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Schematic illustration of idealized fiber arrays and their corresponding unit cells. In the theory of composite materials, the representative elementary volume (REV) (also called the representative volume element (RVE) or the unit cell) is the smallest volume over which a measurement can be made that will yield a value representative of the whole. [1]
Consider the linear subspace of the n-dimensional Euclidean space R n that is spanned by a collection of linearly independent vectors , …,. To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the is the square root of the determinant of the Gramian matrix of the : (), = ….
In physics, a characteristic length is an important dimension that defines the scale of a physical system. Often, such a length is used as an input to a formula in order to predict some characteristics of the system, and it is usually required by the construction of a dimensionless quantity, in the general framework of dimensional analysis and in particular applications such as fluid mechanics.
Vectors are defined in cylindrical coordinates by (ρ, φ, z), where . ρ is the length of the vector projected onto the xy-plane,; φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π),
Diagram showing a section through the centre of a cone (1) subtending a solid angle of 1 steradian in a sphere of radius r, along with the spherical "cap" (2). The external surface area A of the cap equals r2 only if solid angle of the cone is exactly 1 steradian. Hence, in this figure θ = A/2 and r = 1.
The surface-area-to-volume ratio has physical dimension inverse length (L −1) and is therefore expressed in units of inverse metre (m −1) or its prefixed unit multiples and submultiples. As an example, a cube with sides of length 1 cm will have a surface area of 6 cm 2 and a volume of 1 cm 3. The surface to volume ratio for this cube is thus
The local (non-unit) basis vector is b 1 (notated h 1 above, with b reserved for unit vectors) and it is built on the q 1 axis which is a tangent to that coordinate line at the point P. The axis q 1 and thus the vector b 1 form an angle α {\displaystyle \alpha } with the Cartesian x axis and the Cartesian basis vector e 1 .
In general, it can be derived from the divergence theorem that the volume of a polyhedral solid is given by | |, where the sum is over faces of the polyhedron, is an arbitrary point on face , is the unit vector perpendicular to pointing outside the solid, and the multiplication dot is the dot product. [30]