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Encyclopedia > Finite Deformation Tensors

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, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation: the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms εij = 1/2 γij (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. ...

Contents

Deformation gradient tensor

The position (vector) of a particle in the initial, undeformed state of a body is denoted mathbf {X} relative to some coordinate basis. The position of the same particle in the deformed state is denoted mathbf {x}. If d mathbf {X} is a line segment joining two nearby particles in the undeformed state and d mathbf {x} 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. ...

dmathbf{x} = mathbf{F} dmathbf{X}

The quantity mathbf{F} is called the deformation gradient and is given by:

mathbf{F} = nabla_X mathbf {x} =frac {partial mathbf{x}} {partial mathbf {X}}

or, in index notation: Index notation is used in mathematics to refer to the elements of matrices or the components of a vector. ...

F_{ij} = frac {partial x_i} {partial X_j}

It is assumed that mathbf{x} is a differentiable function of mathbf {X} 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. ... Look up Crack in Wiktionary, the free dictionary. ...


mathbf{F} is a second-order tensor and contains information about both the stretch and rotation of the body.


Polar Decomposition

The deformation gradient mathbf{F} 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. ...

mathbf{F} = mathbf{R}mathbf{U} = mathbf{V} mathbf{R}

where mathbf{R} is an proper orthogonal tensor, and mathbf{U} and mathbf{V} 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 mathbf{R} represents a rotation. The tensors mathbf{U} and mathbf{V} represent stretches. mathbf{U} is called the right stretch tensor. mathbf{V} is called the left stretch tensor.


The spectral decompositions of mathbf{U} and mathbf{V} are In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...

mathbf{U} = sum_{i=1..3} lambda_i mathbf{N}_i otimes mathbf{N}_i

and

mathbf{V} = sum_{i=1..3} lambda_i mathbf{n}_i otimes mathbf{n}_i

where


λi are the principal stretches, and mathbf{N}_i, mathbf{n}_i are the directions of the principal stretches (principal directions).


The principal directions are related by

mathbf{n}_i = mathbf{R} mathbf{N}_i

.


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 (mathbf{R}mathbf{R}^T=mathbf{R}^Tmathbf{R}=mathbf{1}) we can exclude the rotation by multiplying mathbf{F} 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 – May 23, 1857) was a French mathematician. ... The title page to George Greens original essay on what is now known as Greens theorem. ...

mathbf{C}=mathbf{F^T}mathbf{F}=mathbf{U}^Tmathbf{U}=mathbf{U}^2

or

C_{ij}=sum_{k=1..3}frac {partial x_k} {partial X_i} frac {partial x_k} {partial X_j}

The spectral decomposition of mathbf{C} is In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...

mathbf{C} = sum_{i=1..3} lambda_i^2 mathbf{N}_i otimes mathbf{N}_j

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:

mathbf{B}=mathbf{F}mathbf{F^T}=mathbf{V}mathbf{V^T}=mathbf{V}^2

In index notation:

B_{ij}=sum_{k=1..3}frac {partial x_i} {partial X_k} frac {partial x_j} {partial X_k}

The spectral decomposition of mathbf{B} is In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...

mathbf{B} = sum_{i=1..3} lambda_i^2 mathbf{n}_i otimes mathbf{n}_j

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 mathbf{alpha=alpha_1}. If the volume remains constant, the contraction in the other two directions is such that mathbf{alpha_1 alpha_2 alpha_3 =1} or mathbf{alpha_2=alpha_3=alpha^{-0.5}}. Then:

mathbf{F}=begin{bmatrix} alpha & 0 & 0  0 & alpha^{-0.5} & 0  0 & 0 & alpha^{-0.5} end{bmatrix}
mathbf{B}=mathbf{C}=begin{bmatrix} alpha^2 & 0 & 0  0 & alpha^{-1} & 0  0 & 0 & alpha^{-1} end{bmatrix}

Simple shear

mathbf{F}=begin{bmatrix} 1 & gamma & 0  0 & 1 & 0  0 & 0 & 1 end{bmatrix} 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...


mathbf{B}=begin{bmatrix} 1+gamma^2 & gamma & 0  gamma & 1 & 0  0 & 0 & 1 end{bmatrix}


mathbf{C}=begin{bmatrix} 1 & gamma & 0  gamma & 1+gamma^2 & 0  0 & 0 & 1 end{bmatrix}


Rigid body rotation

mathbf{F}=begin{bmatrix} cos theta & sin theta & 0  - sin theta & cos theta & 0  0 & 0 & 1 end{bmatrix}


mathbf{B}=mathbf{C}=begin{bmatrix} 1 & 0 & 0  0 & 1 & 0  0 & 0 & 1 end{bmatrix} = mathbf{1}


See also

  • Piola-Kirchhoff stress tensor, the stress tensor for finite deformations.

This article needs to be cleaned up to conform to a higher standard of quality. ...

Source

  • C. W. Macosko Rheology: principles, measurement and applications, VCH Publishers, 1994, ISBN 1-56081-579-5

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