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Mode-I and Mode-II Crack Tip Fields in Implicit Gradient Elasticity Based on Laplacians of Stress and Strain. Part II: Asymptotic Solutions

By Carsten Broese, Jan Frischmann and Charalampos Tsakmakis

Submitted: July 10th 2020Reviewed: August 17th 2020Published: October 10th 2020

DOI: 10.5772/intechopen.93618

Downloaded: 17

Abstract

We develop asymptotic solutions for near-tip fields of Mode-I and Mode-II crack problems and for model responses reflected by implicit gradient elasticity. Especially, a model of gradient elasticity is considered, which is based on Laplacians of stress and strain and turns out to be derivable as a particular case of micromorphic (microstrain) elasticity. While the governing model equations of the crack problems are developed in Part I, the present paper addresses analytical solutions for near-tip fields by using asymptotic expansions of Williams’ type. It is shown that for the assumptions made in Part I, the model does not eliminiate the well-known singularities of classical elasticity. This is in contrast to conclusions made elsewhere, which rely upon different assumptions. However, there are significant differences in comparison to classical elasticity, which are discussed in the paper. For instance, in the case of Mode-II loading conditions, the leading terms of the asymptotic solution for the components of the double stress exhibit the remarkable property that they include two stress intensity factors.

Keywords

  • implicit gradient elasticity
  • mode-I and mode-II crack problems
  • analytical solutions
  • asymptotic expansions of Williams’ type
  • near-tip fields
  • order of singularity
  • stress intensity factors

1. Introduction

The 3-PG-Model, discussed in Part I, is a simple model of implicit gradient elasticity based on Laplacians of stress and strain and has been introduced by Gutkin and Aifantis [1]. It can be derived as a particular case of micromorphic (microstrain) elasticity (see, e. g., Forest and Sievert [2]), so that a free energy function and required boundary conditions are formulated rigorously. In the present paper, we are looking for near-tip asymptotic field solutions for Mode-I and Mode-II crack problems, in the context of plane strain states. The asymptotic solutions are obtained by using expansions of Williams’ type (see Williams [3]).

For the assumptions made in Part I, it is found that both, conventional stress and conventional strain, are singular. This holds also for the nonconventional stress, the so-called double stress. All singular fields have an order of singularity r12. In particular, the leading terms of the asymptotic solutions of the conventional stress are exactly the same as in classical elasticity. Nevertheless, the results are quite interesting, since the two leading terms of the asymptotic solution of the macrostrain are different from the corresponding terms of classical elasticity, and since the form of the asymptotic solution of the double stress exhibits a remarkable feature. To be more specific, the leading term of the asymptotic solution of the double stress includes two stress intensity factors, which are independent of each other. This reflects, from a theoretical point of view, differences in the structure of the asymptotic solutions in comparison to classical elasticity as well as micropolar elasticity, where only one stress intensity factor is present in the solutions of Mode-II crack problems.

There are various works addressing singularities in the field variables. Among others, we mention for couple-stress elasticity the works of Muki and Sternberg [4], Sternberg and Muki [5], Bogy and Sternberg [6, 7], Xia and Hutchinson [8], Huang et al. [9, 10, 11] and Zhang et al. [12]. For micropolar elasticity the works of Paul and Sridharan [13], Chen et al. [14], Diegele et al. [15] and for gradient elasticity the works of Altan and Aifantis [16, 17], Ru and Aifantis [18], Unger and Aifantis [19, 20, 21], Chen et al. [22], Mousavi and Lazar [23], Shi et al. [24, 25], Vardoulakis et al. [26], Karlis et al. [27, 28], Georgiadis [29], Askes and Aifantis [30] and Gutkin and Aifantis [1] are to be mentioned. The latter is an interesting work and proves that use of the 3-PG-Model eliminates singularities from the”elastic stresses of defects” (see also Askes and Aifantis [30] as well as Aifantis [31]). This finding is in contrast to the conclusions of the present paper, but it should be emphasized that the form of the assumed boundary conditions in Gutkin and Aifantis [1] is different from the form assumed here.

The scope of the paper is organized as follows: Mode-I and Mode-II crack problems are considered in the setting of plane strain problems. For the 3-PG-Model, the reduced governing equations for plane strain states have been derived in Part I and are summarized in Section 2. Section 3 provides asymptotic solutions for the near-tip fields by starting from asymptotic expansions of the macrodisplacement and the microdeformation. An alternative and equivalent aproach, starting from asymptotic expansions of the stresses, is sketched in Section 4. The developed asymptotic solutions are summarized and discussed in Section 5. Finally, the paper closes with some conclusions in Section 6.

Throughout the paper, use is made of the notation introduced in Part I.

2. Summary of the governing equations for plane strain problems

Following equations of Part I will be employed to establish asymptotic solutions of the crack tip fields.

Free energy function (see section “The 3-PG-Model as particular case of micro-strain elasticity” in Part I)

ψ=12εαβCαβρζερζ+12c2c1c1γαβCαβρζγρζ+12c2c1kαβγCβγρζkαρζ.E1

Elasticity law for Σ(see section 3.1“The 3-PG-Model as particular case of micro-strain elasticity” in Part I)

Σαβ=c2c1Cαβγρεγρc2c1c1CαβγρΨγρE2

or inversely

εrr=c12μc2ΣrrνΣrr+Σφφ+c2c1c1Ψrr,E3
εφφ=c12μc2ΣφφνΣrr+Σφφ+c2c1c1Ψφφ,E4
ε=c12μc2Σ+c2c1c1Ψ.E5

Elasticity law for σ(see section 3.1“The 3-PG-Model as particular case of micro-strain elasticity” in Part I)

σαβ=c2c1c1CαβρζερζΨρζ.E6

Elasticity law for μ(see section 4.5.1“Elasticity law for double stress” in Part I)

μrrr=c2c1λ+2μrΨrr+λrΨφφ,E7
μrφφ=c2c1λ+2μrΨφφ+λrΨrr,E8
μrzz=c2c1λrΨrr+Ψφφ=νμrrr+μrφφ,E9
μrrφ=c2c12μrΨ,E10
μφrr=c2c1r2μφΨrr2Ψ+λφΨrr+Ψφφ,E11
μφφφ=c2c1r2μφΨφφ+2Ψ+λφΨrr+Ψφφ,E12
μφzz=c2c1rλφΨrr+Ψφφ=νμφrr+μφφφ,E13
μφrφ=c2c1r2μφΨ+ΨrrΨφφ,E14
μrrz=μrφz=μφrz=μφφz=0,E15
μzαβ=0,E16

or inversely

Ψrrr=1c2c1E1ν2μrrrν1+νμrφφ,E17
Ψrφφ=1c2c1E1ν2μrφφν1+νμrrr,E18
Ψrrφ=1+νc2c1Eμrrφ,E19
Ψφrr=1c2c1E1ν2μφrrν1+νμφφφ,E20
Ψφφφ=1c2c1E1ν2μφφφν1+νμφrr,E21
Ψφrφ=1+νc2c1Eμφrφ.E22

Material parameters (see section 2“Preliminaries—Notation” in Part I)

ν=λ2λ+μ,E=2μ1+ν.E23

Strain components (see section 4.1“Kinematics” in Part I)

εrr=rur,εφφ=1rur+φuφ,ε=121rφur+ruφ1ruφ.E24

Microdeformation components (see section 4.1“Kinematics” of Part I)

Ψrrr=rΨrr,Ψrφφ=rΨφφ,Ψrrφ=rΨ,E25
Ψφrr=1rφΨrr2Ψ,E26
Ψφφφ=1rφΨφφ+2Ψ,E27
Ψφrφ=1rφΨ+ΨrrΨφφ,E28
Ψαβz=Ψzαβ=0.E29

Classical equilibrium equations (see section 4.2“Cauchy stress—Classical equilibrium equations” in Part I)

rΣrr+1rφΣ+1rΣrrΣφφ=0,E30
rΣ+1rφΣφφ+2rΣ=0.E31

Nonclassical equilibrium equations (see section 4.5.2“Nonclassical equilibrium conditions” in Part I)

rμrrr+1rφμφrr+1rμrrr2μφrφ+σrr=0,E32
rμrφφ+1rφμφφφ+1rμrφφ+2μφrφ+σφφ=0,E33
rμrzz+1rφμφzz+1rμrzz+σzz=0,E34
rμrrφ+1rφμφrφ+1rμrrφμφφφ+μφrr+σ=0.E35

Field equations for Ψ(see section 4.4“Field equations for Ψ” in Part I)

rrΨrr+1r2φφΨrr+1rrΨrr4r2φΨ2r2+1c2Ψrr+2r2Ψφφ+1ν2μc2Σrrν2μc2Σφφ=0,E36
rrΨφφ+1r2φφΨφφ+1rrΨφφ+4r2φΨ+2r2Ψrr2r2+1c2Ψφφ+1ν2μc2Σφφν2μc2Σrr=0,E37
rrΨ+1r2φφΨ+1rrΨ+2r2φΨrr2r2φΨφφ4r2+1c2Ψ+12μc2Σ=0.E38

Classical compatibility condition (see section 4.3“Classical compatibility condition” in Part I)

χ1Ψαβ+χ2Σαβ=0,E39
χ1Ψαβ c2c1c2rrΨφφ2rΨ+1r2φφΨrr1rrΨrr+2rrΨφφ2r2φΨ,E40
χ2Σαβ 1νc12μc2rrΣrr+Σφφ+1r2φφΣrr+Σφφ+1rrΣrr+Σφφ.E41

Nonclassical compatibility conditions (see section 4.6“Nonclassical compatibility conditions” in Part I)

φμrrφμφrφrrμφrφ+μrrrμrφφ=0,E42
φμrφφ+φμrrrμφφφμφrrrrμφφφrrμφrr=0,E43
φμrφφφμrrrμφφφ+μφrrrrμφφφ+rrμφrr+4μrrφ=0.E44

Classical boundary conditions (see section 4.7“Boundary conditions” in Part I)

Σφ=±π=0,E45
Σφφφ=±π=0.E46

Nonclassical boundary conditions (see section 4.7“Boundary conditions” in Part I)

μφrrφ=±π=μφφφφ=±π=μφrφφ=±π=0,E47

or equivalently

φΨrr2Ψφ=±π=0,E48
φΨφφ+2Ψφ=±π=0,E49
φΨ+ΨrrΨφφφ=±π=0.E50

Symmetry conditions—Mode-I (see section 4.8“Symmetry conditions” in Part I)

Σrrrφ=Σrrrφ,Σφφrφ=Σφφrφ,E51
Σrφ=Σrφ,E52
Ψrrrφ=Ψrrrφ,Ψφφrφ=Ψφφrφ,E53
Ψrφ=Ψrφ,E54
μrrrrφ=μrrrrφ,μφrrrφ=μφrrrφ,E55
μrφφrφ=μrφφrφ,μφφφrφ=μφφφrφ,E56
μrzzrφ=μrzzrφ,μφzzrφ=μφzzrφ,E57
μrrφrφ=μrrφrφ,μφrφrφ=μφrφrφ.E58

Symmetry conditions—Mode-II (see section 4.8“Symmetry conditions” in Part I)

Σrrrφ=Σrrrφ,Σφφrφ=Σφφrφ,E59
Σrφ=Σrφ,E60
Ψrrrφ=Ψrrrφ,Ψφφrφ=Ψφφrφ,E61
Ψrφ=Ψrφ,E62
μrrrrφ=μrrrrφ,μφrrrφ=μφrrrφ,E63
μrφφrφ=μrφφrφ,μφφφrφ=μφφφrφ,E64
μrzzrφ=μrzzrφ,μφzzrφ=μφzzrφ,E65
μrrφrφ=μrrφrφ,μφrφrφ=μφrφrφ.E66

3. Near-tip asymptotic solutions for Mode-I and Mode-II loading conditions

We shall solve the given problems by employing asymptotic expansions of Williams’ type (see Williams [3]).

3.1 Expansions of Williams’ type

As the components of the macrodisplacement and the microdeformation reflect the independent kinematical degrees of freedom, we assume for uαand Ψαβasymptotic expansions of the same form. We fix the crack tip at the origin Oof the coordinate system (see Figure 1 in Part I) and set

uα=rpuα0+rp+12uα1+=k=0rp+k2uαk,E67
Ψαβ=Ψ¯αβ+rpΨαβ0+rp+12Ψαβ1+=Ψ¯αβ+k=0rp+k2Ψαβk,E68

with

uαk=uαkφ,Ψαβk=Ψαβkφ,Ψ¯αβ=Ψ¯αβφ,E69

and pbeing a real number. Since the crack tip is fixed at O, no constant term is present in the expansion of u in Eq. (67). However, we allow a constant term Ψ¯=const., with physical components Ψ¯αβin conjunction with cylindrical coordinates, to be present in the expansion of Ψ. While the Cartesian components Ψ¯ijare constant, the physical components Ψ¯αβare functions of φ. There are the following well known transformation rules between Ψ¯αβand Ψ¯ij(see any textbook)

Ψ¯rr=12Ψ¯11+Ψ¯22+12Ψ¯11Ψ¯22cos2φ+Ψ¯12sin2φ,E70
Ψ¯φφ=12Ψ¯11+Ψ¯2212Ψ¯11Ψ¯22cos2φΨ¯12sin2φ,E71
Ψ¯=12Ψ¯11Ψ¯22sin2φ+Ψ¯12cos2φ.E72

For later reference, we note the relations

φΨ¯rr2Ψ¯=0,φΨ¯φφ+2Ψ¯=0,E73
φΨ¯+Ψ¯rrΨ¯φφ=0,E74

which imply that the physical components Ψ¯αβtrivially obey the nonclassical boundary conditions (32)(35). Anticipating the results in Section 5, we decompose Ψ¯into parts Ψ¯Iand Ψ¯II, reflecting symmetries according to Mode-I and Mode-II:

Ψ¯αβ=Ψ¯αβI+Ψ¯αβII,E75

with

Ψ¯rrIL¯I,1+L¯I,2cos2φ,Ψ¯rrIIL¯IIsin2φ,E76
Ψ¯φφIL¯I,1L¯I,2cos2φ,Ψ¯φφIIL¯IIsin2φ,E77
Ψ¯IL¯I,2sin2φ,Ψ¯IIL¯IIcos2φE78

and

L¯I,112Ψ¯11+Ψ¯22,L¯I,212Ψ¯11Ψ¯22,L¯IIΨ¯12.E79

The main idea in Williams’ approach is to expand each field variable frφin a sum of products as in Eqs. (67) and (68). We say that fis of the order rm, and write frm, whenever rmis the power function of rin the leading term of the expansion of f. It can be deduced, from Eq. (67), that εαβrp1. From this, in turn, together with Eq. (68) and the elasticity laws (3)(5), we can deduce, that Σαβrp1. Thus,

Σαβ=rp1Σαβ0+rp12Σαβ1+=k=0rp1+k2Σαβk,E80

with

Σαβk=Σαβkφ.E81

Expansion (67) suggests that the necessary and sufficient condition for uαto vanish at the crack tip is

p>0.E82

This restriction is in agreement with energetic considerations. To verify, we remark that Ψ¯=0, as Ψ¯is constant. Therefore, from Eq. (68) together with Eqs. (25)(29), we may infer that Ψαβγrp1. For the free energy per unit macrovolume ψit follows that ψr2p2[cf. Eq. (1)]. Now, consider a small circular area rR, enclosing the crack tip. The total free energy (per unit length in z–direction) of this area is

02π0Rψrdrdφ.E83

Since ψrr2p1, restriction (82) is the necessary and sufficient condition for the energy in Eq. (83) to be bounded.

3.2 Consequences of the classical equilibrium equations

Substitute the expansion (80) into Eqs. (30) and (31) and collect coefficients of like powers of r, to obtain

rp2pΣrr0+φΣ0Σφφ0
+rp32p+12Σrr1+φΣ1Σφφ1
+rp1p+1Σrr2+φΣ2Σφφ0
+=0.E84

Similarly, we find from Eq. (31) that

rp2p+1Σ0+φΣφφ0+rp32p+32Σ1+φΣφφ1+rp1p+2Σ2+φΣφφ2+=0.E85

3.3 Consequences of the classical compatibility condition

A look at χ1in Eq. (40) reveals that χ1is a linear differential operator, i. e.,

χ1ΨαβΨ¯αβ=χ1Ψαβχ1Ψ¯αβ.E86

Since Ψ¯αβis independent of r, we infer from Eq. (40) that

χ1Ψ¯αβ=c2c1c21r2φφΨ¯rr2Ψ¯,E87

and by virtue of Eq. (73),

χ1Ψ¯αβ=0.E88

Therefore, from Eq. (86),

χ1Ψαβ=χ1ΨαβΨ¯αβ,E89

and by appealing to expansion (68), we infer from Eq. (40) that

χ1Ψαβ= c2c1c2k=0rp2+k2p+k2p+k21Ψφφk+φφΨrrk2p+k2φΨkp+k2Ψrrk+2p+k2Ψφφk2φΨk.E90

Similarly, by appealing to expansion (80), we infer from Eq. (41) that

χ2Σαβ= 1νc12μc2k=0rp2+k2p1+k2p2+k2Σrrk+Σφφk+φφΣrrk+Σφφk+p1+k2Σrrk+Σφφk.E91

Inserting Eqs. (90) and (91) into Eq. (39) and collecting coefficients of like powers of rgives, after some lengthy but straightforward manipulations,

rp31νc12μc2p12Σrr0+Σφφ0+φφΣrr0+Σφφ0+rp521νc12μc2p122Σrr1+Σφφ1+φφΣrr1+Σφφ1+rp21νc12μc2p2Σrr2+Σφφ2+φφΣrr2+Σφφ2+c2c1c2pp+1Ψφφ0+φφΨrr02p+1Ψ0+rp321νc12μc2p+122Σrr3+Σφφ3+φφΣrr3+Σφφ3+c2c1c2p+12p+32Ψφφ1+φφΨrr12p+32Ψ1+=0.E92

3.4 Consequences of the classical boundary conditions

By invoking the asymptotic expansion (80) in the classical boundary conditions (45) and (46), we conclude that

rp1Σ0φ=±π+rp12Σ1φ=±π+=0,E93
rp1Σφφ0φ=±π+rp12Σφφ1φ=±π+=0.E94

3.5 Cauchy stress

Before going any further, it is convenient to evaluate the results so far. The necessary and sufficient conditions for the equilibrium Eqs. (84) and (85), the compatibility condition (92) and the boundary conditions (93) and (94) to hold for arbitrary rin the vicinity of the crack tip are vanishing coefficients of all powers of r. For Σαβk,k=0,1,2, this leads to the following systems of differential equations and associated boundary conditions.

Terms Σαβ0

φΣ0+pΣrr0Σφφ0=0,E95
φΣφφ0+p+1Σ0=0,E96
φφΣrr0+Σφφ0+p12Σrr0+Σφφ0=0,E97

with boundary conditions

Σ0φ=±π=0,Σφφ0φ=±π=0.E98

Terms Σαβ1

φΣ1+p+12Σrr1Σφφ1=0,E99
φΣφφ1+p+32Σ1=0,E100
φφΣrr1+Σφφ1+p122Σrr1+Σφφ1=0,E101

with boundary conditions

Σ1φ=±π=0,Σφφ1φ=±π=0.E102

Terms Σαβ2

φΣ2+p+1Σrr2Σφφ2=0,E103
φΣφφ2+p+2Σ2=0,E104
1ν2μc2φφΣrr2+Σφφ2+p2Σrr2+Σφφ2+c2c1c2φφΨφφ02p+1φΨ0+pp+1Ψφφ0pΨrr0=0,E105

with boundary conditions

Σ2φ=±π=0,Σφφ2φ=±π=0.E106

It can be recognized that coupling between components of Σand components of Ψarises for the first time in the equations for Σαβ2. Therefore, we shall focus attention only on the terms Σαβ0and Σαβ1.

The solution of the systems of differential equations for Σαβ0and Σαβ1, subjected to the restriction (82), can be established by well known methods (see, e. g., A) and turns out to be identical to the solution of the corresponding problems in classical elasticity. That means that the stress components Σαβare singular, with order of singularity r12, or equivalently,

p=12.E107

The coefficients of the singular terms, Σαβ0, are given by

Σαβ0=K˜I2πfαβIφ+K˜II2πfαβIIφ,E108

where the constants K˜Iand K˜IIare the stress intensity factors. Here and in the following, the indices Iand IIstand for Mode-I and Mode-II, respectively. Moreover, we use the notations K˜Iand K˜II, in order to distinguish the stress intensity factors of the microstrain continuum from the stress intensity factors KIand KIIof classical continua.

The so-called angular functions fαβIand fαβIIare defined through

frrIfφφIfI=145cosφ2cos3φ23cosφ2+cos3φ2sinφ2+sin3φ2,frrIIfφφIIfII=145sinφ2+3sin3φ23sinφ23sin3φ2cosφ2+3cos3φ2,E109

and are normalized by the conditions

fφφIφ=0=1,fIIφ=0=1.E110

The constant terms Σαβ1are given by

Σrr1Σφφ1Σ1=k˜Icos2φsin2φ12sin2φE111

with k˜Ibeing constant. Constant terms for Mode-II are not present. The first two terms of the asymptotic expansion of Σαβare summarized in Section 5.

3.6 Strain

Although the first two terms in the expansion of Σαβare identical to the ones of classical elasticity, the corresponding terms of εαβdiffer from those of classical elasticity. This follows from the fact that the elasticity laws (3)(5) are not classical.

Evidently, the components εαβobey the asymptotic expansion

εαβ=r12εαβ0+εαβ1+.E112

We use this and the asymptotic expansions of Section 3.1, with p=12, in the elasticity laws (3)(5), and collect coefficients of like powers of r. Thus, we derive the following solutions for εαβ0and εαβ1.

Terms εαβ0

εrr0=c12μc2Σrr0νΣrr0+Σφφ0,E113
εφφ0=c12μc2Σφφ0νΣrr0+Σφφ0,E114
ε0=c12μc2Σ0.E115

By taking into account the solutions for Σαβ0of the last section, we find that

εrr0=c1K˜I8μc22π58νcosφ2cos3φ2+c1K˜II8μc22π58νsinφ2+3sin3φ2,E116
εφφ0=c1K˜I8μc22π38νcosφ2+cos3φ2+c1K˜II8μc22π38νsinφ23sin3φ2,E117
ε0=c1K˜I8μc22πsinφ2+sin3φ2+c1K˜II8μc22πcosφ2+3cos3φ2.E118
Termsεαβ1
εrr1=c12μc2Σrr1νΣrr1+Σφφ1+c2c1c2Ψ¯rr,E119
εφφ1=c12μc2Σφφ1νΣrr1+Σφφ1+c2c1c2Ψ¯φφ,E120
ε1=c12μc2Σ1+c2c1c2Ψ¯.E121

Now, we take into account the solutions for Σαβ1, established in the last section, as well as the representations for Ψ¯αβ, given by Eqs. (75)–(79), to obtain

εrr1=k˜I,1ε+k˜I,2εcos2φ+k˜IIεsin2φ,E122
εφφ1=k˜I,1εk˜I,2εcos2φk˜IIεsin2φ,E123
ε1=k˜I,2εsin2φ+k˜IIεcos2φ.E124

The constants k˜I,1ε, k˜I,2εand k˜IIεare defined as follows:

k˜I,1εc1k˜I12ν4μc2+c2c1L¯I,1c2,E125
k˜I,2εc1k˜I4μc2+c2c1L¯I,2c2,E126
k˜IIεc2c1L¯IIc2.E127

The first two terms of the asymptotic expansion of εαβare summarized in Section 5.

3.7 Macrodisplacements

The macrodisplacement components urand uφwill be determined by integrating Eqs. (24). For plane strain elasticity, it is well known that the constants of integration represent rigid body motions. Omitting such motions, we conclude for the radial component urthat

ur=εrrdr=r12εrr0+εrr1+dr,E128

or

r12ur0+rur1+=2r12εrr0+rεrr1+.E129

For the circumferential component uφ, we conclude that

uφ=rεφφurdφ,E130

or

r12uφ0+ruφ1+=r12εφφ0ur0dφ+rεφφ1ur1dφ+.E131

By employing steps similar to those in the last section, we get the following solutions for uα0and uα1.

Terms uα0

ur0=2εrr0,E132
uφ0=εφφ0ur0dφ=εφφ02εrr0dφ.E133

Invoking Eqs. (116) and (117), we get, after some straightforward manipulations,

ur0=c1K˜I4μc22π58νcosφ2cos3φ2
+c1K˜II4μc22π58νsinφ2+3sin3φ2,E134
uφ0=c1K˜I4μc22π78νsinφ2+sin3φ2
+c1K˜II4μc22π78νcosφ2+3cos3φ2.E135

Terms uα1

ur1=εrr1,E136
uφ1=εφφ1ur1dφ=εφφ1εrr1dφ,E137

from which, by virtue of Eqs. (119) and (120),

ur1=k˜I,1ε+k˜I,2εcos2φ+k˜IIεsin2φ,E138
uφ1=k˜I,2εsin2φ+k˜IIεcos2φ.E139

The first two terms of the asymptotic expansion of uαare also summarized in Section 5.

3.8 Microdeformation

We shall derive differential equations for Ψαβ0and Ψαβ1by inserting the asymptotic expansions of Ψαβand Σαβ(see Eqs. (68) and (80), with p=12) into Eqs. (36)(38). Note that, by virtue of Eqs. (73) and (74) and since Ψ¯αβis independent of r, the identity

rrΨ¯rr+1r2φφΨ¯rr+1rrΨ¯rr4r2φΨ¯2r2Ψ¯rr+2r2Ψ¯φφ=0E140

applies. Keeping this in mind and collecting terms of like powers of r, after some lengthy but otherwise straightforward manipulations, Eq. (36) yields

r32φφΨrr04φΨ074Ψrr0+2Ψφφ0+r1φφΨrr14φΨ1Ψrr1+2Ψφφ1+=0.E141

Similarly, from Eqs. (37) and (38), we get

r32φφΨφφ0+4φΨ074Ψφφ0+2Ψrr0+r1φφΨφφ1+4φΨ1Ψφφ1+2Ψrr1+=0,E142
r32φφΨ0+2φΨrr0Ψφφ0154Ψ0+r1φφΨ1+2φΨrr0Ψφφ0154Ψ0+=0.E143

It is worth remarking that if only terms up to order r1are retained in Eqs. (141)(143), then the terms Ψαβ0and Ψαβ1are uncoupled from the terms Ψ¯αβand Σαβk.

In an analogous manner, by substituting the asymptotic expansion (68) into the nonclassical boundary conditions (48)(50), we show that

r12φΨrr02Ψ0φ=±π+rφΨrr12Ψ1φ=±π+=0,E144
r12φΨφφ0+2Ψ0φ=±π+rφΨφφ1+2Ψ1φ=±π+=0,E145
r12φΨ0+Ψrr0Ψφφ0φ=±π+rφΨ1+Ψrr1Ψφφ1φ=±π+=0.E146

3.8.1 Differential equations for Ψαβ0

Equating to zero the coefficients of power r32in Eqs. (141)–(143) leads to the system of ordinary differential equations

φφΨrr04φΨ074Ψrr0+2Ψφφ0=0,E147
φφΨφφ0+4φΨ074Ψφφ0+2Ψrr0=0,E148
φφΨ0+2φΨrr0Ψφφ0154Ψ0=0.E149

Similarly, equating to zero the coefficients of power r12in the boundary conditions (144)(146) leads to

φΨrr02Ψ0φ=±π=0,E150
φΨφφ0+2Ψ0φ=±π=0,E151
φΨ0+Ψrr0Ψφφ0φ=±π=0.E152

Proceeding to solve the system (147)(149), we note that Eqs. (147) and (148) imply the ordinary differential equation

φφΨrr0+Ψφφ0+14Ψrr0+Ψφφ0=0E153

for the sum Ψrr0+Ψφφ0, which has the solution

Ψrr0+Ψφφ0=A0cosφ2+B0sinφ2.E154

For determining the constants of integration A0and B0, we utilize the boundary conditions. From Eqs. (150) and (151), we derive the equation

φΨrr0+Ψφφ0φ=±π=0.E155

By substituting the solution (154), we see that

A0=0.E156

To go further, we notice that Eqs. (147) and (148) imply

φΨ0=18φφΨrr0Ψφφ0154Ψrr0Ψφφ0.E157

Next, we differentiate Eq. (149) with respect to φand use Eq. (157). Rearrangement of terms leads to the ordinary differential equation

12φφφφΨrr0+Ψφφ0+174φφΨrr0+Ψφφ0+22532Ψrr0+Ψφφ0φφφφΨφφ0172φφΨφφ022516Ψφφ0=0.E158

By substituting the solutions (154) and (156), we gain an ordinary differential equation for Ψφφ0,

φφφφΨφφ0+172φφΨφφ0+22516Ψφφ0=6B0sinφ2,E159

which obeyes the solution

Ψφφ0= 12B0sinφ2+E0sin3φ2+F0sin5φ2+C0cos3φ2+D0cos5φ2,E160

with C0,D0,E0and F0being new constants of integration. Further, from Eqs. (154), (156) and (160),

Ψrr0= 12B0sinφ2E0sin3φ2F0sin5φ2C0cos3φ2D0cos5φ2.E161

Finally, using the solutions (161) and (160) in Eq. (157), we obtain the solution Ψ0in the form

Ψ0=C0sin3φ2+D0sin5φ2E0cos3φ2F0cos5φ2+G0,E162

where G0is a further constant of integration. For the constants of integration in the solutions (160)(162) we can verify, by evaluating the boundary conditions (150)(152) that

G0=0,D0=C0,F0=E0.E163

In accordance with the symmetry conditions (53) and (54) for Mode-I as well as (61) and (62) for Mode-II, we set

C0CI0,B0BII0,E0EII0.E164

Then, the solutions (160)(162) become

Ψrr0=CI0cos3φ2cos5φ2+12BII0sinφ2EII0sin3φ2sin5φ2,E165
Ψφφ0=CI0cos3φ2cos5φ2+12BII0sinφ2+EII0sin3φ2sin5φ2,E166
Ψ0=CI0sin3φ2sin5φ2EII0cos3φ2cos5φ2.E167

It is of interest to comment the following issue. Obviously not all constants of integration may be determined, because boundary conditions are prescribed only on the crack faces. Nevertheless, it is remarkable that the solutions of Mode-I include only one unknown constant, whereas the solutions of Mode-II depend on two unknown constants. We shall come back to this specific feature in the next section as well as in Section 5, while discussing the asymptotic solutions of the double stresses.

3.8.2 Differential equations for Ψαβ1

Equating to zero the coefficients of power r1in Eqs. (141)(143) and the coefficients of power rin the boundary conditions (144)(146) furnish the system of ordinary differential equations

φφΨrr14φΨ1Ψrr1+2Ψφφ1=0,E168
φφΨφφ1+4φΨ1Ψφφ1+2Ψrr1=0,E169
φφΨ1+2φΨrr1Ψφφ13Ψ1=0,E170

and corresponding boundary conditions

φΨrr12Ψ1φ=±π=0,E171
φΨφφ1+2Ψ1φ=±π=0,E172
φΨ1+Ψrr1Ψφφ1φ=±π=0.E173

Since the steps for solving the above system of differential equations are quite similar to those in the last section, we omit the details and present only the final solutions

Ψrr1=12A1cosφDI1cosφ+cos3φEII1sinφFII1sin3φ,E174
Ψφφ1=12A1cosφ+DI1cosφ+cos3φ+EII1sinφ+FII1sin3φ,E175
Ψ1=DI1sinφ+sin3φEII112+cosφFII112cos3φ.E176

With regard to the symmetry conditions (53), (54), (61) and (62), the constants AI1,DI1,EII1and FII1are attributed to loading conditions of Mode-I and Mode-II, respectively. The solutions Ψαβ0and Ψαβ1are summarized and discussed in Section 5.

3.9 Double stress

The considerations of Section 3.1, together with p=12(see Eq. (107)), and the elasticity laws for μ[see Eqs. (7)(16)] suggest the asymptotic expansion

μαβγ=r12μαβγ0+μαβγ1+=k=0r12+k2μαβγk,E177

with

μαβγk=μαβγkφ.E178

The goal is to determine μαβγ0and μαβγ1by substituting the asymptotic expansion for Ψαβinto the elasticity laws (7)(16). It is readily verified that in view of the conditions (73) and (74), the terms Ψ¯αβof the expansion (68) will disappear in the subsequent equations. Thus, we conclude from Eqs. (7)–(16), by equating the coefficients of power r12that

μrrr0=c2c1λ+2μ2Ψrr0+λ2Ψφφ0,E179
μrφφ0=c2c1λ+2μ2Ψφφ0+λ2Ψrr0,E180
μrzz0=c2c1λ2Ψrr0+Ψφφ0,E181
μrrφ0=c2c1μΨ0,E182
μφrr0=c2c12μφΨrr02Ψ0+λφΨrr0+Ψφφ0,E183
μφφφ0=c2c12μφΨφφ0+2Ψ0+λφΨrr0+Ψφφ0,E184
μφzz0=c2c1λφΨrr0+Ψφφ0,E185
μφrφ0=c2c12μφΨ0+Ψrr0Ψφφ0,E186

and by equating the coefficients of power r0that

μrrr1=c2c1λ+2μΨrr1+λΨφφ1,E187
μrφφ1=c2c1λ+2μΨφφ1+λΨrr1,E188
μrzz1=c2c1λΨrr1+Ψφφ1,E189
μrφφ1=c2c12μΨ1,E190
μφrr1=c2c12μφΨrr12Ψ1+λφΨrr1+Ψφφ1,E191
μφφφ1=c2c12μφΨφφ1+2Ψ1+λφΨrr1+Ψφφ1,E192
μφzz1=c2c1λφΨrr1+Ψφφ1,E193
μrφφ1=c2c12μφΨ1+Ψrr1Ψφφ1.E194

If we introduce the solutions (160)(162) into Eqs. (179)–(186) and rearrange terms, then, for μαβγ0, we obtain the representations

μrrr0=c2c1μCI0cos3φ2cos5φ2+λ+μ2BII0sinφ2μEII0sin3φ2sin5φ2,E195
μrφφ0=c2c1μCI0cos3φ2cos5φ2+λ+μ2BII0sinφ2+μEII0sin3φ2sin5φ2,E196
μrzz0=c2c1λ2BII0sinφ2,E197
μrrφ0= c2c1μCI0sin3φ2sin5φ2μEII0cos3φ2cos5φ2,E198
μφrr0= c2c1μCI0sin3φ2+sin5φ2+λ+μ2BII0cosφ2+μEII0cos3φ2+cos5φ2,E199
μφφφ0=c2c1μCI0sin3φ2+sin5φ2+λ+μ2BII0cosφ2μEII0cos3φ2+cos5φ2,E200
μφzz0=c2c1λ2BII0cosφ2,E201
μφrφ0=c2c1μCI0cos3φ2+cos5φ2μEII0sin3φ2+sin5φ2.E202

The fact that the solutions μαβγ0depend on two unknown constants in case of Mode-II is a characteristic property. As we shall see in Section 5, this feature leads to the existence of two stress intensity factors for the double stresses in case of Mode-II.

Using steps similar to those above we obtain for μαβγ1the representations

μrrr1= c2c1λ+μAI1cosφ2μDI1cosφ+cos3φ2μEII1sinφ2μFII1sin3φ,E203
μrφφ1= c2c1λ+μAI1cosφ+2μDI1cosφ+cos3φ+2μEII1sinφ+2μFII1sin3φ,E204
μφzz1=c2c1AI1cosφ,E205
μrrφ1= c2c12μDI1sinφ+sin3φ2μEII112+cosφ2μFII112cos3φ,E206
μφrr1=c2c1λ+μAI1sinφ2μDI1sinφsin3φ+2μEII11+cosφ2μFII11+cos3φ,E207
μφφφ1=c2c1λ+μAI1sinφ+2μDI1sinφsin3φ2μEII11+cosφ+2μFII11+cos3φ,E208
μφzz1=c2c1λAI1sinφ,E209
μφrφ1= c2c12μDI1cosφcos3φ2μEII1sinφ+2μFII1sin3φ.E210

Before going to discuss the obtained solutions, it is perhaps of interest to rederive the analytical solutions by an alternative approach, starting from asymptotic expansions of Σand μrather than the asymtptotic expansions of uand Ψused in this section.

4. Alternative approach for the determination of the near-tip fields

In Section 3 we determined the near-tip fields by starting from asymptotic expansions of the same form for the kinematical variables uand Ψ[see Eqs. (67) and (68)]. Alternatively, it is instructive to start from asymptotic expansions of the same type for the stresses Σand μ, i. e.,

Σαβ=rp1Σαβ0+rp12Σαβ1+,E211
μαβγ=rp1μαβγ0+rp12μαβγ1+,E212

where Σαβk=Σαβkφand μαβγk=μαβγkφ, k=0,1,2,. Then, from the elasticity laws (17)(22), we recognize that Ψαβγrp1and hence the components Ψαβare of form (67). It follows that all outcomes of sections 3.2–3.6 apply as well and, in particular, that p=12. Then, it remains to show, how to determine the terms μαβγ0and μαβγ1. The corresponding terms of Ψwill then be established by integrating the elasticity laws (17)(22). For the purposes of the present section, however, it suffices to demonstrate only how to determine the terms μαβγ0. To this end, we shall involve the nonclassical equilibrium Eqs. (32)(35), in conjunction with the elasticity law (6) for σ, as well as the nonclassical compatibility conditions (42)(44). It is necessary to involve the latter for we are directly seeking for solutions of μαβγ.

4.1 Nonclassical equilibrium equations

Since εαβr12and Ψαβr0, we recognize from the elasticity law (6) that σαβr12. On the other hand, by virtue of the expansion (212), rμαβγr32and 1rμαβγr32. Therefore, up to terms of order r1there will be no contributions of σpresent in the nonclassical equilibrium Eqs. (32)(35) and we conclude that

r3212μrrr0+φμφrr0+μrrr02μφrφ0+=0,E213
r3212μrφφ0+φμφφφ0+μrφφ0+2μφrφ0+=0,E214
r3212μrzz0+φμφzz0+μrzz0+=0,E215
r3212μrrφ0+φμφrφ0+μrrφ0μφφφ0+μφrr0+=0.E216

Equating to zero the coefficients of power r32leads to

2φμφrr0+μrrr04μφrφ0=0,E217
2φμφφφ0+μrφφ0+4μφrφ0=0,E218
2φμφrφ0+μrrφ02μφφφ0+2μφrr0=0,E219

and

2φμφzz0+μrzz0=0.E220

The last equation will not be considered further, for it can be established from Eqs. (217) and (218). To see this, we recall Eqs. (9) and (13) to recast Eq. (220) equivalently in the form

2φμφrr0+2φμφφφ0+μrrr0+μrφφ0=0.E221

But this equations can also be obtained by adding up Eqs. (217) and (218).

4.2 Nonclassical compatibility conditions

We insert the asymptotic expansion (212) into the nonclassical compatibility conditions (42)(44) and collect terms of like powers of r, to get

r12φμrrφ0μφrφ0+12μφrφ0+μrrr0μrφφ0+=0,E222
r12φμrφφ0+φμrrr0μφφφ0μφrr0+12μφφφ0+12μφrr0+=0,E223
r12φμrφφ0φμrrr0μφφφ0+μφrr0+12μφφφ012μφrr0+4μrrφ0+=0.E224

Again, equating to zero the coefficients of power r12leads to

φμrrφ012μφrφ0+μrrr0μrφφ0=0,E225
φμrφφ0+φμrrr012μφφφ012μφrr0=0,E226
φμrφφ0φμrrr012μφφφ0+12μφrr0+4μrrφ0=0.E227

4.3 Determination of μαβγ0

Eqs. (217)(219) and (225)(227) are 6differential equations for the 6unknowns μrrr0,μrφφ0,μrrφ0,μφrr0,μφφφ0and μφrφ0. The required boundary conditions can be verified to be [cf. Eq. (47)].

μφrr0φ=±π=μφφφ0φ=±π=μφrφ0φ=±π=0.E228

It can be shown (cf. A) that the solutions are given by

μrrr0=B¯2sinφ2+C¯sin3φ2sin5φ2A¯cos3φ2cos5φ2,E229
μrφφ0=B¯2sinφ2C¯sin3φ2sin5φ2+A¯cos3φ2cos5φ2,E230
μrrφ0=C¯cos3φ2cos5φ2+A¯sin3φ2sin5φ2,E231
μφrr0=B¯2cosφ2C¯cos3φ2+cos5φ2A¯sin3φ2+sin5φ2,E232
μφφφ0=B¯2cosφ2+C¯cos3φ2+cos5φ2+A¯sin3φ2+sin5φ2,E233
μφrφ0=C¯sin3φ2+sin5φ2A¯cos3φ2+cos5φ2.E234

If we define

A¯c2c1μCI0,B¯c2c1λ+μBII0,E235
C¯c2c1μEII0E236

then these are nothing more but the solutions for μαβγ0of Section 3.9.

5. Discussion of the asymptotic solutions

As suggested in Section 3.5, it is common to represent the leading terms of the asymptotic expansion of stresses by introducing stress intensity factors and angular functions. For the Cauchy stress, this is indicated in Eq. (108). Eqs. (108)(110) also reveal that

K˜I2π=Σφφ0φ=0,K˜II2π=Σ0φ=0.E237

To accomplish a representation for μαβγ0similar to the one for Σαβ0in Eq. (108), we remark that there is only one unknown constant for Mode-I, namely CI0, but there are two unknown constants for Mode-II, BII0and EII0[cf. Eqs. (195)–(202)]. Therefore, in analogy to Eq. (108), we set

μαβγ0=L˜I2πgαβγIφ+L˜II,12πgαβγII,1φ+L˜II,22πgαβγII,2φ,E238

and define for Mode–I (cf. Eq. (202))

L˜I2πμφrφ0φ=0=c2c12μCI0,E239

rendering gφrφIφ=0to be normalized,

gφrφIφ=0=1.E240

To define L˜II,1and L˜II,2unambiguously, we note that BII0can be determined by adding Eqs. (199) and (200) while taking φ=0. Similarly, EII0can be determined by substracting Eqs. (199) and (200) from each other while taking φ=0. We intend to normalize the angular functions gαβγII,1and gαβγII,2by

gφrrII,1φ=0=gφrrII,2φ=0=1,E241

and therefore define the stress intensity factors L˜II,1and L˜II,2by (cf. Eqs. (199) and (200))

L˜II,12π12μφrr0+μφφφ0φ=0=12c2c1λ+μBII0,E242
L˜II,22π12μφrr0μφφφ0φ=0=c2c12μEII0.E243

The angular functions will be determined by comparison of Eqs. (238)(243) with Eqs. (195), (196), (198)–(200) ,and (202). Explicitely, we find that

grrrIgrφφIgrrφIgφrrIgφφφIgφrφI=12cos3φ2cos5φ2cos3φ2+cos5φ2sin3φ2+sin5φ2sin3φ2+sin5φ2sin3φ2sin5φ2cos3φ2+cos5φ2,E244
grrrII,1grφφII,1grrφII,1gφrrII,1gφφφII,1gφrφII,1=sinφ2sinφ20cosφ2cosφ20,grrrII,2grφφII,2grrφII,2gφrrII,2gφφφII,2gφrφII,2=12sin3φ2+sin5φ2sin3φ2sin5φ2cos3φ2+cos5φ2cos3φ2+cos5φ2cos3φ2cos5φ2sin3φ2sin5φ2.E245

Some comments addressing Mode-I and Mode-II crack problems are in order at this stage. In classical elasticity, there are two intensity factors in the expansion of the Cauchy stress, one for each mode. In micropolar elasticity (see, e. g., Diegele et al. [15]), there are also two stress intensity factors in the expansion of the Cauchy stress and in addition two nonclassical intensity factors in the expansion of the couple stress, one for each mode. In the present case of microstrain elasticity, there are also two stress intensity factors in the expansion of the Cauchy stress, one for each mode. However, in the expansions of the double stress there is one intensity factor for Mode-I, but there are two intensity factors for Mode-II. Actually, there are no further conditions to relate L˜II,1and L˜II,1and the numerical simulations in Part III confirm this fact.

It is also convenient to replace the constants AI1,DI1,EII1and FII1by the definitions

l˜I,1c2c1λ+μAI1,E246
l˜I,2c2c12μDI1,E247
l˜II,1c2c12μEII1,E248
l˜II,2c2c12μFII1.E249

Evidently, the new constants for Mode-I and Mode-II in the expansions of μαβγ0and μαβγ1can be employed to rewrite Ψαβ0and Ψαβ1. In particular, we can conclude from Eqs. (160)(162) and (239)(243) that

Ψαβ0= L˜Ic2c12μ2πhαβI+L˜II,1c2c1λ+μ2πhαβII,1+L˜II,2c2c12μ2πhαβII,2,E250

with

hrrIhφφIhI=cos3φ2cos5φ2cos3φ2+cos5φ2sin3φ2+sin5φ2.E251

Table 1 summarizes the first two terms of the asymptotic solutions of the near-tip fields. All stresses are singular with order of singularity r12. Especially, the terms Σαβ0and Σαβ1are identical to those of classical elasticity. However, the terms εαβ0and εαβ1are different from the corresponding terms of classical elasticity. In particular, εαβ1includes terms arising from Ψ¯αβ. There are also qualititative differences to micropolar elasticity. For instance, terms of couple stresses corresponding to μαβγ0in Mode-II and to μαβγ1in Mode-I do not exist.

Σαβ=K˜I2πrfαβI+K˜II2πrfαβII+Σαβ1+,E252
ΣrrΣφφΣ=K˜I42πr5cosφ2cos32φ3cosφ2+cos32φsinφ2+sin32φ+K˜II42πr5sinφ2+3sin32φ3sinφ23sin32φcosφ2+3cos32φ+k˜Icos2φsin2φ12sin2φ+.E253
Ψαβ=Ψ¯αβ+r2πL˜Ic2c12μhαβI+r2πL˜II,1c2c1λ+μhαβII,1+r2πL˜II,2c2c12μhαβII,2+rΨαβ1+,E254
ΨrrΨ¯rrΨφφΨ¯φφΨΨ¯=r2πL˜Ic2c12μcos3φ2cos5φ2cos3φ2+cos5φ2sin3φ2+sin5φ2
+r2πL˜II,1c2c1λ+μsinφ2sinφ20
+r2πL˜II,2c2c12μsin3φ2+sin5φ2sin3φ2sin5φ2cos3φ2+cos5φ2
+rl˜I,12c2c1λ+μcosφcosφ0E255
+rl˜I,2c2c12μcosφ+cos3φcosφcos3φsinφsin3φ
+rl˜II,1c2c12μsinφsinφ12+cosφ
+rl˜II,2c2c12μsin3φsin3φ12cos3φ+.
μαβγ=L˜I2πrgαβγI+L˜II,12πrgαβγII,1+L˜II,22πrgαβγII,2+μαβγ1+,E256
μrrrμrφφμrrφμφrrμφφφμφrφ=L˜I2πr12cos3φ2cos5φ2cos3φ2+cos5φ2sin3φ2+sin5φ2sin3φ2+sin5φ2sin3φ2sin5φ2cos3φ2+cos5φ2
+L˜II,12πrsinφ2sinφ20cosφ2cosφ20+L˜II,22πr12sin3φ2+sin5φ2sin3φ2sin5φ2cos3φ2+cos5φ2cos3φ2+cos5φ2cos3φ2cos5φ2sin3φ2sin5φ2
+l˜I,1cosφcosφ0sinφsinφ0+l˜I,2cosφ+cos3φcosφcos3φsinφsin3φsinφsin3φsinφ+sin3φcosφcos3φ
+l˜II,1sinφsinφ12+cosφ1cosφ1+cosφsinφ+l˜II,2sin3φsin3φ12cos3φ1+cos3φ1cos3φsin3φ+,E257
εrrεφφε=K˜I2πrc18μc258νcosφ2cos3φ238νcosφ2+cos3φ2sinφ2+sin3φ2
+K˜II2πrc18μc258νsinφ2+3sin3φ238νsinφ23sin3φ2cosφ2+3cos3φ2
+k˜I,1ε110+k˜I,2εcos2φcos2φsin2φ+k˜IIεsin2φsin2φcos2φ+,E258
uruφ=r2πc1K˜I4μc258νcosφ2cos3φ278νsinφ2+sin3φ2
+r2πc1K˜II4c2μ58νsinφ2+3sin3φ278νcosφ2+3cos3φ2
+rk˜I,1ε10+k˜I,2εcos2φsin2φ+k˜IIεsin2φcos2φ+.E259

Table 1.

Analytical solutions of the fields.

6. Concluding remarks

Closed form analytical solutions, predicted by the 3-PG-Model for Mode-I and Mode-II crack problems, have been developed in the present paper. The solutions are based on asymptotic expansions of Williams’ type of the near-tip fields. The main conclusions, which can be drawn on the basis of the preceding developements, can be briefly stated as follows.

  1. The first two terms in the asymptotic expansion of the components of the Cauchy stress are identical to the ones of classical elasticity. In particular, the Cauchy stress is singular with order of singularity r12.

  2. This is in contrast to statements in other works, which rely upon boundary conditions different from the ones adopted here.

  3. There are, however, significant differences in comparison to classical elasticity, in what concerns the components of macrostrain and macrodisplacement.

  4. There are also significant qualitative differences in comparison to micropolar elasticity concerning the nonclassical stresses.

  5. For instance, the leading terms of the double stress of Mode-II problems include two different stress intensity factors. This is a remarkable feature of the 3-PG-Model.

Acknowledgments

We would like to thank the TU Darmstadt for support of publishing this work in open access.

In order to make the present work self-contained, we sketch briefly how to ascertain the solutions (107)(111) from Eqs. (95)(102). We start with the system of differential Eqs. (95)(97), which can be proved to posses the solutions

Σrr0=Ccosp+1φDsinp+1φ+3p4Acosp1φ+3p4Bsinp1φ,EA1
Σφφ0=Ccosp+1φ+Dsinp+1φ+p+14Acosp1φ+p+14Bsinp1φ,EA2
Σ0=Csinp+1φDcosp+1φ+p14Asinp1φ+p14Bcosp1φ.EA3

Here, A,B,Cand Dare constants of integration. In order to determine these constants, we incorporate the solutions in the boundary conditions (98). After some manipulations, we gain the following two homogeneous systems for the constants A, B, C, and D:

2cosp+1πp12cosp1π2sinp+1πp+12sinp1πDB=00,EA4
2sinp+1πp12sinp1π2cosp+1πp+12cosp1πCA=00.EA5

The conditions for the existence of nontrivial solutions are vanishing determinants of the coefficient matrices of Eqs. (A4) and (A5). It turns out that both conditions lead to the same equation

2cospπsinpπ=sin2pπ=0,EA6

which has the solutions

p=0,±12,±1,±32,.EA7

The smallest value of pcompatible with the restriction (82) is p=12, as stated in Eq. (107). For this case, the systems (A4) and (A5) imply

D=38B,C=18A,EA8

and the solutions (A1)–(A3) become

Σrr0=18A5cosφ2cos3φ2+18B5sinφ2+3sin3φ2,EA9
Σφφ0=18A3cosφ2+cos3φ2+18B3sinφ23sin3φ2,EA10
Σ0=18Asinφ2+sin3φ2+18Bcosφ2+3cos3φ2.EA11

These solutions, in turn, are equivalent to those of Eqs. (108)(109). Moreover, it can be shown that for p=12, the solutions of Eqs. (99)(102) might be expressed in the form (111).

Authors’ contributions

Each one of the authors Broese, Frischmann, and Tsakmakis contributed the same amount of work for the three papers “Mode-I and Mode-II crack tip fields in implicit gradient elasticity based on Laplacians of stress and strain.” Part I–III.

Dr. C. Broese: Theory and numerical simulations.

Dr. J. Frischmann: Analytical solution and numerical simulations.

Prof. Dr. Tsakmakis: Theory and analytical solution.

Therefore, it is a joint work by all three authors.

Funding

The first and second authors acknowledge and thank the Deutsche Forschungsgemeinschaft (DFG) for partial support of this work under Grant TS 29/13–1.

Consent for publication

Not applicable.

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Ethics approval and consent to participate

Not applicable.

Availability of data and material

Not applicable.

List of abbreviations

eq. = equation

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Carsten Broese, Jan Frischmann and Charalampos Tsakmakis (October 10th 2020). Mode-I and Mode-II Crack Tip Fields in Implicit Gradient Elasticity Based on Laplacians of Stress and Strain. Part II: Asymptotic Solutions [Online First], IntechOpen, DOI: 10.5772/intechopen.93618. Available from:

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