Shear Deformation of Compressed Elastic-Plastic Arrays with
Collinear Systems of Cracks
Andriy Boykoa,b, Nadia Kryvac
a
  Ternopil Volodymyr Hnatiuk National Pedagogical University, Maxyma Kryvonosa str.2 , Ternopil, 46027,
  Ukraine
b
  Ternopil Ivan Puluj National Technical University, Ruska str., 56, Ternopil, 46001, Ukraine
c
  Ternopil Ivan Puluj National Technical University, Ruska str., 56, Ternopil, 46001, Ukraine


                Abstract
                The plastic strips propagation in an ideal elastic-plastic body with colinear system of shear
                cracks has been studied. The cracks are opposed to the stable stresses on the faces. Stress-
                and-strain state of the body, dependencies of the strips length on the loading for random
                distance between cracks and the level of their faces friction have been found. The value of
                critical loading has been found where the plastic strips are merging and some plastic fracture
                occurs.

                Keywords 1
                All-round compression, shear cracks, plastic zones, conformal representation.

1. Introduction

    As a rule, the well-known investigations of plastic zones propagation for the bodies with cracks
have dealt with their noninteracting and free of external stresses surfaces [1-3]. Under combined shear
and compression stresses of three-dimensional arrays conditions the cracks opening is accompanied
by the interaction of their faces. It is especially important in case of the cracks of 3d type as the
distance between the surfaces during the crack formation process has been equal to 0 and the presence
of compression has resulted in friction forces occurrence. The above-mentioned situation is typical in
the problems of geomechanics, mechanics of earthquakes and rock formations due to the presence of
big efforts of compression and widely spread shear mechanisms of deformation processes [4-6].
    So, we will study the propagation of plastic deformations within the conception of plastic
deformations location in the cracks plane [1]. The basic reason for this assumption is the interaction
of cracks faces, as we have known that it causes the narrowing of the continuous plastic zone in
perpendicular to the crack direction and contributes to the thin-strip location of plastic deformations
[7]. Moreover, friction also reduces the size and slows down the propagation of plastic zones [7]. On
the contrary, the interaction of cracks of collinear system has accelerated the propagation of plastic
strips. Thus, it is necessary to take into account both of these competing factors of impact on the
plastic strips to determine the conditions of domination of each of them.


2. Problem statement and formalization.

   Let an unbounded ideal elastic-plastic body containing the system of tunnel collinear cracks −𝑙 ≤
|𝑥 + 2𝑛𝑎| ≤ 𝑙, 𝑦 = 0 (𝑛 ∈ 𝑍) −∞ < 𝑧 < ∞ (2𝑙 – length of cracks, 2𝑎 – distances between their

ITTAP’2021: 1nd InternationalWorkshop on Information Technologies: Theoretical and Applied Problems,November 16–18, 2021,
Ternopil, Ukraine
EMAIL: boyko.a111@gmail.com (A. 1); nadja.kryva@gmail.com (A. 2).
ORCID: 0000-0002-1634-3775 (A. 1); 0000-0001-6213-3069 (A. 2).
           ©️ 2021 Copyright for this paper by its authors.
           Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
           CEUR Workshop Proceedings (CEUR-WS.org)
                                                   ∞            ∞
centers), is loaded by the infinite shear efforts 𝜏𝑦𝑧  = 𝜏∞ , 𝜏𝑥𝑧 = 0 (Figure ). Under extra compression
                                                  ∞
by normal stresses conditions on infinities 𝜎𝑦𝑦 = −𝑝 the crack is closed and it can’t excite any
                                                                                   ∞
normal homogeneous field of stresses in the environment: 𝜎𝑦𝑦 (𝑥, 𝑦) = 𝜎𝑦𝑦             = −𝑝 = const. The
interaction of cracks faces is accompanied by their friction resulted in some extra tangent stresses on
the cracks faces which can oppose the shear and whose values are supposed to be stable and the same
𝜏𝑦𝑧 = 𝜏0 = 𝑓0 𝑝 (𝜏0 < 𝑘), where 𝑓0 - static coefficient of the sliding friction. When 𝜏∞ > 𝜏0 the
cracks faces are shifted, some extra tangent stresses are acting on their faces 𝜏𝑦𝑧 = 𝜏0 = 𝑓0 𝑝 and
some plastic strips are developing on the cracks continuation due to the concentration of stresses −𝑙 −
𝑑 ≤ |𝑥 + 2𝑛𝑎| ≤ 𝑙 + 𝑑, 𝑦 = 0 (𝑛 ∈ 𝑍), −∞ < 𝑧 < ∞, whose length 𝑑 should be found. In these
                                                       2      2
plastic strips the yield criterion must be fulfilled: 𝜏𝑥𝑧 + 𝜏𝑦𝑧  = 𝑘 2 , where 𝑘 – shear limit of liquidity.
Unlike linear case, due to the nonlinear character of the problem it cannot be reduced to the study of
                                                                  ∞
similar problems only in case of the given stress on infinities 𝜏𝑦𝑧  = 𝜏∞ − 𝜏0 .




      Figure 1: Cross section of the body.

    Under the defined conditions together with the above-mentioned uniform field of compression
stress some anti flat stress-and-strain state arises in the body which can be found by the shear 𝑤(𝑥, 𝑦)
. Two nonzero components of the stresses tensor are given by the formulae 𝜏𝑥𝑧 = 𝜇 𝜕𝑤⁄𝜕𝑥 and 𝜏𝑦𝑧 =
𝜇 𝜕𝑤⁄𝜕𝑦 (𝜇 – shear modulus of the material). The shear 𝑤(𝑥, 𝑦) is symmetric referred to the lines
𝑥 = 𝑛𝑎 (𝑛 ∈ 𝑍) and is antisymmetric about X-axis. That is why it can be determined only in a half-
strip 0 ≤ 𝑥 ≤ 𝑎, 0 ≤ 𝑦 < ∞ (area 𝐷).
    Due to the balance conditions and Hooke’s law, the function 𝜏(𝜁) = _𝑦𝑧 (𝑥, 𝑦) + 𝑖𝜏𝑥𝑧 (𝑥, 𝑦) is
analytical one in the elastic part of the body. So, to determine stress-and-strain state of the body we
will define a boundary problem for function 𝜏(𝜁) in the area 𝐷, consisting in the necessity to fulfill
four conditions.
    1. Due to the symmetry we have obtained
               Im𝜏(𝜁) = 0 ((𝜁 = 𝑖𝑦, 𝑦 ≥ 0)⋃(𝜁 = 𝑥, 𝑙 + 𝑑 ≤ 𝑥 ≤ 𝑎 )⋃(𝜁 = 𝑎 + 𝑖𝑦, 𝑦 ≥ 0)). (1)
    2. On cracks faces the stress 𝜏𝑦𝑧 = 𝜏0 = const, so
                                Re𝜏(𝜁) = 𝜏0 (𝜁 = 𝑥, 0 ≤ 𝑥 ≤ 𝑙 ).                                     (2)
    3. In the area 𝑙 ≤ 𝑥 ≤ 𝑙 + 𝑑 the yield criterion has been fulfilled, so
                                |𝜏(𝜁)| = 𝑘 (𝜁 = 𝑥, 𝑙 ≤ 𝑥 ≤ 𝑙 + 𝑑).                                   (3)
    4. Stress-and-strain state on infinities is defined by the formula
                                         lim 𝜏(𝜁) = _∞ .                                             (4)
                                       𝜁→∞
    As the function 𝜏(𝜁) in the zone 𝐷 is analytical and one-sheet, it conformally maps 𝐷 to the part of
circle |𝜏| ≤ 𝑘, Re 𝜏 ≥ 𝜏0 , Im 𝜏 ≥ 0 (zone 𝐺 Figure 2). In this case the following points match the
areas boundaries 𝐷 and 𝐺: 𝜁 = ∞ + 𝑖ℎ → 𝜏 = 𝜏∞ , 𝜁 = 0 → 𝜏 = 𝜏0 , 𝜁 = 𝑙 → 𝜏 = 𝜏0 − 𝑖√𝑘 2 − 𝜏02 ,
𝜁 = 𝑙 + 𝑑 → 𝜏 = 𝑘. Section (𝜁 = 𝑖𝑦, 0 ≤ 𝑦 < ∞) within the zone 𝐷 is mapped to the interval (Im𝜏 =
0, 𝜏0 ≤ Re 𝜏 ≤ 𝜏∞ ); interval (𝜁 = 𝑥, 0 ≤ 𝑥 ≤ 𝑙) – to Re 𝜏 = 𝜏0 , − √𝑘 2 − 𝜏02 ≤ Im 𝜏 ≤ 0,
section (𝜁 = 𝑥, 𝑙 + 𝑑 ≤ 〰 ≤ 𝑎)⋃(𝜁 = 𝑎 + 𝑖𝑦, 𝑦 ≥ 0) – to the interval (Im 𝜏 = 0, 𝜏∞ ≤ Re 𝜏 ≤
𝑘). The interval (𝜁 = 𝑥, 𝑙 ≤ 𝑥 ≤ 𝑙 + 𝑑), corresponding to the plastic strip, is mapped to the circle
arch (|𝜏| = 𝑘, −arccos(𝜏0 ⁄𝑘 ) ≤ arg𝜏 ≤ 0 ).


3. Study of plastic strips propagation
   The solution of the boundary problem (1)-(4) is reduced to the construction of the described
conformal mapping [8]. We will introduce some additional complex plane 𝑡, where the areas 𝐷 and 𝐺
match the upper half-plane 𝐻 = {Im𝑡 ≥ 0} (see Figure 2) and we will find the function 𝜏(𝜁) in
parametric form




     Figure 2: Example figure


                             𝜏 = 𝜏(𝑡), 𝜁 = 𝜁(𝑡) (𝑡 ∈ 𝐻)                                              (5)
   Function 𝜏(𝑡) is given as a composition of elementary mappings:
                                               𝑡 (𝑡)exp(𝑖𝜓0 )+exp(−𝑖𝜓0 )
                                    𝜏(𝑡) = 𝑘 6          𝑡 (𝑡)+1
                                                                         ,                           (6)
                                                           6
   where
                                   𝑡 (𝑡)+𝑀            1−(𝑠+1)𝑡
   𝑡6 (𝑡) = 𝑡5 (𝑡)𝜓0 ⁄𝜋 , 𝑡5 (𝑡) = 𝑡3(𝑡)−𝑀, 𝑡3 (𝑡) = √ (𝑠+1)𝑡 ,
                                   3


                             √𝑘 2 −𝜏02
                𝜋                                              𝜏       𝑀2
   𝑀 = −tg (2𝜓 (2arctg 𝜏 −𝜏 − 𝜋)), 𝜓0 = arccos 𝑘0 , 𝑠 = − 𝑀2 +1.
                    0          ∞    0




   As 𝑧 𝑞 (0 < 𝑞 < 1) we consider the analytical in the upper half plane function receiving actual
added values at the same values of 𝑧.
   In the finishing point of the strip 𝜏 = 𝑘. So, from the formula (6) we have obtained the complex
number of the certain point of the additional plane: 𝑡𝐸 = 1/(𝑠 + 1).
   The function determined by composition of elementary mappings 𝜁(𝑡) looks like:
                                               2𝑎               𝜋𝑙
                                       𝜁(𝑡) = 𝜋 arcsin (√𝑡sin 2𝑎).                              (7)
   As 𝜁(𝑡𝐸 ) = 𝑙 + 𝑑, the length of plastic strips can be obtained from the last formula
                                  2𝑎            1      𝜋𝑙
                             𝑑 = 𝜋 arcsin (        sin 2𝑎) − 𝑙.                                 (8)
                                               √𝑠+1
   The length of plastic strips as functions of stresses have been calculated for different stresses and
distance between cracks and for different levels of the faces friction and are shown Figure 3.
   The strips length can’t exceed the half distance between the tops of neighboring cracks. Stress
      ∗
𝜏∞ = 𝜏∞  when 𝑑 = 𝑎 − 𝑙 can cause some ductile fracture of the body. From the formula (8) it is clear




  Figure 3.

                                         ∗
that the value of critical loading 𝜏∞ = 𝜏∞ has satisfied the equation sin(𝜋𝑙/(2𝑎)) = √𝑠 + 1. Hence,
                                                                                            𝜓      𝑙
                                                                 𝜏0∗ = 𝜏0 + √𝑘 2 + 𝜏02 tg ( 20 (1 − 𝑎)).   (9)
                                                                  Under fixed levels of the friction of faces
                                                              conditions (𝜏0 = const) the last par has given
                                                              the dependence of critical loading on the
                                                              distance between cracks (Figure 4). The
                                                              interaction of faces has very strong impact on
                                                              the value of critical loading for the cracks
                                                              locating very close to each other. The bigger
                                                              distance between cracks the more weaken is the
                                                              interaction impact.




                                                                 Without the interaction of faces (𝜏0 = 0) the
                                                              formulae (8), (9) have given the known
                                                              dependencies [9]:
  Figure 4.

                            2𝑎               2
                                        𝑘 2+𝜏∞         𝜋𝑙                       𝜋       𝑙
                                                                      ∗
                       𝑑=        arcsin ( 2 2    sin        ) − 𝑙,   𝜏∞ = 𝑘tg ( (1 − )).
                             𝜋          𝑘 −𝜏∞          2𝑎                       4       𝑎
   In case of large distance between cracks the function stresses has been expressed by the formulae
(5) where 𝜏(𝑡) is the same as for the general case and 𝜁(𝑡) = 𝑙 √𝑡. The length of plastic strips has been
described by the following expression 𝑑 = 𝑙((𝑠 + 1)−1/2 − 1).
   The conducted study has proved a considerable impact of cracks interaction on plastic strips
development near their tops. Therefore, the important problem requiring a special attention is the
study of cracks interaction located in neighbourhood in an elastic-plastic body resulted in possible
plastic strips coalescence and plastic fracture.
4. References
[1] Панасюк В.В. Механика квазихрупкого разрушения материалов – К.: Наук. думка, 1991. –
    411с..
[2] Партон В.З., Морозов Е.М. Механика упругопластического разрушения. – М.: Наука. –
    1974. – 416с.
[3] Атлури С., Кобаяси А. Квазистатическое разрушение упругопластических тел //
    Вычислительные методы механики разрушения. – М.: Мир, 1990. – С.49-82.
[4] He Q. Y., Suorineni F. T., Ma T. H, Oh J. Effect of discontinuity stress shadows on hydraulic
    fracture re-orientation// International Journal of Rock Mechanics and Mining Sciences. – 2017. –
    V. 91 – P. 179–194.
[5] Altammar M. J., Sharma M. M., Manchanda R. The Effect of Pore Pressure on Hydraulic
    Fracture Growth: An Experimental Study // Rock Mechanics and Rock Engineering. – 2018. –V.
    91,No 9 – P. 2709–2732.
[6] Cheng Y. G., Lu Y. Y.,. Ge Z. L, Cheng L,. Zheng ., J. W, Zhang W. F. Experimental study on
    crack propagation control and mechanism analysis of directional hydraulic fracturing // Fuel. –
    2018. –V. 219. – P. 316–324.
[7] Крывень В.А. Влияние трения берегов на локализацию пластических деформаций в
    плоскости трещины продольного сдвига // Динамические системы. – 2001. – Т.17. – С.137-
    142.
[8] Кривень В.А. Новый метод решения одного класса упругопластических задач // Тез.
    Всесоюзн. конференции ”Интегральные уравнения и краевые задачи математической
    физики”. – Владивосток: ДВО АН СССР. – 1990. - С.105
[9] Кривень В.А. Антиплоская деформация упруго-пластического тела, ослабленного
    двоякопериодической системой щелей // Физ. –хим. мех. материалов. – 1979. - №1. - С. 31-
    33.