Stress tensor force expression

Taking into consideration the forces that cause the deformation of a given material, one invokes the relationships for the stresses developed in that solid or liquid material. Following the generalizations of the strain and strain rate to express such quantities in terms of tensorial expressions, the stress tensor is expressed as in Equation... 

In equation [1.1] and [1.2], the stress tensor is expressed in a Cartesian coordinate system (Q, x, y, z). In this coordinate system, the single-column matrices define the normal vector h and force dF. We only have to multiply matrix [Z] by h to calculate the force. 

The shear stress Is uniform throughout the main liquid slab for Couette flow ( ). Therefore, two Independent methods for the calculation of the shear stress are available It can be calculated either from the y component of the force exerted by the particles of the liquid slab upon each reservoir or from the volume average of the shear stress developed Inside the liquid slab from the Irving-Kirkwood formula (JA). For reasons explained In Reference (5) the simpler version of this formula can be used In both our systems although this version does not apply In general to structured systems. The Irvlng-Klrkwood expression for the xy component of the stress tensor used In our simulation Is... 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

In this section, we use the Cartesian force of Section VI to derive several equivalent expressions for the stress tensor of a constrained system of pointlike particles in a flow field with a macroscopic velocity gradient Vv. The excess stress of any system of interacting beads (i.e., point centers of hydrodynamic resistance) in a Newtonian solvent, beyond the Newtonian contribution that would be present at the applied deformation rate in the absence of the beads, is given by the Kramers-Kirkwood expression [1,4,18]... 

Working in cylindrical coordinates, and substituting the force-per-unit-volume expressions that stem from the stress tensor (Eqs. 2.137, 2.138, and 2.139), the Navier-Stokes equations can be written as... 

In Chapter 2 considerable effort is devoted to establishing the relationship between the stress tensor and the strain-rate tensor. The normal and shear stresses that act on the surfaces of a fluid particle are found to depend on the velocity field in a definite, but relatively complex, manner (Eqs. 2.140 and 2.180). Therefore, when these expressions for the forces are substituted into the momentum equation, Eq. 3.53, an equation emerges that has velocities (and pressure) as the dependent variables. This is a very important result. If the forces were not explicit functions of the velocity field, then more dependent variables would likely be needed and a larger, more complex system of equations would emerge. In terms of the velocity field, the Navier-Stokes equations are stated as... 

We notice that stress tensors are not a priori symmetric for (16) and that c)J. symmetric tensors. Further, the 3rd order microstress tensor Ss is normally related to boundary micro tractions, even if, in some cases, it could express weakly non-local internal effects % is interpreted as an externally controlled pore pressure (s includes interactive forces between the gross and fine structures. 

For the momentum conservation of a single-phase fluid, the momentum per unit volume / is equal to the mass flux pU. The momentum flux is thus expressed by the stress tensor i/r = (pi — t). Here p is the static pressure or equilibrium pressure / is a unit tensor and r is the shear stress tensor. Since <1> = —pf where / is the field force per unit mass, Eq. (5.12) gives rise to the momentum equation as... 

Turning back to Eq. 2.5-6, the surface forces Fs can now be expressed in terms of the total stress tensor n as follows ... 

Furthermore, it is convenient to switch to normal co-ordinates (1.13). We can use the expressions for forces (2.26) to rewrite the expression for the stress tensor in normal co-ordinates... 

To obtain the expression for the stress tensor for the set of Brownian particles suspended in a viscoelastic liquid, we use equation (6.7), in which the elastic and internal viscosity forces are specified in Section 3.2... 

Considering the entangled systems in the coarse-grained approximation, we forget about segments the theory contains the effective elastic forces between the fictious adjacent particles, and the stress tensor (equation (6.7)) can be expressed through the variables and u"k in the form... 

When the elastic force and the force of internal viscosity are defined, at N = 1, the expression for the stress tensor directly follows relation (6.7)... 

For a three-dimensional body, discussions of elastic responses in the framework of Hooke s law become more complicated. One defines a 3 x 3 stress tensor P [12], which is the force (with emits of newtons) expressed in a Cartesian coordinate system ... 

Once the structure of a complex fluid has been simulated, computed, or derived by analytic theory, one would like to calculate the stress tensor a and compare it to experimental stress measurements. The appropriate expression for the stress tensor depends on the type of complex fluid. However, if the idealized microstructure is built out of many small, point-like elements located at positions x,, i = 1,2,..., N, and on each such point a nonhydrodynamic force F, is exerted by the rest of the microstructure, then a can be obtained from the general Kirkwood (1949) formula (Doi and Edwards 1986) ... 

To obtain the polymer contribution to the stress tensor for a fluid containing a large number Nc of springs, we define the position of one end of the spring i to be r, j and define the position of the other end to be r,-,2- We then define R, = r,2 — r, i. We note that the spring exerts a force —F. on point 2 and exerts a force F- on point 1. Now we use the Kirkwood Tbrmula for the stress tensor, Eq. (1-42). The summation must be carried out over the locations of the ends of all different springs. This summation can be expressed as... 

With the aim of relating the force per unit area at a point to the components of the stress tensor at that point, let us consider (3) the tetrahedron of Figure 4.4, in which a force per unit area, /, is applied to the oblique surface AS. The other surfaces of the tetrahedron, ASi, ASj, and AS, respectively perpendicular to the Xi, X2, and X3 coordinate axes, can be obtained from AS from the expressions. 

This expression indicates that the force / is a Hnear combination of the components of the stress tensor. The inertial term has been neglected in this equation because the height of the tetrahedron is infinitesimal. 

When a solid elastic body is under the action of an infinitesimal contact force the strain tensor is related to the stress tensor by the expression... 

Equation (8.175) is a generalization of Ehrenfest s theorem (Ehrenfest 1927). This theorem relates the forces acting on a subsystem or atom in a molecule to the forces exerted on its surface and to the time derivative of the momentum density mJ(r). It constitutes the quantum analogue of Newton s equation of motion in classical mechanics expressed in terms of a vector current density and a stress tensor, both defined in real space. 

The term T (Q) is the virial of the forces exerted on the surface of the subsystem, a term expressible in terms of the stress tensor previously defined in eqns (8.173) and (6.12),... 

The expression —gwxw y (SI units N/m2) is an averaged momentum flow per unit area, and so comparable to a shear stress A force in the direction of the y-axis acts at a surface perpendicular to the a -axis. Terms of the general form —gwf-wij are called Reynolds stresses or turbulent stresses. They are symmetrical tensors. In a corresponding manner, the energy equation (3.135), contains a turbulent heat flux of the form... 

The total stress tensor, T, is thus interpreted physically as the surface forces per surface unit acting through the infinitesimal surface on the surrounding fluid with normal unit vector n directed out of the CV. This means that the total stress tensor by definition acts on the surrounding fluid. The counteracting force on the fluid element (CV) is therefore expressed in terms of the total stress tensor by introducing a minus sign in (1.65). 

The structure of the expression for totai is that of a bilinear form it consists of a sum of products of two factors. One of these factors in each term is a flow quantity (heat flux q, mass diffusion flux jc, momentum flux expressed by the viscous stress tensor o, and chemical reaction rate rr)- The other factor in each term is related to a gradient of an intensive state variable (gradients of temperature, chemical potential and velocity) and may contain the external force gc or a difference of thermodynamic state variables, viz. the chemical affinity A. These quantities which multiply the fluxes in the expression for the entropy production are called thermodynamic forces or affinities. 

Alternatively, we may derive a force expression for Txz following the derivation presented in Appendix E.3.2.2 for the stress tensor component r. It follows if we combine Eqs. (E.62) and (E.63) with Eq. (E.49) from which we obtain... 

Regarding the use of Eq. (6.26) in practice we note that the same comments made earlier apply here as well [see discussion after Eq. (6.17)]. A detailed discussion of optimal choices for the Ewald parameters a and for dipolar systems can bo found in Rc fs. 243 and 244. Finally, readers who are interested in performing MD simulations of dipolar fluids are referred to Appendix F.2.2 where we present explicit expressions for forces and torques associated with the three-dimensional Ewald sum [see Eq. (6.26)]. Moreover, explicit expressions for various components of the stress tensor can be found in Appendix F.2.3. 

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