Analytical solutions of the fields.
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.
- 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
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 . It can be derived as a particular case of micromorphic (microstrain) elasticity (see, e. g., Forest and Sievert ), 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 ).
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 . 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 , Sternberg and Muki , Bogy and Sternberg [6, 7], Xia and Hutchinson , Huang et al. [9, 10, 11] and Zhang et al. . For micropolar elasticity the works of Paul and Sridharan , Chen et al. , Diegele et al.  and for gradient elasticity the works of Altan and Aifantis [16, 17], Ru and Aifantis , Unger and Aifantis [19, 20, 21], Chen et al. , Mousavi and Lazar , Shi et al. [24, 25], Vardoulakis et al. , Karlis et al. [27, 28], Georgiadis , Askes and Aifantis  and Gutkin and Aifantis  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  as well as Aifantis ). 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  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)
Elasticity law for (see section “The 3-PG-Model as particular case of micro-strain elasticity” in Part I)
Elasticity law for (see section “The 3-PG-Model as particular case of micro-strain elasticity” in Part I)
Elasticity law for (see section “Elasticity law for double stress” in Part I)
Material parameters (see section “Preliminaries—Notation” in Part I)
Strain components (see section “Kinematics” in Part I)
Microdeformation components (see section “Kinematics” of Part I)
Classical equilibrium equations (see section “Cauchy stress—Classical equilibrium equations” in Part I)
Nonclassical equilibrium equations (see section “Nonclassical equilibrium conditions” in Part I)
Field equations for (see section “Field equations for ” in Part I)
Classical compatibility condition (see section “Classical compatibility condition” in Part I)
Nonclassical compatibility conditions (see section “Nonclassical compatibility conditions” in Part I)
Classical boundary conditions (see section “Boundary conditions” in Part I)
Nonclassical boundary conditions (see section “Boundary conditions” in Part I)
Symmetry conditions—Mode-I (see section “Symmetry conditions” in Part I)
Symmetry conditions—Mode-II (see section “Symmetry conditions” in Part I)
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 ).
3.1 Expansions of Williams’ type
As the components of the macrodisplacement and the microdeformation reflect the independent kinematical degrees of freedom, we assume for and asymptotic expansions of the same form. We fix the crack tip at the origin of the coordinate system (see Figure 1 in Part I) and set
and being a real number. Since the crack tip is fixed at , no constant term is present in the expansion of u in Eq. (67). However, we allow a constant term , with physical components in conjunction with cylindrical coordinates, to be present in the expansion of . While the Cartesian components are constant, the physical components are functions of . There are the following well known transformation rules between and (see any textbook)
For later reference, we note the relations
which imply that the physical components trivially obey the nonclassical boundary conditions (32)–(35). Anticipating the results in Section 5, we decompose into parts and , reflecting symmetries according to Mode-I and Mode-II:
The main idea in Williams’ approach is to expand each field variable in a sum of products as in Eqs. (67) and (68). We say that is of the order , and write , whenever is the power function of in the leading term of the expansion of . It can be deduced, from Eq. (67), that . From this, in turn, together with Eq. (68) and the elasticity laws (3)–(5), we can deduce, that . Thus,
Expansion (67) suggests that the necessary and sufficient condition for to vanish at the crack tip is
This restriction is in agreement with energetic considerations. To verify, we remark that , as is constant. Therefore, from Eq. (68) together with Eqs. (25)–(29), we may infer that . For the free energy per unit macrovolume it follows that [cf. Eq. (1)]. Now, consider a small circular area , enclosing the crack tip. The total free energy (per unit length in z–direction) of this area is
3.2 Consequences of the classical equilibrium equations
Similarly, we find from Eq. (31) that
3.3 Consequences of the classical compatibility condition
A look at in Eq. (40) reveals that is a linear differential operator, i. e.,
Since is independent of , 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
3.4 Consequences of the classical boundary conditions
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 in the vicinity of the crack tip are vanishing coefficients of all powers of . For , 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 and components of arises for the first time in the equations for . Therefore, we shall focus attention only on the terms and .
The solution of the systems of differential equations for and , 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 are singular, with order of singularity , or equivalently,
The coefficients of the singular terms, , are given by
where the constants and are the stress intensity factors. Here and in the following, the indices and stand for Mode-I and Mode-II, respectively. Moreover, we use the notations and , in order to distinguish the stress intensity factors of the microstrain continuum from the stress intensity factors and of classical continua.
The so-called angular functions and are defined through
and are normalized by the conditions
The constant terms are given by
with being constant. Constant terms for Mode-II are not present. The first two terms of the asymptotic expansion of are summarized in Section 5.
Although the first two terms in the expansion of are identical to the ones of classical elasticity, the corresponding terms of differ from those of classical elasticity. This follows from the fact that the elasticity laws (3)–(5) are not classical.
Evidently, the components obey the asymptotic expansion
By taking into account the solutions for of the last section, we find that
Now, we take into account the solutions for , established in the last section, as well as the representations for , given by Eqs. (75)–(79), to obtain
The constants , and are defined as follows:
The first two terms of the asymptotic expansion of are summarized in Section 5.
The macrodisplacement components and 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 that
For the circumferential component , we conclude that
By employing steps similar to those in the last section, we get the following solutions for and .
The first two terms of the asymptotic expansion of are also summarized in Section 5.
We shall derive differential equations for and by inserting the asymptotic expansions of and (see Eqs. (68) and (80), with ) into Eqs. (36)–(38). Note that, by virtue of Eqs. (73) and (74) and since is independent of , the identity
applies. Keeping this in mind and collecting terms of like powers of , after some lengthy but otherwise straightforward manipulations, Eq. (36) yields
3.8.1 Differential equations for
Equating to zero the coefficients of power in Eqs. (141)–(143) leads to the system of ordinary differential equations
for the sum , which has the solution
By substituting the solution (154), we see that
which obeyes the solution
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
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
With regard to the symmetry conditions (53), (54), (61) and (62), the constants and are attributed to loading conditions of Mode-I and Mode-II, respectively. The solutions and are summarized and discussed in Section 5.
3.9 Double stress
The goal is to determine and by substituting the asymptotic expansion for into the elasticity laws (7)–(16). It is readily verified that in view of the conditions (73) and (74), the terms of the expansion (68) will disappear in the subsequent equations. Thus, we conclude from Eqs. (7)–(16), by equating the coefficients of power that
and by equating the coefficients of power that
The fact that the solutions 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.
Using steps similar to those above we obtain for the representations
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 and rather than the asymtptotic expansions of and used 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 and [see Eqs. (67) and (68)]. Alternatively, it is instructive to start from asymptotic expansions of the same type for the stresses and , i. e.,
where and , . Then, from the elasticity laws (17)–(22), we recognize that and hence the components are of form (67). It follows that all outcomes of sections 3.2–3.6 apply as well and, in particular, that . Then, it remains to show, how to determine the terms and . The corresponding terms of 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 . To this end, we shall involve the nonclassical equilibrium Eqs. (32)–(35), in conjunction with the elasticity law (6) for , as well as the nonclassical compatibility conditions (42)–(44). It is necessary to involve the latter for we are directly seeking for solutions of .
4.1 Nonclassical equilibrium equations
Since and , we recognize from the elasticity law (6) that . On the other hand, by virtue of the expansion (212), and . Therefore, up to terms of order there will be no contributions of present in the nonclassical equilibrium Eqs. (32)–(35) and we conclude that
Equating to zero the coefficients of power leads to
4.2 Nonclassical compatibility conditions
Again, equating to zero the coefficients of power leads to
4.3 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 of 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
To accomplish a representation for similar to the one for in Eq. (108), we remark that there is only one unknown constant for Mode-I, namely , but there are two unknown constants for Mode-II, and [cf. Eqs. (195)–(202)]. Therefore, in analogy to Eq. (108), we set
and define for Mode–I (cf. Eq. (202))
rendering to be normalized,
To define and unambiguously, we note that can be determined by adding Eqs. (199) and (200) while taking . Similarly, can be determined by substracting Eqs. (199) and (200) from each other while taking . We intend to normalize the angular functions and by
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. ), 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 and and the numerical simulations in Part III confirm this fact.
It is also convenient to replace the constants and by the definitions
Table 1 summarizes the first two terms of the asymptotic solutions of the near-tip fields. All stresses are singular with order of singularity . Especially, the terms and are identical to those of classical elasticity. However, the terms and are different from the corresponding terms of classical elasticity. In particular, includes terms arising from . There are also qualititative differences to micropolar elasticity. For instance, terms of couple stresses corresponding to in Mode-II and to in Mode-I do not exist.
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.
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 .
This is in contrast to statements in other works, which rely upon boundary conditions different from the ones adopted here.
There are, however, significant differences in comparison to classical elasticity, in what concerns the components of macrostrain and macrodisplacement.
There are also significant qualitative differences in comparison to micropolar elasticity concerning the nonclassical stresses.
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.
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
Here, and are 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 , , , and :
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
which has the solutions
and the solutions (A1)–(A3) become
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.
The first and second authors acknowledge and thank the Deutsche Forschungsgemeinschaft (DFG) for partial support of this work under Grant TS 29/13–1.
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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.
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List of abbreviations
eq. = equation