Invariants of tensors
Encyclopedia
In mathematics
, in the fields of multilinear algebra
and representation theory
, invariants of tensors are coefficients of the characteristic polynomial
of the tensor
A:
,
where
is the identity tensor and 
is the polynomials indeterminate (it is important to bear in mind that a polynomial's indeterminate 
may also be a non-scalar as long as power, scaling and adding make sense for it, e.g., 
is legitimate, and in fact, quite useful).
The first invariant of an n×n tensor A (
) is the coefficient for 
(coefficient for 
is always 1), the second invariant (
) is the coefficient for 
, etc., the nth invariant is the free term.
The definition of the invariants of tensors and specific notations used throughout the article were introduced into the field of Rheology
by Ronald Rivlin
and became extremely popular there. In fact even the trace
of a tensor
is usually denoted as 
in the textbooks on rheology.
The nth invariant is just
, the determinant of 
(up to sign).
The invariants do not change with rotation of the coordinate system (they are objective). Obviously, any function of the invariants only is also objective.
For this case the invariants can be calculated as:
(the sum of principal minors)
where
, 
, 
are the eigenvalues of tensor A.
Because of the Cayley–Hamilton theorem
the following equation is always true:
where E is the second-order identity tensor.
A similar equation holds for tensors of higher order.
is objective, i.e., independent from rotations of the coordinate system. Moreover, every objective tensor function depends only on the tensor's invariants. Thus, objectivity of a tensor function is fulfilled if, and only if, for some function 
we have
A common application to this is the evaluation of the potential energy as function of the strain tensor, within the framework of linear elasticity. Exhausting the above theorem the free energy of the system reduces to a function of 3 scalar parameters rather than 6. Within linear elasticity the free energy has to be quadratic in the tensor's elements, which eliminates an additional scalar. Thus, for an isotropic material only two independent parameters are needed to describe the elastic properties, known as the Lame coefficients. Consequently, experimental fits and computational efforts may be eased significantly.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the fields of multilinear algebra
Multilinear algebra
In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...
and representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, invariants of tensors are coefficients of the characteristic polynomial
Characteristic polynomial
In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
of the tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
A:
,where
is the identity tensor and is the polynomials indeterminate (it is important to bear in mind that a polynomial's indeterminate may also be a non-scalar as long as power, scaling and adding make sense for it, e.g., is legitimate, and in fact, quite useful).The first invariant of an n×n tensor A (
) is the coefficient for (coefficient for is always 1), the second invariant () is the coefficient for , etc., the nth invariant is the free term.The definition of the invariants of tensors and specific notations used throughout the article were introduced into the field of Rheology
Rheology
Rheology is the study of the flow of matter, primarily in the liquid state, but also as 'soft solids' or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applied force....
by Ronald Rivlin
Ronald Rivlin
Ronald Samuel Rivlin was a British-American physicist, mathematician, rheologist and a noted expert on rubber.-Life:Rivlin was born in London in 1915. He studied physics and mathematics at St John's College, Cambridge, being awarded a BA in 1937 and a ScD in 1952...
and became extremely popular there. In fact even the trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
of a tensor
is usually denoted as in the textbooks on rheology.Properties
The first invariant (trace) is always the sum of the diagonal components:The nth invariant is just
, the determinant of (up to sign).The invariants do not change with rotation of the coordinate system (they are objective). Obviously, any function of the invariants only is also objective.
Calculation of the invariants of symmetric 3×3 tensors
Most tensors used in engineering are symmetric 3×3.For this case the invariants can be calculated as:
(the sum of principal minors)
where
, , are the eigenvalues of tensor A.Because of the Cayley–Hamilton theorem
Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation....
the following equation is always true:
where E is the second-order identity tensor.
A similar equation holds for tensors of higher order.
Engineering application
A scalar valued tensor function f that depends merely on the three invariants of a symmetric 3×3 tensor is objective, i.e., independent from rotations of the coordinate system. Moreover, every objective tensor function depends only on the tensor's invariants. Thus, objectivity of a tensor function is fulfilled if, and only if, for some function we haveA common application to this is the evaluation of the potential energy as function of the strain tensor, within the framework of linear elasticity. Exhausting the above theorem the free energy of the system reduces to a function of 3 scalar parameters rather than 6. Within linear elasticity the free energy has to be quadratic in the tensor's elements, which eliminates an additional scalar. Thus, for an isotropic material only two independent parameters are needed to describe the elastic properties, known as the Lame coefficients. Consequently, experimental fits and computational efforts may be eased significantly.
See also
- Symmetric polynomialSymmetric polynomialIn mathematics, a symmetric polynomial is a polynomial P in n variables, such that if any of the variables are interchanged, one obtains the same polynomial...
- Elementary symmetric polynomialElementary symmetric polynomialIn mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial P can be expressed as a polynomial in elementary symmetric polynomials: P can be given by an...
- Newton's identitiesNewton's identitiesIn mathematics, Newton's identities, also known as the Newton–Girard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials...
- Invariant theoryInvariant theoryInvariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...