Search results
Results from the WOW.Com Content Network
A scalar function that depends entirely on the principal invariants of a tensor is objective, i.e., independent of rotations of the coordinate system. This property is commonly used in formulating closed-form expressions for the strain energy density, or Helmholtz free energy, of a nonlinear material possessing isotropic symmetry.
In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product [a] returns a pseudovector.
For example, an element of the tensor product space V ⊗ W is a second-order "tensor" in this more general sense, [29] and an order-d tensor may likewise be defined as an element of a tensor product of d different vector spaces. [30] A type (n, m) tensor, in the sense defined previously, is also a tensor of order n + m in this more
The order parameter is the tensor, which is symmetric, traceless, second-order tensor and vanishes in the isotropic liquid phase. We shall consider a uniaxial Q {\displaystyle \mathbf {Q} } tensor, which is defined by
A second-order fluid is a fluid where the stress tensor is the sum of all tensors that can be formed from the velocity field with up to two derivatives, much as a Newtonian fluid is formed from derivatives up to first order.
A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b.Analogous to vectors, it can be written as a linear combination of the tensor basis e x ⊗ e x ≡ e xx, e x ⊗ e y ≡ e xy, ..., e z ⊗ e z ≡ e zz (the right-hand side of each identity is only an abbreviation, nothing more):
where the coefficients χ (n) are the n-th-order susceptibilities of the medium, and the presence of such a term is generally referred to as an n-th-order nonlinearity. In isotropic media () is zero for even n, and is a scalar for odd n. In general, χ (n) is an (n + 1)-th-rank tensor. It is natural to perform the same expansion for the non ...
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.