Analytical results for a linear hardening elasto-plastic spring investigated via a hemivariational formulation

Year 2024, Volume: 7 Issue: Special Issue: AT&A, 50 - 75, 16.12.2024

Luca Placidi , Anil Misra , Abdou Kandalaft , Mohammad Mahdi Nayeban , Nurettin Yilmaz

https://doi.org/10.33205/cma.1532828

Abstract

We investigate the linear hardening phenomena with a method that is not standard in the literature, i.e. with a hemivariational method. As a result, we do not introduce any flow rules, and the number of assumptions is reduced to the generalized variational principle with proper definition of a new set of kinematic descriptors and, as a function of them, with a new definition of the energy functional. The variational framework guarantees the rationality of the deduction. Analytical derivation of the force displacement hysteretic loop is also derived and, finally, the dissipation energy is furnished with respect to both the final value of the dissipation energy potential or by the corresponding area of the hysteretic loop.

Keywords

Linear hardening , histeretic loop , variational method , Plasticity

References

• B. E. Abali, W. H. Müller and F. dell’Isola: Theory and computation of higher gradient elasticity theories based on action principles, Archive of Applied Mechanics, 87 (2017), 1495–1510.
• I. Ahmad, H. Ahmad, P. Thounthong, Y. Chu and C. Cesarano: Solution of multi-term time-fractional PDE models arising in mathematical biology and physics by local meshless method, Symmetry, 12 (2020), Article ID: 1195.
• E. Aifantis: Pattern formation in plasticity, Internat. J. Engrg. Sci., 33 (1995), 2161–2178.
• J. Alibert, P. Seppecher and F. Dell’Isola: Truss modular beams with deformation energy depending on higher displacement gradients, Math. Mech. Solids, 8 (2003), 51–73
• A. Al-Jaser, C. Cesarano, B. Qaraad and L. Iambor: Second-Order Damped Differential Equations with Superlinear Neutral Term: New Criteria for Oscillation, Axioms, 13 (2024), Article ID: 234.
• U. Andreaus, P. Baragatti: Fatigue crack growth, free vibrations, and breathing crack detection of aluminium alloy and steel beams, J. Strain Anal. Eng. Des., 44 (2009), 595–608.
• E. Artioli, F. Auricchio and L. Veiga: Generalized midpoint integration algorithms for J2 plasticity with linear hardening, Int. J. Numer. Methods Eng., 72 (2007), 422–463.
• N. Auffray, F. Dell’Isola, V. Eremeyev, A. Madeo and G. Rosi: Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids, Math. Mech. Solids, 20 (2015), 375–417.
• E. Barchiesi, A. Misra, L. Placidi and E. Turco: Granular micromechanics-based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations, ZAMM Z. Angew. Math. Mech., 101 (11) (2021), Article ID: e202100059.
• M. Bragaglia, F. Lamastra, P. Russo, L. Vitiello, M. Rinaldi, F. Fabbrocino and F. Nanni: A comparison of thermally conductive polyamide 6-boron nitride composites produced via additive layer manufacturing and compression molding, Polym. Compos., 42 (2021), 2751–2765.
• M. Bragaglia, L. Paleari, F. Lamastra, D. Puglia, F. Fabbrocino and F. Nanni: Graphene nanoplatelet, multiwall carbon nanotube, and hybrid multiwall carbon nanotube–graphene nanoplatelet epoxy nanocomposites as strain sensing coatings, J. Reinf. Plast. Comp., 40 (2021), 632–643.
• M. Bragaglia, L. Paleari, F. Lamastra, P. Russo, F. Fabbrocino and F. Nanni: Oleylamine functionalization of boron nitride nano-platelets for Polyamide-6 thermally conductive injection moulded composites, J. Thermoplast. Compos. Mater., 36 (2023), 2862–2882.
• A. Caporale, R. Luciano and E. Sacco: Micromechanical analysis of interfacial debonding in unidirectional fiber-reinforced composites, Comput. Struct., 84 (2006), 2200–2211.
• A. Cauchy: Sur l’èquilibre et le mouvement d’un système de points matèriels sollicitès par des forces d’attraction ou de rèpulsion mutuelle, In: Oeuvres Complètes: Series 2. Cambridge Library Collection - Mathematics, Cambridge University Press, (2009) 227–252.
• A. Ciallella, I. Giorgio, E. Barchiesi, G. Alaimo, A. Cattenone, B. Smaniotto, A. Vintache, F. D’Annibale, F. Dell’Isola, F. Hild and Others: A 3D pantographic metamaterial behaving as a mechanical shield: experimental and numerical evidence, Mater. Des., 237 (2023), Article ID: 112554.
• L. Contrafatto, M. Cuomo and S. Gazzo: A concrete homogenisation technique at meso-scale level accounting for damaging behaviour of cement paste and aggregates, Comput. Struct., 173 (2016), 1–18.
• L. Contrafatto, M. Cuomo and L. Greco: Meso-scale simulation of concrete multiaxial behaviour, Eur. J. Environ. Civ. En., 21 (7–8) (2016), 896–911.
• F. Cornacchia, F. Fabbrocino, N. Fantuzzi, R. Luciano and R. Penna: Analytical solution of cross-and angle-ply nano plates with strain gradient theory for linear vibrations and buckling, Mech. Adv. Mater. Struct., 28 (2021), 1201–1215.
• M. Cuomo, L. Contrafatto and L. Greco: A variational model based on isogeometric interpolation for the analysis of cracked bodies, Int. J. Eng. Sci., 80 (2014), 173–188.
• C. D’Ambra, G. Lignola, A. Prota, F. Fabbrocino and E. Sacco: FRCM strengthening of clay brick walls for out of plane loads, Compos. B: Eng., 174 (2019), Article ID: 107050.
• F. De Angelis: A comparative analysis of linear and nonlinear kinematic hardening rules in computational elastoplasticity, Tech. Mech., 32 (2012), 164–173.
• G. Del Piero: The variational structure of classical plasticity, Math. Mech. Complex Syst., 6 (3) (2018), 137–180.
• F. Dell’Isola, S. Eugster, R. Fedele and P. Seppecher: Second-gradient continua: From Lagrangian to Eulerian and back, Math. Mech. Solids, 27 (2022), 2715–2750.
• F. Dell’Isola, I. Giorgio and U. Andreaus: Elastic pantographic 2D lattices: a numerical analysis on static response and wave propagation, Proc. Est. Acad. Sci., 64 (2015), 219–225.
• F. Dell’Isola, I. Giorgio, M. Pawlikowski and N. Rizzi: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium, Proc. R. Soc. A, 472 (2016), Article ID: 20150790.
• F. dell’Isola, M. Guarascio and K. Hutter: A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi’s effective stress principle, Arch. Appl. Mech., 70 (5) (2000), 323–337.
• F. Dell’Isola, A. Misra: Principle of Virtual Work as Foundational Framework for Metamaterial Discovery and Rational Design, C. R. Mecanique, 351 (2023), 1–25.
• F. Dell’Isola, A. Madeo and P. Seppecher: Boundary conditions at fluid-permeable interfaces in porous media: A variational approach, Int. J. Solids Struct., 46 (2009), 3150–3164.
• F. Dell’Isola, D. Steigmann: A two-dimensional gradient-elasticity theory for woven fabrics, J. Elast., 18 (2015), 113–125.
• C. Dharmawardhana, A. Misra, S. Aryal, P. Rulis and W. Ching: Role of interatomic bonding in the mechanical anisotropy and interlayer cohesion of CSH crystals, Cem. Concr. Res., 52 (2013), 123–130.
• V. Eremeyev, F. Dell’Isola, C. Boutin and D. Steigmann: Linear pantographic sheets: existence and uniqueness of weak solutions, J. Elast., 132 (2017), 175–196.
• S. Eugster, F. Dell’Isola, R. Fedele and P. Seppecher: Piola transformations in second-gradient continua, Mech. Res. Commun., 120 (2022), Article ID: 103836.
• F. Fabbrocino, G. Carpentieri: Three-dimensional modeling of the wave dynamics of tensegrity lattices, Compos. Struct., 173 (2017), 9–16.
• F. Fabbrocino, I. Farina: Loading noise effects on the system identification of composite structures by dynamic tests with vibrodyne, Compos. B: Eng., 115 (2017), 376–383.
• R. Fedele: Piola’s approach to the equilibrium problem for bodies with second gradient energies. Part I: First gradient theory and differential geometry, Contin. Mech. Thermodyn., 34 (2022), 445–474.
• R. Fedele: Third-gradient continua: nonstandard equilibrium equations and selection of work conjugate variables, Math. Mech. Solids, 27 (2022), 2046–2072.
• R. Fedele, A. Ciani and F. Fiori: X-ray microtomography under loading and 3D-volume digital image correlation A review, Fundam. Inform., 135 (2014), 171–197.
• R. Fedele, M. Filippini and G. Maier: Constitutive model calibration for railway wheel steel through tension-torsion tests, Comput. Struct., 83 (2005), 1005–1020.
• F. Freddi, G. Royer-Carfagni: Regularized variational theories of fracture: a unified approach, J. Mech. Phys. Solids, 58 (2010), 1154–1174.
• I. Giorgio: A variational formulation for one-dimensional linear thermoviscoelasticity, Math. Mech. Complex Syst., 9 (2022), 397–412.
• I. Giorgio, U. Andreaus, D. Scerrato and F. Dell’Isola: A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials, Biomech. Model. Mechanobiol., 15 (2016), 1325–1343.
• I. Giorgio, U. Andreaus, F. Dell’Isola and T. Lekszycki: Viscous second gradient porous materials for bones reconstructed with bio-resorbable grafts, Extreme Mech. Lett., 13 (2017), 141–147.
• I. Giorgio, M. De Angelo, E. Turco and A. Misra: A Biot–Cosserat two-dimensional elastic nonlinear model for a micromorphic medium, Contin. Mech. Thermodyn., 32 (2019), 1357–1369.
• I. Giorgio, F. Dell’Isola, U. Andreaus, F. Alzahrani, T. Hayat and T. Lekszytcki: On mechanically driven biological stimulus for bone remodeling as a diffusive phenomenon, Biomech. Model. Mechanobiol., 18 (2019), 1639–1663.
• I. Giorgio, F. Dell’Isola and A. Misra: Chirality in 2D Cosserat media related to stretch-micro-rotation coupling with links to granular micromechanics, Int. J. Solids Struct., 202 (2020), 28–38.
• I. Giorgio, R. Grygoruk, F. Dell’Isola and D. Steigmann: Pattern formation in the three-dimensional deformations of fibered sheets, Mech. Res. Commun., 69 (2015), 164–171.
• I. Giorgio, F. Hild, E. Gerami, F. Dell’Isola and A. Misra: Experimental verification of 2D Cosserat chirality with stretchmicro-rotation coupling in orthotropic metamaterials with granular motif, Mech. Res. Commun., 126 (2022), Article ID: 104020.
• E. Grande, G. Milani, A. Formisano, B. Ghiassi and F. Fabbrocino: Bond behaviour of FRP strengthening applied on curved masonry substrates: numerical study, Int. J. Mason. Res. Innov., 5 (2020), 303–320.
• L. Greco, M. Cuomo: An implicit G1 multi patch B-spline interpolation for Kirchhoff–Love space rod, Comput. Methods Appl. Mech. Eng., 269 (2014), 173–197.
• L. Greco, M. Cuomo: An isogeometric implicit G1 mixed finite element for Kirchhoff space rods, Comput. Methods Appl. Mech. Eng., 298 (2016), 325–349.
• F. Greco, L. Leonetti, R. Luciano and P. Trovalusci: Multiscale failure analysis of periodic masonry structures with traditional and fiber-reinforced mortar joints, Compos. B: Eng., 118 (2017), 75–95.
• A. Grimaldi, R. Luciano: Tensile stiffness and strength of fiber-reinforced concrete, J. Mech. Phys. Solids, 48 (2000), 1987–2008.
• G. Khoury, C. Majorana, F. Pesavento and B. Schrefler: Modelling of heated concrete, Mag. Concr. Res.", 54 (2002), 77–101.
• D. Kumar, F. Ayant and C. Cesarano: Analytical Solutions of Temperature Distribution in a Rectangular Parallelepiped, Axioms, 11 (2022), Article ID: 488.
• J. Larsen: A new variational principle for cohesive fracture and elastoplasticity, Mech. Res. Commun., 58 (2014), 133-138.
• C. Majorana, V. Salomoni and B. Schrefler: Hygrothermal and mechanical model of concrete at high temperature, Mater. Struct., 31 (1998), 378–386.
• G. Mancusi, F. Fabbrocino, L. Feo and F. Fraternali: Size effect and dynamic properties of 2D lattice materials, Compos. B: Eng., 112 (2017), 235–242.
• A. Misra, P. Poorsolhjouy: Granular micromechanics model for damage and plasticity of cementitious materials based upon thermomechanics, Math. Mech. Solids, 25 (10) (2015), Article ID: 1081286515576821.
• A. Misra, V. Singh: Micromechanical model for viscoelastic materials undergoing damage, Contin. Mech. Thermodyn., 25 (2013), 343–358.
• A. Misra, V. Singh: Thermomechanics-based nonlinear rate-dependent coupled damage-plasticity granular micromechanics model, Contin. Mech. Thermodyn., 27 (4-5) (2015), Article ID: 787.
• M. M. Nava, R. Fedele and M. T. Raimondi: Computational prediction of strain-dependent diffusion of transcription factors through the cell nucleus, Biomech. Model. Mechanobiol., 15 (2016), 983–993.
• C. Navier: Memoire sur les lois de l’equilibre et du mouvement des corps solides elastiques, Academie des Sciences, (1827).
• N. Nejadsadeghi, F. Hild and A. Misra: Parametric experimentation to evaluate chiral bars representative of granular motif, Int. J. Mech. Sci., 221 (2022), Article ID: 107184.
• N. Nejadsadeghi, A. Misra: Extended granular micromechanics approach: a micromorphic theory of degree n, Math. Mech. Solids, 25 (2020), 407–429.
• C. Liu, Y. Zhong: Existence and multiplicity of periodic solutions for nonautonomous second-order discrete Hamiltonian systems, Constr. Math. Anal., 3 (2020), 178–188.
• L. Placidi, E. Barchiesi, A. Misra and D. Timofeev: Micromechanics-based elasto-plastic–damage energy formulation for strain gradient solids with granular microstructure, Contin. Mech. Thermodyn., 33 (2021), 2213–2241.
• P. Poorsolhjouy, A. Misra: Effect of intermediate principal stress and loading-path on failure of cementitious materials using granular micromechanics, Int. J. Solids Struct., 108 (2017), 139–152.
• Y. Rahali, I. Giorgio, J. Ganghoffer and F. Dell’Isola: Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices, Int. J. Eng. Sci., 97 (2015), 148–172.
• G. Ramaglia, G. Lignola, F. Fabbrocino and A. Prota: Numerical investigation of masonry strengthened with composites, Polymers, 10 (2018), Article ID: 334.
• B. Schrefler, P. Brunello, D. Gawin, C. Majorana and F. Pesavento: Concrete at high temperature with application to tunnel fire, Comput. Mech., 29 (2002), 43–51.
• D. Scerrato, I. Giorgio, A. Madeo, A. Limam and F. Darve: A simple non-linear model for internal friction in modified concrete, Int. J. Eng. Sci., 80 (2014), 136–152.
• B. Schrefler, C. Majorana, G. Khoury and D. Gawin: Thermo-hydro-mechanical modelling of high performance concrete at high temperatures, Eng. Comput., 19 (2002), 787–819.
• P. Seppecher, J. Alibert and F. Dell’Isola: Linear elastic trusses leading to continua with exotic mechanical interactions, J. Phys. Conf. Ser., 319 (2011), Article ID: 012018.
• J. Simo, T. Hughes: Computational inelasticity, Springer Science & Business Media, (2006).
• J. Simo, R. Taylor: Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms, Comput. Methods Appl. Mech. Eng., 85 (1991), 273–310.
• R. Sarikaya, Q. Ye, L. Song, C. Tamerler, P. Spencer and A. Misra: Probing the mineralized tissue-adhesive interface for tensile nature and bond strength, J. Mech. Behav. Biomed. Mater., 120 (2021), Article ID:104563.
• M. Spagnuolo, K. Barcz, A. Pfaff, F. Dell’Isola and P. Franciosi: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments, Mech. Res. Commun., 83 (2017), 47–52.
• G. Tocci Monaco, N. Fantuzzi, F. Fabbrocino and R. Luciano: Semi-analytical static analysis of nonlocal strain gradient laminated composite nanoplates in hygrothermal environment, J. Braz. Soc. Mech. Sci. Eng., 43 (2021), Article ID: 274.
• E. Turco, F. Dell’Isola, N. Rizzi, R. Grygoruk,W. Müller and C. Liebold: Fiber rupture in sheared planar pantographic sheets: Numerical and experimental evidence, Mech. Res. Commun., 76 (2016), 86–90.
• E. Turco, M. Golaszewski, I. Giorgio and F. D’Annibale: Pantographic lattices with non-orthogonal fibres: Experiments and their numerical simulations, Compos. B: Eng., 118 (2017), 1–14.
• Y. Yang, A. Misra: Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation, CMES - Comput. Model. Eng. Sci., 64 (1) (2010), 1–36.
• Y. Yang, A. Misra: Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity, Int. J. Solids Struct., 49 (18) (2012), 2500–2514.
• M. E. Yildizdag, L. Placidi and E. Turco: Modeling and numerical investigation of damage behavior in pantographic layers using a hemivariational formulation adapted for a Hencky-type discrete model, Contin. Mech. Thermodyn., 35 (2023), 1481–1494.