In continuum mechanics, **finite deformation tensors** are used when the deformation of a body is sufficiently large to invalidate the assumptions inherent in small strain theory. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ...
For more technical Wiki articles on tensors, see the section later in this article. ...
In engineering mechanics, deformation is a change in shape due to an applied force. ...
The strain tensor, Îµ, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation: the diagonal coefficients Îµii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms Îµij = 1/2 Î³ij (i...
The term elastomer is often used interchangeably with the term rubber, and is preferred when referring to vulcanisates. ...
A subset of the phases of matter, fluids include liquids and gases, plasmas and, to some extent, plastic solids. ...
In medicine, the term soft tissue refers to tissues that connect, support, or surround other structures and organs of the body. ...
## Deformation gradient tensor
The position (vector) of a particle in the initial, undeformed state of a body is denoted relative to some coordinate basis. The position of the same particle in the deformed state is denoted . If is a line segment joining two nearby particles in the undeformed state and is the line segment joining the same two particles in the defomed state, the linear transformation between the two line segments is given by A particle is Look up Particle in Wiktionary, the free dictionary In particle physics, a basic unit of matter or energy. ...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
The quantity is called the **deformation gradient** and is given by: or, in index notation: Index notation is used in mathematics to refer to the elements of matrices or the components of a vector. ...
It is assumed that is a differentiable function of and time *t*, i.e, that cracks and voids do not open or close during the deformation. In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
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is a second-order tensor and contains information about both the stretch and rotation of the body.
### Polar Decomposition The deformation gradient can be decomposed using the polar decomposition theorem into a product of two second-order tensors: In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to polar decomposition of a nonzero complex number z where r is the absolute value of z (a positive real number), and is the complex sign of z. ...
where is an proper orthogonal tensor, and and are both positive definite symmetric tensors of second order. In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...
A tensor A, with components Aij, is said to be symmetric if Aij = Aji for all i, j. ...
The tensor represents a rotation. The tensors and represent stretches. is called the **right stretch tensor**. is called the **left stretch tensor**. The spectral decompositions of and are In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...
and where λ_{i} are the **principal stretches**, and , are the **directions** of the principal stretches (**principal directions**). The principal directions are related by .
## Rotation-Independent Deformation Measures Since a pure rotation should not induce any stresses in a deformable body, it is often convenient to use rotation-independent measures of the deformation in continuum mechanics. Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ...
As a rotation followed its inverse rotation leads to no change () we can exclude the rotation by multiplying by its transpose. In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...
### The Right Cauchy-Green deformation tensor The right Cauchy-Green deformation tensor (named after Augustin Louis Cauchy and George Green) is defined as:: Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 â€“ May 23, 1857) was a French mathematician. ...
The title page to George Greens original essay on what is now known as Greens theorem. ...
or The spectral decomposition of is In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...
Physically, the Cauchy-Green tensor gives us the square of local change in distances due to deformation.
### The Left Cauchy-Green deformation tensor Reversing the order of multiplication in the formula for the Finger tensor leads to the **left Cauchy-Green deformation tensor** which is defined as: In index notation: The spectral decomposition of is In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...
### The Finger deformation tensor The inverse of the left Cauchy-Green tensor is often called the **Finger tensor**. This tensor is named after Josef Finger (1894).
## Examples ### Uniaxial extension of an incompressible material This the case where a specimen is stretched in 1-direction with a stetch ratio of . If the volume remains constant, the contraction in the other two directions is such that or . Then: Simple shear Simple shear is a special case of deformation of a fluid where only one component of velocity vectors has a non-zero value: And the gradient of velocity is perpendicular to it: , where is the shear rate and: The deformation gradient tensor for this deformation has only one...
### Rigid body rotation
## See also - Piola-Kirchhoff stress tensor, the stress tensor for finite deformations.
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## Source - C. W. Macosko
**Rheology: principles, measurement and applications**, VCH Publishers, 1994, ISBN 1-56081-579-5 |