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Vegard's law assumes that both components A and B in their pure form (i.e., before mixing) have the same crystal structure. Here, a A (1-x) B x is the lattice parameter of the solid solution, a A and a B are the lattice parameters of the pure constituents, and x is the molar fraction of B in the solid solution.
where is the volume fraction of the fibers in the composite (and is the volume fraction of the matrix).. If it is assumed that the composite material behaves as a linear-elastic material, i.e., abiding Hooke's law = for some elastic modulus of the composite and some strain of the composite , then equations 1 and 2 can be combined to give
For a (0,2) tensor, [1] twice contracting with the inverse metric tensor and contracting in different indices raises each index: =. Similarly, twice contracting with the metric tensor and contracting in different indices lowers each index:
Sometimes multiplication and division are given equal precedence, or sometimes multiplication is given higher precedence than division; see § Mixed division and multiplication below. If each subtraction is replaced with addition of the opposite (additive inverse), then the associative and commutative laws of addition allow terms to be added in ...
In Einstein notation (implicit summation over repeated index), contravariant components are denoted with upper indices as in = A covector or cotangent vector has components that co-vary with a change of basis in the corresponding (initial) vector space. That is, the components must be transformed by the same matrix as the change of basis matrix ...
The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two. Geometric interpretation
Bond's theory and index brought a greater measure of openness to the calculations for selecting the type, size, and power ratings for ore milling equipment. [ 5 ] Bond was inducted with its first class into the National Mining Hall of Fame posthumously in 1988, [ 6 ] and received a Distinguished Achievement Medal by his alma mater, the Colorado ...
The shear viscosity (or viscosity, in short) of a fluid is a material property that describes the friction between internal neighboring fluid surfaces (or sheets) flowing with different fluid velocities.