ABSTRACT
The governing equations of elastodynamics, be they linear or not, express facts concerning the time and spatial variations of field quantities (Equations 3.57 and 3.75):
δj(R, t) δt
= ∇ · T(R, t) + f(R, t), (4.1) δS(R, t)
δt = I+ : ∇v(R, t) + h(R, t). (4.2)
Apparently, Newton-Cauchy’s equation of motion (4.1) contains field quantities different from the deformation rate equation (4.2) requesting, in the most general form, composition operators j, S according to
δj(R, t) δt
= j [ v(R, t),T(R, t)
] , (4.3)
δS(R, t) δt
= S [ v(R, t),T(R, t)
] , (4.4)
the so-called constitutive equations (de Hoop 1995). They have to be based on physical arguments, in particular, they do not follow from the governing equations. Yet, modeling a solid should satisfy the criteria “close to reality” and “simplicity.” Due to the latter, the dependence of the operators j and S on both field quantities is usually sacrificed. We approximate
δj(R, t) δt
= j [v(R, t)] , (4.5)
δS(R, t) δt
= S [ T(R, t)
] . (4.6)
Considering (3.39), we specify
δj(R, t) δt
= ρ(R) · Dv(R, t) Dt
, (4.7)
K12611 Chapter: 4 page: 113 date: January 18, 2012
K12611 Chapter: 4 page: 114 date: January 18, 2012
δS(R, t) δt
= s(R) : DT(R, t)
Dt (4.8)
and linearize according to δ/δt =⇒ ∂/∂t, D/Dt =⇒ ∂/∂t ∂j(R, t)
∂t = ρ(R) · ∂v(R, t)
∂t , (4.9)
∂S(R, t) ∂t
= s(R) : ∂T(R, t)
∂t (4.10)
with the consequence
j(R, t) = ρ(R) · v(R, t), (4.11) S(R, t) = s(R) : T(R, t). (4.12)
The constitutive equations (4.7) and (4.8) and the linear constitutive equations (4.11) and (4.12), define a second rank mass density tensor ρ(R) and the forth rank compliance tensor s(R). Both tensors characterize a time in-
variant instantaneously reacting inhomogeneous locally reacting anisotropic material: time invariant, because they do not explicitly depend on time, and instantaneously reacting, because j(R, t) and S(R, t) depend on v(R, t) and T(R, t), respectively, only at the same time t. In a similar sense, the material (4.11) and (4.12) is spatially invariant (inhomogeneous) and locally reacting: inhomogeneous, because ρ(R) and s(R) depend on the vector of position R
and locally reacting, because j(R, t) and S(R, t) at point R depend on v(R, t) and T(R, t), respectively, only at the same point. The material is anisotropic because the variation of one (cartesian) component of v and T yields variations of all other components of j and S, that is to say, the relative orientation, for example, of v and j, depends on the direction of v: The material exhibits a macroscopic inner structure.
