Finite strain theory

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 strain theory. 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 elastomers, plastically-deforming materials and other fluids and biological soft tissue.

DisplacementEdit

 
Figure 1. Motion of a continuum body.

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) 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)Edit

The displacement of particles indexed by variable i 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. 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 that define the basis 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 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)Edit

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:

 
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 systemsEdit

  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 configurationsEdit

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

 
Figure 2. Deformation of a continuum body.

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.

Due to the assumption of continuity of  ,   has the inverse  , where   is the spatial deformation gradient tensor. Then, by the implicit function theorem,[1] the Jacobian determinant   must be nonsingular, 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) 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 of   and time  , which implies that cracks and voids do not open or close during the deformation. Thus we have,

 

Relative displacement vectorEdit

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 approximationEdit

For an infinitesimal element  , and assuming continuity on the displacement field, it is possible to use a Taylor series expansion 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 gradientEdit

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[2] 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 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 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 elementEdit

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, and  .

The corresponding formula for the transformation of the volume element is

 
Derivation of Nanson's relation (see also [3])

To see how this formula is derived, we start with the oriented area elements in the reference and current configurations:

 
The reference and current volumes of an element are
 
where  .

Therefore,

 
or,
 
so,
 
So we get
 
or,
 
Q.E.D.

Polar decomposition of the deformation gradient tensorEdit

 
Figure 3. Representation of the polar decomposition of the deformation gradient

The deformation gradient  , like any invertible second-order tensor, can be decomposed, using the polar decomposition 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, 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, i.e.   and   for all non-zero  , and symmetric tensors, 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 or principal stretches, but different eigenvectors 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.

Deformation tensorsEdit

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. As a rotation followed by its inverse rotation leads to no change ( ) we can exclude the rotation by multiplying   by its transpose.

The right Cauchy–Green deformation tensorEdit

In 1839, George Green introduced a deformation tensor known as the right Cauchy–Green deformation tensor or Green's deformation tensor, defined as:[4][5]

 

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 tensorEdit

The IUPAC recommends[5] 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 tensorEdit

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).[5][6][7]

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 tensorEdit

Earlier in 1828,[8] Augustin-Louis Cauchy introduced a deformation tensor defined as the inverse of the left Cauchy–Green deformation tensor,  . This tensor has also been called the Piola tensor[5] and the Finger tensor[9] in the rheology and fluid dynamics literature.

 

Spectral representationEdit

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,
 

ExamplesEdit

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 stretchEdit

Derivatives 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 materials. These derivatives are

 
and follow from the observations that
 

Physical interpretation of deformation tensorsEdit

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  . Its image in the deformed body is  .

The undeformed length of the curve is given by

 
After deformation, the length becomes
 
Note that the right Cauchy–Green deformation tensor is defined as
 
Hence,
 
which indicates that changes in length are characterized by  .

Finite strain tensorsEdit

The concept of strain is used to evaluate how much a given displacement differs locally from a rigid body displacement.[1][10][11] One of such strains for large deformations is the Lagrangian finite strain tensor, also called the Green-Lagrangian strain tensor or Green – St-Venant strain tensor, defined as

 

or as a function of the displacement gradient tensor

 
or
 

The Green-Lagrangian strain tensor is a measure of how much   differs from  .

The Eulerian-Almansi finite strain tensor, referenced to the deformed configuration, i.e. Eulerian description, is defined as

 

or as a function of the displacement gradients we have

 
Derivation of the Lagrangian and Eulerian finite strain tensors

A measure of deformation is the difference between the squares of the differential line element  , in the undeformed configuration, and  , in the deformed configuration (Figure 2). Deformation has occurred if the difference is non zero, otherwise a rigid-body displacement has occurred. Thus we have,

 

In the Lagrangian description, using the material coordinates as the frame of reference, the linear transformation between the differential lines is

 

Then we have,

 

where   are the components of the right Cauchy–Green deformation tensor,  . Then, replacing this equation into the first equation we have,

 
or
 
where  , are the components of a second-order tensor called the Green – St-Venant strain tensor or the Lagrangian finite strain tensor,
 

In the Eulerian description, using the spatial coordinates as the frame of reference, the linear transformation between the differential lines is

 
where   are the components of the spatial deformation gradient tensor,  . Thus we have

 
where the second order tensor   is called Cauchy's deformation tensor,  . Then we have,

 
or
 

where  , are the components of a second-order tensor called the Eulerian-Almansi finite strain tensor,

 

Both the Lagrangian and Eulerian finite strain tensors can be conveniently expressed in terms of the displacement gradient tensor. For the Lagrangian strain tensor, first we differentiate the displacement vector   with respect to the material coordinates   to obtain the material displacement gradient tensor,  

 

Replacing this equation into the expression for the Lagrangian finite strain tensor we have

 
or
 

Similarly, the Eulerian-Almansi finite strain tensor can be expressed as

 

Seth–Hill family of generalized strain tensorsEdit

B. R. Seth from the Indian Institute of Technology Kharagpur was the first to show that the Green and Almansi strain tensors are special cases of a more general strain measure.[12][13] The idea was further expanded upon by Rodney Hill in 1968.[14] The Seth–Hill family of strain measures (also called Doyle-Ericksen tensors)[15] can be expressed as

 

For different values of   we have:

  • Green-Lagrangian strain tensor
     
  • Biot strain tensor
     
  • Logarithmic strain, Natural strain, True strain, or Hencky strain
     
  • Almansi strain
     

The second-order approximation of these tensors is

 
where   is the infinitesimal strain tensor.

Many other different definitions of tensors   are admissible, provided that they all satisfy the conditions that:[16]

  •   vanishes for all rigid-body motions
  • the dependence of   on the displacement gradient tensor   is continuous, continuously differentiable and monotonic
  • it is also desired that   reduces to the infinitesimal strain tensor   as the norm  

An example is the set of tensors

 
which do not belong to the Seth–Hill class, but have the same 2nd-order approximation as the Seth–Hill measures at   for any value of  .[17]

Stretch ratioEdit

The stretch 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.

The stretch ratio for the differential element   (Figure) in the direction of the unit vector   at the material point  , in the undeformed configuration, is defined as

 
where   is the deformed magnitude of the differential element  .

Similarly, the stretch ratio for the differential element   (Figure), in the direction of the unit vector   at the material point  , in the deformed configuration, is defined as

 

The normal strain   in any direction   can be expressed as a function of the stretch ratio,

 

This equation implies that the normal strain is zero, i.e. no deformation, when the stretch is equal to unity. Some materials, such as elastometers can sustain stretch ratios of 3 or 4 before they fail, whereas traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios, perhaps of the order of 1.1 (reference?)

Physical interpretation of the finite strain tensorEdit

The diagonal components   of the Lagrangian finite strain tensor are related to the normal strain, e.g.

 

where   is the normal strain or engineering strain in the direction  .

The off-diagonal components   of the Lagrangian finite strain tensor are related to shear strain, e.g.

 

where   is the change in the angle between two line elements that were originally perpendicular with directions   and  , respectively.

Under certain circumstances, i.e. small displacements and small displacement rates, the components of the Lagrangian finite strain tensor may be approximated by the components of the infinitesimal strain tensor

Derivation of the physical interpretation of the Lagrangian and Eulerian finite strain tensors

The stretch ratio for the differential element   (Figure) in the direction of the unit vector   at the material point  , in the undeformed configuration, is defined as

 

where   is the deformed magnitude of the differential element  .

Similarly, the stretch ratio for the differential element   (Figure), in the direction of the unit vector   at the material point  , in the deformed configuration, is defined as

 

The square of the stretch ratio is defined as

 

Knowing that

 
we have
 
where   and   are unit vectors.

The normal strain or engineering strain   in any direction   can be expressed as a function of the stretch ratio,

 

Thus, the normal strain in the direction   at the material point   may be expressed in terms of the stretch ratio as

 

solving for   we have

 

The shear strain, or change in angle between two line elements   and   initially perpendicular, and oriented in the principal directions   and  , respectively, can also be expressed as a function of the stretch ratio. From the dot product between the deformed lines   and   we have

 

where   is the angle between the lines   and   in the deformed configuration. Defining   as the shear strain or reduction in the angle between two line elements that were originally perpendicular, we have

 
thus,
 
then
 

or

 

Deformation tensors in convected curvilinear coordinatesEdit

A representation of deformation tensors in curvilinear coordinates is useful for many problems in continuum mechanics such as nonlinear shell theories and large plastic deformations. Let   denote the function by which a position vector in space is constructed from coordinates  . The coordinates are said to be "convected" if they correspond to a one-to-one mapping to and from Lagrangian particles in a continuum body. If the coordinate grid is "painted" on the body in its initial configuration, then this grid will deform and flow with the motion of material to remain painted on the same material particles in the deformed configuration so that grid lines intersect at the same material particle in either configuration. The tangent vector to the deformed coordinate grid line curve   at   is given by

 
The three tangent vectors at   form a local basis. These vectors are related the reciprocal basis vectors by
 

Let us define a second-order tensor field   (also called the metric tensor) with components

 
The Christoffel symbols of the first kind can be expressed as
 

To see how the Christoffel symbols are related to the Right Cauchy–Green deformation tensor let us similarly define two bases, the already mentioned one that is tangent to deformed grid lines and another that is tangent to the undeformed grid lines. Namely,

 

The deformation gradient in curvilinear coordinatesEdit

Using the definition of the gradient of a vector field in curvilinear coordinates, the deformation gradient can be written as

 

The right Cauchy–Green tensor in curvilinear coordinatesEdit

The right Cauchy–Green deformation tensor is given by

 
If we express   in terms of components with respect to the basis { } we have
 
Therefore,
 

and the corresponding Christoffel symbol of the first kind may be written in the following form.

 

Some relations between deformation measures and Christoffel symbolsEdit

Consider a one-to-one mapping from   to   and let us assume that there exist two positive-definite, symmetric second-order tensor fields   and   that satisfy

 
Then,
 
Noting that
 
and   we have
 
Define
 
Hence
 
Define
 
Then
 
Define the Christoffel symbols of the second kind as
 
Then
 
Therefore,
 
The invertibility of the mapping implies that