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Part II: Asymptotic Solutions",doi:"10.5772/intechopen.93618",slug:"mode-i-and-mode-ii-crack-tip-fields-in-implicit-gradient-elasticity-based-on-laplacians-of-stress-2",body:'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
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.
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
or inversely
Elasticity law for
Elasticity law for
or inversely
Material parameters (see section
Strain components (see section
Microdeformation components (see section
Classical equilibrium equations (see section
Nonclassical equilibrium equations (see section
Field equations for
Classical compatibility condition (see section
Nonclassical compatibility conditions (see section
Classical boundary conditions (see section
Nonclassical boundary conditions (see section
or equivalently
Symmetry conditions—Mode-I (see section
Symmetry conditions—Mode-II (see section
We shall solve the given problems by employing asymptotic expansions of Williams’ type (see Williams [3]).
As the components of the macrodisplacement and the microdeformation reflect the independent kinematical degrees of freedom, we assume for
with
and
For later reference, we note the relations
which imply that the physical components
with
and
The main idea in Williams’ approach is to expand each field variable
with
Expansion (67) suggests that the necessary and sufficient condition for
This restriction is in agreement with energetic considerations. To verify, we remark that
Since
Substitute the expansion (80) into Eqs. (30) and (31) and collect coefficients of like powers of
Similarly, we find from Eq. (31) that
A look at
Since
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
By invoking the asymptotic expansion (80) in the classical boundary conditions (45) and (46), we conclude that
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
Terms
with boundary conditions
Terms
with boundary conditions
Terms
with boundary conditions
It can be recognized that coupling between components of
The solution of the systems of differential equations for
The coefficients of the singular terms,
where the constants
The so-called angular functions
and are normalized by the conditions
The constant terms
with
Although the first two terms in the expansion of
Evidently, the components
We use this and the asymptotic expansions of Section 3.1, with
Terms
By taking into account the solutions for
Terms
Now, we take into account the solutions for
The constants
The first two terms of the asymptotic expansion of
The macrodisplacement components
or
For the circumferential component
or
By employing steps similar to those in the last section, we get the following solutions for
Terms
Invoking Eqs. (116) and (117), we get, after some straightforward manipulations,
Terms
from which, by virtue of Eqs. (119) and (120),
The first two terms of the asymptotic expansion of
We shall derive differential equations for
applies. Keeping this in mind and collecting terms of like powers of
Similarly, from Eqs. (37) and (38), we get
It is worth remarking that if only terms up to order
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
Similarly, equating to zero the coefficients of power
Proceeding to solve the system (147)–(149), we note that Eqs. (147) and (148) imply the ordinary differential equation
for the sum
For determining the constants of integration
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
By substituting the solutions (154) and (156), we gain an ordinary differential equation for
which obeyes the solution
with
Finally, using the solutions (161) and (160) in Eq. (157), we obtain the solution
where
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.
Equating to zero the coefficients of power
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
The considerations of Section 3.1, together with
with
The goal is to determine
and by equating the coefficients of power
If we introduce the solutions (160)–(162) into Eqs. (179)–(186) and rearrange terms, then, for
The fact that the solutions
Using steps similar to those above we obtain for
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
In Section 3 we determined the near-tip fields by starting from asymptotic expansions of the same form for the kinematical variables
where
Since
Equating to zero the coefficients of power
and
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).
We insert the asymptotic expansion (212) into the nonclassical compatibility conditions (42)–(44) and collect terms of like powers of
Again, equating to zero the coefficients of power
Eqs. (217)–(219) and (225)–(227) are
It can be shown (cf. A) that the solutions are given by
If we define
then these are nothing more but the solutions for
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
and define for Mode–I (cf. Eq. (202))
rendering
To define
and therefore define the stress intensity factors
The angular functions will be determined by comparison of Eqs. (238)–(243) with Eqs. (195), (196), (198)–(200) ,and (202). Explicitely, we find that
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
It is also convenient to replace the constants
Evidently, the new constants for Mode-I and Mode-II in the expansions of
with
Table 1 summarizes the first two terms of the asymptotic solutions of the near-tip fields. All stresses are singular with order of singularity
Analytical solutions of the fields.
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,
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
The smallest value of
and the solutions (A1)–(A3) become
These solutions, in turn, are equivalent to those of Eqs. (108)–(109). Moreover, it can be shown that for
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.
Not applicable.
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.
Not applicable.
Not applicable.
eq. = equation
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Initial biochemical studies have been exclusively analytic: dissecting, purifying, and examining individual components of a biological system; in the apt words of Efraim Racker (1913 –1991), “Don’t waste clean thinking on dirty enzymes.” Today, however, biochemistry is becoming more agglomerative and comprehensive, setting out to integrate and describe entirely particular biological systems. The ‘big data’ metabolomics can define the complement of small molecules, e.g., in a soil or biofilm sample; proteomics can distinguish all the comprising proteins, e.g., serum; metagenomics can identify all the genes in a complex environment, e.g., the bovine rumen. 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\r\n\tThe four topics of this book series - Pollution; Environmental Resilience and Management; Ecosystems and Biodiversity; and Water Science - will address important areas of advancement in the environmental sciences. They will represent an excellent initial grouping of published works on these critical topics.
\r\n\tPollution is caused by a wide variety of human activities and occurs in diverse forms, for example biological, chemical, et cetera. In recent years, significant efforts have been made to ensure that the environment is clean, that rigorous rules are implemented, and old laws are updated to reduce the risks towards humans and ecosystems. However, rapid industrialization and the need for more cultivable sources or habitable lands, for an increasing population, as well as fewer alternatives for waste disposal, make the pollution control tasks more challenging. Therefore, this topic will focus on assessing and managing environmental pollution. It will cover various subjects, including risk assessment due to the pollution of ecosystems, transport and fate of pollutants, restoration or remediation of polluted matrices, and efforts towards sustainable solutions to minimize environmental pollution.
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",annualVolume:11967,isOpenForSubmission:!0,coverUrl:"https://cdn.intechopen.com/series_topics/covers/39.jpg",editor:{id:"137040",title:"Prof.",name:"Jose",middleName:null,surname:"Navarro-Pedreño",fullName:"Jose Navarro-Pedreño",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bRAXrQAO/Profile_Picture_2022-03-09T15:50:19.jpg",institutionString:"Miguel Hernández University of Elche, Spain",institution:null},editorTwo:null,editorThree:null,editorialBoard:[{id:"177015",title:"Prof.",name:"Elke Jurandy",middleName:null,surname:"Bran Nogueira Cardoso",fullName:"Elke Jurandy Bran Nogueira Cardoso",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bRGxzQAG/Profile_Picture_2022-03-25T08:32:33.jpg",institutionString:"Universidade de São Paulo, Brazil",institution:null},{id:"211260",title:"Dr.",name:"Sandra",middleName:null,surname:"Ricart",fullName:"Sandra Ricart",profilePictureURL:"https://mts.intechopen.com/storage/users/211260/images/system/211260.jpeg",institutionString:null,institution:{name:"University of Alicante",institutionURL:null,country:{name:"Spain"}}}]},{id:"40",title:"Ecosystems and Biodiversity",keywords:"Ecosystems, Biodiversity, Fauna, Taxonomy, Invasive species, Destruction of habitats, Overexploitation of natural resources, Pollution, Global warming, Conservation of natural spaces, Bioremediation",scope:"