Victor A. Eremeyev
Abstract Within the unified approach to modelling of media with microstructure we discuss the propagation of acceleration waves. We describe a medium with microstructure as an elastic continuum with strain energy density which depends on deformations and additional internal variable and their first gradients. We use a
Nth-order tensor as a kinematical descriptor of the microstructure. By acceleration
wave we mean an isolated surface propagating in medium across which second deriv- atives of some fields undergo discontinuity jump. Here we formulate the conditions of existence of acceleration waves as algebraic inequality expressed using acoustic tensor.
Keywords Acceleration waves
⋅
Media with microstructure⋅
Acoustic tensor⋅
Micropolar medium1
Introduction
Among many types of nonlinear waves observed in solids and fluids where the ana- lytical solutions are rare, the acceleration waves are exceptional since their condi- tions of propagation can be reduced to algebraic equations. An acceleration wave called also wave of weak discontinuity of order two is a solution of motion equa- tions with discontinuities in the second derivatives on some surfaces that are called singular. It means that the acceleration wave can be represented by an isolated trav- eling smooth enough surface which is a carrier of discontinuity jumps of the second derivatives with respect to the spacial coordinates and time whereas the solution and
V.A. Eremeyev (
✉
)Institute of Mathematics, Mechanics and Computer Science,
Southern Federal University, Milchakova Street 8a, Rostov-on-Don 344090, Russia e-mail: [email protected]
V.A. Eremeyev
The Faculty of Mechanical Engineering, Rzeszów University of Technology, al. Powstańców Warszawy 8, 35–959 Rzeszów, Poland
© Springer Nature Singapore Pte Ltd. 2017
M.A. Sumbatyan (ed.), Wave Dynamics and Composite Mechanics
for Microstructured Materials and Metamaterials,
Advanced Structured Materials 59, DOI 10.1007/978-981-10-3797-9_7
its first derivatives are continuous. Existence conditions of acceleration waves can be reduced to a spectral problem for an acoustic tensor and positivity of its eigen- values. From the mathematical point of view the conditions of existence of accel- eration waves coincide with the condition of strong ellipticity of the equilibrium equations. Ellipticity is a natural property of the equilibrium equations in the case of infinitesimal deformations. On the other hand the violation ellipticity for nonlinear media means that for certain deformations discontinuities may appear. Such discon- tinuous solutions may model such phenomena as shear-bands, phase transitions, slip surfaces, etc. Thus, analysis of conditions of propagation of acceleration waves plays an important role in the mechanics of materials.
Within the nonlinear elasticity including compressible, incompressible and media with constraints, acceleration waves are studied in many works, see, e.g., the original papers by [2,6,7,19,33,34,42–44], see also [21,49,50] where the generalization to thermoelasticity and viscoelasticity is also presented.
For the media with microstructure acceleration waves are considered in a number of papers. In particular, the propagation of acceleration waves is studied in porous media [3, 8–10, 22, 26, 46], in random materials [35, 36], in piezoelectric solids [29]. Acceleration waves are also studied in various types of fluids with complex constitutive equations, see [41,45,47] and the reference therein.
In nonlinear elastic micropolar media acceleration wave are studied in [23]. In [28] a generalization of these results in the case of is presented in elastic and vis- coelastic micropolar media are given. Equivalence of existence of acceleration waves and the condition of strong ellipticity of the equilibrium equations is discussed in [12]. For micropolar thermoelasticity acceleration waves studied in [1,14]. In order to describe the strain localizations in micropolar elastoplasticity the derivation of acoustic tensor and analysis of its properties is performed in [11].
The paper is organized ad follows. First we introduce the motion equations for a medium with microstructure. Then, using Maxwell’s theorem we obtain the condi- tions of propagations of acceleration waves. As an example the acceleration waves in the micropolar medium is considered.
In what follows we use the direct tensor notations [24]. In particular, all vectors and tensors are denoted by semibold Roman font.
2
Basic Equations of a Hyperelastic Media with Internal
Variables
Deformation of a non-linear elastic solid is described by the mapping from known state called initial configuration into another state called actual configuration. The mapping is given by
Here vector𝐱 describes the position of a material point in the actual configura- tion at instant t, whereas𝐗 determines the position of the same material point in the reference configuration, and𝐮 is the displacement vector. In addition to deformation 𝐱 for a medium with microstructure we introduce internal variable 𝐖 which can be scalar, vector, second-order tensor, or even Nth-order tensor. Among examples of such internal variables are porosity in the theory of poroelasticity [4, 46], micro- rotation tensor in micropolar mechanics [13, 16], microdeformations in the micro- morphic media [16–18], damage tensor [5,25, 27]. As a result, the deformation of a solid with microstructure is determined by two fields
𝐱 ≡ 𝐱(𝐗, t), 𝐖 = 𝐖(𝐗, t). (2)
Following the equipresence principle [50] for an hyperelastic medium we intro- duce the strain energy density
W= W(∇𝐱, 𝐖, ∇𝐖; 𝐗). (3)
Here ∇ is the gradient (nabla) operator in Lagrangian coordinates [24]. The spe- cific form of functional dependence (3) depends on the nature of microstructural tensor𝐖. For brevity we omit here also discussion on the objectivity of 𝐖 and form of W consistent with the principle material frame indifference. We assume W to be a twice continuously differentiable function. We use the following notations:
W, ∇𝐱 = 𝜕W 𝜕∇𝐱, W, 𝐖= 𝜕 W 𝜕𝐖, W, ∇𝐱∇𝐱 = 𝜕 2W 𝜕∇𝐱𝜕∇𝐱, W, ∇𝐖∇𝐖= 𝜕 2W 𝜕∇𝐖𝜕∇𝐖.
In addition to (3) we define the kinetic energy density as a positive quadratic form depending on velocity𝐯 ≡ ̇𝐱 and ̇𝐖
K= 1
2 𝜌𝐯 ⋅ 𝐯+ 𝐯 ⋅ 𝐉1∶ ̇𝐖 + ̇𝐖 ∶ 𝐉2∶ ̇𝐖. (4) Here the overdot stands for the derivative with respect to time t,⋅ denotes scalar (inner) product, whereas ∶ denotes the full product in the space of tensors of arbitrary order.𝜌 is the density in the reference configuration, 𝐉1and𝐉2are N + 1th-order and
2Nth-order tensors of microinertia, respectively. The simplest form of K is
K= 1
2 𝜌𝐯 ⋅ 𝐯+ j ̇𝐖 ∶ ̇𝐖, (5)
where j≥ 0 is a scalar measure of microinertia. Considering the principle of least action in the form
𝛿H = 0, H [𝐱, 𝐖] = t2 ∫ t1 ∫ V (K − W)dVdt, (6)
where V is the volume occupied by the medium in the reference configuration,𝛿 is the variation symbol, t1and t2are two instants, we derive motion equations for the
medium with energies (3) and (5) in the following form ∇ ⋅ 𝐏 = 𝜌 ̇𝐯, ∇ ⋅ 𝐆 − 𝜕W
𝜕𝐖= j ̈𝐖, (7)
where we introduced the Lagrangian stress measures of Piola-Kirchhoff type
𝐏 = 𝜕𝜕∇𝐱W , 𝐆 = 𝜕𝜕∇𝐖W . (8)
Let us note that here we neglect any volume loading.
3
Acceleration Waves
We consider such deformations of the medium when discontinuities of considered fields appear at a smooth surface S(t) called singular, see Fig.1. We assume existence of unilateral limit values at S(t) for all considered quantities. We denote a jump of any quantity across S(t) by the double squared brackets, for example,[[𝐟]] = 𝐟+− 𝐟−.
Let us note that S is non-material surface propagating across material points. From (7) it follows the following balance equations on S
𝜌V [[𝐯]] = −𝐍 ⋅ [[𝐏]] , jV[[ ̇𝐖]] = −𝐍 ⋅ [[𝐆]]. (9) where𝐍 is the unit normal to S and V is the intrinsic speed of propagation of S(t) in normal direction, see [49].
An acceleration wave (or weak discontinuity wave, or singular surface of the sec- ond order) is a traveling singular surface S(t) at which the second spatial and time
derivatives of the position vector𝐱 and of the microstructural tensor 𝐖 have jumps, while𝐱 and 𝐖 together with all first derivatives are continuous. So on S(t) we have the following system of equations:
[[∇𝐱]] = 𝟎, [[∇𝐖]] = 𝟎, [[𝐯]] = 𝟎, [[ ̇𝐖]] = 𝟎. (10)
Fig. 1 Propagation of a singular surface
From (9) and (10) it follows that
𝐍 ⋅ [[𝐏]] = 𝟎, 𝐍 ⋅ [[𝐆]] = 𝟎. Equation (10) imply continuity of𝐏 and 𝐆 at S(t):
[[𝐏]] = 𝟎, [[𝐆]] = 𝟎. Obviously, the balance equation (9) are also fulfilled.
In what follows we use Maxwell’s theorem which states that, see [49,50]. Theorem 1 (Maxwell) For a continuously differentiable field𝐘 such that [[𝐘]] = 𝟎
the following relations hold
[[ ̇𝐘]] = −V𝐲, [[∇𝐘]] = 𝐍 ⊗ 𝐲, (11)
where 𝐲 is the tensor amplitude of the jump of the first gradient of 𝐘; the tensor amplitude is a tensor of the order equal to the order of𝐘.
Here⊗ is the tensor product.
Straightforward application of Maxwell’s theorem to the continuous fields of𝐯,
̇𝐖, 𝐏, and 𝐆 results in the system of relations
[[̇𝐯]] = −V𝐚, [[∇𝐯]] = 𝐍 ⊗ 𝐚,
[[ ̈𝐖]] = −V𝐰, [[∇ ̇𝐖]]= 𝐍 ⊗ 𝐰, V[[∇⋅ 𝐏]] = −𝐍 ⋅[[ ̇𝐏]], V[[∇⋅ 𝐆]] = −𝐍 ⋅[[ ̇𝐆]],
(12)
where𝐚 and 𝐰 are the vectorial and tensorial amplitudes of the jumps. With these relations the motion equations transform into
𝐍 ⋅[[ ̇𝐏]] = 𝜌V2𝐚, 𝐍 ⋅[[ ̇𝐆]] =jV2𝐰. (13) Calculating ̇𝐏 and ̇𝐆 we obtain
𝐍 ⋅(W, ∇𝐱∇𝐱∶ (𝐍 ⊗ 𝐚) + W, ∇𝐱∇𝐖∶ (𝐍 ⊗ 𝐰))= 𝜌V2𝐚,
𝐍 ⋅(W, ∇𝐖∇𝐱∶ (𝐍 ⊗ 𝐚) + W, ∇𝐖∇𝐖∶ (𝐍 ⊗ 𝐰))= jV2𝐰.
With matrix notation, we rewrite the latter equations in a more compact form:
Q(𝐍) ⋅ 𝜉 = V2B ⋅ 𝜉, (14)
Q(𝐍) ≡ [W , ∇𝐱∇𝐱{𝐍} W, ∇𝐱∇𝐖{𝐍} W, ∇𝐖∇𝐱{𝐍} W, ∇𝐖∇𝐖{𝐍} ] , B ≡ [𝜌𝐈 𝟎 𝟎 j𝐈(N) ] .
Here 𝐈 and 𝐈(N) are three-dimensional and N-dimensional unit tensors, respec- tively, and we introduced the following operation for arbitrary Mth-order tensor 𝐇 and vector 𝐍. For tensor 𝐇 and vector 𝐍 represented in a Cartesian basis 𝐢k
(k = 1, 2, 3), so that
𝐇 = Hi1i2...iM𝐢i1⊗ 𝐢i2⊗ … ⊗ 𝐢iM ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Mtimes
, 𝐍 = Nk𝐢k,
𝐇{𝐍} denotes the following (M − 2)th-order tensor: 𝐇{𝐍} ≡ Hi1i2…iMNi1NiM−N𝐢i2⊗ … ⊗ 𝐢M
⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ M−2 times
. (15)
Q(𝐍) is the acoustic tensor for the medium with microstructure. From existence
of the strain energy W it follows that Q(𝐍) is symmetric. This implies that the squared velocity of propagation of an acceleration wave is real-valued. For positivity of V we need additional requirement, that isQ(𝐍) has to be positive definite
𝜉 ⋅ Q(𝐍) ⋅ 𝜉 > 0, ∀ 𝜉 ≠ 𝟎, ∀ |𝐍| = 1. (16)
Note that inequality (16) coincides with the condition of strong ellipticity of the equilibrium equations for considered elastic medium with microstructure.
Inequality (16) can be written in the form more convenient for calculations 𝐍 ⋅(W, ∇𝐱∇𝐱 ∶ (𝐍 ⊗ 𝐚) + W, ∇𝐱∇𝐖∶ (𝐍 ⊗ 𝐰))⋅ 𝐚
+𝐍 ⋅(W, ∇𝐖∇𝐱∶ (𝐍 ⊗ 𝐚) + W, ∇𝐖∇𝐖∶ (𝐍 ⊗ 𝐰))⋅ 𝐰 > 0,
∀ 𝐚 ≠ 𝟎, 𝐰 ≠ 𝟎, ∀ |𝐍| = 1. The condition is also equivalent to
d2
d𝜀2W(∇𝐱 + 𝜀𝐍 ⊗ 𝐚, 𝐖, ∇𝐖 + 𝜀𝐍 ⊗ 𝐖)|||| 𝜀=0> 0,
∀ 𝐍 ∶ |𝐍| = 1, 𝐚 ≠ 𝟎, 𝐖 ≠ 𝟎. (17)