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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.
This model has the general form and the isotropic form respectively =: = +. where : is tensor contraction, is the second Piola–Kirchhoff stress, : is a fourth order stiffness tensor and is the Lagrangian Green strain given by = [() + + ()] and are the Lamé constants, and is the second order unit tensor.
It can be shown that the stress tensor is a contravariant second order tensor, which is a statement of how it transforms under a change of the coordinate system. From an x i -system to an x i ' -system, the components σ ij in the initial system are transformed into the components σ ij ' in the new system according to the tensor transformation ...
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.
Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, [2] Naghdi, [3] Simmonds, [4] Green and Zerna, [1] Basar and Weichert, [5] and Ciarlet. [6]
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
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):