Balance of angular momentum

However, if the director field is not uniform, Frank distortional stresses influence the rate of rotation of the director, and a new equation for h (or, equivalently for N) is required to replace Ericksen s equation (10-3). This is obtained from the balance of angular momentum, which gives... 

The stress tensor may be represented as a 3 x 3 matrix, with components Oy, where i and j both go from 1 to 3. The diagonal elements represent normal stresses, whereas the off-diagonal ones represent shear stresses. Positive normal stresses are tensile, while negative ones are compressive (but an opposite sign convention is sometimes used, most notably in the soil mechanics literature). Finally, from the balance of angular momentum (or torque in the static case), it follows that the stress tensor and its matrix representation are symmetric (ay = aji), meaning that only six out of the nine components are in fact independent. 

In addition to the momentum balance equation (6), one generally needs an equation that expresses conservation of mass, but no other balance laws are required for so-called purely mechanical theories, in which temperature plays no role (as mentioned, balance of angular momentum has already been included in the definition of stress). If thermal effects are included, one also needs an equation for the balance of energy (that expresses the first law of thermodynamics energy is conserved) and an entropy inequality (that follows from the second law of thermodynamics the entropy of a closed system cannot decrease). The entropy inequality is, strictly speaking, not a balance law but rather imposes restrictions on the material models. 

Therefore, the balance of moment of momentum or balance of angular momentum related to the fixed point y in aetual configuration in the inertial frame (fixed with distant stars) for (arbitrary part of) body with material volume V and its surface 9V is postulated as... 

Starting with (3.93) in any frame and tracing back the deduction we can obtain the integral form of the balance of angular momentum even in the noninertial frame as may be expected such a result will be more complicated because of the nonobjectivity of X, y, V and objectivity of forces T, b, cf. e.g. [7, 14],... 

The balance of angular momentum (by neglecting any distributed moment loads) results in the symmetry of the stress tensor, i.e.. 

For the mechanical field, the balance equations balance of mass, balance of momentum, and balance of angular momentum are given in the differential form for a fixed reference system. 

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... 

Eq. 26 is commonly referred to as the equation of motion . The balance of angular momentum, in local form, leads to the requirement that the total stress tensor be... 

Balance of angular momentum. The total moment acting on is equal to the rate of change of the angular momentum of expressed in terms of integrals over the reference configuration,... 

An immediate consequence of balance of angular momentum is that the stress tensor is symmetric, i.e. 

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... 

Notice that the constraints (4.118) are automatically satisfied. The remaining governing dynamic equations are derived from (4.152) and (4.153) for the case of a single orientational angle formulation using the above expressions for iuf, D and The balance of angular momentum (4.152) in this example is... 

We now turn to the balance of angular momentum equations (4.120). To first order in the approximations introduced above at (5.515) and (5.517), the nematic energy density (2.50) satisfies (cf. equation (4.131))... 

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