arXiv subjects

Patrizio Neff

Publications and source records attributed to Patrizio Neff.

At least 163 records · Page 9Linked to original sources

arXiv2015

An ellipticity domain for the distortional Hencky-logarithmic strain energy

We describe ellipticity domains for the isochoric elastic energy $ F\mapsto \|{\rm dev}_n\log U\|^2=\bigg\|\log \frac{\sqrt{F^TF}}{(\det F)^{1/n}}\bigg\|^2 =\frac{1}{4}\,\bigg\|\log \frac{C}{({\rm det} C)^{1/n}}\bigg\|^2 $ for $n=2,3$, where $C=F^TF$ for $F\in {\rm GL}^+(n)$. Here, ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n}\, {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the logarithmic strain tensor $\log U$. For $n=2$ we identify the maximal ellipticity domain, while for $n=3$ we show that the energy is Legendre-Hadamard elliptic in the set $\mathcal{E}_3\bigg(W_{_{\rm H}}^{\rm iso}, {\rm LH}, U, \frac{2}{3}\bigg)\,:=\,\bigg\{U\in{\rm PSym}(3) \;\Big|\, \|{\rm dev}_3\log U\|^2\leq \frac{2}{3}\bigg\}$, which is similar to the von-Mises-Huber-Hencky maximum distortion strain energy criterion. Our results complement the characterization of ellipticity domains for the quadratic Hencky energy $ W_{_{\rm H}}(F)=μ\,\|{\rm dev}_3\log U\|^2+ \fracκ{2}\,[{\rm tr} (\log U)]^2 $, $U=\sqrt{F^TF}$ with $μ>0$ and $κ>\frac{2}{3}\, μ$, previously obtained by Bruhns et al.

math.CA↗

arXiv2015

The geometrically nonlinear Cosserat micropolar shear-stretch energy. Part I: A general parameter reduction formula and energy-minimizing microrotations in 2D

In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field $F := \nablaφ: Ω\to \mathrm{GL}^+(n)$ and the microrotation field $R: Ω\to \mathrm{SO}(n)$, the shear-stretch energy is necessarily of the form \begin{equation*} W_{μ,μ_c}(R\,;F) := μ\,\left\lVert{\mathrm{sym}(R^T F - \boldsymbol{1})}\right\rVert^2 + μ_c\,\left\lVert{\mathrm{skew}(R^T F - \boldsymbol{1})}\right\rVert^2\;, \end{equation*} where $μ> 0$ is the Lamé shear modulus and $μ_c \geq 0$ is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations $\mathrm{argmin}_{R\,\in\,\mathrm{SO}(n)}{W_{μ,μ_c}(R\,;F)}$ as a function of $F \in \mathrm{GL}^+(n)$ and weights $μ> 0$ and $μ_c \geq 0$. For $n \geq 2$, we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: $(μ, μ_c) = (1,1)$ and $(μ,μ_c) = (1,0)$. In contrast to Grioli's theorem, we derive non-classical minimizers for the parameter range $μ> μ_c \geq 0$ in dimension $n\!=\!2$. Currently, optimality results for $n \geq 3$ are out of reach for us, but we contribute explicit representations for $n\!=\!2$ which we name $\mathrm{rpolar}^{\pm}_{μ,μ_c}(F) \in \mathrm{SO}(2)$ and which arise for $n\!=\!3$ by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non-classical optimal Cosserat rotations $\mathrm{rpolar}^\pm_{μ,μ_c}(F_γ)$ for simple planar shear.

math.AP↗

arXiv2015

Rank-one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two-dimensional case

We show that in the two-dimensional case, every objective, isotropic and isochoric energy function which is rank-one convex on $\mathrm{GL}^+(2)$ is already polyconvex on $\mathrm{GL}^+(2)$. Thus we negatively answer Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasiconvexity. Our methods are based on different representation formulae for objective and isotropic functions in general as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor.

math.AP↗

arXiv2015

On the generalized sum of squared logarithms inequality

Assume $n\geq 2$. Consider the elementary symmetric polynomials $e_k(y_1,y_2,\ldots, y_n)$ and denote by $E_0,E_1,\ldots,E_{n-1}$ the elementary symmetric polynomials in reverse order \begin{align*} E_k(y_1,y_2,\ldots,y_n):=e_{n-k}(y_1,y_2,\ldots,y_n)=\sum_{i_1<\ldots<i_{n-k}} y_{i_1}y_{i_2}\ldots y_{i_{n-k}}\, , \quad k\in \{0,1,\ldots,n{-}1 \}\, . \end{align*} Let moreover $S$ be a nonempty subset of $\{0,1,\ldots,n{-}1\}$. We investigate necessary and sufficient conditions on the function $f\colon\,I\to\mathbb{R}$, where $I\subset\mathbb{R}$ is an interval, such that the inequality \begin{align} \label{abstract_inequality} f(a_1)+f(a_2)+\ldots+f(a_n)\leq f(b_1)+f(b_2)+\ldots+f(b_n) \tag{*} \end{align} holds for all $a=(a_1,a_2,\ldots,a_n)\in I^n$ and $b=(b_1,b_2,\ldots,b_n)\in I^n$ satisfying $$E_k(a)< E_k(b) \ \hbox{for } k\in S\quad \hbox{and} \quad E_k(a)=E_k(b) \ \hbox{for } k\in \{0,1,\ldots,n{-}1 \}\setminus S\, .$$ As a corollary, we obtain \eqref{abstract_inequality} if $2\leq n\leq 4$, $f(x)=\log^2x$ and $S=\{1,\dotsc,n-1\}$, which is the sum of squared logarithms inequality previously known for $2\le n\le 3$.

math.CA↗

arXiv2015

Wave propagation in pantographic 2D lattices with internal discontinuities

In the present paper we consider a 2D pantographic structure composed by two orthogonal families of Euler beams. Pantographic rectangular 'long' waveguides are considered in which imposed boundary displacements can induce the onset of traveling (possibly non-linear) waves. We performed numerical simulations concerning a set of dynamically interesting cases. The system undergoes large rotations which may involve geometrical non-linearities, possibly opening the path to appealing phenomena such as propagation of solitary waves. Boundary conditions dramatically influence the transmission of the considered waves at discontinuity surfaces. The theoretical study of this kind of objects looks critical, as the concept of pantographic 2D sheets seems to have promising possible applications in a number of fields, e.g. acoustic filters, vascular prostheses and aeronautic/aerospace panels.

physics.comp-ph↗

arXiv2015

Integrability conditions between the first and second Cosserat deformation tensor in geometrically nonlinear micropolar models and existence of minimizers

In this note we extend integrability conditions for the symmetric stretch tensor $U$ in the polar decomposition of the deformation gradient $\nablaφ=F=R\,U$ to the non-symmetric case. In doing so we recover integrability conditions for the first Cosserat deformation tensor. Let $F=\bar R\,\bar U$ with $\bar R:Ω\subset\mathbb{R}^3\longrightarrow\mathrm{SO}(3)$ and $\bar U:Ω\subset\mathbb{R}^3\longrightarrow \mathrm{GL}(3)$. Then $\mathfrak{K}:={\bar R}^T\mathrm{Grad}\,{\bar R}=\mathrm{Anti}\Big( \frac{1}{\mathrm{det} \bar U}\Big[\bar U(\mathrm{Curl} \bar U)^T-\frac{1}{2} \mathrm{tr}(\bar U(\mathrm{Curl} \bar U)^T) 1\!\!1 \Big]\bar U\Big),$ giving a connection between the first Cosserat deformation tensor $\bar U$ and the second Cosserat tensor ${\mathfrak{K}}$. (Here, Anti denotes an isomorphism between $\mathbb{R}^{3\times 3}$ and $\mathfrak{So}(3):=\{\,\mathfrak{A}\in\mathbb{R}^{3\times 3\times 3}\,|\,\mathfrak{A}.u\in\mathfrak{so}(3)\;\forall u\in \mathbb{R}^3\}$.) The formula shows that it is not possible to prescribe $\bar U$ and $\mathfrak{K}$ independent from each other. We also propose a new energy formulation of geometrically nonlinear Cosserat models which completely separate the effects of nonsymmetric straining and curvature. For very weak constitutive assumptions (no direct boundary condition on rotations, zero Cosserat couple modulus, quadratic curvature energy) we show existence of minimizers in Sobolev-spaces.

math-ph↗

arXiv2015

On some fundamental misunderstandings in the indeterminate couple stress model. A comment on recent papers of A.R. Hadjesfandiari and G.F. Dargush

In a series of papers which are either published [A.R. Hadjesfandiari and G.F. Dargush, Couple stress theory for solids, Int. J. Solids Struct. 48, 2496-2510, 2011; A.R. Hadjesfandiari and G.F. Dargush, Fundamental solutions for isotropic size-dependent couple stress elasticity, Int. J. Solids Struct. 50, 1253-1265, 2013] or available as preprints Hadjesfandiari and Dargush have reconsidered the linear indeterminate couple stress model. They are postulating a certain physically plausible split in the virtual work principle. Based on this postulate they claim that the second-order couple stress tensor must always be skew-symmetric. Since they use an incomplete set of boundary conditions in their virtual work principle their statement contains unrecoverable errors. This is shown by specifying their development to the isotropic case. However, their choice of constitutive parameters is mathematically possible and still yields a well-posed boundary value problem.

math-ph↗

arXiv2015

Existence results in dislocation based rate-independent isotropic gradient plasticity with kinematical hardening and plastic spin: The case with symmetric local backstress

In this paper we use convex analysis and variational inequality methods to establish an existence result for a model of infinitesimal rate-independent gradient plasticity with kinematic hardening and plastic spin, in which the local backstress tensor remains symmetric. The model features a defect energy contribution which is quadratic in the dislocation density tensor Curl p, giving rise to nonlocal non-symmetric kinematic hardening. Use is made of a recently established Korn's type inequality for incompatible tensor fields. The solution space for the non-symmetric plastic distortion is naturally H(Curl) together with suitable tangential boundary conditions on the plastic distortion. Connections to other models are established as well.

math.AP↗

arXiv2015

A variant of the linear isotropic indeterminate couple stress model with symmetric local force-stress, symmetric nonlocal force-stress, symmetric couple-stresses and complete traction boundary conditions

In this paper we venture a new look at the linear isotropic indeterminate couple stress model in the general framework of second gradient elasticity and we propose a new alternative formulation which obeys Cauchy-Boltzmann's axiom of the symmetry of the force stress tensor. For this model we prove the existence of solutions for the equilibrium problem. Relations with other gradient elastic theories and the possibility to switch from a {4th order} (gradient elastic) problem to a 2nd order micromorphic model are also discussed with a view of obtaining symmetric force-stress tensors. It is shown that the indeterminate couple stress model can be written entirely with symmetric force-stress and symmetric couple-stress. The difference of the alternative models rests in specifying traction boundary conditions of either rotational type or strain type. If rotational type boundary conditions are used in the partial integration, the classical anti-symmetric nonlocal force stress tensor formulation is obtained. Otherwise, the difference in both formulations is only a divergence--free second order stress field such that the field equations are the same, but the traction boundary conditions are different. For these results we employ a novel integrability condition, connecting the infinitesimal continuum rotation and the infinitesimal continuum strain. Moreover, we provide the complete, consistent traction boundary conditions for both models.

math-ph↗

arXiv2015

Correct traction boundary conditions in the indeterminate couple stress model

In this paper we consider the Grioli-Koiter-Mindlin-Toupin indeterminatecouple stress model. The main aim is to show that the traction boundary conditions were not yet completely deduced. As it turns out, and to our own surprise, restricting the boundary condition framework from the strain gradient models to the couple stress model does not reduce to Mindlin's set of accepted boundary conditions. We present therefore, for the first time the complete, consistent set of traction boundary conditions.

math-ph↗

arXiv2015

Numerical Treatment of a Geometrically Nonlinear Planar Cosserat Shell Model

We present a new way to discretize a geometrically nonlinear elastic planar Cosserat shell. The kinematical model is similar to the general 6-parameter resultant shell model with drilling rotations. The discretization uses geodesic finite elements, which leads to an objective discrete model which naturally allows arbitrarily large rotations. Finite elements of any approximation order can be constructed. The resulting algebraic problem is a minimization problem posed on a nonlinear finite-dimensional Riemannian manifold. We solve this problem using a Riemannian trust-region method, which is a generalization of Newton's method that converges globally without intermediate loading steps. We present the continuous model and the discretization, discuss the properties of the discrete model, and show several numerical examples, including wrinkles of thin elastic sheets in shear.

math.NA↗

arXiv2015

The exponentiated Hencky-logarithmic strain energy. Improvement of planar polyconvexity

In this paper we improve the result about the polyconvexity of the energies from the family of isotropic volumetric-isochoric decoupled strain exponentiated Hencky energies defined in the first part of this series, i.e. $$ W_{_{\rm eH}}(F)= \left\{\begin{array}{lll} \fracμ{k}\,e^{k\,\|{\rm dev}_n\log U\|^2}+\fracκ{2\,\widehat{k}}\,e^{\widehat{k}\,[(\log {\rm det} U)]^2}&\text{if}& {\rm det}\, F>0,\\ +\infty &\text{if} &{\rm det} F\leq 0\,, \end{array}\right. $$ where $F=\nabla φ$ is the gradient of deformation, $U=\sqrt{F^T F}$ is the right stretch tensor and ${\rm dev}_n\log {U}$ is the deviatoric part of the strain tensor $\log U$. The main result in this paper is that in plane elastostatics, i.e. for $n=2$, the energies of this family are polyconvex for $k\geq \frac{1}{4}$, $\widehat{k}\geq \frac{1}{8}$, extending a previous result which proves polyconvexity for $k\geq \frac{1}{3}$, $\widehat{k}\geq \frac{1}{8}$. This leads immediately to an extension of the existence result.

math.CA↗

arXiv2014

Existence theorem for geometrically nonlinear Cosserat micropolar model under uniform convexity requirements

We reconsider the geometrically nonlinear Cosserat model for a uniformly convex elastic energy and write the equilibrium problem as a minimization problem. Applying the direct methods of the calculus of variations we show the existence of minimizers. We present a clear proof based on the coercivity of the elastically stored energy density and on the weak lower semi-continuity of the total energy functional. Use is made of the dislocation density tensor $\bar{\boldsymbol{K}}=\bar{\boldsymbol{R}}^T\,\mathrm{Curl}\,\bar{\boldsymbol{R}}$ as a suitable Cosserat curvature measure.

math.AP↗

arXiv2014

Well-posedness for dislocation based gradient visco-plasticity with isotropic hardening

In this work we establish the well-posedness for infinitesimal dislocation based gradient viscoplasticity with isotropic hardening for general gradient monotone plastic flows. We assume an additive split of the displacement gradient into non-symmetric elastic distortion and non-symmetric plastic distortion. The thermodynamic potential is augmented with a term taking the dislocation density tensor $\operatorname{Curl} p$ into account. The constitutive equations in the models we study are assumed to be of self-controlling type. Based on the generalized version of Korn's inequality for incompatible tensor fields (the non-symmetric plastic distortion) due to Neff/Pauly/Witsch the existence of solutions of quasi-static initial-boundary value problems under consideration is shown using a time-discretization technique and a monotone operator method.

math.AP↗

arXiv2014

Some remarks on the monotonicity of primary matrix functions on the set of symmetric matrices

This note contains some observations on primary matrix functions and different notions of monotonicity with relevance towards constitutive relations in nonlinear elasticity. Focussing on primary matrix functions on the set of symmetric matrices, we discuss and compare different criteria for monotonicity. The demonstrated results are particularly applicable to computations involving the true-stress-true-strain monotonicity condition, a constitutive inequality recently introduced in an Arch. Appl. Mech. article by C.S. Jog and K.D. Patil. We also clarify a statement by Jog and Patil from the same article which could be misinterpreted.

math.CA↗

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↗