Engineering Transactions, 8, 4, pp. 727-761, 1960

### Wytężenie Materiału w Stanach Podkrytycznych

M. Życzkowski
Instytut Podstawowych Problemów Techniki PAN
Poland

Under the name of «exertion » of the material at a given point we shall understand the degree in which the physical state of the material approaches the dangerous state (elastic limit, yield point or strength). The equation of this state describes a limit surface (or hypersurface) in the space of exertion factors (stresses, temperature, etc.). This surface is assumed to be known. As a measure of exertion in subcritical states we assume as a rule the effective stress or the ratio of this stress to the dangerous limit for simple tension K,[Fig. 1, Eqs. (1.3) and (1.5)]. This measure, called by us elementary», implies the possibility of reaching the limit surface at one point, No, only, thus being valid for proportional loading. This measure is subjected to a critical analysis, a number of examples being given in which it leads to erroneous results. A general measure of exertion w is proposed. This is defined by the Eqs. (3.3) and (3.4) and Fig. 2. In this measure the probability various directions of the space of exertion factors of «motion» of a point in is taken into consideration.
Thus it is not a one-valued function of exertion factors. Only such an approach a can have a practical importance, because the same state (prestressing of concrete, for instance) can be considered to be more or less distant from the dangerous state depending on whether tensile or compressive load is expected. In the particular case where the «motion» is likely to take place in only one direction of the space of stresses, the general formula (3.3) becomes the elementary one (1.5). In the measure w other factors, disregarded in the computation of the elementary measure w, are also taken into account, therefore the integrals appearing in the Eq. (3.4) and written using HADAMARD's notation concern a n-dimensional space. In the case where the probability depends on the direction in the space of exertion factors only and does not depend on the location of the point P. the general formula (3.3) takes the form (3.5) and, after iteration in spherical coordinates - the form (3.8). computing the exertion are given, by means In Secs. 4-6 several examples of  the general formula (3.3). The limit surface is assumed in the form of the HUBER-MISES-HENCKY cylinder (5.1) and the BURZYNSKI paraboloid (2.12). The exertion is also investigated in the case of two-point distribution of probability in a one-dimensional space of exertion factors (Sec. 6). The results obtained are represented by means of tables and graphs. The problem of invertigation of optimum initial stresses and minimum material exertion giving a certain appraisal of the advantages of using such stresses, is also treated.

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#### References

[in Russian]

E. BELTRAMI, Sulle condizioni di resistenza dei corpi elastici, Opere Matematiche, Rend. Ist. Lomb., 81 (1885).

M. L.P. BRICE, Etude d'un critère de limmite élastique des solides, Annales des Ponts et Chaussées, 1956, 423-444 i 573-608.

W. BURZYŃSKI, Über die Anstrengungshypothesen, Schweiz. Bauzeitung, nr. 21, 1929.

W. BURZYŃSKI, Teoretyczne podstawy hipotez wytężenia, Czasop. Techn., Lwów 1929.

A. CLEBSCH, Theorie der Elastizität fester Körper, Leipzig 1862.

L. F. COFFIN, The Flow and Fracture of a Brittle Material, J. Appl. Mech., 3, 17 (1950), 233-248.

J. CZECHOWICZ, Zależność wytrzymałości drewna sosny na ściskanie wzdłuż włókien od ciężaru właściwego, Inzyn. Budown., 10, 11 (1954), 303-306.

P. DUWEZ, Materials for High Temperature Aircraft Structures, High Temperature Effects in Aircraft Structures, Pergamon Press, 1958, 58-79.

[in Russian]

A. FLETCHER, Table of Complete Elliptic Integrals, Philos. Magazine, seventh series, A 30 (1940/2), 516-519.

A.M. FREUDENTHAL, Wiley, New York 1950. The Inelastic Behavior of Engineering Materials and Structures,

[in Russian]

B. E. GATEWOOD, Thermal Stresses, McGraw-Hill, New York 1957.

[in Russian]

[in Russian]

J. GUEST, Strength of Ductile Materials under Combined Stresses, Phil. Mag., 50 (1900), 69-132.

B. T. HAIGH, The Strain Energy Function and the Elastic Limit, Engineering, 109 (1920) 158-160.

H. HENCKY, Zur Theorie plastischer Deformationen und der hierdurch im Material hervor-gerufenen achspannungen, Zeitsch. Angew. Math. Mech., 4 (1924), 323-334.

R. HILL, Proc. Roy. Soc., series A, 193 (1948), 281.

L. W. Hu, An Experimental Study on the Fracture of Metals under Hydrostatic Pressure, J. Mech. Phys. Solids, 2, 4 (1956), 96-103.

M. T. HUBER, Właściwa praca odkształcenia jako miara wytężenia materiału, Czasop. Techn. 22 (1904), 38-40, 49-50, 61-62, 80-81; Pisma, t. 2, PWN Warszawa 1956, 3-20.

M. T. HUBER, Kryteria wytrzymałościowe w stereomechanice technicznej, IW SIMP, Warszawa 1948.

[in Russian]

[in Russian]

A. KELLY, C. CHIOU, The Temperature Dependence of Flow Stress of an Age-Hardened Alloy, Acta Metallurgica, 9, 6 (1958), 565-571.

Z. KLĘBOWSKI, Obecny stan wytrzymałościowego obliczenia materiałów o własnościach uogólnionych, Przegl. Techn., 11 (1934).

Z. KLĘBOWSKI, Warunek wytrzymałościowy na tle hipotez wytężenia, PZWS Warszawa 1950.

Z. KLĘBOWSKI, Energetyczne hipotezy wytężenia a możność opracowania ogólnej teorii wytężenia, Ksiega Jubil. M. T. Hubera, Gdańsk 1950, 165-180.

Z. KLĘBOWSKI, O nowej uogólnionej teorii wytrzymałości N. N. Dawidienkowa i J. B. Fridmana, Przegl. Techn., 1952, s. 157.

W. KUNTZE, Zur Frage der Festigkeit bei räumlichen Spannungszuständen, Stahlbau, 10 (1937), 177.

W. LODE, Der Einfluss der mittleren Hauptspannung auf das Fliessen der Metalle, For-schungasrbeiten auf d. Gebiete des Ing., Heft 303, 1928.

J. MAJER, Beitrag zu den dreiachsigen Spannungs-Dehnungs Beziehungen Fester Stoffe, Öster. Ing.-Archiv, 2, 4 (1950), 140-153.

J. MARIN, Failure Theories of Materials Subjected to Combined Stresses, Proc. ASCE 61 (1935), 851-867.

A. MEHLDAHL, Brown Boveri Rev., Zurich 1944, 260.

[in Russian]

[in Russian]

R. MISES, Mechanik der festen Körper im plastisch deformablen Zustand, Göttingen Nachrichten, Math. Phys. Kl., 1913, 582-592.

R. MISES, Mechanik der plastischen Formänderung von Kristallen, Zeit. Angew. Math., 8 (1928), 161-185.

0. MOHR, Welche Umstände bedingen die Elastizitätsgrenze und den Bruch eines Materials, Ver. Deutsch. Ing., 44 (1900), 1524.

J. MURZEWSKI, Probabilistic Theory of Plastic and Brittle Behaviour of Quasi-Homogeneous Materials, Bull. Acad. Polon., Sc. Techn., 7 (1959), 641-650.

A. NADAI, Theory of Flow and Fracture of Solids, Vol. 1, McGraw-Hill, New York 1950.

H. NAKAZAWA, On The Correlation Between Stress Gradient and Elastic Fatigue Failure, Mem. Fac. Technol. Tokyo, Metrop. Univ., 1, 1951, 1-10.

J. ODERFELD, Nomogram wytrzymalosci drewna sosnowego, Zast. Matematyki, 1 (1954), 138-148.

W. OLSZAK, Prestressing Applied to Bound Columns, Arch, Mech. stos., 1 (1949), 80-98.

.W. OLSZAK W. URBANOWSKI, Ortotropia i niejednorodność w teorii plastyczności, Arch, Mech. stos., 8 (1958), 85-110.

W. OLSZAK i W. URBANOWSKI, The Plastic Potential and the Generalized Distortion Energy in the Theory of Non-Homogeneous Anisotropic Elastic-Plastic Bodies, Arch. Mech. stos., 8 (1958), 671-694.

T. PELCZYŃSKI, Wpływ stanu napięcia na przejście materiału w stan plastyczny, Przegl. Mech., 10(1951), 175-179, 204-208.

T. PELCZYÚSKI, Płaszczyzny poślizgów i pękania poślizgowego, Arch, Bud. Maszyn, 1 (1954), 445-457.

[in Russian]

W. J. M. RANKINE, Applied Mechanics, London 1856.

L. RENDULIC, Eine-Betrachtung zur Frage der plastischen, Grenzzustände, Bauingenieur, 19 (1938), 159-164.

E. L. ROBINSON, Effect of Temperature Variation on the Long-Time Rupture Strength of Steels, Ann. Meeting ASME, Atlantic City, Nov. 1951, Pap. 51-A-33.

M. RoS, A. EICHINGER, Versuche zur Klärung der Frage der Bruchgefahr, EMPA Berich- te 14, Zurich 1926.

M. Ros, Sollicitation déterminante d'après Huber des corps solides dont e mécanisme de déformation est engendré par des glissements, Ksiega Jubil. M. T. Hubera, Gdansk 1950, 339-362.

G. D. SANDEL, Schweiz. Bauzeitung, 95 (1930), 335.

F. SCHLEICHER, Der Spannungszustand an der Fliessgrenze, Zeitschr. Angew. Math. Mech. 6 (1926), 199-216.

B.B. COKOJIOBCKMN, Teopua nnacmuunocmu, ToCTEXH3MAT, MOCKBA-JIekHHrpax 1950.

C. TORRE, Die Grenzzustände statisch beanspruchter Stoffe, Schweiz. Archiv Angew. Wiss. Techn., 15 (1949), 116-145.

C. TORRE, Die Mechanik der Grenzbeanspruchungen, Öster. Ing.-Archiv, 1,4 (1950), 93-108.

C. TORRE, Grenzbedingungen für spröden Bruch und plastisches Verhalten bildsamer Metalle, Öster. Ing.-Archiv, 2,4 (1950), 174-189.

H. TRESCA, Mémoire Paris, sur l'écoulement des corps solides, Mémoires par divers savants, 18 (1868), 20 (1872).

[in Russian]

[in Russian]

J. WALCZAK, Nowoczesna miara wytężenia materiału, Arch. Mech, stos., 1, 3 (1951), 5-26.

[in Russian]

W. WEIBULL, A Survey of «Statistical Effects» in the Field of Material Failure, App. Mech. Rev., 11, 5 (1952).

H. M. WESTERGAARD, On the Resistance of Ductile Materials to Combined Stresses in Two or Three Directions Perpendicular to one another, J. Franklin Inst., 189, 1920, 627.

W. WIERZBICKI, Probabilistic and Semi-Probabilistic Method for the Investigation of Structure Safety, Arch. Mech. stos., 9 (1957), 685-694.

[in Russian]

M. ZAKRZEWSKI, Hipoteza złomu kruchego, Prace Wrock. Tow. Nauk., Wrocław 1958.

J. ZAWADZKI, Ciśnienie redukowane jako jeden a parametrów wytężenia, Rozpr. inzyn., 3, 5 (1957), 357-398.

M. ŻYCZKOWSKI, Tablice funkcji Eulera i pokrewnych, PWN Warszawa 1954.