## Abstract

Models of implicit gradient elasticity based on Laplacians of stress and strain can be established in analogy to the models of linear viscoelastic solids. The most simple implicit gradient elasticity model including both, the Laplacian of stress and the Laplacian of strain, is the counterpart of the three-parameter viscoelastic solid. The main investigations in Parts I, II, and III concern the “three-parameter gradient elasticity model” and focus on the near-tip fields of Mode-I and Mode-II crack problems. It is proved that, for the boundary and symmetry conditions assumed in the present work, the model does not avoid the well-known singularities of classical elasticity. Nevertheless, there are significant differences in the form of the asymptotic solutions in comparison to the classical elasticity. These differences are discussed in detail on the basis of closed-form analytical solutions. Part I provides the governing equations and the required boundary and symmetry conditions for the considered crack problems.

### Keywords

- implicit gradient elasticity
- Laplacians of stress
- Laplacians of strain
- micromorphic and micro-strain elasticity
- plane strain state

## 1. Introduction

The most simple constitutive law in explicit gradient elasticity is the model with equation:

Here,

Now, the question arises, if a gradient elasticity model including both, the Laplacian of stress and the Laplacian of strain, could remove both, the singularities of stress and the singularities of strain at the crack tip (cf. Gutkin and Aifantis [6]). The most simple generalization of Eq. (1), including the Laplacians of stress and strain, reads as follows:

where the same notation as in Eq. (1) applies and

Broese et al. [5] proved that Eq. (2) can be derived as a particular case of Mindlins micro-structured elasticity, which arises whenever the micro-deformation of the micromorphic continuum is supposed to be a symmetric tensor. Because the micro-structured elastic continuum of Mindlin and the micromorphic elastic continuum of Eringen (see, e.g., Eringen and Suhubi [8] and Eringen [9]) are essentially equivalent to each other, in the present work we will call both as micromorphic continua. According to Forest and Sievert [10], the resulting micromorphic theory is named micro-strain theory. It is shown in Broese et al. [5] that, in the context of micro-strain elasticity, the 3-PG-Model (2) can be derived as a combination of elasticity constitutive laws and the equilibrium equation for the so-called double stress.

On the other hand, Broese et al. [5] showed that Eq. (2) can be established alternatively by supposing the continuum to be classical, i.e., exhibiting only classical displacement degrees of freedom, but in the framework of the non-conventional thermodynamics proposed in Alber et al. [11]. To be more specific, the micro-deformation variable of the micro-strain approach has to be viewed as an internal state variable analogous to the inelastic strain in linear viscoelasticity. Eq. (2) then turns out to be a constitutive law, which is the counterpart in gradient elasticity of the three-parameter viscoelastic solid. The short hand notation “3-PG-Model” stands for “3-Parameter-Gradient-Elasticity-Model.” A general analogy to the constitutive laws describing viscoelastic solids can be established by using a nonstandard spring in gradient elasticity corresponding to the dashpot element in linear viscoelasticity and the Laplacian operator

Because all resulting governing equations and boundary conditions in the two approaches are equal to each other, we shall proceed further by regarding the 3-PG-Model as a particular case of the micro-strain elasticity. The present work (Parts * I*,

*, and*II

## 2. Preliminaries: notation

Throughout the paper, we largely use the same notation as in Mindlin [3] and Mindlin and Eshel [4], in order to facilitate the comparison with these works. The deformations are assumed to be small, so we do not distinguish, as usually done, between reference and actual configuration. All indices will have the range of integers (

Let

Let

We denote by

where

Since

we have for every second-order tensor

For any tensor

Similar notations hold for any tensor of arbitrary order. The summation convention applies in analogous manner, e.g., we have

The physical components with respect to cylindrical coordinates of

## 3. Governing equations for the 3-PG-Model

This section provides a short overview about the 3-PG-Model in a form which is adequate for developing analytical solutions.

### 3.1 The 3-PG-Model as particular case of micro-strain elasticity

Assume the material body to be a micromorphic continuum. Besides the classical kinematical degrees of freedom, micromorphic continua are characterized by additional degrees of freedom due to the deformations of the micro-continua, which are assumed to be attached at every point of the macro-continuum (see Mindlin [3] and Broese et al. [5]). Therefore, in the micromorphic continuum theory, a nonclassical (double) stress and a nonclassical stress power are introduced in addition to the classical ones, but otherwise the theory is formulated in the framework of classical thermodynamics.

Let

All component representations in Section 3 are referred to the Cartesian coordinate system

This means that

exhibit the symmetry property

Following Forest and Sievert [10], we denote a micromorphic elasticity theory based on Eqs. (16)–(18) as micro-strain elasticity.

According to Broese et al. [5], the 3-PG-model can be established as a particular case of the micro-strain elasticity by assuming the existence of a free energy (per unit macro-volume)

The components

where

Further, there exists a double stress

For static problems, the classical and nonclassical stresses have to satisfy corresponding equilibrium equations. In the absence of body forces and body double forces, these are (see Mindlin [3] or Broese et al. [5])

The concomitant classical and nonclassical boundary conditions are as follows:

have to be prescribed on the boundary

The 3-PG-Model can be obtained from the above equations by first inserting Eq. (24) into Eq. (26), as follows:

Then take the Laplacian of Eq. (21), as follows:

and use Eq. (21) as well as Eq. (22) in Eq. (29), as follows:

The latter together with Eq. (30) yield the following equation:

which is nothing but the 3-PG-Model (2).

#### 3.1.1 A useful equation for Ψ

For later reference, we derive a useful equation for the strain

Further, from Eq. (21), we get the following equation:

By combining the last two equations, we gain the useful relation as follows:

For given _{,} this is a (Helmholtz) partial differential equation for the components of

## 4. Mode-I and mode-II crack problems

In Part

### 4.1 Kinematics

Plane strain state of micro-strain continua in equilibrium is characterized by the assumptions that

and that

On the basis of these assumptions, we conclude (see Section A.2) that the physical components

whereas all other components of

Similarly, we find (see Section 6.3) for the physical components

It is well known (see, e.g., Anderson [12], p. 114) that the nonvanishing physical components of

### 4.2 Cauchy stress: classical equilibrium equations

In view of the assumptions of the last section, we may derive the following results. We conclude from Eqs. (21)–(23), that

and that

Since

where

Quite similar to the case of classical elasticity (see, e.g., Anderson [12], p. 114), the matrix of the components of

### 4.3 Classical compatibility condition

For the analytical solutions in Part

The aim is now to rewrite this equation in terms of the physical components of

By inserting these equations into Eq. (55), we get

This is equivalent to a vanishing sum of two functions of

with

and

The right hand side of Eq. (62) can be simplified by invoking the equilibrium Eqs. (53) and (54). First recast Eq. (54) to solve for

and then take the derivative with respect to

On the other hand, from Eq. (53),

and, after differentiation with respect to

By substituting the latter into Eq. (64),

Finally, by substituting Eqs. (66) and (67) into Eq. (62) and after some rearrangement of terms, we find that

### 4.4 Field equations for Ψ

By expressing

### 4.5 Double stress

#### 4.5.1 Elasticity law for double stress

With respect to physical components, the elasticity law (24) becomes

Keeping in mind that the physical components of

#### 4.5.2 Non-classical equilibrium conditions

The physical components of the non-classical equilibrium condition (26) are

With the aid of the physical components of

or equivalently (cf. Section A.4)

### 4.6 Nonclassical compatibility conditions

Besides the classical compatibility condition for the strain

From these, we obtain useful relations by involving the physical components

and that

By inserting these into Eq. (92), we obtain the following equation:

In a similar way, we conclude from Eqs. (93) and (94) that

In order to involve the components of

where

where in addition use has been made of Eqs. (75) and (80). By inserting these components into Eqs. (97)–(99), we can verify that

The last equation is independent of material parameters. In order to rewrite Eqs. (107) and (108) also in a form independent of material parameters, we add and subtract them from each other to obtain, respectively,

### 4.7 Boundary conditions

As usually, near-tip field solutions rely upon boundary conditions, which are imposed only on the crack faces. Especially, we assume the classical traction

Now, we have from Eq. (27), expressed in physical components, that

Similarly, we get from Eq. (28), expressed in physical components, that

where use has been made of the elasticity law (24). As the isotropic elasticity tensor

Keeping in mind Eqs. (39)–(43), the only nontrivial conditions implied are as follows:

or equivalently

### 4.8 Symmetry conditions

Symmetry conditions are important to classify the near-tip field solutions into types according to Mode-I and Mode-II crack problems. Each type of loading condition is characterized by the following symmetry conditions.

#### 4.8.1 Mode-I

As in classical elasticity (see, e.g., Hellan [14], p. 10), we suppose for the macro-displacement the following symmetry conditions:

i.e., _{,} whereas

implying that

Then, it can be verified, with the help of the elasticity law (21), expressed in physical components, that Eqs. (122)–(125) engender the following conditions for the components of

Further, it can be seen from Eqs. (124) and (125), that

and from the elasticity laws (73)–(82), that

#### 4.8.2 Mode-II

We know from classical elasticity (see, e.g., Hellan [14], p. 10), that the radial component of the displacement vector is an odd function of

It follows for the macro-strain

which suggest to assume the following symmetries for

It can be proved, in a similar fashion to Mode-I, that

and that

Before closing this section, we notice here, that the numerical simulations on the basis of the finite element method in Part

## 5. Concluding remarks

If the implicit gradient elasticity model in Eq. (2), named the 3-PG-Model, is recognized as a particular case of micromorphic (micro-strain) elasticity, a free energy and associated response functions and boundary conditions can be assigned. Part

## Acknowledgments

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

This section provides the component representations with respect to cylindrical coordinates of some space derivatives of a second-order tensor

## A.1 Cylindrical coordinates

We denote by

## A.2 The gradient of a symmetric second-order tensor

In the case of a second-order tensor _{,} we have (cf. Section 2)

where

When

This symmetry also applies with respect to physical components,

It can be seen that Eq. (A2) furnishes the following physical components of

## A.3 The Laplacian of a symmetric second-order tensor

The Laplacian of a second-order tensor

This may be written as

where

and

We can calculate the physical components

## A.4. The divergence of a third-order tensor

Let

where

and

## References

- 1.
Altan BS, Aifantis EC. On some aspects in the special theory of gradient elasticity. Journal of the Mechanical Behavior of Materials. 1997; 8 :231-282. DOI: 10.1007/s00161-014-0406-1 - 2.
Georgiadis HG. The mode III crack problem in microstructured solids governed by dipolar gradient elasticity: Static and dynamic analysis. Journal of Applied Mechanics. 2003; 70 :517-530 - 3.
Mindlin RD. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis. 1964; 16 :51-78 - 4.
Mindlin RD, Eshel NN. On first strain-gradient theories in linear elasticity. International Journal of Solids and Structures. 1968; 4 :109-124 - 5.
Broese C, Tsakmakis C, Beskos DE. Gradient elasticity based on Laplacians of stress and strain. Journal of Elasticity. 2018; 131 :39-74 - 6.
Gutkin M, Aifantis EC. Dislocations in the theory of gradient elasticity. Scripta Materialia. 1999; 40 :559-566 - 7.
Askes H, Gutiérrez MA. Implicit gradient elasticity. International Journal for Numerical Methods in Engineering. 2006; 67 :400-416 - 8.
Eringen AC, Suhubi ES. Nonlinear theory of simple micro-elastic solids–I. International Journal of Engineering Science. 1964; 2 :189-203 - 9.
Eringen AC. Theory of micropolar elasticity. In: Fracture – An Advanced Treatise. Vol. 1. Microscopic and Macroscopic Fundamentals. New York: Academic Press; 1968. pp. 621-729 - 10.
Forest S, Sievert R. Nonlinear microstrain theories. International Journal of Solids and Structures. 2006; 43 (24):7224-7245 - 11.
Alber H-D, Hutter K, Tsakmakis C. Nonconventional thermodynamics, indeterminate couple stress elasticity and heat conduction. Continuum Mechanics and Thermodynamics. 2014. DOI: 10.1007/s00161-014-0406-1 - 12.
Anderson TL. Fracture Mechanics - Fundamentals and Applications. CRC Press; 1991 - 13.
Malvern LE. Introduction to the Mechanics of a Continuous Medium. Englewood Cliffs, New Jersey: Prentice Hall, Inc.; 1969 - 14.
Hellan K. Introduction to Fracture Mechanics. New York: McGraw-Hill; 1984