Email the author CASTIGLIANO’S THEOREM IN THE STUDY OF NONLINEAR ELASTIC STRUCTURAL ELEMENTS WITH POWER-LAW CONSTITUTIVE MODEL
Abstract
The strong development of the industry in the last period has led to the strong diversification of the materials used in engineering applications and also to the increase of the forces and speeds that appear in these systems. As a result, numerous researches were undertaken that took into account the non-linear behavior of materials during the operation of mechanical systems. Constituent laws can no longer be considered linear laws. Numerous forms of constitutive laws have been proposed to cover the existing non-linearities. Among them, the power law type laws were noted, in which the dependence is of an exponential type. In the work, results were obtained regarding strain and stress appearing in standard elements used in engineering. Also, the possibility of using Castigliano's theorem was studied in the case of application to materials with non-linear behavior, with a power law constitutive law. The obtained results can be useful to engineers because they offer quick results and with good precision.
Full Text:
PDFReferences
Drapaca, C.S.; Sivaloganathan, S.; Tenti, G. Nonlinear constitutive laws in viscoelasticity. Mathematics and Mechanics of Solids, Vol. 12, No. 5, pp. 475-501, 2007.
Nigam, H.; Dvorak, G.J.; Baheieldin, Y.A. An Experimental Investigation of Elastic-Plastic Behavior od a Fibrous Boron Aluminum Composite.Matrix-Dominated Mode. International Journal of Plasticity , 1994, 10 (1) , pp.23-48.
Gil, C.M.; Lissenden, C.J.;Lerch, B.A. Unusual nonlinear response of some metallic materials. Mechanics of Materials, 1999, 31 (9) , pp.565-577
Teferra, K.; Deodatis, G. Variability response functions for beams with nonlinear constitutive laws. Probabilistic Engineering Mechanics, Vol. 29, pp.139-148, 2012.
Michel, J.C.; Suquet, P. The Constitutive Law of Nonlinear Viscous and Porous Materials. Journal of the Mechanics and Physics of Solids, Vol. 40, No. 4, pp.783-812, 1992.
Cantón-Sánchez, R.; Romero, I. Dimen-sionally reduced nonlinear solids with general loads and constitutive laws: Theory and finite element formulation for rod-like bodies. International Journal of Solids and Structures, 2021, 210, pp.273-288.
Wackerfuss, J.; Gruttmann, F. A nonlinear Hu-Washizu variational formulation and related finite-element implementation for spatial beams with arbitrary moderate thick cross-sections. Computer Methods in Applied Mechanics and Engineering, 2011, 200 (17-20) , pp.1671-1690
Lainé, E; Vallée, C and Fortuné, D. Nonlinear isotropic constitutive laws: choice of the three invariants, convex potentials and constitutive inequalities. International Journal of Engineering Science, Vol. 37, No. 15, pp.1927-1941, 1999.
Ehrhardt, M. Absorbing Boundary Conditions for Hyperbolic Systems. Numerical Mathematics-Theory Methods and Applications, 2010, 3 (3) , pp.295-337
Yang, Y.B.; Leu, L.J. Constitutive Laws and Force Recovery Procedures in Nonlinear-Analysis of Trusses. Computer Methods in Applied Mechanics and Engneering, Vol. 92, No. 1 , pp. 121-131, 1991.
Mrówczynski, D.; Gajewski, T. Garbowski, T. Application of the genera-lized nonlinear constitutive law in 2D shear flexible beam structures. Archives of Civil Engineering, Vol. 67, No. 3, pp. 157-176, 2021.
Pedersen, P.; Taylor, J.E. Optimal Design Based on Power-Law Nonlinear Elasticity. IUTAM Symposium on Optimal Design with Advanced Materials, 1993, p.51-66.
Steigemann, M. Power-Law Solutions of Anisotropic Multi-material Elasticity Problems. Journal of Elasticity, 2015, 118 (1) , pp.63-87
Marin, M; Carrera, E and Vlase, S. An extension of the Hamilton variational principle for piezoelectric bodies with dipolar structure. Mechanics of Advanced Materials and Structures, Vol. 30, No. 12, pp. 2453-2457, 2023.
Nami, M.R. ; Janghorban, M. Free Vibration of Functionally Graded Size Dependent Nanoplates Based on Second Order Shear Deformation Theory using Nonlocal Elasticity Theory. Iranian Journal of Sciences and Technology-Transaction of Mechanical Engineering, 2015, 39 (M1) , pp.15-28.
Codarcea-Munteanu, L.; Marin, M.; Vlase, S. The study of vibrations in the context of porous micropolar media thermoelasticity and the absence of energy dissipation. Journal of Computational Applied Mechanics, Vol. 54, No. 3, pp. 437-454, 2023.
Giménez-Palomares, F; Jiménez-Mocholí, AJ; Espinós-Capilla, A. Obtaining Stress Distribution in Bending through Virtual Laboratories. 8th International Technology, Education and Development Conference (INTED) 2014, pp.6209-6218.
Katouzian, M.;, Vlase, S.; Marin,M. Elastic moduli for a rectangular fibers array arrangement in a two phases composite. Journal of Computational Applied Mechanics, Vol. 55, No. 3, pp. 538-551, 2024.
Korepanov, VV; Matveenko, VP; Sharda-kov, I.N. Numerical analysis of singular solutions of two-dimensional problems of asymmetric elasticity. Mechanics of Solids, 2013, 48 (4) , pp.397-404.
Elishakoff, I; Gentilini, C.; Viola, E. Three-dimensional analysis of an all-round clamped plate made of functionally graded materials. Acta Mechanica, 2005, 180, (1-4) , pp.21-36.
Braun, M. Bishoff, M.; Ramm, E. Nonlinear Shell Formulations for Complete 3-Dimensional Constitutive Laws including Composites and Laminates. Computational Mechanics, Vol. 15, No. 1, pp.1-18, 1994.
Kim, J.H.; Lee, M.G.; Kang, T.J. Development of nonlinear constitutive laws for anisotropic and asymmetric fiber reinforced composites. Polymer Composites, Vol. 29, No. 2, pp. 216-228, 2008.
Bardella, L. A phenomenological constitutive law for the nonlinear viscoelastic behaviour of epoxy resins in the glassy state. European Journal of Mechanics A-Solids, Vol. 20, No. 6, pp. 907-924, 2001.
Cai, W.; Wang, P.; Wang, Y.J. Linking fractional model to power-law constitutive model with applications to nonlinear polymeric stress-strain responses. Mechanics of Time-Dependent Materials, Vol. 27, No. 3, pp. 875-888, 2023.
Katouzian, K.; Vlase, S.; Scutaru, M.L. Finite Element Method-Based Simulation Creep Behavior of Viscoelastic Carbon-Fiber Composite. Polymers, Vol. 13, No. 7, Art. No. 1017, 2021.
Katouzian, M.; Vlase, S.; Calin, M.R. Experimental procedures to determine the viscoelastic parameters of laminated composites. Journal of Optoelectronics and Advanced Materials, 2011, 13 (9-10) , pp.1185-1188.
Launay, B. A Simplified Nonlinear Model for Describing the Viscoelastic Porperties of Wheat-Flour Doughs at High Shear Strain. Cereal Chemistry, 1990, 67 (1) , pp.25-31.
Suárez-Antola, R. Power Law and Stretched Exponential Responses in Composite Solids. International Conference on Materials Science, Machinery and Energy Engineering (MSMEE 2013), Materials Science, Machinery and Energy Engineering, 2014, 853 , pp.9-16.
Ponte Castaneda, P.; Suquet, P. Nonlinear composites. Advanced in Applied Mechanisc, Vol 34, 1998, 34 , pp.171-302.
Bowyer, E.P.; Krylov, V.V. Experimental investigation of damping flexural vibrations in glass fibre composite plates containing one- and two-dimensional acoustic black holes. Composite Structures, 2014, 107 , pp.406-415.
Xu, YJ; Wu, QL; Yao, F. Creep behavior of bagasse fiber reinforced polymer composites. Bioresource Technology, 2010, 101 (9) , pp.3280-328.
Teodorescu Draghicescu Horatiu, Scutaru M.L., Rosu D, Calin M.R., Griogore P, New Advanced Sandwich Composite with twill weave carbon and EPS, Journal of Optoelectronics and Advanced Materials-Rapid Ccommunications, 2013, Vol 15, (3-4), pg 199-203
Herve, E.; Dendievel, R..; Bonnet, G. Steady-State Power-Law Creep in Inclusion Matrix Composite-Materials. Acta Metallurgica Materialia, 1995, 43 (11) , pp.4027-4034.
Ionita, D.; Arghir, M. Consideration on the Lagrange Inhterpolation Method Applicable in Mechanical Engineering. Acta Technica Napocensis. Series Applied Mathematics, Mechanics and Engineering, 2021, 64 (2) , pp.329-332.
Vlase, S.; Teodorescu-Draghicescu, H.; Calin, M.R.; Serbina, L. Simulation of the elastic properties of some fibre-reinforced composite laminates under off-axis loading system.Optoelectronics and Advanced Materials-Rapid Communications, 2011, 5 (3-4) , pp.424-429
Wang, C.; Zhang, W.; Kassab, G.S. The validation of a generalized Hooke's law for coronary arteries. American Journal of Physiology-Heart and Circulatory Physiology, Vol. 294, No.1, pp. H66-H73, 2008, DI 10.1152/ajpheart.00703.2007
May-Newman, K.; Yin, F.C.P. A constitutive law for mitral valve tissue. Journal of Biomechanical Engineering-Transactions of the ASME, Vol. 120, No. 1, pp. 38-47, 1998, DI 10.1115/1.2834305.
Caggiano, L.R.; Holmes, J.W. A Comparison of Fiber Based Material Laws for Myocardial Scar. Journal of Elasticity. Vol 145, No. 1-2, pp. 321-337, 2021, DI 10.1007/s10659-021-09845-5.
Marcucci, L.; Bondi, M.; Randazzo, G.; Reggiani, C.; Natali, A.; Pavan, P.G. Fibre and extracellular matrix contributions to passive forces in human skeletal muscles: An experimental based constitutive law for numerical modelling of the passive element in the classical Hill-type three element model. PLOS ONE, Vol. 14, No. 11, Art. No. e0224232,2019, DI 10.1371/journal.pone. 0224232.
Agarwal, A. An improved parameter estimation and comparison for soft tissue constitutive models containing an exponential function. Biomechanics and Modeling in Mechanobiology, Vol. 16, No. 4, pp. 1309-1327, 2017, DI 10.1007/s10237-017-0889-3.
Nemani, A.V.; Ghaffari, M.; Nasiri, A. Advancements in Additive Manufacturing for Copper-Based Alloys and Composites: A Comprehensive Review. Journal of Manufacturing and Materials Processing, 2024, 8 (2).
Ishfaq, K.; Asad, M.; Harris, M.; Alfaify, A.; Anwar, S. ; Lamberti, L.; Scutaru, M.L. EDM of Ti-6Al-4V under Nano-Graphene Mixed Dielectric: A Detailed Investigation on Axial and Radial Dimensional Overcuts
NANOMATERIALS, 2022, 12 (3)
Seshagirirao, D.V.; Raju, S.; Mantrala, K.M. Tensile, hardness and metallurgical analysis of laser cured Co-Cr-Mo alloy samples. Advances in Materials and Processing Technologies, 2023, 9 (4) , pp.1448-1458.
Vlase,S.; Teodorescu-Draghicescu, H.; Calin, M.R. Behavior of multiphase fiber-reinforced polymers under short time cyclic loading. Optoelectronics and Advanced Materials-Rapid Ccommunications, 2011, 5 (3-4) , pp.419-423
Vlase, S.; Teodorescu-Draghicescu, H.; Scutaru, M.L. Advanced Polylite composite laminate material behavior to tensile stress on weft direction. Journal of Optoelectronics and Advanced Materials, 2012, 14 (7-8) , pp.658-663.
Noll, W. On the Continuity of the Solid and Fluid States. Indiana University Mathematics Journal, 1955, 4(1):3-81. doi: 10.1512/IUMJ.1955.4.54001
Luo, J.; Gao, J.J.; Qi, S.Y. Study on microstructure and mechanical properties of Ni60+WC/ Ni35/ AISI1040 functional surface gradient structure of remanufacturing chute plate for the mining scraper by a low cost high power CO2laser cladding technique. Materials Research Express, 2020, 7(8)
Ziaee, S. Dynamic stability of porous functionally graded nanotubes via nonlocal strain and velocity gradient theory. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2023, 45 (12)
Costa, A.; Miranda, R.; Yapp, D. Analysis of beam material interaction in welding of titanium with fiber lasers. Materials and Manufacturing Processes, 2007, 22 (7-8) , pp.798-803
Teodorescu-Draghicescu, H.; Vlase, S.; Mihalcica, M. Advanced Pultruded Glass Fibers-Reinforced Isophtalic Polyester Resin. MATERIALE PLASTICE, 2015, 52 (1) , pp.62-64.
Ali, S.Z.; Ho, W.O; Udrea, F. A High Temperature SOI CMOS NO2 Sensor. Proceedings of the 14th International Symposium on Olfaction and Electronic Nose, 2011, 1362 , pp.53-+
Flannery, A.F.; Mourlas, N.J.; Kovacs, G.T.A. PECVD silicon carbide as a chemically resistant material for micromachined transdu-cers. 9th International Conference on Solid-State Sensors and Actuators, Sensors and Actuators A-Physical, 1998, 70 (1-2) , pp.48-55.
Chen, Y.H.; Uchino, K .; Viehland, D. Substituent effects in 0.65Pb(Mg1/3Nb2/3) O3-0.35PbTiO3 piezoelectric ceramic. Journal of Electroceramics, 2001, 6 (1) , pp.13-19.
Tao, H.C.; Fan, L.Z.; Qu, X.H. In situ synthesis of TiO2-graphene nanosheets composites as anode materials for high-power lithium ion batteries. Electrochimica Acta, 2012, 69 , pp.328-333
Loewke, K.; Meyer, D.; Nemat-Nasser, S. Signal identification in smart composite materials using the two-dimensional fast Fourier transform. Smart Materials and Structures, 2005¸ 14 (5) , pp.895-903
Sawada, K.; Tabuchi, M.; Kimura, K. Analysis of long-term creep curves by constitutive equations. 11th International Conference of Creep and Fracture of Engineering Materials and Structures, May 04-09, 2008, Bad Berneck, Germany, 2009, Vol. 510-11, pp. 190-194, DI 10.1016/j.msea.2008.06.057
Aharonov, E.; Scholz, C.H. The Brittle-Ductile Transition Predicted by a Physics-Based Friction Law. Journal of Geophysical Research Solid Earth, Vol.124, No. 3, pp. 2721-2737, 2019, DI 10.1029/2018JB016878
Raj, S.V.; Freed, A.D. A phenomenological description of primary creep in class M materials. Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing, Vol. 283, No. 1-2, pp.196-202, 2000, DI 10.1016/S0921-5093(99)00790-X.
Fabrizio, M.; Lazzari, B.; Rivera, J.M. Asymptotic behaviour for a two-dimensional thermoelastic model. Mathematical Methods in the Applied Sciences, Vol. 30, No. 5, pp. 549-566, 2007, DI 10.1002/mma.798
Zhang, W.; Wang, C.; Kassab, G.S. The mathematical formulation of a generalized Hooke's law for blood vessels. Biomaterials, Vol. 28, No. 24, pp. 3569-3578, 2007, DI 10.1016/j.biomaterials.2007.04.030
Vlase, S.; Marin, M. Standard Deformations of Nonlinear Elastic Structural Elements with Power-Law Constitutive Model. Mathematics 2024, 12, 3992. https://doi.org/10.3390/math12243992.
Timoshenko, S. Elements of Strength of Materials. Van Nostrand; 5th edition,1968.
Timoshenko, S.P.; Woinowsky-Krieger, S. Theory of Plates and Shells, McGraw-Hill, New York, 1959
Refbacks
- There are currently no refbacks.
