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The stress vectors acting on the faces of the tetrahedron are denoted as T (e 1), T (e 2), and T (e 3), and are by definition the components σ ij of the stress tensor σ. This tetrahedron is sometimes called the Cauchy tetrahedron .
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):
This way the tensor representing the scalar product, taking two vectors and resulting in a scalar has order 2 + 0 = 2, the same as the stress tensor, taking one vector and returning another 1 + 1 = 2.
In continuum mechanics, the most commonly used measure of stress is the Cauchy stress tensor, often called simply the stress tensor or "true stress". However, several alternative measures of stress can be defined: [1] [2] [3] The Kirchhoff stress (). The nominal stress ().
In tensor calculus, is classified as a second-order tensor of type (0,2) or (1,1) depending on convention. Like any linear map between vectors, the stress tensor can be represented in any chosen Cartesian coordinate system by a 3×3 matrix of real numbers.
The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor. A rank-one tensor may be defined as expressible as the outer product of the number of nonzero vectors needed to obtain the correct order. Dyadic tensor A dyadic tensor is a tensor of order two, and may be represented as a square matrix.
Traction (mechanics), adhesive friction or force; Traction vector, in mechanics, the force per unit area on a surface, including normal and shear components; Traction motor, an electric motor used for propulsion of a vehicle, for example a car or a locomotive; Railway electric traction, the use of electric motors to propel rail cars
where t = t(n) is called the surface traction, integrated over the surface of the body, in turn n denotes a unit vector normal and directed outwards to the surface S. Let the coordinate system ( x 1 , x 2 , x 3 ) be an inertial frame of reference , r be the position vector of a point particle in the continuous body with respect to the origin of ...