Vector internal forces

Deformation is caused by stress from either an external force or an imbalance of internal forces. Quantitatively, a stress a on an area of a specimen is equal to the force applied per unit area. Since a force is a vector with three components, the stress component from the normal component of the force is called normal stress it causes elongation or contraction of the material depending on the direction of the force. The stress components from the two tangential components of the force are called shear stresses they are responsible for the shear deformation. 

F. Mueller-Plathe and D. Brown, Comput. Phys. Commun., 64, 7 (1991). Multi-Color Algorithms m Molecular Simulation Vectorization and Parallelization of Internal Forces and Constraints. 

If Eq. (58) is multiplied from the left by bn, then one will obtain q = (bn Vn )qn -Because of Eq. (57) bn Vn = 1, which ensures that qn and qn are the same during an internal vibration. This is of crucial importance for the calculation of internal force constants. If Vn = an, v will be properly normalized in the sense that bn an = 1 (see Eq. 31b). The term (bn Vn) in the denominator of (55) is important only when qn is not equal to qn. This is the case for c-vectors calculated with redundant sets of parameters [19]. 

Our analysis will be based on the simple configuration shown in Figure 4.2. This figure depicts a serial-link manipulator chain with no internal closed loops. The joints are arbitrary, and they are modelled using the general joint model of Chapter 2. The base member is fixed to the inertial firame. The spatial f ce vector, f, represents the vector of forces and moments applied by the tip of the chain to the environment. For an open chain, f is identically zero. For a constrained chain, however, f is unknown and in general nonzero. 

The solution process and time integration algorithm is based on identifying a common framework for the parts (finite elements, spring/damper connectors, rigid body motion, constraint equations and boundary conditions). The internal force vector F is seeked as ... 

Equation (8.10) can be expressed in a compact matrix vector form suitable for programming. The first term in (8.10) is often called the vector of internal forces, because it is derived from the internal stresses arising in the body. This vector contains the left-hand side of the equations with unknown velocities v. The second term and third term together are called the right-hand side, or vector forces external forces, with contributions from the surface tractions applied to the deformed body from the body forces distributed in the domain. In addition, to solve Eq. (8.10), the displacement boundary conditions have to be imposed at the boundary nodes. 

The derivatives here are of the vector of internal forces with respect to the unknown displacements, d. Iterations are performed by solving the system of linear equations to get displacement increments KAd = F ". Then, the vector of displace-... 

Figure 11 Analysis of reaction path vector t(s) and gradient for the reaction CH3 - - H2 —> CH4 - - H. (a) Characterization of t(.s) in terms of amplitudes A. j(f y) (see equation 38). (b) Decomposition of the gradient in terms of attractive and repulsive internal forces. Forces corresponding to the internal coordinate that dominates t(s) in a given range of the RP are indicated by thick solid lines...

A finite body is still under load when it is in statical equilibrium while subject to external forces. The conditions for equilibrium are that the vector sum of all the external forces acting on the body is zero and the sum of all the external moments acting on it is zero. These conditions must hold for any portion of the body so that the internal force system must also be in equilibrium. Any arrangement of applied forces that is in equilibrium to give zero translational and rotational accelerations will be called a set of balanced forces. 

The procedure Merge transforms the internal displacement coordinates and momenta, the coordinates and velocities of centers of masses, and directional unit vectors of the molecules back to the Cartesian coordinates and momenta. Evolve with Hr = Hr(q) means only a shift of all momenta for a corresponding impulse of force (SISM requires only one force evaluation per integration step). 

In contrast to the flexibiUty method, the stiffness method considers the displacements as unknown quantities in constmcting the overall stiffness matrix (K). The force vector T is first calculated for each load case, then equation 20 is solved for the displacement D. Thermal effects, deadweight, and support displacement loads are converted to an equivalent force vector in T. Internal pipe forces and stresses are then calculated by applying the displacement vector [D] to the individual element stiffness matrices. 

There are two possible kinds of force acting on a fluid cell internal stresses, by which an element of fluid is acted on by forces across its surface by the rest of the fluid, and external forces, such as gravity, that exert a force per unit volume on the entire volume of fluid. We define an ideal fluid to be a fluid such that for any motion of the fluid there exists a pressure p(x, t) such that if 5 is a surface in the fluid with unit normal vector n, the stress force that is exerted across S per unit area at x at time t is equal to —p x,t)h. An ideal fluid is therefore one for which the only forces are internal ones, and are orthogonal to 5 i.e. there are no tangential forces. ... 

Vectors and matrices are given as boldface symbols throughout.) E is the unit matrix G is a matrix which depends, only although not in a simple fashion, on the geometry and the atomic masses of the molecule. F is the matrix of force constants expressed in the 3 N — 6 independent internal coordinates. Its elements are ... 

Table 1.1 Conjugate pairs of variables in work terms for the fundamental equation for the internal energy U. Here/is force of elongation, Z is length in the direction of the force, <7 is surface tension, As is surface area, , is the electric potential of the phase containing species i, qi is the contribution of species i to the electric charge of a phase, E is electric field strength, p is the electric dipole moment of the system, B is magnetic field strength (magnetic flux density), and m is the magnetic moment of the system. The dots indicate scalar products of vectors.

The second approach used in first-principles tribological simulations focuses on the behavior of the sheared fluid. That is, the walls are not considered and the system is treated as bulk fluid, as discussed. These simulations are typically performed using ab initio molecular dynamics (AIMD) with DFT and plane-wave basis sets. A general tribological AIMD simulation would be run as follows. A system representing the fluid would be placed in a simulation cell repeated periodically in all three directions. Shear or load is applied to the system using schemes such as that of Parrinello and Rahman, which was discussed above. In this approach, one defines a (potentially time-dependent) reference stress tensor aref and alters the nuclear and cell dynamics, such that the internal stress tensor crsys is equal to aref. When crsys = aref, the internal and external forces on the cell vectors balance, and the system is subject to the desired shear or load. 

Here, v is the velocity vector field, p is the mass density of the fluid, D/Dt = S/Sf + V V is the material derivative, Vp is the gradient of the pressure, r[j is the shear viscosity, and F is the external force acting on the fluid volume. The right-hand side of Eq. (1) is a momentum balance between the internal pressure and viscous stress and the external forces on the fluid body. Any excess momentum contributes to the material acceleration of the fluid volume, on the left-hand side. 

The element force vector (fe) is formed by the internal heat generation term and the temperatures in the previous time steps as... 

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