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)
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, 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,
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 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.
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 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 but we avoid those issues in this article.
The time derivative of is
where is the velocity. The derivative on the right hand side represents a material velocity gradient. It is common to convert that into a spatial gradient, i.e.,
where is the spatial velocity gradient. If the spatial velocity gradient is constant, 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 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, and .
The corresponding formula for the transformation of the volume element is
Polar decomposition of the deformation gradient tensor
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 , 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
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 tensor
In 1839, George Green introduced a deformation tensor known as the right Cauchy–Green deformation tensor or Green's deformation tensor, defined as:
Physically, the Cauchy–Green tensor gives us the square of local change in distances due to deformation, i.e.
where 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 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).
where is the determinant of the deformation gradient.
For incompressible materials, a slightly different set of invariants is used:
The Cauchy deformation tensor
Earlier in 1828,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 and the Finger tensor in the rheology and fluid dynamics literature.
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,
Derivatives of stretch
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 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 . 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
which indicates that changes in length are characterized by .
Finite strain tensors
The concept of strain is used to evaluate how much a given displacement differs locally from a rigid body displacement. 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
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
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:
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 .
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 tensor
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
Deformation tensors in convected curvilinear coordinates
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
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 coordinates
The problem of compatibility in continuum mechanics involves the determination of allowable single-valued continuous fields on bodies. These allowable conditions leave the body without unphysical gaps or overlaps after a deformation. Most such conditions apply to simply-connected bodies. Additional conditions are required for the internal boundaries of multiply connected bodies.
Compatibility of the deformation gradient
The necessary and sufficient conditions for the existence of a compatible field over a simply connected body are
Compatibility of the right Cauchy–Green deformation tensor
The necessary and sufficient conditions for the existence of a compatible field over a simply connected body are
We can show these are the mixed components of the Riemann–Christoffel curvature tensor. Therefore, the necessary conditions for -compatibility are that the Riemann–Christoffel curvature of the deformation is zero.
Compatibility of the left Cauchy–Green deformation tensor
No general sufficiency conditions are known for the left Cauchy–Green deformation tensor in three-dimensions. Compatibility conditions for two-dimensional fields have been found by Janet Blume.
^Belytschko, Ted; Liu, Wing Kam; Moran, Brian (2000). Nonlinear Finite Elements for Continua and Structures (reprint with corrections, 2006 ed.). John Wiley & Sons Ltd. pp. 92–94. ISBN978-0-471-98773-4.
^T.C. Doyle and J.L. Eriksen (1956). "Non-linear elasticity." Advances in Applied Mechanics 4, 53–115.
^Z.P. Bažant and L. Cedolin (1991). Stability of Structures. Elastic, Inelastic, Fracture and Damage Theories. Oxford Univ. Press, New York (2nd ed. Dover Publ., New York 2003; 3rd ed., World Scientific 2010).
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