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

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.


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 r À 1 2 . 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.
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.

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.

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]).

Expansions of Williams' type
As the components of the macrodisplacement and the microdeformation reflect the independent kinematical degrees of freedom, we assume for u α and Ψ αβ asymptotic expansions of the same form. We fix the crack tip at the origin O of the coordinate system (see Figure 1 in Part I) and set with and p being 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 ¼ const:, with physical components Ψ αβ in conjunction with cylindrical coordinates, to be present in the expansion of Ψ. While the Cartesian components Ψ ij are constant, the physical components Ψ αβ are functions of φ. There are the following well known transformation rules between Ψ αβ and Ψ ij (see any textbook) For later reference, we note the relations which imply that the physical components Ψ αβ trivially obey the nonclassical boundary conditions (32)-(35). Anticipating the results in Section 5, we decompose Ψ into parts Ψ I and Ψ II , reflecting symmetries according to Mode-I and Mode-II: with and The main idea in Williams' approach is to expand each field variable f r, φ ð Þ in a sum of products as in Eqs. (67) and (68). We say that f is of the order r m , and write f $ r m , whenever r m is the power function of r in the leading term of the expansion of f . It can be deduced, from Eq. (67), that ε αβ $ r pÀ1 . From this, in turn, together with Eq. (68) and the elasticity laws (3)-(5), we can deduce, that Σ αβ $ r pÀ1 . Thus, with Expansion (67) suggests that the necessary and sufficient condition for u α to vanish at the crack tip is This restriction is in agreement with energetic considerations. To verify, we remark that ∇Ψ ¼ 0, as Ψ is constant. Therefore, from Eq. (68) together with Eqs. (25)-(29), we may infer that ∇Ψ ð Þ αβγ $ r pÀ1 . For the free energy per unit macrovolume ψ it follows that ψ $ r 2 pÀ2 [cf. Eq. (1)]. Now, consider a small circular area r ≤ R, enclosing the crack tip. The total free energy (per unit length in z-direction) of this area is Since ψ r $ r 2 pÀ1 , restriction (82) is the necessary and sufficient condition for the energy in Eq. (83) to be bounded.

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 Similarly, we find from Eq. (31) that

Consequences of the classical compatibility condition
A look at χ 1 Á ð Þ in Eq. (40) reveals that χ 1 is a linear differential operator, i. e., Since Ψ αβ is independent of r, we infer from Eq. (40) that and by virtue of Eq. (73), Therefore, from Eq. (86), and by appealing to expansion (68), we infer from Eq. (40) that Similarly, by appealing to expansion (80), we infer from Eq. (41) that Inserting Eqs. (90) and (91) into Eq. (39) and collecting coefficients of like powers of r gives, after some lengthy but straightforward manipulations,

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

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 r in the vicinity of the crack tip are vanishing coefficients of all powers of r. For Σ k ð Þ αβ , k ¼ 0, 1, 2, this leads to the following systems of differential equations and associated boundary conditions.
with boundary conditions with boundary conditions with boundary conditions It can be recognized that coupling between components of Σ and components of Ψ arises for the first time in the equations for Σ  (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 Σ αβ are singular, with order of singularity r À 1 2 , or equivalently, The coefficients of the singular terms, Σ 0 ð Þ αβ , are given by where the constantsK I andK II are the stress intensity factors. Here and in the following, the indices I and II stand for Mode-I and Mode-II, respectively. Moreover, we use the notationsK I andK II , in order to distinguish the stress intensity factors of the microstrain continuum from the stress intensity factors K I and K II of classical continua.
The so-called angular functions f I αβ and f II αβ are defined through and are normalized by the conditions The constant terms Σ 1 ð Þ αβ are given by withk I being constant. Constant terms for Mode-II are not present. The first two terms of the asymptotic expansion of Σ αβ are summarized in Section 5.

Strain
Although the first two terms in the expansion of Σ αβ are identical to the ones of classical elasticity, the corresponding terms of ε αβ differ from those of classical elasticity. This follows from the fact that the elasticity laws (3)-(5) are not classical.
Evidently, the components ε αβ obey the asymptotic expansion We use this and the asymptotic expansions of Section 3.1, with p ¼ 1 2 , in the elasticity laws (3)-(5), and collect coefficients of like powers of r. Thus, we derive the following solutions for ε By taking into account the solutions for Σ 0 ð Þ αβ of the last section, we find that Terms ε Now, we take into account the solutions for Σ The constantsk ε I,1 ,k ε I,2 andk ε II are defined as follows: The first two terms of the asymptotic expansion of ε αβ are summarized in Section 5.

Macrodisplacements
The macrodisplacement components u r and u φ will 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 u r that or For the circumferential component u φ , we conclude that or By employing steps similar to those in the last section, we get the following solutions for u Invoking Eqs. (116) and (117), we get, after some straightforward manipulations, Terms u from which, by virtue of Eqs. (119) and (120), The first two terms of the asymptotic expansion of u α are also summarized in Section 5.

Microdeformation
We shall derive differential equations for Ψ applies. Keeping this in mind and collecting terms of like powers of r, after some lengthy but otherwise straightforward manipulations, Eq. (36) yields Similarly, from Eqs. (37) and (38), we get In an analogous manner, by substituting the asymptotic expansion (68) into the nonclassical boundary conditions (48)-(50), we show that Equating to zero the coefficients of power r À 3 2 in Eqs. (141)-(143) leads to the system of ordinary differential equations Similarly, equating to zero the coefficients of power r 1 2 in the boundary conditions (144)-(146) leads to Proceeding to solve the system (147)-(149), we note that Eqs. (147) and (148) imply the ordinary differential equation For determining the constants of integration A 0 ð Þ and B 0 ð Þ , we utilize the boundary conditions. From Eqs. (150) and (151), we derive the equation By substituting the solution (154), we see that To go further, we notice that Eqs. (147) and (148) imply Next, we differentiate Eq. (149) with respect to φ and use Eq. (157). Rearrangement of terms leads to the ordinary differential equation By substituting the solutions (154) and (156), we gain an ordinary differential equation which obeyes the solution with C 0 ð Þ , D 0 ð Þ , E 0 ð Þ and F 0 ð Þ being new constants of integration. Further, from Eqs. (154), (156) and (160), Finally, using the solutions (161) and (160) in Eq. (157), we obtain the solution where G 0 ð Þ is 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 In accordance with the symmetry conditions (53) and (54) for Mode-I as well as (61) and (62) for Mode-II, we set Then, the solutions (160)-(162) become 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.

Differential equations for Ψ 1 ð Þ αβ
Equating to zero the coefficients of power r À1 in Eqs. (141)-(143) and the coefficients of power r in the boundary conditions (144)-(146) furnish the system of ordinary differential equations and corresponding boundary conditions 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 αβ are summarized and discussed in Section 5.

Double stress
The considerations of Section 3.1, together with p ¼ 1 2 (see Eq. (107)), and the elasticity laws for μ [see Eqs. (7)- (16)] suggest the asymptotic expansion The goal is to determine μ 0 ð Þ αβγ and μ 1 ð Þ αβγ by substituting the asymptotic expansion for Ψ αβ into the elasticity laws (7)- (16). It is readily verified that in view of the conditions (73) and (74), the terms Ψ αβ of the expansion (68) will disappear in the subsequent equations. Thus, we conclude from Eqs. (7)- (16), by equating the coefficients of power r À 1 2 that and by equating the coefficients of power r 0 that If we introduce the solutions (160)-(162) into Eqs. (179)-(186) and rearrange terms, then, for μ 0 ð Þ αβγ , we obtain the representations The fact that the solutions μ 0 ð Þ αβγ depend 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. particular, that p ¼ 1 2 . Then, it remains to show, how to determine the terms μ 0 ð Þ αβγ and μ 1 ð Þ αβγ . The corresponding terms of Ψ will 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 μ 0 ð Þ αβγ . 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 μ αβγ .

Nonclassical equilibrium equations
Since ε αβ $ r À 1 2 and Ψ αβ $ r 0 , we recognize from the elasticity law (6) that σ αβ $ r À 1 2 . On the other hand, by virtue of the expansion (212), ∂ r μ αβγ $ r À 3 2 and 1 r μ αβγ $ r À 3 2 . Therefore, up to terms of order r À1 there will be no contributions of σ present in the nonclassical equilibrium Eqs. (32)-(35) and we conclude that Equating to zero the coefficients of power r À 3 2 leads to 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 But this equations can also be obtained by adding up Eqs. (217) and (218).

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 Again, equating to zero the coefficients of power r À 1 2 leads to

Determination of
It can be shown (cf. A) that the solutions are given by If we define then these are nothing more but the solutions for μ 0 ð Þ αβγ of Section 3.9.

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 thatK To accomplish a representation for μ and define for Mode-I (cf. Eq. (202)) and therefore define the stress intensity factorsL II,1 andL II,2 by (cf. Eqs. (199) and (200))L The angular functions will be determined by comparison of Eqs.
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 relateL II,1 andL II,1 and the numerical simulations in Part III confirm this fact.

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 r À 1 2 .
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. These solutions, in turn, are equivalent to those of Eqs. (108)-(109). Moreover, it can be shown that for p ¼ 1 2 , the solutions of Eqs. (99)-(102) might be expressed in the form (111).

Author details
Carsten Broese, Jan Frischmann and Charalampos Tsakmakis* Faculty of Civil Engineering, Department of Continuum Mechanics, TU Darmstadt, Darmstadt, Germany *Address all correspondence to: tsakmakis@mechanik.tu-darmstadt.de © 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.