A Variational Approach to Fracture and Other Inelastic Phenomena

Piero, Gianpietro Del

doi:10.1007/978-94-007-7226-7_2

Cited by 7 publications

(4 citation statements)

References 102 publications

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“…• for damage and fracture phenomena (see e.g. Francfort and Marigo [66], Yang and Misra [110,111], Contrafatto and Cuomo [31,32,33], [36], Rinaldi and Lai [128] and Del Piero [55]) ;…”

Section: Bernoulli Constant Of Motion Along Flow Curvesmentioning

confidence: 99%

Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids

Auffray

Dell’Isola

Eremeyev

et al. 2013

Mathematics and Mechanics of Solids

237

195

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On ne trouvera point de Figures dans cet Ouvrage. Les méthodes que j'y expose ne demandent ni constructions, ni raisonnemens géométriques ou méchaniques, mais seulement des opérations algébriques, assujetties à une marche réguliere et uniforme. Ceux qui aiment l'Analyse, verront avec plaisir la Méchanique en divenir une nouvelle branche, et me sauront gré d'en avoir étendu ansi le domaine." From the Avertissement of the Méchanique Analitique by Lagrange [87]1 AbstractIn this paper a stationary action principle is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments, for instance by Cahn and Hilliard [15,16]. We remark that these fluids are sometimes also called Korteweg-de Vries or Cahn-Allen fluids. In general continua whose deformation energy depends on the second gradient of placement are called second gradient (or Piola-Toupin or Mindlin or Green-Rivlin or Germain or second grade) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both the material and spatial descriptions and the corresponding Euler-Lagrange equations and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and ∇C or on C −1 and ∇C −1 , where C is the Cauchy-Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions (see e.g. Casal [25] or Seppecher [140,143] for an alternative deduction based on thermodynamic arguments) are recovered. A version of Bernoulli's law valid for capillary fluids is found and, in Appendix B, useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola's contribution to analytical continuum mechanics are also presented. In this context the reader is also referred to Capecchi and Ruta [17]. Part Ithe author obtained the evolution equations for capillary fluids by combining the principle of virtual works in the Eulerian description with the first principle of thermodynamics (also in the case of isothermal motions). This shows that it can be sometimes useful to use an heuristic procedure in which the principle of virtual powers is reinforced by additionally requiring also the validity of the balance of mechanical energy. Also interesting in this context are the results presented in Casal [25], Gavrilyuk and Gouin [68].In the opinion of the present authors the methods of analytical continuum mechanics are the most effective ones (see also [100]), at least when formulating models for mechanical phenomena involving multiple time and length scales. The reader is invited to consider, with respect to this particular class of phenomena, the difficulties which are to be confronted when using continuum thermodynamics, for instance, to describe interfacial phenomena in phase transition (see e.g. d...

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“…• for damage and fracture phenomena (see e.g. Francfort and Marigo [66], Yang and Misra [110,111], Contrafatto and Cuomo [31,32,33], [36], Rinaldi and Lai [128] and Del Piero [55]) ;…”

Section: Bernoulli Constant Of Motion Along Flow Curvesmentioning

confidence: 99%

Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids

Auffray

Dell’Isola

Eremeyev

et al. 2013

Mathematics and Mechanics of Solids

237

195

View full text Add to dashboard Cite

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“…For a system characterized by these energies, the displacement and damage evolution, when certain loads are applied and increased, is determined by solving a constrained incremental minimization problem. Incremental minimization is a powerful mathematical tool to capture local minima in evolution problems, and it has been applied to problems of fracture [35], plasticity [36,37], and crystal plasticity [38,39]. Here it is solved numerically by implementing a Sequential Quadratic Programming algorithm.…”

Section: Introductionmentioning

confidence: 99%

Fabric-reinforced cementitious matrix behavior at high-temperature: Experimental and numerical results

Donnini

Basalo

Corinaldesi

et al. 2017

Composites Part B: Engineering

117

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The use of externally applied composite systems to upgrade, strengthen or rehabilitate masonry or concrete structures is well established. However, structural strengthening with organic type composites, such as fiber-reinforced polymer (FRP) systems, may be impractical when the element is exposed to high-temperature service conditions, due to significant degradation of the organic resin. Instead, the use of an inorganic matrix, as in the case of fabric-reinforced cementitious matrix (FRCM) composites, may overcome this problem. The purpose of this study is to evaluate the mechanical behavior under high-temperature conditions of FRCM systems. Different FRCM composites are evaluated and include carbon fabrics ranging from dry to highly-impregnated with an organic resin. The experimental spectrum is comprised of uniaxial tensile and double-shear bond tests performed under temperatures ranging from 20 to 120 °C to determine the influence of temperature over the FRCM mechanical properties. Furthermore, SEM analysis was used to study the damage processes at the fiber-matrix interface post tensile testing. Experimental results show variations in the FRCM mechanical properties if tested at high temperature conditions (caused by the deterioration of the resin coating at the interface fiber-matrix) while residual performance after exposure to elevated temperatures remains unchanged. FRCM reinforced with dry fabrics has proven not to be affected by temperatures up to 120 °C. A numerical model using a fracture variational approach, based on incremental energy minimization, was also developed to simulate the FRCM behavior in double shear tests under different temperatures exposition

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“…An integral part of phase-field models is the length-scale parameter l 0 . Originally introduced as a purely numerical parameter, which recovers the discrete nature of fracture in the limit l 0 → 0, several contributions interpret l 0 as a material parameter [11,18,43]. Instead of choosing l 0 as small as possible the length-scale parameter is calibrated to match the critical strength of the material.…”

Section: Calibrationmentioning

confidence: 99%

Predicting Fracture in the Proximal Humerus using Phase Field Models

Hug¹,

Dahan²,

Kollmannsberger³

et al. 2022

Preprint

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Proximal humerus impacted fractures are of clinical concern in the elderly population. Prediction of such fractures by CT-based finite element methods encounters several major obstacles such as heterogeneous mechanical properties and fracture due to compressive strains. We herein propose to investigate a variation of the phase field method (PFM) embedded into the finite cell method (FCM) to simulate impacted humeral fractures in fresh frozen human humeri. The force-strain response, failure loads and the fracture path are compared to experimental observations for validation purposes. The PFM (by means of the regularization parameter 0 ) is first calibrated by one experiment and thereafter used for the prediction of the mechanical response of two other human fresh frozen humeri. All humeri are fractured at the surgical neck and strains are monitored by Digital Image Correlation (DIC). Experimental strains in the elastic regime are reproduced with good agreement (R 2 = 0.726), similarly to the validated finite element method [9]. The failure pattern and fracture evolution at the surgical neck predicted by the PFM mimic extremely well the experimental observations for all three humeri. The maximum relative error in the computed failure loads is 3.8%. To the best of our knowledge this is the first method that can predict well the experimental compressive failure pattern as well as the force-strain relationship in proximal humerus fractures.

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A Variational Approach to Fracture and Other Inelastic Phenomena

Cited by 7 publications

References 102 publications

Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids

Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids

Fabric-reinforced cementitious matrix behavior at high-temperature: Experimental and numerical results

Predicting Fracture in the Proximal Humerus using Phase Field Models

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