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Research Papers

# A Generalized Continuum Formulation for Composite Microcracked Materials and Wave Propagation in a Bar

[+] Author and Article Information
Patrizia Trovalusci

Department of Structural Engineering and Geotechnics, “Sapienza” University of Rome, Via Gramsci 97, 00197 Rome, Italypatrizia.trovalusci@uniroma1.it

Valerio Varano

Department of Structures, University “RomaTre”, Via Segre 4/6, 00146 Rome, Italy

Giuseppe Rega

Department of Structural Engineering and Geotechnics, “Sapienza” University of Rome, Via Gramsci 97, 00197 Rome, Italy

Our model refers to materials with a periodic microstructure and is developed within a purely mechanical deterministic framework, tacitly assuming an RVE with a given size. A scale-dependent homogenization should be involved if randomness were taken into account (6).

Dot symbol indicates the time derivative of a variable.

We use the representation of infinitesimal rotations and moments through skew-symmetric tensors instead of axial vectors. It is worth noting that there exists a one-to-one correspondence between the two notations: $Θv=θ×v$ and $∀v∊V$, where $Θ$ is a skew-symmetric tensor, $θ$ is its axial vector, and $V$ is a vector space.

The beamlike interactions between particle-slits entail a multiple interaction $(AHB)$ in the moment balance equations. Of course, various kinds of interactions can be considered but with the assumed no-central force scheme, in analogy to the Voigt and Poincaré approaches to molecular linear elasticity, we expect to identify a correct number of material constants (32). Moreover, this assumption will allow us to identify a symmetric macrostress tensor.

Note that this definition could be refined by distinguishing between the different elementary volumes of the lattice elements, as in rigorous Voronoi tessellations of atomistic models of matter (39).

In the following, the same symbols in the summations indicate the same ranges for indexes.

From now on, the dependence of the fields on the position $x$ will be undertaken.

The circumstance of having the symmetric part of the displacement gradient in the generalized internal work formula depends on the internal constraint 7.

Such an assumption agrees with the standard assumptions of continuum theories based on lattice mechanics (33-35).

In some earlier papers of the authors (12,28), according to the formulation of the classical molecular theory of elasticity (31,34), the identification procedure was based on the equivalence of the intermolecular potential and the continuum strain energy. This approach requires the selection of the response functions for the lattice internal actions to derive the stress measures of the macromodel.

A mechanical theory of such continua has been presented by Capriz (41). Micromorphic continua, among them second gradient, microstretch, micropolar material, etc. (42) are special cases of multifield materials.

The apex “T” stands for the major transposition index such that: $AA⋅B=A⋅ATB$ for any fourth order tensor $A$ and any pair of second order tensors $A$ and $B$.

In general for multifield continua the standard invariance theorems cannot be used to derive the full set of balance equations (53).

In a previous paper of the authors (28), the stiffer internal constraint $W=0$ was considered instead of Eq. 7. This implies that the skew-symmetric part of the stress $S$ is equal to the left hand term of Eq. 25, so that it cannot be selected independently of the nonstandard stress measures $z$ and $Z$ as it occurs herein. For a discussion see (54).

In the framework of the pursued formulation, fibers with the same local rigid rotation are considered.

The balance of angular momentum implies that the mass couple $B$ is null.

J. Appl. Mech 77(6), 061002 (Aug 16, 2010) (11 pages) doi:10.1115/1.4001639 History: Received May 30, 2009; Revised February 05, 2010; Posted April 20, 2010; Published August 16, 2010; Online August 16, 2010

## Abstract

A multifield continuum is adopted to grossly describe the dynamical behavior of composite microcracked solids. The constitutive relations for the internal and external (inertial) actions are obtained using a multiscale modeling based on the hypotheses of the classical molecular theory of elasticity and the ensuing overall elastodynamic properties allow us to take properly into account the microscopic features of these materials. Referring to a one-dimensional microcracked bar, the ability of such a continuum to reveal the presence of internal heterogeneities is investigated by analyzing the relevant dispersive wave propagation properties. Scattering of traveling waves is shown to be associated with the microcrack density in the bar.

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## Figures

Figure 3

Lattice statical interactions: (a) particle-particle, slit-particle, and (b) slit-slit

Figure 4

Waves traveling along the bar: (a) low microcrack density and (b) high microcrack density. Elastic wave (thin line); resulting wave (thick line).

Figure 5

Phase-velocity versus microcrack density: elastic bar (thin); microcracked bar (thick)

Figure 6

Macrowave in the elastic bar (thin line) and resulting wave in the damaged bar (thick line) for low (a), medium (b), and high (c) microcrack density

Figure 7

Mean resultant amplitude ratio versus microcrack density

Figure 8

Macrowave in the elastic bar (thin line) and resulting wave in the damaged bar (thick line) for lower (a) and higher (b) microcrack density

Figure 1

An orthotropic module with particles, slits, and relevant mass portions. For the sake of simplicity, one slit for each pair of particles is represented.

Figure 2

Lattice kinematical descriptors

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