechanics [show]Rheology [show]Scientists v · d · e·

[edit] Strain

A strain is a normalized measure of deformation representing the displacement between particles in the body relative to a reference length.

A general deformation of a body can be expressed in the form where is the reference position of material points in the body. Such a measure does not distinguish between rigid body motions (translations and rotations) and changes in shape (and size) of the body. A deformation has units of length.

We could, for example, define strain to be




.

Hence strains are dimensionless and are usually expressed as a decimal fraction, a percentage or in parts-per notation. Strains measure how much a given deformation differs locally from a rigid-body deformation.[3]

A strain is in general a tensor quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along a material line elements or fibers is the normal strain, and the amount of distortion associated with the sliding of plane layers over each other is the shear strain, within a deforming body.[4] This could be applied by elongation, shortening, or volume changes, or angular distortion.[5]

The state of strain at a material point of a continuum body is defined as the totality of all the changes in length of material lines or fibers, the normal strain, which pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other, the shear strain, radiating from this point. However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions.

If there is an increase in length of the material line, the normal strain is called tensile strain, otherwise, if there is reduction or compression in the length of the material line, it is called compressive strain.


[edit] Strain measures

Depending on the amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories:

· Finite strain theory, also called large strain theory, large deformation theory, deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and deformed configurations of the continuum are significantly different and a clear distinction has to be made between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue.

· Infinitesimal strain theory, also called small strain theory, small deformation theory, small displacement theory, or small displacement-gradient theory where strains and rotations are both small. In this case, the undeformed and deformed configurations of the body can be assumed identical. The infinitesimal strain theory is used in the analysis of deformations of materials exhibiting elastic behavior, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel.

· Large-displacement or large-rotation theory, which assumes small strains but large rotations and displacements.

In each of these theories the strain is then defined differently. The engineering strain is the most common definition applied to materials used in mechanical and structural engineering, which are subjected to very small deformations. On the other hand, for some materials, e.g. elastomers and polymers, subjected to large deformations, the engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%,[6] thus other more complex definitions of strain are required, such as stretch, logarithmic strain, Green strain, and Almansi strain.


[edit] Engineering strain

The Cauchy strain or engineering strain is expressed as the ratio of total deformation to the initial dimension of the material body in which the forces are being applied. The engineering normal strain or engineering extensional strain or nominal strain e of a material line element or fiber axially loaded is expressed as the change in length ΔL per unit of the original length L of the line element or fibers. The normal strain is positive if the material fibers are stretched or negative if they are compressed. Thus, we have

where is the engineering normal strain, L is the original length of the fiber and is the final length of the fiber.

The true shear strain is defined as the change in the angle (in radians) between two material line elements initially perpendicular to each other in the undeformed or initial configuration. The engineering shear strain is defined as the tangent of that angle, and is equal to the length of deformation at its maximum divided by the perpendicular length in the plane of force application which sometimes makes it easier to calculate.


[edit] Stretch ratio

The stretch ratio or extension ratio is a measure of the extensional or normal strain of a differential line element, which can be defined at either the undeformed configuration or the deformed configuration. It is defined as the ratio between the final length ℓ and the initial length L of the material line.

The extension ratio is approximately related to the engineering strain by

This equation implies that the normal strain is zero, so that there is no deformation when the stretch is equal to unity.

The stretch ratio is used in the analysis of materials that exhibit large deformations, such as elastomers, which can sustain stretch ratios of 3 or 4 before they fail. On the other hand, traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios.


[edit]

True strain

The logarithmic strain ε, also called natural strain, true strain or Hencky strain. Considering an incremental strain (Ludwik)

the logarithmic strain is obtained by integrating this incremental strain:

where e is the engineering strain. The logarithmic strain provides he correct measure of the final strain when deformation takes place in a series of increments, taking into account the influence of the strain path.[4]

[edit] Green strain

Main article: Finite strain theory

The Green strain is defined as:

[edit] Almansi strain

Main article: Finite strain theory

The Euler-Almansi strain is defined as

[edit] Normal strain

Two-dimensional geometric deformation of an infinitesimal material element.

As with stresses, strains may also be classified as 'normal strain' and 'shear strain' (i.e. acting perpendicular to or along the face of an element respectively). For an isotropic material that obeys Hooke's law, a normal stress will cause a normal strain. Normal strains produce dilations.

Consider a two-dimensional infinitesimal rectangular material element with dimensions , which after deformation, takes the form of a rhombus. From the geometry of the adjacent figure we have

and

For very small displacement gradients the squares of the derivatives are negligible and we have

The normal strain in the -direction of the rectangular element is defined by

Similarly, the normal strain in the -direction, and -direction, becomes

[edit] Shear strain

The engineering shear strain is defined as (γxy) is the change in angle between lines                 and                   . Therefore,

From the geometry of the figure, we have

For small displacement gradients we have

For small rotations, i.e.    and   are     we         have                    . Therefore,

thus

By interchanging                     and                       and                   and             , it can be shown that

Similarly, for the                             and                           planes, we have

The tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition, ,                as

[edit] Metric tensor

A strain field associated with a displacement is defined, at any point, by the change in length of the tangent vectors representing the speeds of arbitrarily parametrized curves passing through that point.

A basic geometric result, due to Fréchet, von Neumann and Jordan, states that, if the lengths of the tangent vectors fulfill the axioms of a norm and the parallelogram law, then the length of a vector is the square root of the value of the quadratic form associated, by the polarization formula, with a positive definite bilinear map called the metric tensor.


[edit]Description of deformation

Deformation is the change in the metric properties of a continuous body, meaning that a curve drawn in the initial body placement changes its length when displaced to a curve in the final placement. If all the curves do not change length, it is said that a rigid bodydisplacement occurred.

It is convenient to identify a reference configuration or initial geometric state of the continuum body which all subsequent configurations are referenced from. The reference configuration need not to be one the body actually will ever occupy. Often, the configuration at t = 0 is considered the reference configuration, κ0(B). The configuration at the current time t is the current configuration.

For deformation analysis, the reference configuration is identified as undeformed configuration, and the current configuration as deformed configuration. Additionally, time is not considered when analyzing deformation, thus the sequence of configurations between the undeformed and deformed configurations are of no interest.

The components Xi of the position vector X of a particle in the reference configuration, taken with respect to the reference coordinate system, are called the material or reference coordinates. On the other hand, the components xi of the position vector x of a particle in the deformed configuration, taken with respect to the spatial coordinate system of reference, are called the spatial coordinates

There are two methods for analysing the deformation of a continuum. One description is made in terms of the material or referential coordinates, called material description or Lagrangian description. A second description is of deformation is made in terms of the spatial coordinates it is called the spatial description or Eulerian description.

There is continuity during deformation of a continuum body in the sense that:

· The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.

· The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.


[edit]Affine deformation

A deformation is called an affine deformation, if it can be described by an affine transformation. Such a transformation is composed of a linear transformation (such as rotation, shear, extension and compression) and a rigid body translation. Affine deformations are also called homogeneous deformations.[7]

Therefore an affine deformation has the form

where X is the position of a point in the deformed configuration, X is the position in a reference configuration, t is a time-like parameter, F is the linear transformer and C is the translation. In matrix form, where the components are with respect to an orthonormal basis,

The above deformation becomes non-affine or inhomogeneous if                                                   or

[edit] Rigid body motion

A rigid body motion is a special affine deformation that does not involve any shear, extension or compression. The transformation matrix F is proper orthogonal in order to allow rotations but no reflections.

A rigid body motion can be described by

where

In matrix form,

[edit] Displacement

Figure 1. Motion of a continuum body.

A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement consist of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration to a current or deformed configuration (Figure 1).

If after a displacement of the continuum there is a relative displacement between particles, a deformation has occurred. On the other hand, if after displacement of the continuum the relative displacement between particles in the current configuration is zero, then there is no deformation and a rigid-body displacement is said to have occurred.

The vector joining the positions of a particle P in the undeformed configuration and deformed configuration is called the displacement vector u(X,t) = uiei in the Lagrangian description, or U(x,t) = UJEJ in the Eulerian description.

A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field, In general, the displacement field is expressed in terms of the material coordinates as

or in terms of the spatial coordinates as

where αJi are the direction cosines between the material and spatial coordinate systems with unit vectors             and             , respectively. Thus

and the relationship between ui and UJ is then given by

Knowing that

then

It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in b = 0, and the direction cosines become Kronecker deltas:

Thus, we have

or in terms of the spatial coordinates as

[edit] Displacement gradient tensor

The partial differentiation of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor                           . Thus we have:

where F is the deformation gradient tensor.

Similarly, the partial differentiation of the displacement vector with respect to the spatial coordinates yields the spatial displacement gradient tensor .

Thus we have,

[edit] Examples of deformations

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