CC BY
51
Wnuk M.P. / Физическая мезомеханика 12 4 (2009) 71-77
71
New mathematical models pertinent to material fracture at meso- and nanoscales
M.P. Wnuk
University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
Novel properties of the present cohesive crack models provide a better insight and an effective tool to explain multiscale nature of fracture process and the associated transitions from macro- to meso- and nanolevels of material response to deformation and fracture. These multiscale features of any real material appear to be inherent defense mechanisms provided by nature. New parameters introduced into the theory described here are: fracture quantum a0 and dimension D of the fractal used to represent the crack surface, or equivalent-ly — the fractal exponent a = (2- D)/2, where dimension D varies between 1 (smooth crack) and 2 (very rough crack).
As the degree of fractality increases, the characteristic material length is shown to rapidly grow to the levels around three orders of magnitude higher than those predicted for the classic case. Such effect is helpful in explaining an unusual size-sensitivity of fracture testing in materials with cementitious bonding such as concrete and certain types of ceramics, where fractal cracks are commonly observed. It turns out that while the effects due to quantization of the fracture model tend to disappear for longer cracks, the influence of the fractal geometry remains significant, or even becomes more pronounced for the long cracks.
In the limit of vanishing fracture quantum and/or reduced degree of fractality the quantized cohesive model of a fractal crack, as presented here, reduces to the well-known classic models of Dugdale-Barenblatt or to the linear elastic fracture mechanics or the quantized fracture mechanics fracture theories. Therefore, the basic concepts of linear elastic fracture mechanics, quantized fracture mechanics and fractal geometry are all incorporated into the present theory.
Keywords: fracture, deformation, cohesion, fractals, discrete process, multiscale, nano, meso, macro
Новые математические модели разрушения материалов на мезо- и наноуровне
М.П. Внук
Университет Виcкoнcин-Милуoки, Милуоки, WI 53201, США
Новые свойства представленных моделей когезионной трещины позволяют глубже понять и объяснить многомасштабную природу процесса разрушения и связанных с ним процессов перехода от макро- к мезо- и наноскопическим уровням отклика материала на деформацию и разрушение. Эти многомасштабные особенности любого реального материала присущи механизмам сопротивления разрушению, созданным самой природой. В данной работе в разработанную теорию введены новые параметры: квант разрушения a0 и размерность фрактала D, используемая для описания поверхности трещины, или эквивалентная величина — фрактальная экспонента а = (2-D)/2, где D равно 1 (гладкая трещина) или 2 (очень шероховатая трещина).
Показано, что с ростом степени фрактальности характерная длина материала быстро увеличивается до уровня, на три порядка величины превышающего значения, получаемые в классическом случае. Этот эффект позволяет объяснить необычайную размерную чувствительность в экспериментах на разрушение материалов с вяжущим связующим, таких как цемент и некоторые виды керамик, в которых обычно наблюдаются фрактальные трещины. Оказалось, что несмотря на то что для очень длинных трещин эффекты квантизации модели почти несущественны, для длинных трещин влияние фрактальной геометрии остается значительным или даже становится более выраженным.
Продемонстрировано, что при стремящемся к нулю квантуме разрушения и/или малой степени фрактальности квантовая когезионная модель фрактальной трещины сводится к хорошо известным моделям Дагдейла-Баренблата, или к линейной упругой механике разрушения, или к квантовой механике разрушения. Поэтому основные положения механики линейного разрушения, квантовой механики разрушения и фрактальной геометрии являются составными частями представленной теории.
Ключевые слова: разрушение, деформация, когезия, фракталы, дискретный процесс, многомасштабность, нано, мезо, макро
1. Introduction
Cohesive models of cracks (Fig. 1) have been remarkably successful in explaining certain essential features of
© Wnuk M.P., 2009
fracture process such as a finite stress at the crack tip, a non-zero crack opening displacement at the tip of the crack, and a certain equilibrium length of cohesive zone. This
length increases with the applied load up to the point of incipient fracture and it serves as a measure of the material resistance to crack propagation.
We shall begin with re-visiting this model and then go on to incorporate fractal and discrete nature of fracture occurring in real materials. Recent research on nanocracks [1, 2] indicate that the classic failure criteria break down for very small cracks. Such examples show the need for novel non-local failure criteria for design of structures, in which multiscale fracture mechanisms are present. In order to refine available mathematical tools and to extend their validity for nanoscale and for fractal (rough) cracks we will merge here the basic concepts of the cohesive crack model with the fractal view of the decohesion process. At the same time we shall employ the non-local criteria, which remain valid at atomistic or nanoscale levels. The present model is used to predict the upper and the lower bounds for the cohesion modulus, which is used as a measure of material resistance to initiation and to propagation of fracture.
The concept of discrete nature of fracture has been investigated by many researchers, beginning with H. Neuber, V.V. Novozhilov, M.P. Wnuk, A. Seweryn, N. Pugno and R.S. Ruoff [3-7]. Comparison of various fracture criteria used to describe discrete fracture process suggests that a certain finite length parameter, determined either by the microstructural or atomistic considerations, must be introduced into the basic equations underlying the theory of fracture. Even though various names and symbols have been used in describing this entity, such as "Neuber particle" by M.L. Williams [8], "unit growth step" by M.P. Wnuk [5] and "fracture quantum" by V.V. Novozhilov [4] and N. Pugno and R.S. Ruoff [7], the physical meaning of these length parameters is the same, and it can be accounted for mathematically by a discretization technique, cf. [9, 10].
A discretization procedure for the cohesive model of a fractal crack requires that all pertinent entities describing the influence of the cohesive stress that restrains opening of the crack, such as effective stress intensity factor, the modulus of cohesion, extent of the end zone and the opening displacement within the high-strain region adjacent to the crack tip are re-visited and replaced by certain averages
Fig. 1. Cohesive crack model created by extending the crack to a new length, which incorporates the length of the physical crack and the length the equilibrium cohesive zone. For quantized fracture mechanics the unit growth step or the fracture quantum A is embedded within the cohesive zone and located next to the crack tip. This quantity enters all equations pertinent to the quantized fracture mechanics representation of fracture
over a finite length referred to as either "unit growth step" [5] or "fracture quantum" [4, 7]. Thus, two novel aspects of the model enter the theory: (1) degree of fractality related to the roughness of the newly created surface, and (2) discrete nature of the propagating crack.
The fractal crack model described here is based on a simplifying assumption, according to which the original problem is approximated by considerations of a smooth crack embedded in the stress field generated by a fractal crack [11]. The well-known concepts of the stress intensity factor and the Barenblatt cohesive modulus, which is a measure of material toughness, have been re-defined to accommodate the fractal view of fracture. Specifically, the cohesion modulus, in addition to its dependence on the distribution of the cohesion forces, is shown now to be a function of the "degree of fractality" reflected by the fractal dimension D, or by the Hausford fractal roughness parameter H. It is also a function of the unit growth step, the so-called fracture quantum and depends on the form of the "decohesion law". As expected in the limit of the fractal dimension D approaching 1 and for the disappearing magnitude of the fracture quantum, one recovers the cohesive crack model known from the fracture mechanics of smooth cracks.
In what follows we shall solve the problem assuming that we are dealing with a fractal crack represented by a cohesive model and that propagation of fracture is not continuous but discrete. Naturally, when the problem is reduced to that of a smooth crack and when the discrete nature of decohesion act is neglected; the present model yields the results identical to those known in classical fracture mechanics. This is analogous to the "correspondence principle" used in quantum mechanics to recover the solutions pertinent to continuum.
2. Essential results
Cohesion modulus for the Dugdale model
Sdx
Kcoh = 2
Hc+R
V(c+R)
2 2 2 - x2
= Syl n(c + R)
2 -i —cos n I c + R
= Sjm00yl X + R
2 -i —cos
n [ X + R
X
(1)
has been compared with the analogous result obtained from the discrete cohesive model ("Dugdale discrete"). Here, in addition to the quantities c, R and S a new variable appears; and it is the fracture quantum a0. The final result is
(Kcoh) = Kcdoh = S4na0 X
"|2 1V2
-cos • |-| dY8 . (2)
Y + R 11
The length variables X and R represent the nondimensi-onal crack length X = c/a0 and the equilibrium length of the cohesive zone R = R/a0, respectively. The expression (1) simplifies substantially for the physically important case of the relaxation length R being much smaller than the crack length X. This is so-called Barenblatt condition. Under an assumption of R << X equation (1) reduces to an expression, which does not depend on the crack length, and in what follows will be used as the normalizing constant for the cohesion modulus.
If the inverse cosine function in (1) is denoted by y, then both sides of the resulting equality 1
-;-= COS y (3)
1 + R/X (3)
can be expanded into the Maclaurin series yielding a simple result y = tJ 2 Rj X, or
cos
(4)
Using this expression and neglecting R vs. X under the square root in (1) gives the desired formula for the Dugdale-Barenblatt cohesion modulus valid in the limiting case of R << X, namely
Ks = [KcohW = - ^VnOOVzR. (5)
n
Next, the equilibrium length R is evaluated as a function of the loading parameter Q (=tcct/2S). When the stress intensity factors due to the applied load Kc and (K^) are calculated according to both models
(X + R f,
V/2 (6)
(Kc) = aJnaO
and set equal to the respective cohesion moduli given in equations (1) and (2), the following relationships defining the equilibrium length R as a function of the loading parameter Q are obtained for both models
X
Qd = cos
-1
X + R
qD =
X+1
I (Y + R)
X
cos
Y + R
(7)
,/X + R +12 '
It follows that for the same load the length of the equilibrium cohesive zone calculated from the discrete Dugdale model is greater that the one predicted by the classic model of Dugdale, namely
Rd > RD at an arbitrary Q. (8)
This length serves as a measure of material ability to relax stresses prior to the onset of fracture.
It is of interest to compare these results against analogous equation expressing the relation between the loading
parameter Q and the equilibrium length of the relaxation zone R, as predicted from the cohesive discrete model with fractal geometry of the crack taken into account. The mathematics involved here is rather cumbersome, and the details were shown by M. Wnuk and A. Yavari [10]. To distinguish the quantities pertinent to so stated problem additional subscripts (or superscripts) D, d and f will be used; meaning "Dugdale", "discrete" and "fractal", correspondingly. With auxiliary notation defined below let us state the essential results. To describe the discreteness and the roughness of the crack, we shall need the following functions
, , 1 1(1 - ^ )2a + (1+ ^ )2a X(a) = -2TI-
n o
-d
(1 - s2)a
H(X,R,a) = I (1 -¿I+^Sf ds.
X/ ( X +R )
n.2«
(9)
„2o.ii 2\ a
n (1 - S )
With such notation the relationship between the applied load and the equilibrium relaxation zone R associated with the crack of length X and characterized by the fractality exponent a reads
Qf =
2x(a)
X+1
1/2
I (Y + R)2aH2(Y,R,a)dY
X
"|1/2
2a + 1
(X + R + 1)2a+1 - (X + R)2a+1
(10)
Now, when formula is used in conjunction with two equations in (7) we are able to compare the equilibrium length of the cohesive zone R associated with three models discussed in the present work, namely — classic Dugdale model (D), discrete cohesive model (Dd) and the discrete cohesive model which incorporates the fractal geometry (Ddf or just f). Due to the nature of the equations (7) and the equation (10) the functions QD, QDd and Qf are plot-
Q 1.0
0.5
0.0
J-—-—"— ^^ ................. „ __________' 3, - " " ' ■-
- /</*' !'' '".'■'"*'4
// y ' /s ^ • 1 -Qd(1. R)
// * 2-QM(1,R)
3 - Qf(1, R, .4)
4 - Qj(1, R, .3)
0.0
0.4
0.8
R
Fig. 2. Graphs representing the relationships between the applied loading parameter Q and the equilibrium length of the relaxation zone R. The top curve corresponds to the classic cohesive crack model of Dugdale-Barenblatt. The curve below the top one corresponds to the discretized cohesive crack model, cf. equations (7), while the other two curves were obtained from equation (10) pertinent to the fractal geometry of a discrete cohesive crack. In all four cases shown the nondimensional crack length X is assumed to be one
X
Fig. 3. The inverse functions corresponding to the same four cases shown in Fig. 2. Now the equilibrium lengths of the relaxation zone R are shown as functions of the applied load Q, at a fixed crack length X = 1. It is seen that the length R, observed for any constant load, significantly increases when the discrete nature of fracture and/or fractal geometry of the crack is accounted for. Top two curves correspond to fractal discrete model at a equal 0.3 and 0.4. The next curve resulted from discretized smooth cohesive crack model (it would be identical with the curve obtained for a fractal crack when a = 0.5). The lowest curve represents Dugdale's classic result. It can be shown that for X >> 1 the last two curves merge
ted first against the length R (Fig. 2). Then, the inverse functions are determined numerically (Fig. 3). We note that only the first of the equations in (7) can be inverted in a closed form; this is the classic result of Dugdale. The other two results, second equation in (7) and equation (10) can only be manipulated numerically to find the inverse functions R = R(Q). This has been done and the results are shown in Fig. 3.
As can be seen from Fig. 3, one obtains different predictions from each model. The equilibrium length R, when considered for a fixed load Q, tends to increase in the following manner: the smallest value is obtained for the classic Dugdale model, a larger value is predicted by the discretized cohesive model, while incorporating the fractal geometry via index a leads to yet larger values of R. Since the length R and it is indicative of material ability to relax stresses prior to the onset of crack propagation it can be used as a measure of material fracture toughness. The physical interpretation of these results confirms a supposition that nature itself provides intricate mechanisms for enhancing material resistance to propagation of fracture; it is accomplished by a transition from continuous to a discrete mode of crack growth, and further enhanced by presence of roughness of the crack surface as reflected through the fractal geometry. These conclusions are illustrated graphically in the last section (cf. Fig. 7). It is noteworthy that the phenomena discussed here are essential for explaining fracture in the nano-range. The parametric study indicates that while the effects of the quantization procedure disappear for the longer cracks (much larger than the quantum fracture), the influence of the fractal geometry remains significant also for long cracks.
One might ask a question — could the relations between load Q and the equilibrium length R be simplified for the limiting case ofR being much smaller than the crack length
X? For the first two models considered here, classic Dugdale and quantized Dugdale, see equations (7), the answer is "yes". Yet, for the discrete cohesive fractal crack, the answer is "no". Applying the condition R << X in equations (7), one obtains the approximations valid for the R << X case, namely
Q? =
Qm =
2R
1/2
X
2R
12
(11)
X +1/2_
They both can be readily inverted resulting in the R vs. Q relation. It is noted, however, that for the nanoscale fracture the Barenblatt condition R << X is usually not satisfied, and then the calculations must be performed through application of the complete equations (7) and (10).
If the length of the relaxation zone R present in the equations (11) is replaced by its critical value, occurring at the onset of fracture, Rcr
(12)
K2
R-=i (1 -,
then the equations (11) can be used to express the critical stresses predicted by both models
QD =
QDrd =
X
1/2
2Rcr
1/2
(13)
X +1/2
The expression (12) resulted from setting the cohesion modulus defined by (5) to the material fracture toughness K c. Graphical representations of these results are provided in Fig. 4. It is seen that the first of these equations predicts an infinite critical stress for vanishing crack length; as expected this is identical with the Griffith result. However, when the model is subjected to discretization, the situation changes drastically, see the curved lines shown in Figs. 4 and 5. In the nanocracks range these curves diverge substantially from
Fig. 4. Critical stresses predicted by the Griffith theory (straight lines) and the quantized cohesive crack model, shown for two different levels of material toughness, Rcr = a0 and Rcr = 10a0. All graphs show the critical stress as a function of crack length using the double logarithmic scale
0.1
1 -sD(X)
2-sDd(X)
2........
0.01
0.1
10 X
Fig. 5. Critical stresses normalized through the inherent material strength shown as functions of the crack length. The straight line corresponds to the Griffith theory, while the curve (shown in a double logarithmic scale) represents the prediction of the quantized cohesive crack model, as defined by the second equation in (16). The differences between the two results are significant only within the nanocracks range. For a material with no initial pre-existing defects the new model predicts finite critical stress, which is identified with the material "inherent strength"
the straight lines produced by the Griffith theory. The curves with finite values attained at X = 0 were generated by the second equation in (13), and they represent the predictions for the critical stresses valid for the nanocracks range. By "nanocracks" we understand the cracks with sizes comparable to the fracture quantum a0. Most important is the finite strength predicted from the discrete fracture model for a virgin material with no pre-existing defects (X = 0). In a nondimensional form suggested by the definition of the parameter Q, the inherent strength reads
Qo = [QDdS^o = [2VR ]R=Sa = 2VRT- (14)
We shall refer to this quantity as a dimensionless inherent material strength. With substitution of (12) and invoking
Fig. 6. Function Fcoh defined by equation (17) demonstrates the enhancement of the cohesion modulus when the crack is represented by the discrete cohesive model and when the crack length is less or about equal the fracture quantum a0. Note that this phenomenon is particularly visible when the Barenblatt condition of R being much less than X is not satisfied; that is when the length of the relaxation zone R may assume arbitrary values compared to the crack length X. This is often the case for the nanocracks accompanied by their associated relaxation zones
the definition Q duce to
=
-nn/ 2S, it can be readily shown to re-
(15)
This expression for the inherent material strength is identical to the quantized fracture mechanics result presented by N. Pugno and R.S. Ruoff [7]. This was to be expected, as despite very different physical interpretations of the near-tip conditions, the Dugdale-Barenblatt model in its classic form generates the same critical stress as the Griffith theory. Such property of the model carries over to its quantized mode, since the quantized fracture mechanics (quantized Griffith model) is compared with the quantized cohesive crack model, or quantized Dugdale-Barenblatt model.
Finally, if both sides of the expressions (13) are divided by Qo defined in (14), the following simple results for the normalized critical stresses are obtained
sD =
sDd =
QDr ^D =
Qo
QDd c = aDd
Qo
1
(16)
2X +1
This is perhaps the most concise and most appropriate way to normalize the critical stresses (note that the inherent material strength ct0 is used here as the normalization constant). With such a choice of the normalization constant, all pertinent graphs representing dependence of the critical stresses on the crack length reduce to just two curves, shown in Fig. 5. The differences between the Griffith theory and the quantized cohesive crack model are significant only in the nanocrack range. Note that in (16) the dependence on the toughness level disappears, as Kc is hidden now in the normalization constant ct0 .
Another result pertinent specifically to nanofracture initiated by small cracks, i.e., cracks on the order of magnitude of fracture quantum a0. For this nanorange the nondi-mensional cohesion modulus for a discrete cohesive crack
^coh( X, R) =
K
coh
Ks
X+1
11/2
j (Y + R)[2/ncos-1(Y/(Y + R))]2dY i
_ X_L_ (17)
2/ n S^na0\[2R
shows a visible enhancement for X > 0. This effect becomes more pronounced when the Barenblatt condition R << X is not satisfied, as is often the case in the nanocracks range. On the other hand, when the length R is much smaller than the length X, then the ratio (17) does not depend on the crack length and can be shown to approach unity (as then the integral inside the curly bracket of (17) reduces to (4/ n2 )2R). This effect and the phenomenon of the enhancement of toughness for the nanocracks are illustrated in Fig. 6.
Fig. 7. Near tip region of a crack represented by four different models. The size of the equilibrium cohesive zone associated with a crack serves as a measure of the degree of stress relaxation occurring in the crack vicinity. It appears that the two features of the model considered here, discreteness and roughness of the newly created surface are inherent mechanisms of stress relaxation provided by nature
3. Conclusions
Relations between applied load and the equilibrium length of the cohesive zone have been established for three different mathematical representations of the crack, and these are
a) cohesive crack model of Dugdale-Barenblatt for a smooth crack described by two ^-factors, Kc and KS corresponding to the applied stress and the cohesive stress, respectively,
b) discrete cohesive crack model described by the averages (Ka)c+a0 and (Ks)c+a0> and
c) discrete and fractal cohesive crack model involving the fractal equivalents of the averages used in second part, namely and (kS) .
\ / c+flo * 'c+flo
For a classic linear elastic fracture mechanics crack
model there is no cohesive zone, and the crack itself provides a mechanism for relaxing the high stresses in the vicinity of a stress concentrator. Therefore, the classic representation may be thought of as a limiting case of a more general and refined mathematical formulation involving cohesive zone associated with a crack. For such a representation the equilibrium between the driving force Kc and
material resistance KS2 defines a unique relation between the length of the equilibrium cohesion zone, say R, and the applied load, say Q. The equilibrium between R (= R/a0) and Q is maintained during the loading process up to the point of incipient fracture.
Cohesive zone generated prior to fracture is related to the material fracture toughness. It also provides an important mechanism for relaxing stresses prior to fracture initiated in the immediate vicinity of a stress concentrator.
Figure 7 shows schematically four sketches of a crack represented by four mathematical models; (a) the linear elastic fracture mechanics (LEFM) concept of a Griffith crack embedded in a linear elastic solid, (b) Dugdale-Barenblatt cohesive model, (c) discretetized cohesive model and, finally, by (d) a fractal cohesive and discrete model. The very first crack shown in the figure has no cohesive zone at all, and the stresses are singular at the tip of the crack. In this case fracture toughness must be measured by employing the ASTM standards that make no mention of cohesion. The second crack in the figure corresponds to a cohesive model suggested independently by G.I. Barenblatt [12] and D.S. Dugdale [13]. With cohesion included we gain a better insight into material response to fracture by visualizing the effect of the cohesive zone. In the next two models considered we have generalized the cohesive crack model by incorporating material mesomechanical features related to discrete growth of the crack and with its fractal geometry taken into account. Therefore, in the most advanced model considered here two new variables enter the theory: fracture quantum a0 and degree of fractality measured either by the fractal dimension D or by the fractal exponent a. Accounting for the discrete and fractal nature of fracture leads to a conclusion that the equilibrium length of the cohesive zone is indeed significantly influenced by both the quantum (discrete) and fractal aspects of the subcritical as well as the propagating crack.
At the micro- and nanoscale the size of the Neuber particle, or process zone in a more updated nomenclature, becomes important not only for mathematical treatment of the problem, but also for the physical interpretation of the decohesion phenomenon at the atomistic scale.
It is noteworthy that each of the successive models listed in Fig. 7 predicts for a given fixed level of the applied load successively larger cohesive zone, namely
RDd ^ rd ^ rd ^ RLEFM. The interpretation of the subscripts is as follows: f— fractal; Dd — Dugdale discrete, D — Dugdale. Of course the LEFM value of R is zero, but using the Kc value obtained from the tests specified by ASTM one could, in hindsight, define an equivalent length R that could be associated with a LEFM crack. It is noted that all equations given here reduce to the LEFM results when a fractal crack is replaced by a smooth crack and when the fracture quantum is allowed to vanish.
Wnuk M.P. / Физическая мезомеханика 12 4 (2009) 71-77
77
References
1. Isupov L.P., Mikhailov S.E. A comparative analysis of several nonlocal
fracture criteria // Archive of Applied Mechanics. - 1998. - V. 68. -P. 597-612.
2. Ippolito M., Mattoni A., Colombo L., Pugno N. Role of lattice discreteness on brittle fracture: Atomistic simulations versus analytical models // Phys. Rev. B. - 2006. - V. 73. - P. 104111 (6 pages).
3. NeuberH. Theory of notch stresses. - Berlin: Springer-Verlag, 1958. -
180 p.
4. Novozhilov V.V. On a necessary and sufficient criterion for brittle strength // J. Appl. Math. Mech. - 1969. - V. 33. - P. 212-222 (in Russian).
5. Wnuk M.P. Quasi-static extension of a tensile crack contained in a viscoelastic-plastic solid // J. Appl. Mech. - 1974. - V. 41. - P. 234242.
6. Seweryn A. Brittle-fracture criterion for structures with sharp notches // Eng. Fract. Mech. - 1994. - V. 47. - P. 673-681.
7. Pugno N., Ruoff R.S. Quantized fracture mechanics // Phil. Mag. -2004. - V. 84(27). - P. 2829-2845.
8. Williams M.L. Dealing with singularities in elastic mechanics of fracture // 11th Polish Symposium on Mechanics of Solids, Krynica, Poland, 1965.
9. Wnuk M.P., Yavari A. Discrete fractal fracture mechanics // Eng. Fract. Mech. - 2008. - V. 75(5) .- P. 1127-1142.
10. Wnuk M, Yavari A. A discrete cohesive model for fractal cracks // Eng. Fract. Mech. - 2009. - V. 76. - P. 548-559.
11. Wnuk M.P., Yavari A. On estimating stress intensity factors and modulus of cohesion for fractal cracks // Eng. Fract. Mech. - 2003. - V. 70. -P. 1659-1674.
12. Barenblatt G.I. The mathematical theory of equilibrium of crack in brittle fracture // Advances Appl. Mech. - 1962. - V. 7. - P. 55-129.
13. Dugdale D.S. Yielding of steel sheets containing slits // J. Mech. Phys. Solids. - 1960. - V. 8. - P. 100-104.
Поступила в редакцию 16.04.2009 г.
Сведения об авторе
Michael P. Wnuk, Professor, University of Wisconsin-Milwaukee, mpw@uwm.edu
