Applied Mathematics and Mechanics >
Thermo-mechanically coupled compatibility conditions in orthogonal curvilinear coordinates: equivalent temperature variation of initially stressed elastomers
Received date: 2024-11-26
Revised date: 2025-01-16
Online published: 2025-03-03
Supported by
Project supported by the National Natural Science Foundation of China (Nos. 12241205 and 12032019), the National Key Research and Development Program of China (No. 2022YFA1203200), and the Strategic Priority Research Program of Chinese Academy of Sciences (Nos. XDB0620101 and XDB0620103)
Copyright
The initial stresses widely exist in elastic materials. While achieving a continuum stress-free configuration through compatible unloading is desirable, mechanical unloading alone frequently proves insufficient, posing challenges in avoiding virtual stress-free configurations. In this paper, we introduce a novel concept of equivalent temperature variation to counteract the incompatible initial strain. Our focus is on initially stressed cylindrical and spherical elastomers, where we first derive the Saint-Venant, Beltrami-Michell, and Volterra integral conditions in orthogonal curvilinear coordinates using the exterior differential form theory. It is shown that for any given axially or spherically distributed initial stress, an equivalent temperature variation always exists. Furthermore, we propose two innovative initial stress forms based on the steady-state heat conduction. By introducing an equivalent temperature variation, the initial stress can be released through a compatible thermo-mechanical unloading process, offering valuable insights into the constitutive theory of initially stressed elastic materials.
Mengru ZHANG, Mingzhu XU, Weiting CHEN, Yapu ZHAO . Thermo-mechanically coupled compatibility conditions in orthogonal curvilinear coordinates: equivalent temperature variation of initially stressed elastomers[J]. Applied Mathematics and Mechanics, 2025 , 46(3) : 423 -446 . DOI: 10.1007/s10483-025-3230-9
| [1] | BEATTY, M. F. Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues — with examples. Applied Mechanics Reviews, 40(12), 1699–1734 (1987) |
| [2] | MARTINEZ, R. V., GLAVAN, A. C., KEPLINGER, C., OYETIBO, A. I., and WHITESIDES, G. M. Soft actuators and robots that are resistant to mechanical damage. Advanced Functional Materials, 24(20), 3003–3010 (2014) |
| [3] | ROGERS, J. A., SOMEYA, T., and HUANG, Y. G. Materials and mechanics for stretchable electronics. Science, 327(5973), 1603–1607 (2010) |
| [4] | CARPI, F., ANDERSON, I., BAUER, S., FREDIANI, G., GALLONE, G., GEI, M., GRAAF, C., JEAN-MISTRAL, C., KAAL, W., KOFOD, G., KOLLOSCH, M., KORNBLUH, R., LASSEN, B., MATYSEK, M., MICHEL, S., NOWAK, S., O'BRIEN, B., PEI, Q., PELRINE, R., RECHENBACH, B., ROSSET, S., and SHEA, H. Standards for dielectric elastomer transducers. Smart Materials and Structures, 24(10), 105025 (2015) |
| [5] | BUSTAMANTE, R., SHARIFF, M. H. B. M., and HOSSAIN, M. Mathematical formulations for elastic magneto-electrically coupled soft materials at finite strains: time-independent processes. International Journal of Engineering Science, 159, 103429 (2021) |
| [6] | SHARIFF, M. H. B. M., MERODIO, J., and BUSTAMANTE, R. Nonlinear elastic constitutive relations of residually stressed composites with stiff curved fibres. Applied Mathematics and Mechanics (English Edition), 43(10), 1515–1530 (2022) https://doi.org/10.1007/s10483-022-2910-7 |
| [7] | GAO, Y., LI, B., WANG, J. S., and FENG, X. Q. Fracture toughness analysis of helical fiber-reinforced biocomposites. Journal of the Mechanics and Physics of Solids, 146, 104206 (2021) |
| [8] | YIN, S. F., LI, B., and FENG, X. Q. Bio-chemo-mechanical theory of active shells. Journal of the Mechanics and Physics of Solids, 152, 104419 (2021) |
| [9] | WANG, X. H., HUANG, X. F., LIN, K., and ZHAO, Y. P. The constructions and pyrolysis of 3D kerogen macromolecular models: experiments and simulations. Global Challenges, 3(5), 1900006 (2019) |
| [10] | WANG, X. H. and ZHAO, Y. P. The time-temperature-maturity relationship: a chemical kinetic model of kerogen evolution based on a developed molecule-maturity index. Fuel, 278, 118264 (2020) |
| [11] | WANG, X. H., HUANG, X. F., GAO, M. N., and ZHAO, Y. P. Mechanical response of kerogen at high strain rates. International Journal of Impact Engineering, 155, 103905 (2021) |
| [12] | CHEN, W. T. and ZHAO, Y. P. Thermo-mechanically coupled constitutive equations for soft elastomers with arbitrary initial states. International Journal of Engineering Science, 178, 103730 (2022) |
| [13] | CHEN, W. T. and ZHAO, Y. P. Hyperelastic constitutive relations for soft elastomers with thermally-induced residual stress. International Journal of Engineering Science, 195, 103991 (2024) |
| [14] | CHEN, W. T., HUANG, X. F., YUAN, Q. Z., and ZHAO, Y. P. Initially stressed strain gradient elasticity: a constitutive model incorporates size effects and initial stresses. International Journal of Engineering Science, 205, 104166 (2024) |
| [15] | DING, L. Q., WANG, Z. Q., LV, J. G., WANG, Y., and LIU, B. L. A new model for real-time prediction of wellbore stability considering elastic and strength anisotropy of bedding formation. Energies, 15(1), 251 (2022) |
| [16] | TRUESDELL, C. Das ungel?ste hauptproblem der endlichen elastizit?tstheorie. Journal of Applied Mathematics and Mechanics, 36(3-4), 97–103 (1956) |
| [17] | ZHAO, Y. P. Modern Continuum Mechanics, Science Press, Beijing (2018) |
| [18] | MOONEY, M. A theory of large elastic deformation. Journal of Applied Physics, 11(9), 582–592 (1940) |
| [19] | RIVLIN, R. S. Large elastic deformations of isotropic materials, IV, further developments of the general theory. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 241(835), 379–397 (1948) |
| [20] | RAJAGOPAL, K. R. and SACCOMANDI, G. A novel approach to the description of constitutive relations. Frontiers in Materials, 3, 36 (2016) |
| [21] | DAL, H., A?IKG?Z, K., and BADIENIA, Y. On the performance of isotropic hyperelastic constitutive models for rubber-Like materials: a state of the art review. Applied Mechanics Reviews, 73(2), 020802 (2021) |
| [22] | ZHU, F. W., DUI, G. S., and REN, Q. W. A continuum model of jointed rock masses based on micromechanics and its integration algorithm. Science China Technological Sciences, 54(3) 581–590 (2011) |
| [23] | CHUONG, C. J. and FUNG, Y. C. Three-dimensional stress distribution in arteries. Journal of Biomechanical Engineering, 105(3), 268–274 (1983) |
| [24] | WITHERS, P. J. and BHADESHIA, H. K. D. H. Residual stress, part 2: nature and origins. Materials Science and Technology, 17(4), 366–375 (2001) |
| [25] | HOLZAPFEL, G. A. and OGDEN, R. W. Mechanics of Biological Tissue, Springer-Verlag, Berlin (2006) |
| [26] | HOSFORD, W. F. Mechanical Behavior of Materials, Cambridge University Press, Cambridge (2010) |
| [27] | ZANG, A. and STEPHANSSON, O. Stress Field of the Earth's Crust, Springer Science & Business Media, Berlin (2010) |
| [28] | CHEN, Y. C. and EBERTH, J. F. Constitutive function, residual stress, and state of uniform stress in arteries. Journal of the Mechanics and Physics of Solids, 60(6), 1145–1157 (2012) |
| [29] | STYLIANOPOULOS, T., MARTIN, J. D., CHAUHAN, V. P., JAIN, S. R., DIOP-FRIMPONG, B., BARDEESY, N., SMITH, B. L., FERRONE, C. R., HORNICEK, F. J., BOUCHER, Y., MUNN, L. L., and JAIN, R. K. Causes, consequences, and remedies for growth-induced solid stress in murine and human tumors. Proceedings of the National Academy of Sciences, 109(38), 15101–15108 (2012) |
| [30] | HEIDBACH, O., RAJABI, M., CUI, X. F., FUCHS, K., MüLLER, B., REINECKER, J., REITER, K., TINGAY, M., WENZEL, F., XIE, F. R., ZIEGLER, M. O., ZOBACK, M. L., and ZOBACK, M. The world stress map database release 2016: crustal stress pattern across scales. Tectonophysics, 744, 484–498 (2018) |
| [31] | WEBSTER, G. A. and EZEILO, A. N. Residual stress distributions and their influence on fatigue lifetimes. International Journal of Fatigue, 23, 375–383 (2001) |
| [32] | SHEN, W. H. and ZHAO, Y. P. Combined effect of pressure and shear stress on penny-shaped fluid-driven cracks. Journal of Applied Mechanics, 85(3), 031003 (2018) |
| [33] | SUN, F. Q., SHEN, W. H., and ZHAO, Y. P. Deflected trajectory of a single fluid-driven crack under anisotropic in-situ stress. Extreme Mechanics Letters, 29, 100483 (2019) |
| [34] | MONAJEMI, A. A. and MOHAMMADIMEHR, M. Effects of residual stress and viscous and hysteretic dampings on the stability of a spinning micro-shaft. Applied Mathematics and Mechanics (English Edition), 41(8), 1251–1268 (2020) https://doi.org/10.1007/s10483-020-2640-8 |
| [35] | CHEN, J., WANG, Y. W., and LI, X. F. Antiplane shear crack in a prestressed elastic medium based on the couple stress theory. Applied Mathematics and Mechanics (English Edition), 44(4), 583–602 (2023) https://doi.org/10.1007/s10483-023-2977-6 |
| [36] | FUNG, Y. C. What are the residual stresses doing in our blood vessels? Annals of Biomedical Engineering, 19(3), 237–249 (1991) |
| [37] | AMAR, M. B. and GORIELY, A. Growth and instability in elastic tissues. Journal of the Mechanics and Physics of Solids, 53(10), 2284–2319 (2005) |
| [38] | LI, B., CAO, Y. P., FENG, X. Q., and GAO, H. J. Surface wrinkling of mucosa induced by volumetric growth: theory, simulation and experiment. Journal of the Mechanics and Physics of Solids, 59(4), 758–774 (2011) |
| [39] | NAM, K., PARK, I., and KO, S. Patterning by controlled cracking. nature, 485(7397), 221–224 (2012) |
| [40] | DU, Y. K., Lü, C. F., DESTRADE, M., and CHEN, W. Q. Influence of initial residual stress on growth and pattern creation for a layered aorta. Scientific Reports, 9(1), 1–9 (2019) |
| [41] | HOGER, A. On the residual stress possible in an elastic body with material symmetry. Archive for Rational Mechanics and Analysis, 88(3), 271–289 (1985) |
| [42] | HOGER, A. On the determination of residual stress in an elastic body. Journal of Elasticity, 16(3), 303–324 (1986) |
| [43] | HUANG, X. F., LIU, Z. W., and XIE, H. M. Recent progress in residual stress measurement techniques. Acta Mechanica Solida Sinica, 26(6), 570–583 (2013) |
| [44] | GRECO, A., SGAMBITTERRA, E., and FURGIUELE, F. A new methodology for measuring residual stress using a modified Berkovich nano-indenter. International Journal of Mechanical Sciences, 207, 106662 (2021) |
| [45] | LI, X. W., LIU, J. W., WU, H., MIAO, K. S., WU, H., LI, R. G., LIU, C. L., FANG, W. B., and FAN, G. H. Research progress of residual stress measurement methods. Heliyon, 10(7), e28348 (2024) |
| [46] | LI, Z. Q., CUI, C. C., ARTEAGA, O., BIAN, S. B., TONG, H., LU, J., and XU, X. P. Full-field measurement of residual stress in single-crystal diamond substrates based on Mueller matrix microscopy. Measurement, 234, 114790 (2024) |
| [47] | MOON, S., CHOI, M., HONG, S., KIM, S. W., and YOON, M. Non-equibiaxial residual stress evaluation methodology using simulated indentation behavior and machine learning. Nuclear Engineering and Technology, 56(4), 1347–1356 (2024) |
| [48] | ZHOU, H. M., SUN, Q., XI, G. D., and LI, D. Q. Numerical prediction of process-induced residual stresses in glass bulb panel. Applied Mathematics and Mechanics (English Edition), 27(9), 1197–1206 (2006) https://doi.org/10.1007/s10483-006-0906-z |
| [49] | CHEN, Y. C. and HOGER, A. Constitutive functions of elastic materials in finite growth and deformations. Journal of Elasticity, 59, 175–193 (2000) |
| [50] | SARAVANAN, U. Representation for stress from a stressed reference configuration. International Journal of Engineering Science, 46(11), 1063–1076 (2008) |
| [51] | RODRIGUEZ, E. K., HOGER, A., and MCCULLOCH, A. D. Stress-dependent finite growth in soft elastic tissues. Journal of Biomechanics, 27(4), 455–467 (1994) |
| [52] | HOLZAPFEL, G. A., GASSER, T. C., and OGDEN, R. W. A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of Elasticity, 61, 1–48 (2000) |
| [53] | DU, Y. K., Lü, C. F., CHEN, W. Q., and DESTRADE, M. Modified multiplicative decomposition model for tissue growth: beyond the initial stress-free state. Journal of the Mechanics and Physics of Solids, 118, 133–151 (2018) |
| [54] | RAUSCH, M. K. and KUHL, E. On the effect of prestrain and residual stress in thin biological membranes. Journal of the Mechanics and Physics of Solids, 61(9), 1955–1969 (2013) |
| [55] | CHESTER, S. A. and ANAND, L. A thermo-mechanically coupled theory for fluid permeation in elastomeric materials: application to thermally responsive gels. Journal of the Mechanics and Physics of Solids, 59(10), 1978–2006 (2011) |
| [56] | ZHANG, M. R., CHEN, W. T., HUANG, X. F., YUAN, Q. Z., and ZHAO, Y. P. Hyperelastic constitutive relations for porous materials with initial stress. Journal of the Mechanics and Physics of Solids, 193, 105886 (2024) |
| [57] | REINA, C., DJODOM, L. F., ORTIZ, M., and CONTI, S. Kinematics of elasto-plasticity: validity and limits of applicability of F = FeFp for general three-dimensional deformations. Journal of the Mechanics and Physics of Solids, 121, 99–113 (2018) |
| [58] | YANG, F. Q., LI, Y., and ZHANG, K. A multiplicative finite strain deformation for diffusion-induced stress: an incremental approach. International Journal of Engineering Science, 187, 103841 (2023) |
| [59] | MUKHERJEE, S. and RAVINDRAN, P. Representation of stress and free energy for a viscoelastic body from a stressed reference. Journal of the Mechanics and Physics of Solids, 184, 105544 (2024) |
| [60] | SADAGOAPAN, T. S., RAVINDRAN, P., and MURTHY, H. S. N. A continuum model for predicting strain evolution in carbon fiber-reinforced composites subjected to cyclic loading. Sādhanā, 47(3), 1–9 (2022) |
| [61] | YAVARI, A. and SOZIO, F. On the direct and reverse multiplicative decompositions of deformation gradient in nonlinear anisotropic anelasticity. Journal of the Mechanics and Physics of Solids, 170, 105101 (2023) |
| [62] | SU, H. D., YAN, H. X., and JIN, B. Finite element method for coupled diffusion-deformation theory in polymeric gel based on slip-link model. Applied Mathematics and Mechanics (English Edition), 39(4), 581–596 (2018) https://doi.org/10.1007/s10483-018-2315-7 |
| [63] | DU, Y. K., SU, Y. P., Lü, C. F., CHEN, W. Q., and DESTRADE, M. Electro-mechanically guided growth and patterns. Journal of the Mechanics and Physics of Solids, 143, 104073 (2020) |
| [64] | HOGER, A. Virtual configurations and constitutive equations for residually stressed bodies with material symmetry. Journal of Elasticity, 48(2), 125–144 (1997) |
| [65] | JOHNSON, B. E. and HOGER, A. The dependence of the elasticity tensor on residual stress. Journal of Elasticity, 33(2), 145–165 (1993) |
| [66] | JOHNSON, B. E. and HOGER, A. The use of a virtual configuration in formulating constitutive equations for residually stressed elastic materials. Journal of Elasticity, 41(3), 177–215 (1995) |
| [67] | JOHNSON, B. E. and HOGER, A. The use of strain energy to quantify the effect of residual stress on mechanical behavior. Mathematics and Mechanics of Solids, 3(4), 447–470 (1998) |
| [68] | LEV, Y., FAYE, A., and VOLOKH, K. Y. Thermoelastic deformation and failure of rubberlike materials. Journal of the Mechanics and Physics of Solids, 122, 538–554 (2019) |
| [69] | FUHG, J. N., JADOON, A., WEEGER, O., SEIDL, D. T., and JONES, R. E. Polyconvex neural network models of thermoelasticity. Journal of the Mechanics and Physics of Solids, 192, 105837 (2024) |
| [70] | YAVARI, A. Compatibility equations of nonlinear elasticity for non-simply-connected bodies. Archive for Rational Mechanics and Analysis, 209(1), 237–253 (2013) |
| [71] | FOSDICK, R. L. Modern Developments in the Mechanics of Continua, Academic Press, London (1966) |
| [72] | CESàRO, E. Sulle formule del Volterra, fondamentali nella teoria delle distorsioni elastiche. Il Nuovo Cimento (1901-1910), 12(1), 143–154 (1906) |
| [73] | VOLTERRA, V. Sur l'équilibre des corps élastiques multiplement connexes. Annales scientifiques de l'école Normale Supérieure, 24, 401–517 (1907) |
| [74] | CIARLET, P. G., GRATIE, L., and MARDARE, C. A generalization of the classical Cesàro-Volterra path integral formula. Comptes Rendus. Mathématique, 347(9-10), 577–582 (2009) |
| [75] | CIARLET, P. G., GRATIE, L., and MARDARE, C. A Cesàro-Volterra formula with litte regularity. Journal de Mathématiques Pures et Appliquées, 93(1), 41–60 (2010) |
| [76] | BARRETTA, R. On Cesàro-Volterra method in orthotropic Saint-Venant beam. Journal of Elasticity, 112(2), 233–253 (2013) |
| [77] | FOSDICK, R. and ROYER-CARFAGNI, G. Hadamard's conditions of compatibility from Cesaro's line-integral representation. International Journal of Engineering Science, 146, 103174 (2020) |
| [78] | SKALAK, R., ZARGARYAN, S., JAIN, R. K., NETTI, P. A., and HOGER, A. Compatibility and the genesis of residual stress by volumetric growth. Journal of Mathematical Biology, 34(8), 889–914 (1996) |
| [79] | LANIR, Y. Mechanisms of residual stress in soft tissues. Journal of Biomechanical Engineering, 131(4), 044506 (2009) |
| [80] | ZAZA, D., CIAVARELLA, M., and ZURLO, G. Strain incompatibility as a source of residual stress in welding and additive manufacturing. European Journal of Mechanics-A/Solids, 85, 104147 (2021) |
| [81] | AMROUCHE, C., CIARLET, P. G., GRATIE, L., and KESAVAN, S. On Saint Venant's compatibility conditions and Poincare's lemma. Comptes Rendus Mathématique, 342(11), 887–891 (2006) |
| [82] | BORODACHEV, N. M. The equations of compatibility of deformations. Journal of Applied Mathematics and Mechanics, 65(6), 1021–1024 (2001) |
| [83] | LEE, C. W. Thermoelastic stresses in thick-walled cylinders under axial temperature gradient. Journal of Applied Mechanics, 33(2), 467–469 (1966) |
| [84] | MUTI, S. and DOKUZ, M. S. Two-dimensional Beltrami-Michell equations for a mixture of two linear elastic solids and some applications using the Airy stress function. International Journal of Solids and Structures, 59, 140–146 (2015) |
| [85] | TEODORESCU, P. P. On Beltrami-Michell and Beltrami-Michell type equations. Mechanics Research Communications, 3(6), 475–482 (1976) |
| [86] | VALLéE, C. and FORTUNé, D. Compatibility equations in shell theory. International Journal of Engineering Science, 34(5), 495–499 (1996) |
| [87] | ARGATOV, I. I., JIN, X. Q., and KEER, L. M. Depth-sensing spherical indentation of an elastic sphere on an elastic substrate. Journal of the Mechanics and Physics of Solids, 149, 104297 (2021) |
| [88] | WAHYUDI, H., CHU, K. W., and YU, A. B. 3D particle-scale modeling of gas-solids flow and heat transfer in fluidized beds with an immersed tube. International Journal of Heat and Mass Transfer, 97, 521–537 (2016) |
| [89] | OLSEN, T., KASPER, T., and DE WIT, J. Immersed tunnels in soft soil conditions experience from the last 20 years. Tunnelling and Underground Space Technology, 121, 104315 (2022) |
| [90] | HELMS, F., HAVERICH, A., B?ER, U., and WILHELMI, M. Transluminal compression increases mechanical stability, stiffness and endothelialization capacity of fibrin-based bioartificial blood vessels. Journal of the Mechanical Behavior of Biomedical Materials, 124, 104835 (2021) |
| [91] | ENOMOTO, T., MAO, X., and SATAKE, U. Cutting performance by surgical scissors of tubular soft tissues such as blood vessels. CIRP Annals, 70(1), 69–72 (2021) |
| [92] | CHENG, Y., YANG, H. W., XU, Z. Y., and LU, C. P. Cavity expansion analysis of normal indention of rocks with lateral confinement. Computers and Geotechnics, 145, 104693 (2022) |
| [93] | FARQUHAR, M. E., BURRAGE, K., SANTOS, R. W. D., BUENO-OROVIO, A., and LAWSON, B. A. J. Graph-based homogenisation for modelling cardiac fibrosis. Journal of Computational Physics, 459, 111126 (2022) |
| [94] | BELHAMADIA, Y., BRIFFARD, T., and FORTIN, A. Efficiency of parallel anisotropic mesh adaptation for the solution of the bidomain model in cardiac tissue. Journal of Computational Science, 61, 101656 (2022) |
| [95] | CIARLETTA, P., DESTRADE, M., GOWER, A. L., and TAFFETANI, M. Morphology of residually stressed tubular tissues: beyond the elastic multiplicative decomposition. Journal of the Mechanics and Physics of Solids, 90, 242–253 (2016) |
| [96] | MERODIO, J. and OGDEN, R. W. Extension, inflation and torsion of a residually stressed circular cylindrical tube. Continuum Mechanics and Thermodynamics, 28, 157–174 (2016) |
| [97] | MELNIKOV, A., OGDEN, R. W., DORFMANN, L., and MERODIO, J. Bifurcation analysis of elastic residually-stressed circular cylindrical tubes. International Journal of Solids and Structures, 226-227, 111062 (2021) |
| [98] | HOLZAPFEL, G. A. and OGDEN, R. W. Biomechanics of Soft Tissue in Cardiovascular Systems, Springer-Verlag, Vienna (2003) |
| [99] | YUCESOY, A. and PENCE, T. J. On the inflation of residually stressed spherical shells. Journal of Elasticity, 151(1), 107–126 (2022) |
| [100] | KACHANOV, M., SHAFIRO, B., and TSUKROV, I. Handbook of Elasticity Solutions, Springer Science & Business Media, Dordrecht (2003) |
| [101] | SEDOV, L. I. Foundations of the Non-Linear Mechanics of Continua, Pergamon Press, Oxford (1966) |
| [102] | CIARLET, P. G., CRISTINEL, M., and SHEN, M. Saint Venant compatibility equations in curvilinear coordinates. Analysis and Applications, 5(3), 231–251 (2007) |
| [103] | VLASOV, V. Z. The equations of continuity of deformations in curvilinear coordinates. Prikladnaya Matematika i Mekhanika, 8, 301 (1944) |
| [104] | BRDI?KA, M. The equations of compatibility and stress functions in tensor form. Czechoslovak Journal of Physics, 3(1), 52 (1953) |
| [105] | BARREKETTE, E. S. Stress compatibility equations in cylindrical coordinates. AIAA Journal, 6(4), 767–767 (1968) |
| [106] | TUBA, I. S. Compatibility equations for arbitrary orthogonal curvilinear coordinates. AIAA Journal, 4(9), 1695–1696 (1966) |
| [107] | ERINGEN, A. C. Mechanics of Continua, Robert E. Krieger Publishing Company, New York (1980) |
| [108] | JOG, C. S. The equations of equilibrium in orthogonal curvilinear reference coordinates. Journal of Elasticity, 104, 385–395 (2011) |
| [109] | LUBARDA, V. A. Remarks on axially and centrally symmetric elasticity problems. International Journal of Engineering Science, 47(4), 642–647 (2009) |
| [110] | BUSTAMANTE, R. and RAJAGOPAL, K. R. Modelling residual stresses in elastic bodies described by implicit constitutive relations. International Journal of Non-Linear Mechanics, 105, 113–129 (2018) |
| [111] | CIARLETTA, P., DESTRADE, M., and GOWER, A. On residual stresses and homeostasis: an elastic theory of functional adaptation in living matter. Scientific Reports, 6(1), 24390 (2016) |
| [112] | OGDEN, R. W. and SCHULZE-BAUER, C. A. J. Phenomenological and structural aspects of the mechanical response of arteries, ASME International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers, New York, 125–140 (2000) |
/
| 〈 |
|
〉 |
Email Alert
RSS