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Paolo Maria Mariano

Publications and source records attributed to Paolo Maria Mariano.

17 recordsLinked to original sources
arXiv2025

Spectra and pseudospectra in the evaluation of material stability in phase field schemes

We consider the dynamics of bodies with "active" microstructure described by vector-valued phase fields. For waves with time-varying amplitude, the associated evolution equation involves a matrix that can be non-normal, depending on the constitutive choices adopted for the microstructural actions associated with the considered phase field. The occurrence of non-normality requires to look at the pseudospectrum of the considered matrix, namely the set of all possible eigenvalues of matrices in a $\varepsilon$-neighborhood of the matrix itself, because the eigenvalues of non-normal matrices can be very sensitive to small perturbations and therefore the spectral analysis alone would not be sufficient to distinguish with certainty between table and unstable behavior. We develop the relevant analyses in the case of quasicrystals for which the values of some constitutive parameters are not known or are uncertain from an experimental point of view, a circumstance suggesting parametric analyses. We find circumstances in which the pseudospectra obtained by means of the so-called structured perturbations predict instability when, instead, the spectral analysis indicates stability.

math-ph↗
arXiv2023

Continuum Kinematics with Incompatible-Compatible Decomposition

Abstract. We present a framework for the kinematics of a material body undergoing anelastic deformation. For such processes, the material structure of the body, as reflected by the geometric structure given to the set of body points, changes. The setting we propose may be relevant to phenomena such as plasticity, fracture, discontinuities, and non-injectivity of the deformations. In this framework, we construct an unambiguous decomposition into incompatible and compatible factors which includes the standard elastic-plastic decomposition in plasticity.

math-ph↗
arXiv2021

Crack occurrence in bodies with gradient polyconvex energies

Energy minimality selects among possible configurations of a continuous body with and without cracks those compatible with assigned boundary conditions of Dirichlet-type. Crack paths are described in terms of curvature varifolds so that we consider both \textquotedblleft phase" (cracked or non-cracked) and crack orientation. The energy considered is gradient polyconvex: it accounts for relative variations of second-neighbor surfaces and pressure-confinement effects. We prove the existence of minimizers for such an energy. They are pairs of deformations and varifolds. The former ones are taken to be $SBV$ maps satisfying an impenetrability condition. Their jump set is constrained to be in the varifold support.

math.AP↗
arXiv2015

Existence results in the linear dynamics of quasicrystals with phason diffusion and non-linear gyroscopic effects

Quasicrystals are characterized by quasi-periodic arrangements of atoms. The description of their mechanics involves deformation and a (so called phason) vector field accounting at macroscopic scale of local phase changes, due to atomic flips necessary to match quasi-periodicity under the action of the external environment. Here we discuss the mechanics of quasicrystals, commenting the shift from its initial formulation, as standard elasticity in a space with dimension twice the ambient one, to a more elaborated setting avoiding physical inconveniences of the original proposal. In the new setting we tackle two problems. First we discuss the linear dynamics of quasicrystals including a phason diffusion. We prove existence of weak solutions and their uniqueness under rather general boundary and initial conditions. We then consider phason rotational inertia, non-linearly coupled with the curl of the macroscopic velocity, and prove once again existence of weak solutions to the pertinent balance equations.

math-ph↗
arXiv2010

Physical significance of the curvature varifold-based description of crack nucleation

The nucleation and/or growth of cracks in elastic-brittle solids has been recently described in [14] in terms of a special class of measures and with a variational technique requiring the minimization of a certain energy over classes of bodies. Here, the physical foundations of the theory and the basic ideas leading to it are described and commented further on. A view on certain possible developments and shifts toward different settings is also given. This article has expository character.

math-ph↗
arXiv2009

The relative power and its invariance

The relative power of actions in a Cauchy body suffering mutations due to defect evolution is introduced. It is shown that its invariance under the action of the Euclidean group over the ambient space and the material space allows one to obtain (i) the balance of standard and configurational actions and (ii) the identification of configurational ingredients from a unique source.

math-ph↗
arXiv2008

Complex bodies with memory: linearized setting

The mechanics of complex bodies with memory effects is discussed in linearized setting. The attention is focused on the characterization of free energies in terms of minimum work and maximum recoverable work in the bulk and along a discontinuity surface endowed with its own surface energy, a surface internal to the body. To this aim, use is made of techniques proposed by Del Piero. Consequences of the Clausius-Duhem inequality are investigated for complex bodies with instantaneous linear elastic response.

math-ph↗
arXiv2008

Representation of Material Elements and Geometry of Substructural Interactions

This paper collects some characteristic aspects of the general model-building framework of the mechanics of complex bodies, that are bodies in which the material substructure influences prominently the gross behavior through interactions conjugated with substructural changes. The emphasis in on issues that are foundational in nature: the geometrica description of the material elements, their energetic characterization, the representation of interactions, conditions for the existence of ground states in conservative setting.

math-ph↗
arXiv2008

Ground states in complex bodies

A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappinngs and Cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and Cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance of configurational actions follows. After describing a list of possible applications of the general results collected here, a concrete discussion of the existence of ground states in thermodynamically stable quasicrystals is presented at the end.

math-ph↗
arXiv2004

Non-linear hydrodynamics of incommensurate intergrowth compounds and quasicrystals

Hamiltonian structures for non-linear hydrodynamics of incommensurate intergrowth compounds (IIC) and quasicrystals (IQ) are constructed. We discuss also the way to account for internal friction of phason nature. We show that the existence of a self-force in IIC and IQ is not only matter of constitutive issues, rather it is related with questions of SO(3) invariance. The covariant mechanics of discontinuity surfaces in quasiperiodic structures is also analyzed. The attention is mainly focused on the interaction between `diffuse' grain boundaries and sharp discontinuity (moving possibly) surfaces.

math-ph↗
arXiv2004

SO(3) invariance and covariance in mixtures of simple bodies

We adapt to mixtures the procedure of invariance of external power under the action of SO(3) to deduce balance equations. The two classical axioms about the growths of momentum and moment of momentum are derived with the help of a rule on the structure of the total power. We discuss also the covariance of the balance equations of each constituent and the nature of the constituent stress.

math-ph↗
arXiv2004

Geometry of interactions in complex bodies

We analyze geometrical structures necessary to represent bulk and surface interactions of standard and substructural nature in complex bodies. Our attention is mainly focused on the influence of diffuse interfaces on sharp discontinuity surfaces. In analyzing this phenomenon, we prove the covariance of surface balances of standard and substructural interactions.

math-ph↗
arXiv2003

Cancellation of vorticity in steady-state non-isentropic flows of complex fluids

In steady-state non-isentropic flows of perfect fluids there is always thermodynamic generation of vorticity when the difference between the product of the temperature with the gradient of the entropy and the gradient of total enthalpy is different from zero. We note that this property does not hold in general for complex fluids for which the prominent influence of the material substructure on the gross motion may cancel the thermodynamic vorticity. We indicate the explicit condition for this cancellation (topological transition from vortex sheet to shear flow) for general complex fluids described by coarse-grained order parameters and extended forms of Ginzburg-Landau energies. As a prominent sample case we treat first Korteweg's fluid, used commonly as a model of capillary motion or phase transitions characterized by diffused interfaces. Then we discuss general complex fluids. We show also that, when the entropy and the total enthalpy are constant throughout the flow, vorticity may be generated by the inhomogeneous character of the distribution of material substructures, and indicate the explicit condition for such a generation. We discuss also some aspects of unsteady motion and show that in two-dimensional flows of incompressible perfect complex fluids the vorticity is in general not conserved, due to a mechanism of transfer of energy between different levels.

math-ph↗
arXiv2003

Influence of the material substructure on crack propagation: a unified treatment

The influence of the material texture (substructure) on the force driving the crack tip in complex materials admitting Ginzburg-Landau-like energies is analyzed in a three-dimensional continuum setting. The theory proposed accounts for finite deformations and general coarse-grained order parameters. A modified expression of the J-integral is obtained together with other path-integrals which are necessary to treat cases where the process zone around the tip has finite size. The results can be applied to a wide class of material substructures. As examples, cracks in ferroelectrics and in materials with strain-gradient effects are discussed: in these cases the specializations of the general results fit reasonably experimental data.

math-ph↗
arXiv2003

Symmetries and Hamiltonian formalism for complex materials

Preliminary results toward the analysis of the Hamiltonian structure of multifield theories describing complex materials are mustered: we involve the invariance under the action of a general Lie group of the balance of substructural interactions. Poisson brackets are also introduced in the material representation to account for general material substructures. A Hamilton-Jacobi equation suitable for multifield models is presented. Finally, a spatial version of all these topics is discussed without making use of the notion of paragon setting.

math-ph↗