Loss Of Convexity Research Articles

Relaxed incremental variational approach for damage in arteries

AbstractIn this contribution, several extensions of the relaxed incremental variational damage model by BALZANI & ORTIZ [1] are provided, in order to appropriately account for the description of arterial tissues. In particular, the one‐dimensional formulation from [1] is combined with a numerical homogenization scheme. Moreover, hysteresis behavior is included and a new optimization strategy is used to determine the bifurcation parameters when checking for a loss of convexity. The performance of the proposed model is demonstrated in a three‐dimensional finite element simulation of a overstretched simplified atherosclerotic artery. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Relaxed incremental variational approach for the modeling of damage-induced stress hysteresis in arterial walls

In this paper, a three-dimensional relaxed incremental variational damage model is proposed, which enables the description of complex softening hysteresis as observed in supra-physiologically loaded arterial tissues, and which thereby avoids a loss of convexity of the underlying formulation. The proposed model extends the relaxed formulation of Balzani and Ortiz [2012. Relaxed incremental variational formulation for damage at large strains with application to fiber-reinforced materials and materials with truss-like microstructures. Int. J. Numer. Methods Eng. 92, 551–570], such that the typical stress-hysteresis observed in arterial tissues under cyclic loading can be described. This is mainly achieved by constructing a modified one-dimensional model accounting for cyclic loading in the individual fiber direction and numerically homogenizing the response taking into account a fiber orientation distribution function. A new solution strategy for the identification of the convexified stress potential is proposed based on an evolutionary algorithm which leads to an improved robustness compared to solely Newton-based optimization schemes. In order to enable an efficient adjustment of the new model to experimentally observed softening hysteresis, an adjustment scheme using a surrogate model is proposed. Therewith, the relaxed formulation is adjusted to experimental data in the supra-physiological domain of the media and adventitia of a human carotid artery. The performance of the model is then demonstrated in a finite element example of an overstretched artery. Although here three-dimensional thick-walled atherosclerotic arteries are considered, it is emphasized that the formulation can also directly be applied to thin-walled simulations of arteries using shell elements or other fiber-reinforced biomembranes.

Hadamard instability analysis of “negative creep” in coupled chemo-thermo-mechanical systems

The coupled system of nonlinear partial differential equations for momentum, diffusion and energy is examined in terms of Hadamard instability, which in a unified way provides the conditions of both “negative creep” and “spinodal decomposition” (loss of convexity of thermodynamic functions) (Markenscoff in Quart Appl Math 59:147–151, 2001; Quart Appl Math 59:471–477, 2001) by balancing terms of different orders in the eigenvalue equation. It is shown here that instabilities of “negative creep” occur in both infinite and finite domains.

Electromechanical instabilities in fiber-constrained, dielectric-elastomer composites subjected to all-around dead-loading

In this work we investigate the possible development of instabilities in a certain class of dielectric-elastomer composites (DECs) subjected to all-around dead electromechanical loading. The DECs consist of a dielectric elastomer matrix phase constrained to plane strain deformations by means of aligned, long, rigid dielectric fibers of elliptical cross-section that are also aligned but randomly distributed in the transverse plane. Two types of instabilities are considered: loss of positive definiteness (LPD), and loss of strong ellipticity (LE). LPD simply corresponds to the loss of local convexity of the homogenized electroelastic stored-energy function for the DECs and can be of two types depending on the resulting instability modes. When the modes are aligned with the ‘principal’ solution, the instability corresponds to a maximum in the nominal electric field, possibly followed by snapping behavior. Alternatively, when the modes are orthogonal to the principal solution, the instability corresponds to a bifurcation from the principal solution. The LE, on the other hand, corresponds to LPD of the electromechanical acoustic tensor and manifests itself by the onset of highly localized shear band instabilities. Our results show that the stability, as well as the type of instability (when stability is lost), of the DECs depends sensitively on the loading conditions and is also affected by the microstructure of the DECs (i.e. the volume fraction of the fibers and their aspect ratio). In particular, it is found that the non-aligned LPD instabilities typically precede aligned LPD and LE instabilities, especially for in-plane isotropic microstructures, but aligned (limit load) LPD and LE instabilities are also possible for composites with anisotropic microstructures. In this context, it should be noted that the possible development of non-aligned LPD bifurcation instabilities appears to have been ignored by prior work. Fortunately, however, such instabilities can usually be avoided by orienting the long (in-plane) axis of the fibers parallel to the tensile stress direction and orthogonal to the applied electric field.

A Continuum Damage–Breakage Faulting Model and Solid-Granular Transitions

We present a thermodynamically-based formulation for mechanical modeling of faulting processes in the seismogenic brittle crust using a continuum damage–breakage rheology. The model combines previous results of a continuum damage framework for brittle solids with continuum breakage mechanics for granular flow. The formulation accounts for the density of distributed cracking and other internal flaws in damaged rocks with a scalar damage parameter, and addresses the grain size distribution of a granular phase in a failure slip zone with a breakage parameter. The stress–strain relation and kinetics of the damage and breakage processes are governed by the total energy function of the system, which combines the energy of the damaged solid with the energy of the granular material. A dynamic brittle instability is associated with a critical level of damage in the solid, leading to loss of convexity of the solid energy function and transition to a granular phase associated with lower energy level. A non-local formulation provides an intrinsic length scale associated with the internal damage structure, which leads to a finite length scale for damage localization that eliminates the unrealistic singular localization of local models. Shear heating during deformation can lead to a secondary finite-width internal localization. The formulation provides a framework for studying multiple aspects of brittle deformation, including potential feedback between evolving elastic moduli and properties of the slip localization zone and subsequent rupture behavior. The model has a more general transition from slow deformation to dynamic rupture than that associated with frictional sliding on a single pre-existing failure zone, and gives time and length scales for the onset of the dynamic fracturing process. Several features including the existence of finite localization width and transition from slow to rapid dynamic slip are illustrated using numerical simulations. A configuration having an existing narrow slip zone with localized damage produces for appropriate loading conditions an overall cyclic stick–slip motion. The simulated frictional response includes transitions from friction coefficient of ~0.7 at low slip velocity to dynamic friction below 0.4 at slip rates above ~0.1 m/s, followed by rapidly increasing friction for slip rates above ~1 m/s, consistent with laboratory observations.

Edelen’s dissipation potentials and the visco-plasticity of particulate media

Following is an elaboration on D. G. B. Edelen’s (1972–1973) nonlinear generalization of the classical Rayleigh-Onsager dissipation potentials and the implications for the models of viscoplasticity. A brief derivation is given via standard vector calculus of Edelen’s potentials and the associated non-dissipative or “gyroscopic” forces and fluxes. It is also shown that certain extensions of Edelen’s formulae can be obtained by means of a recently proposed source-flux relation or “inverse divergence,” a generalization of the classical Gauss-Maxwell construct. The Legendre–Fenchel duality of Edelen’s potentials is explored, with important consequences for rate-independent friction or plasticity. The use of dissipation potentials serves to facilitate the development of viscoplastic constitutive equations, a point illustrated here by the special cases of Stokesian fluid-particle suspensions and granular media. In particular, we consider inhomogeneous systems with particle migration coupled to gradients in particle concentration, strain rate, and fabric. Employing a mixture-theoretic treatment of Stokesian suspensions, one is able to identify particle stress as the work conjugate of the global deformation of the particle phase. However, in contrast to past treatments, this stress is not assumed to be a privileged driving force for particle migration. A comparison is made with models based on extremal dissipation or entropy production. It is shown that such models yield the correct dissipative components of force or flux but generally fail to capture certain non-dissipative, but mechanically relevant components. The significance of Edelen’s gyroscopic forces and their relation to reactive constraints or other reversible couplings is touched upon. When gyroscopic terms are absent, one obtains a class of strongly dissipative or hyperdissipative materials whose quasi-static mechanics are governed by variational principles based on dissipation potential. This provides an interesting analog to elastostatic variational principles based on strain energy for hyperelastic materials and to the associated material instabilities arising from loss of convexity.

Variation-Aware Geometric Programming Models for the Clock Network Buffer Sizing Problem

In this paper, we present and analyze four efficient models that produce significantly improved results by optimizing conflicting power and skew objectives in the clock network buffer sizing problem. Each model is in geometric programming format and has certain advantages, such as maximum reduction in power, robustness to process variation, and striking a balance between skew and power optimization. The buffer sizing problem is formulated as a geometric programming problem to provide globally optimal solutions to the four models. We also show that a geometric programming multiobjective model can be used to optimize both power and skew without requiring any tuning from a designer. The presented self-tuning multiobjective formulation not only provides optimal solutions for buffer sizes, but also finds the tuning parameters that result in overall combined reduction in power and skew without loss of convexity. The effectiveness of the models are illustrated on several publicly available benchmarks. The models provide on average 40% to 60% improvement in power while reducing skew in several cases. We have also proposed a smart heuristic for discretization of the continuous geometric programming solution that preserves skew and power. Finally, we provide a guideline for designers to decide which one of the proposed models is the most appropriate for their needs.

Damage–breakage rheology model and solid-granular transition near brittle instability

We develop a continuum-based theoretical framework that describes brittle instability and localization of deformation into a narrow slip zone as a phase transition between damaged solid and granular material. The formulation is based on irreversible thermodynamics of damage and breakage processes, each associated with a single key state variable, and corresponding energy functions for the damaged solid and granular material. Dynamic instability is associated with a critical level of damage in the solid, leading to loss of convexity of the solid energy function and transition to a granular phase associated with lower energy level. Depending on the confining stress and other conditions, the failure process in the generated granular phase may be associated with mode I and fragmentation or mode II and granular flow. The developed model provides a new approach for analyzing in a unified way various aspects of brittle failure and localization of deformation, with evolving elastic moduli, evolving slip rates and evolving material phases. Numerical simulations indicate that the key parameters governing the evolution from a slow failure process to dynamic slip, and the related transition from damaged solid to granular material, can be constrained by laboratory and seismological observations.

Localization in heterogeneous materials: A variational approach and its application to polycrystalline solids

In the case when macroscopic instability in a heterogeneous material is defined as the loss of rank-one convexity of the homogenized tangent modulus (namely, the propensity to develop bands of localization), the stability domain is the one of positive definiteness of a potential function on a representative volume element defining the unit cell of the microsctructure. It is possible to bound this domain by minimizing the functional on a set of kinematically admissible velocity fields which intend to catch the localization of deformation in the microstructure.The approach is applied to polycrystal plasticity in a bidimensional geometry: the plane double-slip model of Asaro is retained for the local behavior. The stability domain of a polycrystalline aggregate, schematized by a bidimensional pavement, is displayed as a function of the heterogeneity of crystal orientations, latent hardening and stress distributions. The localization velocity field is estimated in each case. In particular, the destabilizing effect of the coupling between shear strain and rotation at the microscopic level is exhibited.

Relaxed Incremental Variational Formulation for Damage in Fiber‐Reinforced Materials

AbstractAn incremental variational formulation for damage at finite strains is proposed based on the classical continuum damage mechanics. Since loss of convexity is obtained at some critical deformations a relaxed incremental stress potential is constructed which convexifies the original non‐convex problem. The resulting model can be interpreted as the homogenization of a micro‐heterogeneous material bifurcated into a strongly and weakly damaged phase at the microscale. A one‐dimensional relaxed formulation is derived and based thereon, a model for fiber‐reinforced materials is given. Finally, some numerical examples illustrate the performance of the model. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Relaxed incremental variational formulation for damage at large strains with application to fiber‐reinforced materials and materials with truss‐like microstructures

SUMMARYIn this paper, an incremental variational formulation for damage at finite strains is presented. The classical continuum damage mechanics serves as a basis where a stress‐softening term depending on a scalar‐valued damage function is prepended an effective hyperelastic strain energy function, which describes the virtually undamaged material. Because loss of convexity is obtained at some critical deformations, a relaxed incremental stress potential is constructed, which convexifies the original nonconvex problem. The resulting model can be interpreted as the homogenization of a microheterogeneous material bifurcated into a strongly and weakly damaged phase at the microscale. A one‐dimensional relaxed formulation is derived, and a model for fiber‐reinforced materials based thereon is given. Finally, numerical examples illustrate the performance of the model by showing mesh independency of the model in an extended truss, analyzing a numerically homogenized microtruss material and investigating a fiber‐reinforced cantilever beam subject to bending and an overstretched arterial wall. Copyright © 2012 John Wiley & Sons, Ltd.

Analytical study of the asymptotic behavior of a thin plate with temperature-dependent elastic modulus under cyclic thermomechanical loadings

In a recent paper (Hasbroucq et al., 2010 [1]), we have analyzed the long-term behavior of a discrete mechanical system with temperature-dependent elastic properties under cyclic thermomechanical loadings. In particular, we have shown that the residual stress and strain fields are time-dependent when shakedown occurs and thus the Halphen's (2005) [2] conjecture is not a necessary shakedown condition. Also, we have shown that there is loss of convexity of elastic and shakedown domains in the Bree diagram. In order to examine if these facts are related to the finite dimensional character of the structure and to the uniaxial path load in Hasbroucq et al. (2010) [1], we consider here a continuum media, namely a thin plate undergoing a biaxial loading. Linear and quadratic dependence of Young's modulus and Poisson's coefficient with respect to the temperature are considered. Closed-form expressions of the asymptotic responses of the plate under a given thermomechanical history loads are presented for shakedown, alternating plasticity and ratcheting regimes. In addition, the present study shows that in the Bree diagram, the elastic domain is convex while the shakedown zone is not so and, contrary to Hasbroucq et al. (2010) [1], the residual (elastic) strain field is time-independent for shakedown. Therefore, the present example falls in the Halphen's shakedown conjecture.

Full-Thickness Grafting From the Ulnar Border of the Palm

Full-thickness grafting from the ulnar border of the palm is an acceptable choice to cover appropriately sized defects of the volar surface of the hand. The outcome and donor-site morbidity are discussed. Twenty-two patients with variably located defects were treated with this technique. Mean age was 26.6 years and mean graft size was 3.8 × 1.6 cm. A number 15 scalpel was used to harvest the graft in a subdermal plane. The graft was secured with a tie-over dressing. The donor-site was closed primary with a continuous intradermal or "over-and-over" suture with few interrupted stitches. Color and texture match were adequate in 91% of the grafts. A contour defect with loss of the normal convexity of the ulnar border of the palm was noticed in 77% of donor sites. Hypertrophic scarring occurred in 3 patients (14%). Sensation in the 4 grafts covering fingertip areas was adequate (mean static 2 point discrimination: 5.5 mm). Mean follow-up was 12.7 months. The technique is useful in appropriately sized defects of the volar hand (within 6 × 3 cm). Donor-site morbidity and alternative techniques should be adequately discussed with the patient preoperatively.

A non-local visco-elastic damage model and dynamic fracturing

We present an extended formulation of a non-linear continuum visco-elastic damage rheology that accounts for non-local damage accumulation, dynamic fracturing, and transition from solid to granular state in the slip zone. Generalizing the standard Hookean strain energy, the model has three additional energy terms: a non-analytic second-order function of the first and second strain invariants, a term proportional to the strain rate, and a term proportional to the spatial gradient of a damage state variable. The first term leads to non-linear stress–strain relation, with abrupt changes in the effective elastic moduli upon stress reversal from compression to tension along with damage- and stress-induced anisotropy. The second term gives rise to Kelvin–Voigt viscous relaxation. The resulting formulation combines Kelvin–Voigt and Maxwell visco-elasticity, accounting for both long-term relaxation and short-term dissipation stabilizing the damage evolution. The third term produces a finite length scale for damage diffusion that eliminates the unrealistic singular localization of the local damage model. An equation for damage evolution derived from basic thermodynamic considerations quantifies the kinetics of damage under different conditions, including quasi-static and dynamic degradation and gradual healing. In the vicinity of macroscopic failure, at a critical level of damage associated with loss of convexity of the energy function, the formulation includes a transition from damaged solid to granular flow dynamics. The formulation provides a framework for studying multiple aspects of brittle deformation, including potential feedback mechanisms between evolving elastic and related properties of the slip localization zone and subsequent rupture behavior. Several features of the model including existence of finite localization width and transition from slow to rapid dynamic slip are illustrated using numerical simulations. The analysis clarifies the dependency of some damage parameters on basic properties of the examined domain and boundary conditions, and the conditions for which simpler local damage descriptions are appropriate.

ON THE PENALTY FUNCTION AND ON CONTINUITY PROPERTIES OF RISK MEASURES

We discuss two issues about risk measures: we first point out an alternative interpretation of the penalty function in the dual representation of a risk measure; then we analyze the continuity properties of comonotone convex risk measures. In particular, due to the loss of convexity, local and global continuity are no more equivalent and many implications true for convex risk measures do not hold any more.

Loss of convexity and embeddedness for geometric evolution equations of higher order

We show that for a large class of geometric evolution equations of immersed surfaces in the Euclidean space, there are compact embedded surfaces that lose their embeddedness and compact strictly convex surfaces that lose their convexity under these evolution equations.

Bifurcation structures and dominant modes near relative equilibria in the one-dimensional discrete nonlinear Schrödinger equation

We investigate the bifurcation structure of a family of relative equilibria of a ring of seven oscillators described by the discrete nonlinear Schrödinger equation (DNLSE) when the period of these orbits and a suitable defect act as bifurcation parameters. We find a reduced Hamiltonian that gives substantial insight into the dynamics of this system. The convexity of this Hamiltonian at given nonresonant equilibria supports the stability of nearby quasiperiodic solutions. We show that the local loss of convexity in the reduced Hamiltonian is determined by the Hessian of its integrable part in the family of relative equilibria under study. Stable quasiperiodic solutions are studied by considering the power spectral densities of a set of suitable fast and slow actions, whose origin is suggested by the averaging principle. We also show that the return times form an optimal embedding to characterize the system dynamics. We show that the power spectral density of a suitable interference signal, arising from a ring of Bose–Einstein condensates and described by the DNLSE, has a single prominent peak at the breather-like relative equilibria.

Dissipative issue of high-order shock capturing schemes with non-convex equations of state

It is well known that, closed with a non-convex equation of state (EOS), the Riemann problem for the Euler equations allows non-standard waves, such as split shocks, sonic isentropic compressions or rarefaction shocks, to occur. Loss of convexity then leads to non-uniqueness of entropic or Lax solutions, which can only be resolved via the Liu-Oleinik criterion (equivalent to the existence of viscous profiles for all admissible shock waves). This suggests that in order to capture the physical solution, a numerical scheme must provide an appropriate level of dissipation. A legitimate question then concerns the ability of high-order shock capturing schemes to naturally select such a solution. To investigate this question and evaluate modern as well as future high-order numerical schemes, there is therefore a crucial need for well-documented benchmarks. A thermodynamically consistent C ∞ non-convex EOS that can be easily introduced in Eulerian as well as Lagrangian hydrocodes for test purposes is here proposed, along with a reference solution for an initial value problem exhibiting a complex composite wave pattern (the Bizarrium test problem). Two standard Lagrangian numerical approaches, both based on a finite volume method, are then reviewed (vNR and Godunov-type schemes) and evaluated on this Riemann problem. In particular, a complete description of several state-of-the-art high-order Godunov-type schemes applicable to general EOSs is provided. We show that this particular test problem reveals quite severe when working on high-order schemes, and recommend it as a benchmark for devising new limiters and/or next-generation highly accurate schemes.

Scaling relations of earthquakes and aseismic deformation in a damage rheology model

SUMMARY We perform analytical and numerical studies of scaling relations of earthquakes and partition of elastic strain energy between seismic and aseismic components using a thermodynamically based continuum damage model. Brittle instabilities occur in the model at critical damage level associated with loss of convexity of the strain energy function. A new procedure is developed for calculating stress drop and plastic strain in regions sustaining brittle instabilities. The formulation connects the damage rheology parameters with dynamic friction of simpler frameworks, and the plastic strain accumulation is governed by a procedure that is equivalent to Drucker–Prager plasticity. The numerical simulations use variable boundary forces proportional to the slip-deficit between the assumed far field plate motion and displacement of the boundary nodes. These boundary conditions account for the evolution of elastic properties and plastic strain in the model region. 3-D simulations of earthquakes in a model with a large strike-slip fault produce scaling relations between the scalar seismic potency, rupture area, and stress drop values that are in good agreement with observations and other theoretical studies. The area and potency of the simulated earthquakes generally follow a linear log–log relation with a slope of 2/3, and are associated with stress drop values between 1 and 10 MPa. A parameter-space study shows that the area-potency scaling is shifted to higher stress drops in simulations with parameters corresponding to lower dynamic friction, more efficient healing, and higher degree of seismic coupling.

Analysis of elastic stability and structural response of face-centered cubic crystals subject to [110] loading

The elastic stability and structural response of face-centered cubic crystals subject to uniaxial [110] loading are investigated systematically based on lattice-statics analysis, molecular-dynamics (MD) simulations, and symmetry and bifurcation considerations. The onset of instability in isostress MD simulations is found to be associated with loss or diminution of energy convexity (Born’s stability criterion) as determined from computed strain and temperature dependent elastic moduli. Atomic mechanisms of crystal destabilization beyond the onset of instability are consistent with theoretical eigendeformations and are precursors to interatomic slip, which, in turn, may lead to failure or mechanical stacking faults.