Balance of linear momentum

We use the principle of virtual power to derive the balance of linear momentum and the boundary conditions for each constituent. That is,... 

The generic equations of balance are statements of truth, which is a priori self-evident and which must apply to all continuum materials regardless of their individual characteristics. Constitutive relations relate diffusive flux vectors to concentration gradients through phenomenological parameters called transport coefficients. They describe the detailed response characteristics of specific materials. There are seven generic principles (1) conservation of mass, (2) balance of linear momentum, (3) balance of ro-... 

General physical laws often state that quantities like mass, energy, and momentum are conserved. In computational mechanics, the most important of these balance laws pertains to linear momentum (when reckoned per unit volume, linear momentum may be expressed as the material density p times velocity v). The balance equation for linear momentum may be considered as a generalization of Newton s second law, which states that mass times acceleration equals total force. As we saw in the previous section, stresses in a material produce tractions, which may be considered as internal forces. In addition, external forces such as gravity may contribute to the total force. These are commonly reckoned per unit mass and are usually referred to as body forces to distinguish them from tractions, which may be considered as surface forces. For a one-dimensional motion, balance of linear momentum requires that (37,38)... 

Taking tp = me, (4.33) yields the local form of the balance of linear momentum ... 

It is important to emphasize that Q. itself is time-dependent since the material volume element of interest is undergoing deformation. Note that in our statement of the balance of linear momentum we have introduced notation for our description of the time derivative which differs from the conventional time derivative, indicating that we are evaluating the material time derivative. The material time derivative evaluates the time rate of change of a quantity for a given material particle. Explicitly, we write... 

The role of constitutive equations is to instruct us in the relation between the forces within our continuum and the deformations that attend them. More prosaically, if we examine the governing equations derived from the balance of linear momentum, it is found that we have more unknowns than we do equations to determine them. Spanning this information gap is the role played by constitutive models. From the standpoint of building effective theories of material behavior, the construction of realistic and tractable constitutive models is one of our greatest challenges. In the sections that follow we will use the example of linear elasticity as a paradigm for the description of constitutive response. Having made our initial foray into this theory, we will examine in turn some of the ideas that attend the treatment of permanent deformation where the development of microscopically motivated constitutive models is much less mature. 

Cauchy Tetrahedron and Equilibrium In this problem, use the balance of linear momentum to deduce the Cauchy stress by writing down the equations of continuum dynamics for the tetrahedral volume element shown in fig. 2.7. Assign an area AS to the face of the tetrahedron with normal n and further denote the distance from the origin to this plane along the direction why h. Show that the dynamical equations may be written as... 

Solid sub-problem. The balance of linear momentum for the solid is given by... 

If we denote by T, Cauchy s stress tensor of our material and by b, the density of body forces, then by Truesdell s third principle the balances of linear momentum and of moment of momentum for the whole mixture in local form turn out to be... 

The Balance of momentum or balance of linear momentum for an arbitrary part of a body in actual configuration and in inertial frame (fixed on distant stars) is postulated as " ... 

Traction t is here expressed through the stress tensor by (3.72). We also note that postulating (3.89) for one fixed point y the form (3.89) is valid for arbitrary but fixed point (say yo as follows from the balance of linear momentum (3.74) multiplied by constant (y - yo) a (i.e., as outer product in Rem. 16) and by summation with (3.89), of course all in our inertial frame). For this reason the origin y = o is often used in formulations of this postulate, e.g., [16], without loss of generality. 

When the flow is isothermal and the liquid has a viscosity that can be taken to be independent of pressure (ie, rj is constant over the entire flow field), substitution of equations 1 and 2 into the balance of linear momentum leads to the Navier-Stokes equations, which have been the subject of intense study in classical fluid mechanics for more than a century. 

Eq. 24 is usually referred to as the continuity equation . If s n,r,t) is the surface traction and b(r,t) the body force per unit volume, then we need to satisfy the balance of linear momentum and the balance of angular momentum. The former can be stated as... 

Equations (11) and (12) are the linear momentum balances in the flow direction. Here the total pressure head has been partitioned into partial pressure components through the use of the solid volume distribution function . Equation (13) is the balance of linear momentum for the solid phase (the fluid phase is the same but of opposite sign) in the y direction. In this equation, we have allowed for an interphase pressure effect (incorporated in the parameter E) to ensure that the mixture remains saturated at all times [8]. Equations (14) and (15) account for the angular momentum balance in the binormal direction for the two constituents. Finally, the primes imply differentiation with respect to 3 ... 

BALANCE OF LINEAR MOMENTUM. The total force acting on an arbitrary part fit of the body is equal to the rate of change of the linear momentum of Ll t expressed in terms of integrals over the reference configuration. 

An immediate consequence of balance of linear momentum is that the stress satisfies the equation of motion ... 

The divergence is again taken with respect to the second index and y is an arbitrary scalar which absorbs all contributions parallel to the director n. Noting the first of EQNS (35) this is effectively the Euler-Lagrange equation of static theory with a dynamic term, g, added. We can also rewrite the first of EQNS (S) representing balance of linear momentum by substituting expression (17) for the stress tensor, and adding an inner product of EQN (38) with Vn to obtain... 

The balance of linear momentum (5.59) integrates at once to reveal that p may take the form of an arbitrary constant, as could be expected given the set-up of this particular problem and the incompressibility constraint. The scalar Lagrange multiplier in (5.60) is evaluated in the usual way by taking the scalar product with n. This gives A = Xa(n H) so that the balance of angular momentum becomes... 

Following Clark et al [46], the fluid inertial contribution pvi = pdv/dt may be neglected because experimental sample depths are often very thin, or it may be omitted for the reasons suggested by Pieranski et al [220]. Therefore we can solve for the balance of linear momentum by setting the pressure p to be... 

It is considered that the inertia of the liquid crystal can be ignored in typical cells having small depths (cf. References [45, p.483] and [174, 220]), in which case the term on the left-hand side of equation (5.446) can be omitted. The balance of linear momentum equations then reduce to... 

Prom these quantities it is seen that Ujj = 0, because a and c have no spatial dependence. Further, the material time derivative of the velocity satisfies = 0. In the absence of any external body force Fi and generalised body forces Gf and GJ, the balance of linear momentum (6.211) consequently reduces to... 

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