Velocity Strain or Deformation
Rate
The strain rate is derived from the spatial velocity derivative:
(4)
or in matrix form:
(5)
Where, the velocity gradient in the current configuration is:
(6)
The velocity of a material particle is:
(7)
Where, the partial differentiation with respect to time
means the rate of change of the spatial position
of a given particle. The velocity difference between two particles
in the current configuration is given by:
(8)
In matrix form:
(9)
On the other hand, it is possible to write the velocity difference directly
as:
(10)
One has as a result:
(12)
Now,
is composed of a rate of deformation and a rate of rotation or
spin:
(13)
Since these are rate quantities, the spin can be treated as a vector. It is thus possible to
decompose
into a symmetric strain rate matrix and an anti symmetric rotation
rate matrix just as in the small motion theory the infinitesimal displacement gradient is
decomposed into an infinitesimal strain and an infinitesimal rotation. The symmetric part of the
decomposition is the strain rate or the rate of deformation and is:
(14)
The anti symmetric part of the decomposition is the spin matrix:
(15)
The velocity-strain measures the current rate of deformation, but it gives no information
about the total deformation of the continuum. In general,
式 13 is not integral
analytically; except in the unidimensional case, where one obtains the true
strain:
(16)
and
are respectively the dimensions in the deformed and initial
configurations. Furthermore, the integral in time for a material point does not yield a
well-defined, path-independent tensor so that information about phenomena such as total
stretching is not available in an algorithm that employs only the strain velocity. Therefore, to
obtain a measure of total deformation, the strain velocity has to be transformed to some other
strain rate.
The volumetric strain is calculated from density. For one dimensional
deformation:
(17)
Green Strain Tensor
The square of the distance which separates two points in the final configuration is given in
matrix form by:
(18)
Subtracting the square or the initial distance, you have:
(19)
(20)
and
are called respectively right and left Cauchy-Green tensor.
Using
式 8 in
Vicinity Transformation:
(21)
(22)
In the unidimensional case, the value of the strain is:
(23)
Where,
and
are respectively the dimensions in the deformed and initial
configurations.
It can be shown that any motion
can always be represented as a pure rigid body rotation followed by
a pure stretch of three orthogonal directions:
(24)
with the rotation matrix
satisfying the orthogonality condition:
(25)
and
symmetric.
The polar decomposition theorem is important because it will enable to distinguish the
straining part of the motion from the rigid body rotation.
From
式 20 and
式 24:
(26)
(27)
式 26 allows the computation of
, and 式 27 of
.
As the decomposition of the Jacobian matrix
exists and is unique,
is a new measure of strain which is sometimes called the
Jaumann strain. Jaumann strain requires the calculation of principal
directions.
If rotations are small,
(28)
(29)
(30)
if second order terms are neglected.
As a result, for the Jacobian matrix:
(31)
leading, if the second order terms are neglected, to the classical linear
relationships:
(32)
(33)
(34)
So for 式 32 and 式 33, when rigid body
rotations are large, the linear strain tensor becomes non-zero even in the absence of
deformation.
The preceding developments show that the linear strain measure is appropriate if rotations can
be neglected; that means if they are of the same magnitude as the strains and if these are of
the order of 10-2 or less. It is also worth noticing that linear strains can be used
for moderately large strains of the order of 10-1 provided that the rotations are
small. On the other hand, for slender structures which are quite in extensible, nonlinear
kinematics must be used even when the rotations are order of 10-2 because, if you are
interested in strains of 10-3 - 10-4, using linear strain the error due to
the rotations would be greater than the error due to the strains.
Large deformation problems in which nonlinear kinematics is necessary, are those in which
rigid body rotation and deformation are large.