Continuum mechanics  

Laws


Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration.^{[1]} A configuration is a set containing the positions of all particles of the body.
A deformation may be caused by external loads,^{[2]} body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc.
Strain is a description of deformation in terms of relative displacement of particles in the body that excludes rigidbody motions. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body 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 dislocation mechanisms 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.
Strain is a 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 x = F(X) where X 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
where I is the identity tensor. Hence strains are dimensionless and are usually expressed as a decimal fraction, a percentage or in partsper notation. Strains measure how much a given deformation differs locally from a rigidbody 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 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.
Depending on the amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories:
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.
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 and negative if they are compressed. Thus, we have
where e is the engineering normal strain, L is the original length of the fiber and l is the final length of the fiber. Measures of strain are often expressed in parts per million or microstrains.
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.
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.
The logarithmic strain ε, also called, true strain or Hencky strain ^{[7]}. 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 the 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]}
The Green strain is defined as:
The EulerAlmansi strain is defined as
Strains are classified as either normal or shear. A normal strain is perpendicular to the face of an element, and a shear strain is parallel to it. These definitions are consistent with those of normal stress and shear stress.
For an isotropic material that obeys Hooke's law, a normal stress will cause a normal strain. Normal strains produce dilations.
Consider a twodimensional, infinitesimal, rectangular material element with dimensions dx × dy, 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 xdirection of the rectangular element is defined by
Similarly, the normal strain in the y and zdirections becomes
Shear strain  

Common symbols  γ or ε 
SI unit  1, or radian 
Derivations from other quantities  γ = τ/G 
The engineering shear strain (γ_{xy}) is defined as the change in angle between lines AC and AB. Therefore,
From the geometry of the figure, we have
For small displacement gradients we have
For small rotations, i.e. α and β are ≪ 1 we have tan α ≈ α, tan β ≈ β. Therefore,
thus
By interchanging x and y and u_{x} and u_{y}, it can be shown that γ_{xy} = γ_{yx}.
Similarly, for the yz and xzplanes, we have
The tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition, γ, as
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 fulfil 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.
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 none of the curves changes length, it is said that a rigid body displacement 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 be one the body actually will ever occupy. Often, the configuration at t = 0 is considered the reference configuration, κ_{0}(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 X_{i} 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 x_{i} 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:
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.^{[8]}
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 timelike 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 nonaffine or inhomogeneous if F = F(X,t) or c = c(X,t).
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,
A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigidbody displacement and a deformation. A rigidbody displacement consists 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 κ_{0}(B) to a current or deformed configuration κ_{t}(B) (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 rigidbody 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) = u_{i}e_{i} in the Lagrangian description, or U(x,t) = U_{J}E_{J} 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 α_{Ji} are the direction cosines between the material and spatial coordinate systems with unit vectors E_{J} and e_{i}, respectively. Thus
and the relationship between u_{i} and U_{J} 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
The partial differentiation of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor ∇_{X}U. Thus we have:
or 
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 ∇_{x}U. Thus we have,
or 
Homogeneous (or affine) deformations are useful in elucidating the behavior of materials. Some homogeneous deformations of interest are
Plane deformations are also of interest, particularly in the experimental context.
A plane deformation, also called plane strain, is one where the deformation is restricted to one of the planes in the reference configuration. If the deformation is restricted to the plane described by the basis vectors e_{1}, e_{2}, the deformation gradient has the form
In matrix form,
From the polar decomposition theorem, the deformation gradient, up to a change of coordinates, can be decomposed into a stretch and a rotation. Since all the deformation is in a plane, we can write^{[8]}
where θ is the angle of rotation and λ_{1}, λ_{2} are the principal stretches.
If the deformation is isochoric (volume preserving) then det(F) = 1 and we have
Alternatively,
A simple shear deformation is defined as an isochoric plane deformation in which there is a set of line elements with a given reference orientation that do not change length and orientation during the deformation.^{[8]}
If e_{1} is the fixed reference orientation in which line elements do not deform during the deformation then λ_{1} = 1 and F·e_{1} = e_{1}. Therefore,
Since the deformation is isochoric,
Define
Then, the deformation gradient in simple shear can be expressed as
Now,
Since
we can also write the deformation gradient as