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Fitting of the strength hypotheses

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Abstract

The equivalent stress approach allows the comparison of arbitrary multiaxial stress states with a uniaxial one. Based on the concept of the equivalent stress, several strength hypotheses (limit surfaces) were formulated. These hypotheses take into account not only existing information from the tests. They can also describe a experiences and different expectations concerning the material behavior mathematically. Due to the simplicity and clarity, the equivalent stress concept is widely used.

Collected experimental data, applications, and know-how make this approach an extremely powerful tool for engineering design. The method can also be easily applied in the case of new materials. As an example, polymethacrylimide (PMI) hard foam ROHACELL® 110IG, from manufacturer Evonik Röhm GmbH (Darmstadt), will be in the focus of this study.

Modern strength hypotheses are functions of several parameters. The application of these hypotheses requires in addition to tension, compression, and torsion test data further experimental results. Such data are often not reliable or widely scattered. Hence, the adjustment of the parameters of the chosen hypothesis is not unique. Some extrapolations with respect to the hydrostatic stress states can lead to unacceptable results and, therefore, the parameters must be restricted.

In this work, several restrictions based on the principles of conservative material description are introduced. They adjust a shape of the limit surface in the principal stress space. These restrictions are geometrically justified. The influence of various restrictions on the limit surface is analyzed. The effectiveness is tested with the help of own measured data. Reliable material descriptions for hard foams will be suggested.

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References

  1. Altenbach H., Altenbach J., Zolochevsky A.: Erweiterte Deformations-modelle und Versagens-kriterien der Werkstoff-mechanik. Deutscher Verlag für Grundstoffindustrie, Stuttgart (1995)

    Google Scholar 

  2. Altenbach, H., Bolchoun, A., Kolupaev, V.A.: Phenomenological yield and failure criteria. In: Altenbach H., Öchsner A. (eds.) Plasticity of Pressure-Sensitive Materials, Engineering Materials. Berlin Heidelberg. pp. 49–152 (2014)

  3. Altenbach, H., Kolupaev, V.A. : Classical and non-classical failure criteria. In: Altenbach, H., Sadowski, T. (eds.) Failure and Damage Analysis of Advanced Materials, International Centre for Mechanical Sciences CISM, Courses and Lectures Vol. 560., pp. 1–66. Springer, Heidelberg (2014)

  4. Annin B.D.: Theory of ideal plasticity with a singular yield surface. J. Appl. Mech. Tech. Phys. 40(2), 347–353 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Awaji H., Sato S.: A statistical theory for the fracture of brittle solids under multi-axial stresses. Int. J. Fract. 14(1), R13–R16 (1978)

    Article  Google Scholar 

  6. Backhaus G.: Deformations gesetze. Akademie-Verlag, Berlin (1983)

    Google Scholar 

  7. Benvenuto E.: An Introduction to the History of Structural Mechanics. Springer, New York (1991)

    Book  MATH  Google Scholar 

  8. Betten J.: Kontinuumsmechanik, 2nd ed. Springer, Berlin (2001)

    Book  Google Scholar 

  9. Bigoni D., Piccolroaz A.: Yield criteria for quasibrittle and frictional materials. Int. J. Solids Struct. 41(11), 2855–2878 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Birger I.A., Shopp B.F., Iosilevich G.B.: Strength Computations for Machine Components. Handbook (in Russ.: Raschet na prochnost’ detalej mashin. Spravochnik). Mashinostroenie, Moscow (1993)

    Google Scholar 

  11. Blatz P.J.: Application of finite elastic theory to the behavior of rubber-like materials. Rubber Chem. Technol. 36(5), 1459–1496 (1963)

    Article  Google Scholar 

  12. Blumenauer H.: Werkstoffprüfung. Dt. Verl. für Grund-stoff-industrie, Leipzig (1996)

    Google Scholar 

  13. Bolchoun A., Kolupaev V.A., Altenbach H.: Convex and non-convex flow surfaces (in German: Konvexe und nichtkonvexe Fließflächen). Forsch. Ing. 75(2), 73–92 (2011)

    Article  Google Scholar 

  14. Burzyński W.: Study on Material Effort Hypotheses, (in Polish: Studjum nad Hipotezami Wytężenia). Akademia Nauk Technicznych, Lwów (1928)

    Google Scholar 

  15. Burzyński W.: Über die Anstrengungshypothesen. Schweiz. Bauztg. 94(21), 259–262 (1929)

    Google Scholar 

  16. Burzyński W.: Über die Anstrengungshypothesen (Reply). Schweiz. Bauztg. 95(7), 87–88 (1930)

    Google Scholar 

  17. Burzyński, W.: Selected passages from Włodzimierz Burzyński’s doctoral dissertation “Study on Material Effort Hypotheses” printed in Polish by the Academy of Technical Sciences, Lwów, (1928), 1–192. Engng Trans. Polish Academy of Sciences, 57:3–4, 127–157 (2009)

  18. Chen W.F., Zhang H.: Structural Plasticity: Theory, Problems, and CAE Software. Springer, New York (1991)

    Book  MATH  Google Scholar 

  19. Christensen R.M.: The Theory of Materials Failure. University Press, Oxford (2013)

    Book  Google Scholar 

  20. Christensen R.M., Freeman D.C., DeTeresa S.J.: Failure criteria for isotropic materials, applications to low-density types. Int. J. Solids Struct. 39(4), 973–982 (2002)

    Article  MATH  Google Scholar 

  21. Coffin L.F., Schenectady N.Y.: The flow and fracture of a brittle material. J. Appl. Mech. 17, 233–248 (1950)

    Google Scholar 

  22. Coulomb C.A.: Essai sur une application des regles des maximis et minimis a quelques problemes de statique relatifs, a la architecture. Mem. Acad. Roy. Div. Sav 7, 343–387 (1776)

    Google Scholar 

  23. de Araújo F.C.: Elasticidade e Plasticidade. Imprensa Portuguesa, Porto (1962)

    Google Scholar 

  24. de Saint-Venant A.J.C.B.: Theorie du mouvement non permanent des eaux, avec application aux crues des rivieres et a l’introduction de marees dans leurs lits. C. R. Seances Acad. Sci. 73, 237–240 (1871)

    MATH  Google Scholar 

  25. DeRuntz J.A., Hoffman O.: The static strength of syntactic foams. Trans. ASME. J. Appl. Mech. 36, 551–557 (1969)

    Article  Google Scholar 

  26. Droste, A.: Beschreibung und Anwendung eines elastisch-plastischen Materialmodells mit Schädigung für hochporöse Metallschäume. Bericht Nr. II-9, Inst. für Mechanik (Bauwesen), Stuttgart, (2004)

  27. Drucker, D.C.: Stress-strain relations for strain hardening materials: Discussion and proposed experiments. In: Reissner, E., Prager, W., Stoker, R.R. (eds.) Non-Linear Problems in Mechanics of Continua. Proceedings of the First Symposium in Applied Mathematics, vol. 1, pp. 181–187. Brown University, American Mathematical Society, New York (1949)

  28. Drucker D.C., Prager W.: Soil mechanics and plastic analysis or limit design. Q. Appl. Math. 10, 157–165 (1952)

    MathSciNet  MATH  Google Scholar 

  29. Ehlers,W.: Constitutive equations for granularmaterials in geomechanical context. In: Hutter, K. (ed.) Continuum Mechanics in Environmental Science and Geophysics, Number 337 in CISM, pp. 313–402. Springer, Wien (1993)

  30. Ehlers W.: A single-surface yield function for geomaterials. Arch. Appl. Mech. 65(4), 246–259 (1995)

    Article  MATH  Google Scholar 

  31. Fahlbusch, N.-C.: Entwicklung und Analyse mikromechanischer Modelle zur Beschreibung des Effektivverhaltens von geschlossenzelligen Polymerschäumen. PhD thesis, Fachbereich Maschinenbau der Technischen Universität Darmstadt, (2015)

  32. Fahlbusch N.-C., Becker W., Kolupaev V.A., Geertz G.: Non-linear material behaviour and failure of closed-cell polymer foams. Acta Mech. 226(12), 1–9 (2015)

    MathSciNet  Google Scholar 

  33. Fahlbusch, N.-C., Kolupaev, V.A., Becker,W. : Generalized limit surfaces with an example of hard foams. In: Naumenko, K. (ed.) Advanced Methods of Continuum Mechanics for Materials and Structures., pp. 1–28. Springer, Berlin (2016)

  34. Finnie I., Heller W.R.: Creep of Engineering Materials. McGraw-Hill, New York (1959)

    Google Scholar 

  35. Föppl A., Föppl L.: Drang und Zwang: Eine höhere Festigkeitslehre für Ingenieure. R. Oldenbourg, München (1920)

    MATH  Google Scholar 

  36. Fromm H.: Grenzen des elastischen Verhaltens beanspruchter Stoffe. In: Auerbach, F., Hort, W. (eds.) Statik und Dynamik elastischer Körper nebst Anwendungsgebieten. II., pp. 359–435. Teil. Zum Gebrauch für Ingenieure, Phys. Math. (1931)

    Google Scholar 

  37. Gol’denblat I.I., Kopnov V.A.: Yield and Strength Criteria for Structural Materials (in Russ.: Kriterii prochnosti i plastichnosti konstrukzionnych materialov). Mashinostroenie, Moscow (1968)

    Google Scholar 

  38. Göldner H., Holzweißig F.: Leitfaden der Technischen Mechanik: Statik, Festigkeitslehre, Kinematik, Dynamik. Fachbuchverlag, Leipzig (1989)

    Google Scholar 

  39. Green R.J.: A plasticity theory for porous solids. Int. J. Mech. Sci. 14(4), 215–224 (1972)

    Article  MATH  Google Scholar 

  40. Hayhurst D.R.: Creep rupture under multi-axial states of stress. J. Mech. Phys. Solids 20(6), 381–390 (1972)

    Article  Google Scholar 

  41. Haythornthwaite R.M.: Range of yield condition in ideal plasticity. Proc. ASCE. J. Eng. Mech. Div. EM 6(87), 117–133 (1961)

    Google Scholar 

  42. Hencky H.: Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen. ZAMM 4(4), 323–334 (1924)

    Article  MATH  Google Scholar 

  43. Hill R.: LXVI. On the inhomogeneous deformation of a plastic lamina in a compression test. Philos. Mag. Ser. 7 41(319), 733–744 (1950)

    Article  MATH  Google Scholar 

  44. Huang, P.C.: Fracture criterion of isotropic materials. Technical report, Naval Surface Warfare Center, DTIC Document, NAVSWC TR 90-76. Dahlgren, Virginia (1986)

  45. Huber, M.T.: Specific strain work as a measurement of material effort (in Polish: Właściwa praca odkształcenia jako miara wytężenia materyału). Czas. Tech. 22, 34–40, 49–50, 61–62, 80–81 (1904)

  46. Ishlinsky A.Yu.: Hypothesis of strength of shape change (in Russ.: Gipoteza prochnosti formoizmenenija). Uchebnye Zapis. Mosk. Univ. Mekhanika 46, 104–114 (1940)

    Google Scholar 

  47. Ismar H., Mahrenholtz O.: Über Beanspruchungshypothesen für metallische Werkstoffe. Konstruktion 34, 305–310 (1982)

    Google Scholar 

  48. Ivlev D.D.: The theory of fracture of solids (in Russ.: K teorii razrusheniia tverdykh tel). J. Appl. Math. Mech. 23(3), 884–895 (1959)

    Article  MathSciNet  Google Scholar 

  49. Ko, W.L.: Application of the Finite Elastic Theory to the Behavior of Rubber-like Materials. PhD thesis, California Institute of Technology, Pasadena (1963)

  50. Kolupaev, V.A.: 3D-Creep Behaviour of Parts Made of Non-Reinforced Thermoplastics (in German: Dreidimensionales Kriechverhalten von Bauteilen aus unverstärkten Thermoplasten). PhD thesis, Martin-Luther-Universität Halle-Wittenberg, Halle, (2006)

  51. Kolupaev V.A., Altenbach H.: Considerations on the Unified Strength Theory due to Mao-Hong Yu (in German: Einige Überlegungen zur Unified Strength Theory von Mao-Hong Yu). Forsch. Ing. 74(3), 135–166 (2010)

    Article  Google Scholar 

  52. Kolupaev V.A., Becker W., Massow H.: Failure of hard foams under multiaxial loading. In: Grellmann, W. (ed.) Internationale wissenschaftliche Tagung Polymerwerkstoffe polymertech 14, 25–27., pp. 183–186. Juni, Merseburg Kunststoff-Kompetenzzentrum Halle-Merseburg, Institut für Polymerwerkstoffe (2014)

    Google Scholar 

  53. Kolupaev V.A., Becker W., Massow H., Dierkes D.: Design of test specimens from hard foams for the investigation of biaxial tensile strength (in German: Auslegung von Probekörpern aus Hartschaum zur Ermittlung der biaxialen Zugfestigkeit). Forsch. Ing. 78(3–4), 69–86 (2014)

    Article  Google Scholar 

  54. Kolupaev V.A., Becker W., Massow H., Kiegelmann E.M.: Reliable designs in foam (in German: Mit Schaumstoffen zuverlässig konstruieren). Mag. Plast. Kunstst. Int. 105(1-2), 32–35 (2015)

    Google Scholar 

  55. Kolupaev V.A., Bolchoun A., Altenbach H.: New trends in application of strength hypotheses (in German: Aktuelle Trends beim Einsatz von Festigkeitshypothesen). Konstruktion 61(5), 59–66 (2009)

    Google Scholar 

  56. Kolupaev, V.A., Bolchoun, A., Altenbach, H.: Geometrical-mechanical model applied to PVC-foams. In: Radusch, H.J., Fiedler, L., (eds.) 14. International Scientific Conference on Polymeric Materials 2010, 15–17 September. Martin-Luther-Universität Halle-Wittenberg, Halle (Saale), 31 p, (2011)

  57. Kolupaev V.A., Bolchoun A., Altenbach H.: Strength hypothesis applied to hard foams, advances in experimental mechanics VIII. Appl. Mech. Mater. 70, 99–104 (2011)

    Article  Google Scholar 

  58. Kolupaev, V.A., Fahlbusch, N.-C., Massow, H., Becker, W.: Multi-axial tests on hard foams. In: Cellular Materials - CellMAT 2014 October 22–24, Dresden, 2014. 6 p, Deutsche Gesellschaft für Materialkunde DGM.

  59. Kolupaev V.A., Yu M.-H., Altenbach H.: Visualisation of the unified strength theory. Arch. Appl. Mech. 83(7), 1061–1085 (2013)

    Article  MATH  Google Scholar 

  60. Lagzdin’ A.Zh., Tamuzh V.P.: Construction of a phenomenological theory of the fracture of an anisotropic medium (in Russ.: K postroeniju fenomenologicheskoj teorii razrushenija anizotropnoj sredy). Mekhanika Polim. 7(4), 563–571 (1971)

    Google Scholar 

  61. Lebedev A.A.: Development of the theories of strength in the mechanics of materials. Strength Mater. 43(5), 578–592 (2010)

    Article  Google Scholar 

  62. Lemaitre J., Chaboche J.L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  63. Marciniak Z.: Graphical representation of states of stress and strain. Arch. Mech. 3, 261–274 (1971)

    MathSciNet  MATH  Google Scholar 

  64. Mariotte, M.: Traité du Mouvement des Eaux et des Autres Corps Fluides. J. Jambert, Paris (1700)

  65. Matsuoka H., Nakai T.: Stress-deformation and strength characteristics of soil under three different principal stresses. Proc. Jpn. Soc. Civ. Eng. JSCE 232, 59–70 (1974)

    Article  Google Scholar 

  66. McAdam D.J.J.: Fracture of metals under combined stresses. Trans. Am. Soc. Met. 37, 538–566 (1946)

    Google Scholar 

  67. Mendera Z.: Wytężenie spoiny czołowej w interpretacji powierzchni granicznych. Prz. Spaw. SIMP XVIII(1), 6–13 (1966)

    Google Scholar 

  68. Mirolyubov, I.N.: On the generalization of the strength theory based on the octahedral stresses in the case of brittle materials (in Russ.: K voprosu ob obobshenii teorii prochnosti oktaedricheskikh kasatelnyh naprjazhenij na khrupkie materialy). Trudy Leningradskogo Technologicheskogo Instituta, pp. 42–52 (1953)

  69. Münch, M.: Mechanisches Kurzzeitverhalten von thermoplastischen Konstruktionsschaumstoffen unter mehrachsiger Beanspruchung. PhD thesis, Institut für Werkstofftechnik, Universität Kassel, Kassel, (2005)

  70. Murzewski, J.: Une theorie statistique du corps fragile quasihomogene. In IXe Congrès International de Mécanique Appliquée, ICAM-1956, vol. 5, pp. 313–320, Université de Bruxelles, (1957)

  71. Murzewski J.: A probabilistic theory of plastic and brittle behaviour of quasi-homogeneous materials. Arch. Mech. Stosow. 3(12), 203–227 (1960)

    MathSciNet  MATH  Google Scholar 

  72. Novozhilov V.V.: On the connection between stresses and strains in a nonlinear-elastic continuum (in Russ.: O svjazi mezhdu naprjazhenijami i deformazijami v nelinejno-uprugoj srede). Prikl. Mat. Mekhanika XV(2), 183–194 (1951)

    Google Scholar 

  73. Novozhilov V.V.: On the principles of the statical analysis of the experimental results for isotropic materials (in Russ.: O prinzipakh obrabotki rezultatov staticheskikh ispytanij izotropnykh materialov). Prikl. Mat. Mekhanika XV(6), 709–722 (1951)

    Google Scholar 

  74. Ottosen N.S., Ristinmaa M.: The Mechanics of Constitutive Modeling. Elsevier Science, London (2005)

    MATH  Google Scholar 

  75. Pae K.D.: The macroscopic yielding behaviour of polymers in multiaxial stress fields. J. Mater. Sci. 12, 1209–1214 (1977)

    Article  Google Scholar 

  76. Paul B.: Macroscopic plastic flow and brittle fracture. In: Liebowitz, H. (ed.) Fracture: An Advanced Treatise, vol II., pp. 313–496. Academic, New York (1968)

    Google Scholar 

  77. Pełczyński T.: Wpływ stanu napięcia na przejście materiału w stan plastyczny. Prz. Mech. 7, 204–208 (1951)

    Google Scholar 

  78. Penasa, M., Piccolroaz, A., Argani, L., Bigoni, D.: Integration algorithms of elastoplasticity for ceramic powder compaction. J. Eur. Ceram. Soc. 34(11), 2775–2788 (2014)

  79. Piccolroaz A., Bigoni D.: Yield criteria for quasibrittle and frictional materials: a generalization to surfaces with corners. Int. J. Solids Struct. 46(20), 3587–3596 (2009)

    Article  MATH  Google Scholar 

  80. Pisarenko G.S., Lebedev A.A.: Deformation and Strength of Materials under Complex Stress State (in Russ.: Deformirovanie i prochnost’ materialov pri slozhnom naprjazhennom sostojanii). Naukowa Dumka, Kiev (1976)

    Google Scholar 

  81. Prager W., Hodge P.: Theorie ideal plastischer Körper. Springer, Wien (1954)

    Book  MATH  Google Scholar 

  82. Rankine W.J.M.: Manual of Applied Mechanics. Griffin, London (1876)

    MATH  Google Scholar 

  83. Reckling K.: Plastizitätstheorie und ihre Anwendung auf Festigkeitsprobleme. Springer, Berlin (1967)

    Book  MATH  Google Scholar 

  84. Reuss A.: Vereinfachte Beschreibung der plastischen Formänderungsgeschwindigkeiten bei Voraussetzung der Schubspannungsfließ bedingung. ZAMM 13(5), 356–360 (1933)

    Article  MATH  Google Scholar 

  85. Rohacell. Product information ROHACELL® IG/IG-F. Evonik Industries, Evonik Röhm GmbH, Performance Polymers Business Unit, http://www.rohacell.com, Darmstadt, (2010)

  86. Sähn S., Göldner H., Nickel J., Fischer K.: Bruch- und Beurteilungskriterien in der Festigkeitslehre. Fachbuch verlag, Leipzig (1993)

    Google Scholar 

  87. Sauter, J., Winterger, N.: Neue und alte statische Festigkeitshypothesen. VDI, Reihe 1: Konstruktionstechnik / Maschinenelemente Nr. 191, Düsseldorf, (1990)

  88. Sayir M.: Zur Fließbedingung der Plastizitätstheorie. Ing. Arch. 39, 414–432 (1970)

    Article  MATH  Google Scholar 

  89. Sayir M., Ziegler H.: Der Verträglichkeitssatz der Plastizitätstheorie und seine Anwendung auf räumlich unstetige Felder. Z. Angew. Math. Phys. ZAMP 20(1), 78–93 (1969)

    Article  MATH  Google Scholar 

  90. Schleicher F.: Der Spannungszustand an der Fließgrenze (Plastizitätsbedingung). ZAMM 6(3), 199–216 (1926)

    Article  MATH  Google Scholar 

  91. Schleicher F.: Über die Sicherheit gegen Überschreiten der Fliessgrenze bei statischer Beanspruchung. Bauingenieur 9(15), 253–261 (1928)

    Google Scholar 

  92. Schlimmer M.: Zeitabhängiges mechanisches Werkstoffverhalten: Grundlagen, Experimente, Rechenverfahren für die Praxis. Springer, Berlin (1984)

    Book  Google Scholar 

  93. Schmidt R.: Über den Zusammenhang von Spannungen und Formänderungen im Verfestigungsgebiet. Ing. Arch. 3(3), 215–235 (1932)

    Article  MATH  Google Scholar 

  94. Shanley F.R.: Strength of Materials. McGraw-Hill, New York (1957)

    Google Scholar 

  95. Shaw M.C., Sata T.: The plastic behavior of cellular materials. Int. J. Mech. Sci. 8, 469–478 (1966)

    Article  Google Scholar 

  96. Skrzypek J.J.: Plasticity and Creep: Theory, Examples and Problems. CRC Press, Boca Raton (1993)

    MATH  Google Scholar 

  97. Timoshenko S.P.: History of Strength of Materials: With a Brief Account of the History of Theory of Elasticity and Theory of Structure. McGraw-Hill, New York (1953)

    Google Scholar 

  98. Tresca H.: Mémoire sur l’ecoulement des corps solides. Mém. Pres. Div. Savants 18, 733–799 (1868)

    Google Scholar 

  99. Tschoegl N.W.: Failure surfaces in principal stress space. J. Polym. Sci. C Polym. Symp. 32, 239–267 (1971)

    Article  Google Scholar 

  100. von Mises, R.: Mechanik des festen K örpers im plastischen deformablen Zustand. Nachrichten der Königlichen Gesellschaft der Wissenschaften Göttingen, Mathematisch-physikalische Klasse, pp. 589–592, (1913)

  101. von Mises R.: Mechanik der plastischen Formänderung von Kristallen. ZAMM 8, 161–185 (1928)

    Article  MATH  Google Scholar 

  102. Wang D.A., Pan J.: A non-quadratic yield function for polymeric foams. Int. J. Plast. 22(3), 434–458 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  103. Wolfram S.: The Mathematica Book: The Definitive Best-Selling Presentation of Mathematica by the Creator of the System. Wolfram Media, Champaign (2003)

    Google Scholar 

  104. Yagn Yu.I.: New methods of strength prediction (in Russ.: Novye metody pascheta na prochnost’). Vestn. Inzhenerov Tekhnikov 6, 237–244 (1931)

    Google Scholar 

  105. Yagn Yu.I.: Strength of Materials: Theory and Problems (in Russ.: Soprotivlenie materialov: teorja i zadachnik). Kubuch, Leningrad (1933)

    Google Scholar 

  106. Yu, M.-H.: General behaviour of isotropic yield function (in Chinese). Scientific and Technological Research Paper of Xi’an Jiaotong University, pp. 1–11, (1961)

  107. Yu M.-H.: Twin shear stress yield criterion. Int. J. Mech. Sci. 25(1), 71–74 (1983)

    Article  Google Scholar 

  108. Yu M.-H.: Twin shear stress yield criterion, Reply to Prof. Hill’s comments. Int. J. Mech. Sci. 25(11), 845–846 (1983)

    Article  Google Scholar 

  109. Yu M.-H.: Advances in strength theories for materials under complex stress state in the 20th century. Appl. Mech. Rev. 55(5), 169–218 (2002)

    Article  Google Scholar 

  110. Yu M.-H.: Unified Strength Theory and its Applications. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  111. Zhang J., Kikuchi N., Li V., Yee A., Nusholtz G.: Constitutive modeling of polymeric foam material subjected to dynamic crash loading. Int. J. Impact Eng. 21(5), 369–386 (1998)

    Article  Google Scholar 

  112. Zhang J., Lin Z., Wong A., Kikuchi N., Li V.C., Yee A.F., Nusholtz G.S.: Constitutive modeling and material characterization of polymeric foams. J. Eng. Mater. Technol. 119(3), 284–291 (1997)

    Article  Google Scholar 

  113. Zhang T.: A general constitutive relation for linear elastic foams. Int. J. Mech. Sci. 50, 1123–1132 (2008)

    Article  MATH  Google Scholar 

  114. Ziegler H.: Zum plastischen Potential der Bodenmechanik. Z. Angew. Math. Phys. ZAMP 20, 659–675 (1969)

    Article  MATH  Google Scholar 

  115. Źyczkowski M.: Combined Loadings in the Theory of Plasticity. PWN-Polish Scientific Publ., Warszawa (1981)

    MATH  Google Scholar 

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  1. Fraunhofer Institute for Structural Durability and System Reliability LBF, Schloßgartenstr. 6, 64289, Darmstadt, Germany

    V. A. Kolupaev

  2. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an, 710049, People’s Republic of China

    M.-H. Yu

  3. Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    H. Altenbach

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  1. V. A. Kolupaev
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  3. H. Altenbach
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Kolupaev, V.A., Yu, MH. & Altenbach, H. Fitting of the strength hypotheses. Acta Mech 227, 1533–1556 (2016). https://doi.org/10.1007/s00707-016-1566-9

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