Acceleration wave

In the mechanics of continuous media, the behaviour of a material body is described by a number of field variables which are required to satisfy a set of governing partial differential equations arising from balance laws, and from kinematic and constitutive considerations. The variables are generally assumed to have the requisite degree of smoothness consistent with the governing equations, except possibly on surfaces in the body across which some of the variables may suffer jump discontinuities (cf. also Smooth function).

Suppose that a material body occupies a region $ \Omega $ in $ \mathbf R ^ {d} $ at time $ t = 0 $, and at some later time $ t $ occupies, in its deformed state, a region $ \Omega _ {t} $. The motion of the body is described by the function $ \phi : \Omega \rightarrow {\Omega _ {t} } $, $ x = \phi ( X,t ) $, in which $ X $ denotes the position at time $ t = 0 $ of a material particle, and $ x $ its position at time $ t $. The function $ \phi $ is assumed to be invertible, and both $ \phi $ and its inverse are assumed to be continuously differentiable with respect to the spatial and temporal variables on which they depend, except possibly on specified surfaces in the body. A propagating smooth surface divides the body $ \Omega _ {0} $ or $ \Omega _ {t} $ into two regions, forming a common boundary between them. The unit normal $ n $ to the surface $ S ( t ) $ is considered to be in the direction in which $ S ( t ) $ propagates. The region ahead of the surface is denoted by $ \Omega _ {0} ^ {+} $ and the region behind the surface is denoted by $ \Omega _ {0} ^ {-} $. Let $ f ( x,t ) $ be an arbitrary scalar-, vector- or tensor-valued function which is continuous in both $ \Omega _ {0} ^ {+} $ and $ \Omega _ {0} ^ {-} $. This function has definite limits $ f ^ {+} $ and $ f ^ {-} $ at a point on $ S ( t ) $, as the point is approached from $ \Omega _ {0} ^ {+} $ and $ \Omega _ {0} ^ {-} $. The jump of $ f $ at $ X \in S ( t ) $ is defined by

$$ \tag{a1 } [ f ( X ) ] \equiv f ^ {+} ( X ) - f ^ {-} ( X ) . $$

The surface $ S ( t ) $ is called a singular surface with respect to $ f $ at time $ t $ if $ [ f ] \neq 0 $. A singular surface that has a non-zero normal velocity called a wave. An acceleration wave is a propagating singular surface across which the motion $ \phi $, velocity $ {\dot \phi } ( X,t ) $ and (hence) the deformation gradient $ F \equiv { \mathop{\rm grad} } \phi $, are continuous; however, quantities involving second-order derivatives of the motion, such as the acceleration $ a $ and the time rate of deformation gradient $ {\dot{F} } $, are discontinuous. Various kinematical and geometrical conditions of compatibility involving the variables $ a $ and $ {\dot{F} } $, the normal $ n $ to the surface, and the speed $ U $ of the surface, may be derived with the aid of Hadamard's lemma (see, for example, [a2], [a4]). These considerations lead to the propagation condition

$$ \tag{a2 } [ {\dot{T} } ] n = - \rho U [ a ] , $$

which is a statement of balance of linear momentum across the surface; $ T $ is the Cauchy stress, $ \rho $ is the mass density, and $ U $ is the intrinsic speed of the surface. To make further progress it is necessary to introduce information about the constitution of the material; in the case of an elastic material, for example, (a2) becomes the eigenvalue problem

$$ \tag{a3 } Q s = \rho U ^ {2} s, $$

in which $ s $ is referred to as the amplitude vector of the acceleration jump, and $ Q $ is the acoustic tensor. This leads to the Fresnel–Hadamard theorem: The amplitude $ s $ of an acceleration wave travelling in the direction $ n $ must be an eigenvector of the acoustic tensor $ Q $; the corresponding eigenvalue is $ \rho U ^ {2} $. It follows that, for real wave speeds to exist, $ Q $ must possess at least one real and positive eigenvalue. The acoustic tensor is symmetric, and consequently its eigenvalues are real, if and only if the material is hyperelastic. In addition, $ Q $ possesses three positive eigenvalues if and only if it is positive definite; in the context of elasticity, positive definiteness of $ Q $ implies that the material is strongly elliptic. Further information on acceleration waves may be found in the references cited.

References

[a1]	A.C. Eringen, E.S. Suhubi, "Elastodynamics" , I , Acad. Press (1975)
[a2]	J. Hadamard, "Leçons sur la propagation des ondes et les équations de l'hydrodynamique" , Dunod (1903)
[a3]	M.F. McCarthy, "Singular surfaces and waves" A.C. Eringen (ed.) , Continuum Physics II: Continuum Mechanics of Single Surface Bodies , Acad. Press (1975)
[a4]	C.-C. Wang, C. Truesdell, "Introduction to rational elasticity" , Noordhoff (1973)