Torque the vector cross product, p is the particle's linear momentum, and r is the displacement vector from the origin (the origin is assumed to be a fixed location anywhere in space). The time-derivative of this is: This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definition of force formula_17 (whether or not mass is constant) and the definition of velocity formula_18 The cross product of momentum formula_20 with its associated velocity formula_21 is zero because velocity and momentum are parallel, so the second term vanishes. By definition, torque τ
Linear elasticity fourth-order stiffness tensor, formula_8 is the body force per unit volume, formula_9 is the mass density, formula_10 represents the nabla operator, formula_11 represents a transpose, formula_12 represents the second derivative with respect to time, and formula_13 is the inner product of two second-order tensors (summation over repeated indices is implied). Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are: An elastostatic boundary value problem for an isotropic-homogeneous media is a system of 15 independent equations and equal number of unknowns (3 equilibrium equations, 6 strain-displacement equations, and 6 constitutive