Neuber 1961

H. NEUBER Professor of Engineering Mechanics, a n d Director of the L a b o r a t o r y for Strength o f Materials, Technical University, Munich, G e r m a n y

Theory of Stress Concentration for Shear-Strained Prismatical Bodies With Arbitrary Nonlinear Stress-Strain Law Knowledge of stress concentration is very important for engineers in many practical cases. But the stress-concentration factors for Hooke's law often cannot be used because most technical materials have stress-strain laws deviating from Hooke's law. In the second edition of his book "Kerbspannungslehre" the author gave a calculation method for a special nonlinear deformation law. In the present paper a general theory for arbitrary stress-strain laws is established which leads to a calculation method for the real values of the concentrated stresses in the material.

In the second edition of his book "Kerbspannungslehre" (Springer, Berlin, 1958), in the following referred to as KSL-2, the author investigated the behavior of stress and strain concentrations in the case of deviations from Hooke's law for the range of greater deformations. In that work the problem of shear distribution in prismatical bodies served as a suitable mechanical model to encounter the mathematical difficulties. Extensive calculations based on a special nonlinear deformation law resulted in a relation between the "ideal" maximum stress TH for Hooke's law and the real maximum stress r as well as the nominal stress TN for all notch shapes. In the meantime, a further exploration of the mathematical circumstances seemed to be desirable in order to attain a general calculation method applicable to arbitrary nonlinear deformation laws. Some hints were found in the fact that in the limit case of strong stress concentration at small nominal stresses, i.e., very great Hookian stress-concentration factors (SCF), there exists an identical relationship for flat as well as deep notches. In this paper the Hookian stress of the sharply curved notch is denoted as "leading-function" N(r). A hypothesis is established by which the connection between real stress and nominal stress as well as Hookian (elastic) SCF can be represented by a simple relation using the leading function. The relationship found in this way is applicable without contradiction to arbitrary nonlinear deformation laws. Herewith the problem is reduced to the determination of the leading function corresponding to a given (e.g., measured) deformation law. By means of geometrical conditions for orthogonal nets, general relations for A'(r) are established in this paper. For any deformation law F(T) the leading function is reducible to the integration of a linear second-order differential equation the coefficients of which are connected with F(R). For the special nonlinear deformation law as used in KSL-2 the leading function is derived with exact consideration of the notch-angle influence. In the case of arbitrary deformation law the leading function of 0-deg notch is proved to be the geometrical mean value of stress and deformation. As the notch-

angle influence indeed is insignificant, this relation can be used approximately for any deformation law and any notch angle. Further, it is shown that the geometrical mean valtie of the stress and strain-concentration factors at any stress-strain law is equal to the Hookian stress-concentration factor. This theory, which for the present is derived for two-dimensional shear only, could be generalized with good approximation to arbitrary two or three-dimensional states of stress by means of one of the well-known theories of failure.

The Basic Equations The complicated behavior of stress and strain concentration under any nonlinear deformation law can be studied by means of the shear flow in prismatic bodies. Fig. 1 shows a symmetrically notched prismatic body which serves as a mechanical model. The following notation is used: r y(r) G F = Gy

= = = =

shear stress shear deformation shear modulus in Hookian range deformation function

The general deformation law is F = F(r)

(1)

with the restricting condition lim F(T) r->-0

= r

(la)

Presented at the Tenth International Congress of Applied Mechanics, Stresa, Italy, August 31-September 7, 1960. Discussion of this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th Street, New York 17, N. Y., and will be accepted until January 10, 1962. Discussion received after the closing date will be returned. Manuscript received by ASME Applied Mechanics Division, September 26, 1960; revised draft, Feb. 22, 1961. 544

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i.e., the material satisfies Hooke's law in the case of small loading. The first step in the region of nonlinear deformation laws was realized in KSL-2 with regard to the special deformation law F(r) = [1 -

(16)

(T/T*)«]'/'

where r * represents a mathematical limitr-stress value which can be chosen for good correspondence with experimental results in the interesting range, Fig. 2.

r

The solution for the stress function can be represented in this case by a superposition of two conformal mappings. The results are applicable for materials with similar characteristic stressstrain curves (for instance, some steels) but not for all metals. As isotropy is presumed, the main directions of shear stress and shear deformation coincide along the stress lines. The latter may be denoted by v = const (» stress function). Let u be the only displacement, which goes in the direction of the axis of the prism (warping of the cross section) so that the gradient of u yields the shear deformation and the curves u = const are orthogonal to the stress lines. dsu = hudu and dsv — hvdv are the line elements, Fig. 3. From Fig. 4 there follows the condition of equilibrium TK = C

(2)

The shear deformation is equal to the gradient of the warping function du

(3)

7 = Corresponding to equation (1) F = Gy = Fig. 2

lines

G

(4)

Specioi deformation l a w as used in KSL-2

of constant

For simplification the warping function is normalized by

warping stress

G = C

lines

(v=const.)

(5)

and with equation (1) it follows that C

(6)

Geometrical Conditions Considering the components of the line elements, Fig. 5, one obtains ^

1 dy he dv

J ^ dx huu du du

(7)

1 dy sin