Microcontinuum field theories: foundations and solids by A. Cemal Eringen

By A. Cemal Eringen

Microcontinuum box theories represent an extension of classical box theories --- of elastic solids, viscous fluids, electromagnetism, and so forth --- to microscopic size and time scales. fabric our bodies are seen as collections of a giant variety of deformable debris (sub-continua), appropriate for modeling blood, porous media, polymers, liquid crystals, slurries, and composite fabrics. This quantity is anxious with the basics and the dynamics of microcontinua. It offers an intensive dialogue of the kinematics of microcontinua, pressure, tension tensors, stability legislation, thermodynamics, and constitutive equations for solids and fluids topic to electromagnetic fields and mechanical so much. The dialogue contains boundary- and initial-value difficulties and provides variational and specialty theorems. options are given for plenty of utilized difficulties related to micropolar, microstretch and micromorphic our bodies, and electromagnetic solids.

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Since vorticity is a vector, it is independent of the coordinate frame used. Our definition agrees with the usual “right-hand-rule” sign convention of angular rotation. Vorticity also can be represented as a second-order tensor. 3) vy v − z z y note that × v = 2 zy i + xz j + yx k . Thus, the vector and the second-order tensor contain the same information. 3) only changes the sign, the vorticity (second-order) tensor is said to be skew-symmetric. 11 31 Vorticity and Circulation Flows with vorticity are said to be rotational flows; flows without vorticity are said to be irrotational flows.

Looking first at the time rate of change of lengths, it is seen that after a time interval t the rate of change of length along the x-axis per unit length, which will be denoted by dxx , is the final length minus the original length divided by the original length, all divided by t. 3) dzz = lim t→0 z t z The dxx dyy , and dzz are the normal rates of deformation and can loosely be thought of as rates of normal, or extensional, strain. The term “loosely” is used, since the definitions of strain you might be familiar with from the study of solid mechanics are for infinitesimal strains.

The last one, the requirement that the stress-rate of deformation equation be linear in the rate of deformation, can be broadened to include quadratic terms in the rate of deformation. 1) is an additional viscosity coefficient. Note that its dimensions differ from the where standard viscosity by an additional unit of time. This constitutive law describes what are called Stokesian fluids, after George Stokes, or sometimes Reiner-Rivlin fluids, after Markus Reiner of Israel and Ronald Rivlin of the United States.

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