Stress dyadic

The fundamental quantities in elasticity are second-order tensors, or dyadicx the deformation is represented by the strain thudte. and the internal forces are represented by Ihe stress dyadic. The physical constitution of the defurmuble body determines ihe relation between the strain dyadic and the stress dyadic, which relation is. in the infinitesimal theory, assumed lo be linear and homogeneous. While for anisotropic bodies this relation may involve as much as 21 independent constants, in the euse of isotropic bodies, the number of elastic constants is reduced lo two. 

Calculations have thus far been performed for the three standard cubic arrays, namely simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fee). As a result of this geometric symmetry, the couple N and particle stress dyadic A are given by the configuration-specific relations... 

Equation (3) for the stress dyadic, applied to the translational motion, gives... 

In this equation, q represents a conductive heat flux, E the total energy per unit mass including internal, kinetic, and gravitational potential energy contributions, and T is the viscous stress dyadic. The reference states for the various internal energies are chosen in snch a way that energy effects of phase changes and chemical reactions are acconnted for in this formulation. 

This equation shows that the stress contribution tensor is essentially a dyadic product of the end-to-end vector r and the statistical force /, which is exerted by the chain on the considered end-point. The angular brackets indicate the averaging with the aid of the mentioned distribution function. Eq. (2.25) can be explained as follows Factor rt in the brackets gives the probability that the mentioned statistical force actually contributes to the stress. This factor gives the projection of the end-to-end vector of the chain on the normal of the considered sectional plane. If a unit area plane is considered, as is usual in stress-analysis, the said projection gives that part of the unit of volume, from which molecules possessing just this projection, actually contribute to the stress on the sectional plane. 

The stress tensor is a symmetric dyadic and its mathematical properties were reviewed at the end of Chapter 5. It has the dimensions of pressure, force/unit area, or, equivalently, of an energy density. The quantum stress tensor plays a dominant role in the description of the mechanical properties of an atom in a molecule and in the local mechanics of the charge density. 

The derivatives with respect to x sample the off-diagonal behaviour of F > and generate terms related to the current density j and the quantum stress tensor er. The first-order term is proportional to the current density, and this vector field is the x complement of the gradient vector field Vp. The second-order term is proportional to the stress tensor. Considered as a real symmetric matrix, its eigenvalues and eigenvectors will characterize the critical points in the vector field J and its trace determines the kinetic energy densities jK(r) and G(r). The cross-term in the expansion is a dyadic whose trace is the divergence of the current density. 

Here the divergence of the stress tensor is the net force per unit volume acting on a fluid element. Note that is not a simple divergence because is a dyadic and not a vector, although the term is interpretable physically as a rate of momentum change. 

The static component of the stress tensor [8, 9] is defined as the dyadic product of the force f acting at contact c with the corresponding branch vector, where every contact contributes with ist force and its branch vector, if the particle lies in the averaging volume... 

Here, the stress tensor is denoted by a, body forces (per unit volume) by F, and forces on constituent i (per unit volume) solely due to interactions with constituent j by /. represents the dyadic cross product. Summing the momentum balance equation over the components in the direction normal to the flow, the y-direction as in Figure 10.20, assuming normal accelerations are negligible and gradients are zero in the x- and z-directions and that the only body force is gravity, yields... 

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