Distance or deformation tensors are unit-free second-level tensors that measure local changes in distance between material elements when a body is deformed. Changes in distance of matter elements correspond to the stretching or compression of the material lines that connect the matter elements under consideration. These changes in the internal arrangement correspond to a change in the external shape of the solid and are visible , for example, as expansion or compression .
The distance sensors are an essential parameter in the description of the kinematics of deformation and in continuum mechanics a number of different distance sensors are defined, which in turn serve to define the strain tensors . In some material models of hyperelasticity , distance sensors are used directly.
Extension of line elements
Stretching and shearing of the tangents (red and blue) on material lines (black) in the course of a deformation
In the quantitative assessment of a deformation of a body, material lines of the body before and after deformation are compared with one another. In practice, strain gauges can be stuck to the body for this purpose. The direction of the strain gage is described mathematically with a material line element in the undeformed initial configuration and in the deformed current configuration , see figure on the right. These line elements are related in a linear approximation via the deformation gradient :
The stretching of a line element in the direction
is the relationship
The stretch tensor
is called the right Cauchy-Green tensor and is therefore a measure of the stretching of line elements. The superscript " " stands for the transposition . The stretchings are extremal in the direction of the eigenvectors of . The stretching is calculated in the deformed position
The Cauchy Stretches Ensor
is therefore a spatial measure for the stretching of line elements.
Stretching of normal vectors
Stretching and shearing of the normals (red and blue) on material surfaces (gray) in the course of a deformation
The extension of normal vectors can also be determined with distance sensors . A family of surfaces can be represented by a scalar function
and an area parameter can be defined.
The normal vectors to these surfaces are the gradients
These are related to the normal in the reference configuration as follows:
The arithmetic symbol denotes the dyadic product . The stretching of the normal vectors in the deformed and undeformed position in a material point leads to the finger tensor
which is a measure for the stretching of the material surface normals. The finger tensor operates in the initial configuration.
Its counterpart in the current configuration is the left Cauchy-Green tensor
for the
can be derived.
Main invariants of the right Cauchy-Green tensor
In the event of a deformation, the material line, surface and volume elements are transformed with the deformation gradient from the initial configuration to the current configuration
The cofactor of the deformation gradient is its transposed adjoint :
It turns out that the main invariants of the Cauchy-Green tensor on the right are measures for the change in line, surface and volume elements:
The Frobenius norm is defined with the Frobenius scalar product ":" of tensors:
Physical interpretation
The connection between the right Cauchy-Green tensor and the change in the line, surface and volume elements can be seen macroscopically.
Be
the motion function of the particles of a material body. The material coordinates are taken by the particles at a certain time when the body is in its undeformed starting position. The time-dependent vector describes the spatial coordinates that the particles assume during their movement - including deformation - at time t.
Lengths of lines
If a material line is marked with the curve parameter in the undeformed initial state , the length of the line results to
Therein is the unit tensor . In the deformed position, this length changes to
The change in the length of the marked line is determined by the stretch tensor .
Areas
If, in the same way, in the undeformed initial state, a material surface is designated with the surface parameters , the content of the surface results in
In the deformed position, this area changes to
The change in the content of the marked area is determined by the cofactor of the distance sensor .
Volumes
In the undeformed initial state, a material volume is marked with the location parameters . The volume is then calculated to
In the deformed position, this volume changes to
wherein the determinant product theorem has been exploited. The change in volume can thus be expressed with the stretch tensor, as with material lines and surfaces.
Left and right extension tensor and turns
In the case of non-deformation , the distance sensors are equal to the unit tensor and are independent of any rotations of the body that may occur. The reason for this lies in the polar decomposition of the deformation gradient
which splits the deformation locally into a rotation, mediated by the orthogonal rotation tensor (with and the determinant ), and a pure stretching, mediated by the symmetrical, positive-definite right and left stretch tensor or , respectively . By multiplying the deformation gradient with its transposed one , the rotations and "reverse rotations" cancel each other out:
which of course also applies to the inverse of the right and left Cauchy-Green tensor. The right and left Cauchy-Green tensor and their inverses are therefore unaffected by rotations of the body.
Principal Axis Transformations
The right and left stretch tensor as well as the right and left Cauchy-Green tensor are therefore similar , which is why they have the same eigenvalues and therefore the same main invariants . The eigenvalues of the distance sensors are called main extensions. All distance sensors are symmetrically positive definite and therefore all three eigenvalues are positive and the three eigenvectors are mutually perpendicular (or orthogonalizable) in pairs, so that they form an orthonormal basis . Let be the eigenvectors of , the eigenvectors of and its eigenvalues. Then the major axis transformations are:
From it follows:
and further:
Derivation of the elongations
Some material models of hyperelasticity contain functions of the eigenvalues of the left distance tensor and the stresses result from the derivation of these functions according to the left Cauchy-Green tensor . It is therefore worthwhile to provide the derivative of the eigenvalues according to the stretch tensor . It turns out
Is calculated accordingly
proof
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First the eigenvalues of the left Cauchy-Green tensor are considered. The eigenvalues solve the characteristic polynomial of the left Cauchy-Green tensor:
The coefficients of this polynomial are the three main invariants of the left Cauchy-Green tensor. Implicit differentiation of the characteristic polynomial using the chain rule and the derivatives of the main invariants for symmetric tensors
,
inserting the formulas of Vieta,
yields
the main axis transformation of the left Cauchy-Green tensor
with its pairwise orthogonal eigenvectors normalized to magnitude one, which agree with the eigenvectors of the left range tensor , and the form of the unit tensor results, for example, for i = 1:
For i = 2 and i = 3, a corresponding calculation is made, which is a
fixed. The desired derivation of the eigenvalues of the left distance tensor after the left Cauchy-Green tensor is finally determined
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example
A square (black) with an inscribed circle (red) is deformed into a rectangle (blue) with an inscribed ellipse (purple)
A square with side length one is stretched to a rectangle with width and height and rotated by an angle , see the illustration on the right. Let the square be positioned at the origin of a Cartesian coordinate system, so that for the points of the square
applies. Then is in the deformed state
This calculates the deformation gradient and the distance sensors
In the middle of the square a straight line of length ½ is marked at an angle to the x-axis, see illustration. In the starting position, the points on the line then have the following coordinates:
By definition, the length of the line is independent of its direction:
In the deformed position the points have the coordinates
hence increasing the length of the line
changed. The result is again independent of the angle of rotation . If the square and the rectangle are equal, then
and the lengths of the deformed line form a curve in a polar diagram as in the figure on the right. There is .
See also
Mechanics:
Mathematics:
Individual evidence
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↑ The Fréchet derivative of a scalar function with respect to a tensor
is the tensor for which - if it exists - applies:
There is and ":" the Frobenius scalar product . Then will too
written.
literature