arXiv subjects

Ionel-Dumitrel Ghiba

Publications and source records attributed to Ionel-Dumitrel Ghiba.

61 records · Page 4Linked to original sources

arXiv2014

The exponentiated Hencky-logarithmic strain energy. Part III: Coupling with idealized isotropic finite strain plasticity

We investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric-isochoric decoupled strain energies \begin{align*} F\mapsto W_{_{\rm eH}}(F):=\hat{W}_{_{\rm eH}}(U):=\{\begin{array}{lll} \fracμ{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\fracκ{\text{}{2\, {\hat{k}}}}\,e^{\hat{k}\,[{\rm tr}(\log U)]^2}&\text{if}& {\rm det}\, F>0,\\ +\infty &\text{if} &{\rm det} F\leq 0, \end{array}.\quad \end{align*} based on the Hencky-logarithmic (true, natural) strain tensor $\log U$. Here, $μ>0$ is the infinitesimal shear modulus, $κ=\frac{2μ+3λ}{3}>0$ is the infinitesimal bulk modulus with $λ$ the first Lamé constant, $k,\hat{k}$ are dimensionless fitting parameters, $F=\nabla φ$ is the gradient of deformation, $U=\sqrt{F^T F}$ is the right stretch tensor and ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n}\, {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the strain tensor $\log U$. Based on the multiplicative decomposition $F=F_e\, F_p$, we couple these energies with some isotropic elasto-plastic flow rules $F_p\,\frac{\rm d}{{\rm d} t}[F_p^{-1}]\in-\partial χ({\rm dev}_3 Σ_{e})$ defined in the plastic distortion $F_p$, where $\partial χ$ is the subdifferential of the indicator function $χ$ of the convex elastic domain $\mathcal{E}_{\rm e}(W_{\rm iso},{Σ_{e}},\frac{1}{3}{\boldsymbolσ}_{\!\mathbf{y}}^2)$ in the mixed-variant $Σ_{e}$-stress space and $Σ_{e}=F_e^T D_{F_e} W_{\rm iso}(F_e)$. While $W_{_{\rm eH}}$ may loose ellipticity, we show that loss of ellipticity is effectively prevented by the coupling with plasticity, since the ellipticity domain of $W_{_{\rm eH}}$ on the one hand, and the elastic domain in $Σ_{e}$-stress space on the other hand, are closely related.

math-ph↗

arXiv2014

The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers

We consider a family of isotropic volumetric-isochoric decoupled strain energies $$ F\mapsto W_{\rm eH}(F):=\widehat{W}_{\rm eH}(U):=\left\{\begin{array}{lll} \fracμ{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\fracκ{2\hat{k}}\,e^{\hat{k}\,[{\rm tr}(\log U)]^2}&\text{if}& {\rm det}\, F>0,\\ +\infty &\text{if} &{\rm det} F\leq 0, \end{array}\right.\quad $$ based on the Hencky-logarithmic (true, natural) strain tensor $\log U$, where $μ>0$ is the infinitesimal shear modulus, $κ=\frac{2μ+3λ}{3}>0$ is the infinitesimal bulk modulus with $λ$ the first Lamé constant, $k,\hat{k}$ are dimensionless parameters, $F=\nabla φ$ is the gradient of deformation, $U=\sqrt{F^T F}$ is the right stretch tensor and ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n} {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part (the projection onto the traceless tensors) of the strain tensor $\log U$. For small elastic strains the energies reduce to first order to the classical quadratic Hencky energy $$ F\mapsto W{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U):=μ\,\|{\rm dev}_n\log U\|^2+\fracκ{2}\,[{\rm tr}(\log U)]^2, $$ which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family $W_{_{\rm eH}}$ are polyconvex for $k\geq \frac{1}{3}$, $\widehat{k}\geq \frac{1}{8}$, extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann's polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor $U$. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations and we prove the existence of minimizers by the direct methods of the calculus of variations.

math.CA↗

arXiv2014

The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank-one convexity

We investigate a family of isotropic volumetric-isochoric decoupled strain energies $$ F\mapsto W_{_{\rm eH}}(F):=\widehat{W}_{_{\rm eH}}(U):=\left\{\begin{array}{lll} \fracμ{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\fracκ{2\, {\widehat{k}}}\,e^{\widehat{k}\,[{ \rm tr}(\log U)]^2}&\text{if}& { \rm det} F>0,\\ +\infty &\text{if} &{ \rm det} F\leq 0, \end{array}\right.\quad $$ based on the Hencky-logarithmic (true, natural) strain tensor $\log U$, where $μ>0$ is the infinitesimal shear modulus, $κ=\frac{2μ+3λ}{3}>0$ is the infinitesimal bulk modulus with $λ$ the first Lamé constant, $k,\widehat{k}$ are dimensionless parameters, $F=\nabla φ$ is the gradient of deformation, $U=\sqrt{F^T F}$ is the right stretch tensor and ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n} {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the strain tensor $\log U$. For small elastic strains, $W_{_{\rm eH}}$ approximates the classical quadratic Hencky strain energy $$ F\mapsto W_{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U):=μ\,\|{\rm dev}_n\log U\|^2+\fracκ{2}\,[{\rm tr}(\log U)]^2, $$ which is not everywhere rank-one convex. In plane elastostatics, i.e. $n=2$, we prove the everywhere rank-one convexity of the proposed family $W_{_{\rm eH}}$, for $k\geq \frac{1}{4}$ and $\widehat{k}\geq \frac{1}{8}$. Moreover, we show that the corresponding Cauchy (true)-stress-true-strain relation is invertible for $n=2,3$ and we show the monotonicity of the Cauchy (true) stress tensor as a function of the true strain tensor in a domain of bounded distortions. We also prove that the rank-one convexity of the energies belonging to the family $W_{_{\rm eH}}$ is not preserved in dimension $n=3$.

math.CA↗

arXiv2014

The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations

In this paper we consider the equilibrium problem in the relaxed linear model of micromorphic elastic materials. The basic kinematical fields of this extended continuum model are the displacement $u\in \mathbb{R}^3$ and the non-symmetric micro-distortion density tensor $P\in \mathbb{R}^{3\times 3}$. In this relaxed theory a symmetric force-stress tensor arises despite the presence of microstructure and the curvature contribution depends solely on the micro-dislocation tensor ${\rm Curl}\, P$. However, the relaxed model is able to fully describe rotations of the microstructure and to predict non-polar size-effects. In contrast to classical linear micromorphic models, we allow the usual elasticity tensors to become positive-semidefinite. We prove that, nevertheless, the equilibrium problem has a unique weak solution in a suitable Hilbert space. The mathematical framework also settles the question of which boundary conditions to take for the micro-distortion. Similarities and differences between linear micromorphic elasticity and dislocation gauge theory are discussed and pointed out.

math-ph↗

arXiv2013

A unifying perspective: the relaxed linear micromorphic continuum

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict non-polar size-effects. It is intended for the homogenized description of highly heterogeneous, but non polar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. It unifies and simplifies the understanding of the linear micromorphic models.

math-ph↗

arXiv2013

The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics

We study well-posedness for the relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. In contrast to classical micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. Another interesting feature concerns the prescription of boundary values for the micro-distortion field: only tangential traces may be determined which are weaker than the usual strong anchoring boundary condition. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes.

math.AP↗

arXiv2013

Wave propagation in relaxed micromorphic continua: modelling metamaterials with frequency band-gaps

In this paper the relaxed micromorphic model proposed in [Patrizio Neff, Ionel-Dumitrel Ghiba, Angela Madeo, Luca Placidi, Giuseppe Rosi. A unifying perspective: the relaxed linear micromorphic continuum, submitted, 2013, arXiv:1308.3219; and Ionel-Dumitrel Ghiba, Patrizio Neff, Angela Madeo, Luca Placidi, Giuseppe Rosi. The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics, submitted, 2013, arXiv:1308.3762] has been used to study wave propagation in unbounded continua with microstructure. By studying dispersion relations for the considered relaxed medium, we are able to disclose precise frequency ranges (band-gaps) for which propagation of waves cannot occur. These dispersion relations are strongly nonlinear so giving rise to a macroscopic dispersive behavior of the considered medium. We prove that the presence of band-gaps is related to a unique elastic coefficient, the so-called Cosserat couple modulus $μ_{c}$, which is also responsible for the loss of symmetry of the Cauchy force stress tensor. This parameter can be seen as the trigger of a bifurcation phenomenon since the fact of slightly changing its value around a given threshold drastically changes the observed response of the material with respect to wave propagation. We finally show that band-gaps cannot be accounted for by classical micromorphic models as well as by Cosserat and second gradient ones. The potential fields of application of the proposed relaxed model are manifold, above all for what concerns the conception of new engineering materials to be used for vibration control and stealth technology.

cond-mat.mtrl-sci↗