In
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, the finite strain theory—also called large strain theory, or large deformation theory—deals with
deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in
infinitesimal strain theory
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with
elastomer
An elastomer is a polymer with viscoelasticity (i.e. both viscosity and elasticity) and with weak intermolecular forces, generally low Young's modulus and high failure strain compared with other materials. The term, a portmanteau of ''elastic p ...
s,
plastically-deforming materials and other
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s and
biological soft tissue
Soft tissue is all the tissue in the body that is not hardened by the processes of ossification or calcification such as bones and teeth. Soft tissue connects, surrounds or supports internal organs and bones, and includes muscle, tendons, ligam ...
.
Displacement
The displacement of a body has two components: a
rigid-body displacement and a deformation.
* A rigid-body displacement consists of a simultaneous
translation (physics)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every ...
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).
A change in the configuration of a continuum body can be described by a
displacement field. 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. The distance between any two particles changes if and only if deformation has occurred. If displacement occurs without deformation, then it is a rigid-body displacement.
Material coordinates (Lagrangian description)
The displacement of particles indexed by variable may be expressed as follows. The vector joining the positions of a particle in the undeformed configuration
and deformed configuration
is called the
displacement vector
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
. Using
in place of
and
in place of
, both of which are vectors from the origin of the coordinate system to each respective point, we have the
Lagrangian description of the displacement vector:
where
are the orthonormal
unit vectors
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vec ...
that define the
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of the spatial (lab-frame) coordinate system.
Expressed in terms of the material coordinates, i.e.
as a function of
, the displacement field is:
where
is the displacement vector representing rigid-body translation.
The
partial derivative
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). Part ...
of the displacement vector with respect to the material coordinates yields the ''material displacement gradient tensor''
. Thus we have,
where
is the
''deformation gradient tensor''.
Spatial coordinates (Eulerian description)
In the
Eulerian description, the vector extending from a particle
in the undeformed configuration to its location in the deformed configuration is called the
displacement vector
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
:
where
are the unit vectors that define the basis of the material (body-frame) coordinate system.
Expressed in terms of spatial coordinates, i.e.
as a function of
, the displacement field is:
The partial derivative of the displacement vector with respect to the spatial coordinates yields the ''spatial displacement gradient tensor''
. Thus we have,
Relationship between the material and spatial coordinate systems
are the
direction cosines between the material and spatial coordinate systems with unit vectors
and
, respectively. Thus
The relationship between
and
is then given by
Knowing that
then
Combining the coordinate systems of deformed and undeformed configurations
It is common to superimpose the coordinate systems for the deformed and undeformed configurations, which results in
, and the direction cosines become
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
s, i.e.,
Thus in material (undeformed) coordinates, the displacement may be expressed as:
And in spatial (deformed) coordinates, the displacement may be expressed as:
Deformation gradient tensor
The deformation gradient tensor
is related to both the reference and current configuration, as seen by the unit vectors
and
, therefore it is a ''
two-point tensor
Two-point tensors, or double vectors, are tensor-like quantities which transform as Euclidean vectors with respect to each of their indices. They are used in continuum mechanics to transform between reference ("material") and present ("configurat ...
''.
Due to the assumption of continuity of
,
has the inverse
, where
is the ''spatial deformation gradient tensor''. Then, by the
implicit function theorem,
the
Jacobian determinant
must be
nonsingular
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplica ...
, i.e.
The ''material deformation gradient tensor''
is a
second-order tensor that represents the gradient of the mapping function or functional relation
, which describes the
motion of a continuum. The material deformation gradient tensor characterizes the local deformation at a material point with position vector
, i.e., deformation at neighbouring points, by transforming (
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
) a material line element emanating from that point from the reference configuration to the current or deformed configuration, assuming continuity in the mapping function
, i.e.
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
of
and time
, which implies that
cracks and voids do not open or close during the deformation. Thus we have,
Relative displacement vector
Consider a
particle or material point with position vector
in the undeformed configuration (Figure 2). After a displacement of the body, the new position of the particle indicated by
in the new configuration is given by the vector position
. The coordinate systems for the undeformed and deformed configuration can be superimposed for convenience.
Consider now a material point
neighboring
, with position vector
. In the deformed configuration this particle has a new position
given by the position vector
. Assuming that the line segments
and
joining the particles
and
in both the undeformed and deformed configuration, respectively, to be very small, then we can express them as
and
. Thus from Figure 2 we have
where
is the relative displacement vector, which represents the relative displacement of
with respect to
in the deformed configuration.
Taylor approximation
For an infinitesimal element
, and assuming continuity on the displacement field, it is possible to use a
Taylor series expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
around point
, neglecting higher-order terms, to approximate the components of the relative displacement vector for the neighboring particle
as
Thus, the previous equation
can be written as
Time-derivative of the deformation gradient
Calculations that involve the time-dependent deformation of a body often require a time derivative of the deformation gradient to be calculated. A geometrically consistent definition of such a derivative requires an excursion into
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
[A. Yavari, J.E. Marsden, and M. Ortiz]
On spatial and material covariant balance laws in elasticity
Journal of Mathematical Physics, 47, 2006, 042903; pp. 1–53. but we avoid those issues in this article.
The time derivative of
is
where
is the (material) velocity. The derivative on the right hand side represents a material velocity gradient. It is common to convert that into a spatial gradient by applying the chain rule for derivatives, i.e.,
where
is the spatial velocity gradient and where
is the spatial (Eulerian) velocity at
. If the spatial velocity gradient is constant in time, the above equation can be solved exactly to give
assuming
at
. There are several methods of computing the
exponential
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
* Exponential decay, decrease at a rate proportional to value
*Exp ...
above.
Related quantities often used in continuum mechanics are the rate of deformation tensor and the spin tensor defined, respectively, as:
The rate of deformation tensor gives the rate of stretching of line elements while the spin tensor indicates the rate of rotation or
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wit ...
of the motion.
The material time derivative of the inverse of the deformation gradient (keeping the reference configuration fixed) is often required in analyses that involve finite strains. This derivative is
The above relation can be verified by taking the material time derivative of
and noting that
.
Transformation of a surface and volume element
To transform quantities that are defined with respect to areas in a deformed configuration to those relative to areas in a reference configuration, and vice versa, we use Nanson's relation, expressed as
where
is an area of a region in the deformed configuration,
is the same area in the reference configuration, and
is the outward normal to the area element in the current configuration while
is the outward normal in the reference configuration,
is the
deformation gradient
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
, and
.
The corresponding formula for the transformation of the volume element is
Polar decomposition of the deformation gradient tensor
The deformation gradient
, like any invertible second-order tensor, can be decomposed, using the
polar decomposition
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive se ...
theorem, into a product of two second-order tensors (Truesdell and Noll, 1965): an orthogonal tensor and a positive definite symmetric tensor, i.e.,
where the tensor
is a
proper orthogonal tensor
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
, i.e.,
and
, representing a rotation; the tensor
is the ''right stretch tensor''; and
the ''left stretch tensor''. The terms ''right'' and ''left'' means that they are to the right and left of the rotation tensor
, respectively.
and
are both
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite f ...
, i.e.
and
for all non-zero
, and
symmetric tensor
In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments:
:T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_)
for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of orde ...
s, i.e.
and
, of second order.
This decomposition implies that the deformation of a line element
in the undeformed configuration onto
in the deformed configuration, i.e.,
, may be obtained either by first stretching the element by
, i.e.
, followed by a rotation
, i.e.,
; or equivalently, by applying a rigid rotation
first, i.e.,
, followed later by a stretching
, i.e.,
(See Figure 3).
Due to the orthogonality of
so that
and
have the same
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
or ''principal stretches'', but different
eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
or ''principal directions''
and
, respectively. The principal directions are related by
This polar decomposition, which is unique as
is invertible with a positive determinant, is a corrolary of the
singular-value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is relate ...
.
Deformation tensors
Several rotation-independent deformation tensors are used in mechanics. In solid mechanics, the most popular of these are the right and left Cauchy–Green deformation tensors.
Since a pure rotation should not induce any strains in a deformable body, it is often convenient to use rotation-independent measures of deformation in
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
. As a rotation followed by its inverse rotation leads to no change (
) we can exclude the rotation by multiplying
by its
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
.
The right Cauchy–Green deformation tensor
In 1839,
George Green introduced a deformation tensor known as the ''right Cauchy–Green deformation tensor'' or ''Green's deformation tensor'', defined as:
[The ]IUPAC
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
recommends that this tensor be called the Cauchy strain tensor.
Physically, the Cauchy–Green tensor gives us the square of local change in distances due to deformation, i.e.
Invariants of
are often used in the expressions for
strain energy density functions. The most commonly used
invariants are
where
is the determinant of the deformation gradient
and
are stretch ratios for the unit fibers that are initially oriented along the eigenvector directions of the right (reference) stretch tensor (these are not generally aligned with the three axis of the coordinate systems).
The Finger deformation tensor
The
IUPAC
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
recommends
[ that the inverse of the right Cauchy–Green deformation tensor (called the Cauchy tensor in that document), i. e., , be called the Finger tensor. However, that nomenclature is not universally accepted in applied mechanics.
]
The left Cauchy–Green or Finger deformation tensor
Reversing the order of multiplication in the formula for the right Green–Cauchy deformation tensor leads to the ''left Cauchy–Green deformation tensor'' which is defined as:
The left Cauchy–Green deformation tensor is often called the ''Finger deformation tensor'', named after Josef Finger (1894).[The ]IUPAC
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
recommends that this tensor be called the Green strain tensor.
Invariants of are also used in the expressions for strain energy density functions. The conventional invariants are defined as
where is the determinant of the deformation gradient.
For compressible materials, a slightly different set of invariants is used:
The Cauchy deformation tensor
Earlier in 1828, Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
introduced a deformation tensor defined as the inverse of the left Cauchy–Green deformation tensor, . This tensor has also been called the Piola tensor and the Finger tensor in the rheology and fluid dynamics literature.
Spectral representation
If there are three distinct principal stretches , the spectral decompositions of and is given by
Furthermore,
Observe that
Therefore, the uniqueness of the spectral decomposition also implies that . The left stretch () is also called the ''spatial stretch tensor'' while the right stretch () is called the ''material stretch tensor''.
The effect of acting on is to stretch the vector by and to rotate it to the new orientation , i.e.,
In a similar vein,
Examples
; Uniaxial extension of an incompressible material
: This is the case where a specimen is stretched in 1-direction with a stretch ratio of . If the volume remains constant, the contraction in the other two directions is such that or . Then:
; Simple shear
:
; Rigid body rotation
:
Derivatives of stretch
Derivatives
The derivative of a function is the rate of change of the function's output relative to its input value.
Derivative may also refer to:
In mathematics and economics
* Brzozowski derivative in the theory of formal languages
* Formal derivative, an ...
of the stretch with respect to the right Cauchy–Green deformation tensor are used to derive the stress-strain relations of many solids, particularly hyperelastic material
A hyperelastic or Green elastic materialR.W. Ogden, 1984, ''Non-Linear Elastic Deformations'', , Dover. is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density ...
s. These derivatives are
and follow from the observations that
Physical interpretation of deformation tensors
Let be a Cartesian coordinate system defined on the undeformed body and let be another system defined on the deformed body. Let a curve in the undeformed body be parametrized using